tag:blogger.com,1999:blog-20360140533897516962024-10-04T08:35:04.327+01:00Colin Foster's Mathematics Education BlogColin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.comBlogger26125tag:blogger.com,1999:blog-2036014053389751696.post-25164525534506884922023-03-30T07:00:00.014+01:002023-05-16T17:12:16.732+01:00Teaching specific tactics for problem solving<p><i>This is my final blogpost, as my year as President of the Mathematical Association draws to a close, so I've allowed myself to go on at slightly longer length than usual. I wanted to conclude this series by thinking about problem solving, which for me is always the ultimate goal of learning mathematics. </i><i>How can we help students not just to 'do problem solving' but actually learn to get better at it?</i></p><h4>What I mean by problem solving</h4><p>The term <i>problem solving</i> is used in different ways in different strands of the research literature (Note 1, see Foster, under review). Sometimes 'a problem' just means any mathematics question at all, such as a 'word problem', which might be an ordinary mathematics question dressed up in some 'real-life' context. But when I talk about a mathematics 'problem' I mean a <i>non-routine task</i> - in other words, a problem which the student doesn't have a ready-made method for solving - and this is the usual definition within the mathematics education literature. This means that whether something is a problem or not for any particular person depends on what methods they happen to have at their disposal at that particular time (see Foster, 2019, 2021).</p><p>The illustration that I like to use is of driving into a tunnel (Figure 1). Sometimes, before you enter a tunnel, you can see daylight out of the other side. The tunnel might still be quite long, but it is straight enough for you to see the entire route through before you begin. This is analogous to a <i>routine</i> task (Note 2), or exercise. Such tasks can be important for developing fluency with useful procedures - I am not saying 'routine is bad'. But with a routine task there is no challenge in deciding <i>what</i> to do, as that's clear from the outset.</p><p>The alternative scenario is a tunnel which you <i>cannot</i> see right through. You don't know before you enter if you will need to turn left or right - there might even be a dead-end and you could have to turn around and try a different approach. You might have some ideas for <i>starting</i>, but at the outset you don't know exactly how you're going to proceed - you will have to be flexible and respond to what happens; the first thing you try may not work. This is what I call a <i>non-routine </i>task, or <i>problem</i>.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiiRPrR_lSrRT6tlko2Xx6wdV3fTDAAdLuBe5-CSteCumb1Kz7HFVmYr4mjklD5ClfXScghvHK_79ubfnOO1YoSPdS40tnAyiAG6cT2a-wztvA15uMirO_VJ0-N8jd6eX8AmEbT5idYtoQXhka0E6bD0mbPaOTVI_mJ4O1VWubDSmMeZ2oMZkkTcsmtTg=s800" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="544" data-original-width="800" height="435" src="https://blogger.googleusercontent.com/img/a/AVvXsEiiRPrR_lSrRT6tlko2Xx6wdV3fTDAAdLuBe5-CSteCumb1Kz7HFVmYr4mjklD5ClfXScghvHK_79ubfnOO1YoSPdS40tnAyiAG6cT2a-wztvA15uMirO_VJ0-N8jd6eX8AmEbT5idYtoQXhka0E6bD0mbPaOTVI_mJ4O1VWubDSmMeZ2oMZkkTcsmtTg=w640-h435" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. A <i>Routine</i> task (left) versus a <i>Non-routine</i> task (right)</td></tr></tbody></table><p>It's important to realise that routine tasks are not necessarily <i>quick and</i> <i>easy</i> - they may be multi-step and require serious thought. When I say 'routine' I am not implying 'trivial'. For example, multiplying two 5-digit numbers together is a routine task for a mathematics teacher, because we know algorithms to use for this (whereas it wouldn't be for a child who hadn't learned a multiplication algorithm). However, even though a 5-digit by 5-digit multiplication is routine, I might easily make a mistake and get the wrong answer. But, even if I mess it up, it's still a routine task, because I know beforehand exactly how I should do it (Note 3).</p><p>So, if we accept that being able to tackle non-routine tasks (i.e., problems) is an important outcome of education, how do we help students get to the point where they are powerful problem solvers? It can't be enough simply to throw non-routine problems at them and watch them struggle. If we do that, a tiny minority may somehow discover the necessary problem-solving strategies to succeed, while the vast majority simply struggle to get anywhere and conclude that they must be not natural problem solvers. Instead, I think we need to explicitly teach these problem-solving strategies to everyone. But do these 'strategies' exist, and, if so, what are they?</p><h4>A story about chess</h4><p>When I was at primary school, my father taught me to play chess. What I mean is that he taught me the object of the game and the rules for how the pieces move. As far as either of us knew, that is what it meant to ‘learn chess’. I assumed that any further improvement would just come the more I practised, meaning the more games I played. At primary school, some children played chess at lunchtime, and I was considered to be good at chess, because I knew all the rules, and would try to think more than a move ahead, and anticipate what my opponent might do. With hindsight, I also suspect that people <i>assumed</i> I was good at chess because I was good at mathematics.</p><p>At secondary school, there was a chess club, which I joined, and, as before, I was one of the ones who knew the rules of chess, so I was treated as an expert, and I helped to teach others. We played in competitions against other schools, who had similar kinds of chess clubs, and sometimes we won games and sometimes they did – it was pretty random. Until one day we played against a local independent (private) school, and we all lost all our games within just a few minutes. What was going on? Were these ‘posh’ students just ‘clever’? Apparently the teacher who ran their chess club had played chess himself in national competitions, so perhaps that had something to do with it. Whatever the reason, we never lost against that school again - because we never played them again!</p><p>It wasn’t until I went to university, where nearly everyone had come from an independent school, that I discovered that people who had no interest in chess at all could beat me, which seemed very odd. I had assumed that those students in the chess team that beat us at school must have all been obsessively committed chess nerds, but actually I'm sure now that for most of them chess was just one of a hundred things they did fairly half-heartedly. The difference wasn’t, as we had assumed, that they had practised super-intensively, or were somehow smarter at thinking more moves ahead than we were. It was, of course, that they had been taught specific moves to use, and strategies for different points in the game (openings, endings, etc.), and had maybe even been shown some famous games. I had never realised that there were books about chess (e.g., Fischer, Margulies, & Mosenfelder, 1982) – and, if I had seen such books on the shelf, I would probably have assumed that they would not be interesting to me, and would be merely explaining the <i>rules of the game</i>, which I already knew. And if I had realised that such books taught <i>strategies</i> I might even have concluded that reading such a book was tantamount to cheating. You have to come up with your own moves, surely, otherwise that isn’t ‘playing the game’?</p><h4>Teaching the rules but not the strategies</h4><p>The reason for telling this story is that I think something a bit similar to this situation goes on in the teaching of mathematics in schools. Every teacher teaches students ‘the rules of the game’, such as that the angle sum of a plane triangle is <span style="text-align: center;">$180^\circ$, </span>and then we give students increasingly challenging problems to solve that <i>depend</i> on these rules. We might even tell ourselves that the problems “only require knowledge of” such-and-such short list of ‘angle facts’, and therefore that the students 'should' be able to solve them. When they get stuck, we might say, “Keep persevering – you know <i>everything</i> you need to know to solve it – you just need to keep thinking!” – but is that really true? This feels like saying that if you know the rules of chess then you know everything you need to know to win any game against any opponent, which I guess is what I thought as a child.</p><p>I began thinking back on all of this when I heard some teachers discussing what may be the most notorious example of a ‘hard but elementary’ angle problem. It is known as “Langley's Adventitious Angles”, and was posed by Edward Langley (Note 4) in <i>The Mathematical Gazette</i> in 1922. It involves what is sometimes referred to as the 80-80-20 triangle (see Figure 1). The task is to show that $x = 30$.</p><p></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjfxIn_1jdoaSPSm9zuNcjypRp-1y8ZSBtHLEhtobXUuvuC0kh0GHFXAt3wqi4iEAd9QhRTWtKLFyhEiBFMcZEAuWr-ciLZFxQ09ZrISSggDuzAT_oq14udkFfr2IwQ3sKAOy2gpAPToURtuWKcIaMF7sKQhDfrLK2KzU2ip7zDebMUPViKBRA4QDOQlA=s1341" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1341" data-original-width="890" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjfxIn_1jdoaSPSm9zuNcjypRp-1y8ZSBtHLEhtobXUuvuC0kh0GHFXAt3wqi4iEAd9QhRTWtKLFyhEiBFMcZEAuWr-ciLZFxQ09ZrISSggDuzAT_oq14udkFfr2IwQ3sKAOy2gpAPToURtuWKcIaMF7sKQhDfrLK2KzU2ip7zDebMUPViKBRA4QDOQlA=s320" width="212" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">$ABC$ is an isosceles triangle.<br />$B = C = 80^\circ$.<br />$CF$ at $30^\circ$ to $AC$ cuts $AB$ in $F$.<br />$BE$ at $20^\circ$ to $AB$ cuts $AC$ in $E$.<br />Prove $BEF = 30^\circ$.<br />Figure 1. Edward Langley’s original problem (Langley, 1922, p. 173).</td></tr></tbody></table><p></p><p>Since the problem was posed, over 40 different solutions have been produced (Chen, 2019; Rike, 2002; see also related problems in Leikin, 2001). However, most people find it extremely difficult to obtain a solution, and simply ‘angle chasing’ around, adding and subtracting angles, doesn't get you anywhere. On the other hand, the heavy machinery of things like trigonometry is not necessary either. So, this has been called <a href="https://www.duckware.com/tech/worldshardesteasygeometryproblem.html" target="_blank">“The World’s Hardest Easy Geometry Problem”</a> – easy in the sense that it requires only elementary facts from geometry; hard because it’s extremely difficult to see how to use those elementary facts to solve it!</p><p>This raises the question 'Why is it that people can't solve this problem?' Similar questions are sometimes asked about the lovely geometry problems that Catriona Agg tweets (as <a href="https://twitter.com/Cshearer41">@Cshearer41</a>, see Shearer & Agg, 2019). I have heard plenty of highly competent mathematics teachers saying that, although they also really like her problems, they can't actually do them. So it's interesting to ask why not. Is it because the problems depend on knowledge of highly-advance geometrical theorems that these teachers have never studied? Of course not. All of Catriona's problems depend on simple school-level geometry - but they are hard nonetheless!</p><p>I think the missing ingredient is problem-solving <i>tactics</i> (finer grained than 'strategies'). Of course, content knowledge of geometry (e.g., the angle sum of a plane triangle is <span style="text-align: center;">$180^\circ$) is essential. It is <i>necessary</i> but <i>not sufficient</i>. You can have encyclopaedic knowledge of all the geometry theorems in the world, but still be unable to apply them. So, I think success depends on having access to </span>problem-solving tactics, and by this I don't mean high-level Polya-style generic strategies like 'draw a diagram' or 'be systematic'. Those are true, but hard to apply in any particular situation of being stuck (Schoenfeld, 1985). I mean much more <i>topic-specific strategies </i>(Foster, under review). In the case of <span style="text-align: center;">Langley’s problem, the key strategy </span>turns out to be to add an <i>auxiliary line</i> to the diagram, choosing the position wisely, so that it is parallel to a line that is already there, so creating corresponding or alternate angles or similar triangles (Note 5 - spoiler alert there!).</p><p>My recent Economic and Social Research Council (ESRC) project, <i>Exploring socially-distributed professional knowledge for coherent curriculum design</i>, carried out in collaboration with Professor Geoff Wake, Dr Fay Baldry and Professor Keiichi Nishimura in Japan, explored how the mathematics curriculum is designed and taught in Japan. In Japan, teachers explicitly teach problem-solving strategies, such as 'add an auxiliary line' (see Baldry et al., 2022). In the problem-solving strand of the <a href="https://www.lboro.ac.uk/services/lumen/curriculum/" target="_blank">LUMEN Curriculum resources</a>, which we are currently designing at Loughborough University, the lessons explicitly and systematically teach problem-solving strategies like these, using problems which are dramatically unlocked by that strategy. The aim is for all students to build up a toolbox of these strategies, along with the knowledge of which one is likely to be useful for which problems. We are hoping that this will leave less to chance, and be a more effective way of helping <i>all</i> students become powerful problem solvers.</p><h4 style="text-align: left;">Concluding thoughts</h4><p>Writing these 26 blogposts over the course of my year as President of the <a href="https://www.m-a.org.uk/" target="_blank">Mathematical Association</a> has been a great pleasure, and I have particularly appreciated the many people who have got in touch with comments and reactions. Please continue to follow my work on Twitter <a href="https://twitter.com/colinfoster77" target="_blank">@colinfoster77</a> and through my website <a href="https://www.foster77.co.uk/">https://www.foster77.co.uk/</a>, and I hope to see many of you at the <a href="https://www.m-a.org.uk/conference-2023" target="_blank">Joint Conference of Mathematics Subject Associations 2023</a> next week!</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p><span style="font-family: inherit;">1. Do you agree about the value of teaching problem-solving tactics, in addition to 'content'? Why / why not?</span></p><p><span style="font-family: inherit;">2. Where in your teaching/curriculum do students encounter strategies such as 'Draw in an auxiliary line'?</span></p><p><span style="font-family: inherit;">3. How might you plan to teach other problem-solving tactics explicitly?</span></p><h3>Notes</h3><p>1. I talked about many of the ideas in this post in my conversation with Ben Gordon on his podcast (BAGs to Learn Podcast by Ben Gordon, 2021).</p><p>2. I use 'task' to refer to anything mathematical a student is asked to do: it could be written, oral or practical.</p><p>3. It is actually a bit more subtle than this, because if the two 5-digit numbers that you asked me to multiply together happened to be, say, 11111 and 11111, then that might turn it into a non-routine task - i.e., a problem - because I might wish to avoid plodding through a standard algorithm and instead exploit the <a href="https://en.wikipedia.org/wiki/Repdigit">repdigit</a> nature of these two numbers. However, on the other hand, if I had played around with such numbers before, I might know how strings of 1s behave when multiplied, and even know exactly how to write down the answer immediately, and so it would be back to being a non-routine exercise. So, whether something is routine or not depends in detail on what you happen to know.</p><p>4. Langley was the founding editor of <i><a href="https://www.m-a.org.uk/the-mathematical-gazette">The Mathematical Gazette</a></i>. A curious fact is that he apparently had a blackberry named after him – not a lot of people can say that!</p><p>5. One possible solution is given in the diagrams below:<a href="https://blogger.googleusercontent.com/img/a/AVvXsEgxiO1tFHkp2MYFvsAZZU_3n2DbSOzWQDERh5gm4azBX8DyvlwDX0aIP8rEWdYmzCozh_UekKH8N2AJHYc76Vibr500GC2jBe34ILEEFfv4E08JoUJtb2xBYY78HiV9zacsMi6JVtcw-fDdRrvQWrYQU6-bNTo74dHiwpLRAx1h0PEeEto3Z1MPGLp7bw=s4835" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="1358" data-original-width="4835" height="180" src="https://blogger.googleusercontent.com/img/a/AVvXsEgxiO1tFHkp2MYFvsAZZU_3n2DbSOzWQDERh5gm4azBX8DyvlwDX0aIP8rEWdYmzCozh_UekKH8N2AJHYc76Vibr500GC2jBe34ILEEFfv4E08JoUJtb2xBYY78HiV9zacsMi6JVtcw-fDdRrvQWrYQU6-bNTo74dHiwpLRAx1h0PEeEto3Z1MPGLp7bw=w640-h180" width="640" /></a>Adding in the auxiliary line $DE$, parallel to $BC$, and joining $D$ to $C$, creates $60^\circ$ angles (all shaded in red), and thus equilateral triangles. Since $BFC = 50^\circ$, $BCF$ is isosceles, so the purple line segments are equal, and since $BCG$ is equilateral, the yellow line segments are equal to the purple line segments. This means that triangle $BFG$ is isosceles, and so the pink angles must both be $80^\circ$, which means that the brown angles must both be $40^\circ$, which means that the two green triangles $DEF$ and $GEF$ are congruent. And so $x$ is half of angle $DEG$, which is $30^\circ$. </p><h3>References</h3><p>BAGs to Learn Podcast by Ben Gordon (2021, December 2). Episode 4 – Colin Foster – Problem Solving in the mathematics curriculum [Audio podcast]. <a href="https://anchor.fm/ben-gordon83/episodes/Episode-4---Colin-Foster---Problem-Solving-in-the-mathematics-curriculum-e1b5ic3">https://anchor.fm/ben-gordon83/episodes/Episode-4---Colin-Foster---Problem-Solving-in-the-mathematics-curriculum-e1b5ic3</a></p><p>Baldry, F., Mann, J., Horsman, R., Koiwa, D., & Foster, C. (2021). The use of carefully-planned board-work to support the productive discussion of multiple student responses in a Japanese problem-solving lesson. <i>Journal of Mathematics Teacher Education</i>. Advance online publication. <a href="https://doi.org/10.1007/s10857-021-09511-6">https://doi.org/10.1007/s10857-021-09511-6</a></p><p>Chen, Y. (2019). 103.39 A lemma to solve Langley’s problem. <i>The Mathematical Gazette, 103</i>(558), 521-524. <a href="https://doi.org/10.1017/mag.2019.121">https://doi.org/10.1017/mag.2019.121</a></p><p>Fischer, B., Margulies, S., & Mosenfelder, D. (1982). <i>Bobby Fischer teaches chess</i>. Bantam Books.</p><p>Foster, C. (2019). The fundamental problem with teaching problem solving. <i>Mathematics Teaching, 265</i>, 8–10. <a href="https://www.atm.org.uk/write/MediaUploads/Journals/MT265/MT26503.pdf">https://www.atm.org.uk/write/MediaUploads/Journals/MT265/MT26503.pdf</a></p><p>Foster, C. (2021). Problem solving and prior knowledge. <i>Mathematics in School, 50</i>(4), 6–8. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Problem%20solving%20and%20prior%20knowledge.pd">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Problem%20solving%20and%20prior%20knowledge.pd</a>f</p><p>Foster, C. (2023). Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics. <i>The Curriculum Journal</i>. Advance online publication. <a href="https://doi.org/10.1002/curj.213">https://doi.org/10.1002/curj.213</a></p><p>Langley, E. M. (1922). Problem 644. <i>The Mathematical Gazette, 11</i>(160), 173. <a href="https://doi.org/10.2307/3604747">https://doi.org/10.2307/3604747</a> </p><p>Leikin, R. (2001). Dividable triangles—what are they? <i>The Mathematics Teacher, 94</i>(5), 392-398. <a href="https://doi.org/10.5951/MT.94.5.0392">https://doi.org/10.5951/MT.94.5.0392</a></p><p>Quadling, D. A. (1978). Last words on adventitious angles. <i>The Mathematical Gazette</i>, 174-183. <a href="https://doi.org/10.2307/3616686">https://doi.org/10.2307/3616686</a></p><p>Rike, T. (2002). An intriguing geometry problem. <i>Berkeley Math Circle</i>, 1-4.</p><p>Schoenfeld, A. H. (1985). <i>Mathematical problem solving</i>. Elsevier.</p><p>Shearer, C. & Agg, K. (2019). <i>Geometry puzzles in felt tip: A compilation of puzzles from 2018</i>. Independent.</p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com2tag:blogger.com,1999:blog-2036014053389751696.post-31683898660867173392023-03-16T07:00:00.002+00:002023-03-16T07:00:00.152+00:00Crocodiles and inequality signs<p><i>Teachers of mathematics seem not to be particularly fond of crocodiles/alligators when they are used to give meaning to the inequality signs $<$ and $>$. What are the problems with doing this, and should we resist all anthropomorphising or zoomorphising of mathematical symbols?</i></p><p>It's long been noticed that a crocodile's mouth looks a little bit like an inequality symbol, $<$. Furthermore, crocodiles are (apparently?) greedy and, when given the choice, always eat the larger object (Figure 1). And so we can use this as a rationale for writing $3<4$ and $4>3$.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixyYQKkdGvd75W1LMM2dXcIB8a62LEltnoeMuXKcL1FtCVXOTYHmFxTZqXbJX1Ijd5gDxapnSbCOw0KDz9t8w8icZdflFcMAMPSOIcFRr6nGFskBkGxb9nJ1ftSpl9_v6xTsFlV80ova_IxD055MFtkP7R-abbDQMbbIn_wQ-mrkk2k-YhflDPKNFAPA/s4616/crocodile%20image.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1780" data-original-width="4616" height="246" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixyYQKkdGvd75W1LMM2dXcIB8a62LEltnoeMuXKcL1FtCVXOTYHmFxTZqXbJX1Ijd5gDxapnSbCOw0KDz9t8w8icZdflFcMAMPSOIcFRr6nGFskBkGxb9nJ1ftSpl9_v6xTsFlV80ova_IxD055MFtkP7R-abbDQMbbIn_wQ-mrkk2k-YhflDPKNFAPA/w640-h246/crocodile%20image.png" width="640" /></a></div><p style="text-align: center;"><i>Figure 1. Will a hungry crocodile eat a mouse or an elephant?</i></p><p>Now, framed in this way, clearly this is a bit silly. Even very young children are likely to wonder:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>What if the crocodile isn't hungry?</i></p><p><i>What if the smaller animal is tastier, more nutritious, easier to catch, or less likely to attack than the larger </i><i>animal</i><i>?</i></p></blockquote><p style="text-align: left;">Is this 'just a bit of fun', not to be taken too seriously? Are we showing a 'sense of humour failure' by making a fuss? Or is this perhaps where, even at a really young age, children begin to subtly experience being asked to suspend all common sense - and, indeed, their age-appropriate knowledge of biology - when doing mathematics? Perhaps this is an example of when children begin to get used to the idea that success in mathematics means accepting nonsensical statements and claims? Before long, learners will talk as though some things are "true in maths but not in real life". Is this <i>zoomorphism</i> of an inequality symbol the thin end of a $<$-shaped wedge that we would just be much better off without?</p><p>I tend to think that the crocodiles are really unnecessary here. Certainly, the $<$ symbol has a 'small end' and a 'large end'. The greater quantity goes at the larger end, and that's all you need to know. There is no need to bring animals with long jaws into it at all. But are my observations on the shape of the $<$ symbol also playing with symbols in an unhelpful way and part of the same kind of problem - trying to make the 'abstract' symbol more 'iconic' than it really is?</p><p>What does the research say? Wege, Batchelor, Inglis, Mistry and Schlimm (2020) found that mathematical signs that visually
resembled the concepts they represent were easier to use than those that didn't, and they advised, for instance, "choosing symmetric symbols for commutative operations and asymmetric symbols for non-commutative operations" (p. 388). It seems that even experienced mathematicians find it easier to work with symbols whose properties mirror those of the concept being expressed. Imagine being forced to use $<$ to represent 'greater than' and $>$ to represent 'less than'. There is more than unfamiliarity to overcome there, but an obstinate reversal of the 'natural' way round that fits the meaning of these symbols. Although symbols like this are clearly arbitrary, in that they could have been otherwise, that doesn't mean that they were created totally at random, without rhyme or reason.</p><p>The equals sign $=$, for example, originates from the idea of expressing ‘equality’ of the left-hand side (LHS) and the right-hand side (RHS) by using two, parallel, ‘equal’ lines:</p><p style="text-align: center;">LHS $=$ RHS</p><p>Understood this way, one of those lines can be thought of as representing the value of the LHS of the equation and the other one the value of the RHS of the equation. But it is unclear which is which. This doesn’t really matter, I suppose, but perhaps this symbol would be more transparent if if it were rotated through $90^\circ$:</p><p style="text-align: center;">LHS $\lvert \rvert$ RHS</p><p>Now, the left-hand vertical line represents the value of the LHS and the right-hand vertical line represents the value of the RHS, and these being the same length represents equality of the two sides.</p><p>I suppose there could be the danger with this of thinking that <span style="text-align: center;">$\lvert \rvert$</span> was the number eleven, but we get round that kind of ambiguity all the time with mathematical symbols. For instance, we use modulus symbols to write things like</p><p style="text-align: center;">$\lvert -11 \rvert=11,$</p><p>carefully making the modulus lines a little longer that the $1$s, and we seem to get away with having at least five different meanings for a short line segment in this equation. </p><p>But the really nice thing about using <span style="text-align: center;">$\lvert \rvert$</span> for equality would be that we could use very natural symbols for greater than and less than:</p><p style="text-align: center;">$3 < 4$ becomes $3 $ <span style="font-size: x-small;">|</span>$\rvert \; 4 $</p><p style="text-align: center;"><span style="text-align: left;">$4 > 3$ becomes $4 \; \lvert$<span style="font-size: x-small;">|</span>$ \; 3$</span></p><p>The ‘rule’ (if you even need to call it that) is that the shorter line refers to the smaller side and the longer line refers to the greater side. But it hardly even needs saying. You could just start using symbols with this level of transparency, and learners would quickly infer what was going on.</p><p>I am not saying this would be worth doing. This post is not really making any practical suggestions; rather, it is a thought experiment. When might there be a benefit in replacing something arbitrary with something a little bit less arbitrary (though still arbitrary!)? Or in involving learners in discussing what kinds of symbols they think might appropriately represent various operations or relations? Should we always go straight to the conventional, correct symbol that 'mathematicians' use? Or are there times when it might be worth using more transparent but informal alternatives, while learners are in the process of getting to grips with the concepts, and then later transition to the more formal symbols, perhaps after appreciating some of the inconveniences with the less formal versions?</p><p>After all, writing <span style="text-align: center;">$3 $ </span><span style="font-size: x-small; text-align: center;">|</span><span style="text-align: center;">$\rvert \; 4 $ is all very well, but how would you express $3 \le 4$, and how would you handle double (or more) inequalities, like </span><span style="text-align: center;">$3 < 4 < 5$? I think it would get quite messy and awkward. In our usual notation, I do</span><span style="text-align: center;"> like the way in which we show 'approximately' by making the straight lines wiggly in $3 \approx 4$ and I like how we can show 'much less than' as $3 \ll 4000$. I like how in $\LaTeX$ we can even extend this by using the code '\lll' and write $3 \lll 10^{100}$, with a symbol composed of </span><i style="text-align: center;">three</i><span style="text-align: center;"> less-than symbols.</span></p><p>I am left wondering whether it is any worse to call the $<$ symbol 'a crocodile' than it is to refer to $\bar{x}$ as '$x$ bar' or $\hat{x}$ as '$x$ hat'? Can these really be 'informal' names if they are what 'everyone' calls them? I notice as I type these in <span style="text-align: center;">$\LaTeX$ that 'bar' and 'hat' are precisely the words I need to type to produce them, so knowing these names is actually quite useful.</span></p><h3 style="text-align: left;">Questions to reflect on </h3><p>1. When is anthropomorphising or zoomorphising mathematical symbols OK and when is it not?</p><p>2. When are 'informal' names for symbols OK and when should they be avoided?</p><p>2. Is it ever worth introducing made-up, informal versions of symbols (or names for them), with learners? Can they be a useful stepping stone towards formal symbols, or are they just extra things that learners will have to 'unlearn' later? </p><h3 style="text-align: left;">Reference</h3><p>Wege, T. E., Batchelor, S., Inglis, M., Mistry, H., & Schlimm, D. (2020). Iconicity in mathematical notation: Commutativity and symmetry. <i>Journal of Numerical Cognition, 6</i>(3), 378-392. <a href="https://doi.org/10.5964/jnc.v6i3.314">https://doi.org/10.5964/jnc.v6i3.314</a> </p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com3tag:blogger.com,1999:blog-2036014053389751696.post-27807995046870126302023-03-02T07:00:00.077+00:002023-03-02T07:48:40.334+00:00Are probabilities and inequalities approximate?<p><i>If mathematics is about being certain and precise, then how can probability be part of mathematics, because probability is about <b>not</b> being sure?</i></p><p>Probabilities are all about measuring and quantifying uncertainty. But I think that students are often a bit confused about what this means. One thoughtful student began writing all her answers to probability questions using the $\approx$ symbol. When asked why she was doing this, she said, “Well, probabilities are just probabilities – they’re not exact”.</p><p>It struck me that there are a few different things that she might have meant by this. She might have meant that, when flipping a £1 coin, say, p(Heads) $\approx\frac{1}{2}$, because no coin toss in the real world can ever be perfectly balanced, with precisely equal probability of landing on either side. Any <i>real</i> coin, undergoing any <i>real</i> throw, will be at least a little bit biased one way or the other (Note 1). So, maybe the $\approx$ symbol is communicating this approximate feature. However, that would seem to apply to <i>all</i> real-world measurements, of any kind, since no measurement can be made with absolute precision. If we say that the diameter of the coin is 22.5 mm, this will have to be $\pm$ some margin of error. So, on this basis, all lengths (and, indeed all measurements) would have to use the $\approx$ symbol too, and she wasn't doing that.</p><p>Alternatively, the approximate aspect that the student was thinking about might have been the uncertainty of the outcome on <i>any single coin flip</i>. On a frequentist view, probabilities are about long-run averages of relative frequencies, not individual instances. Even if we knew for some hypothetical coin that p(Heads) were precisely equal to $\frac{1}{2}$, that wouldn't help us to predict on <i>any given flip </i>whether the coin would come down Heads or Tails. There is still uncertainty, so perhaps it was this uncertainty that the student was wishing to capture in her use of the $\approx$ symbol. </p><p>Although $\frac{1}{2}$ is exactly in the middle of the probability scale that runs from $0$ to $1$, in a sense it represents <i>maximum</i> uncertainty, since if the probability were to take <i>any other value</i> we would stand a better chance of being able to predict the outcome on a single throw of the coin. If p(Heads) were 0.6, we could bet on Heads, and we'd expect to be right more than half of the time; if p(Heads) were 0.4, we could bet on Tails, and we'd expect to be right more than half of the time. But with p(Heads) at precisely 0.5 no strategy in the long-run will enable us to predict outcomes with better than 50% accuracy.</p><p>It can be hard to help students see that an uncertain outcome does not necessarily imply an approximate probability. We may be able to state a perfectly precise probability for an event, but, unless that probability is $0$ or $1$, we will still have uncertainty over what outcome we will obtain in any particular instance. I think I have often skated over such issues when teaching probability, and inadvertently left students thinking that the topic of probability is all about guesswork and approximation (e.g., subjective probabilities, such as that a particular football team will win a particular match).</p><h4 style="text-align: left;">Inequalities</h4><p>I have seen similar reactions from students to work on solving inequalities - it feels like it isn't proper mathematics, because we are not getting 'a definite answer'.</p><p>When we solve an equation like $2x+5=11$, we obtain an <i>exact</i> solution, $x=3$. We find that $x$ takes this one specific value, and no other, and that is that. But, when we solve an <i>inequality</i> like $2x+5>11$, we obtain a solution expressed as <i>another</i> inequality, $x>3$, and this may seem to students to be expressing some <i>uncertainty</i>, perhaps a bit like a probability. We've just replaced one vague inequality with another vague inequality; we still don’t exactly know what value $x$ takes! It might be $4$, it might be $3.1$, it might be $4$ $000$ $000$. There are infinitely many possibilities, just as there were before we began solving it, so it seems as though we have made little progress. "So we <i>still</i> don't know what $x$ is!" a student might complain.</p><p>'Solving an inequality' feels like a contradiction in terms. For the students, 'Solving' means 'Finding the answer'. They might concede to saying 'Or answers', perhaps for a quadratic equation, where they know that they haven't solved the equation until they've stated <i>all</i> the possible answers. Or, with simultaneous equations, where the values of <i>both</i> unknowns need to be found before someone can claim to have solved it. But here there are <i>infinitely many</i> possible answers, so we seem to know very little indeed about what the value of $x$ is!</p><p>However, infinitely many possible value have also been <i>ruled out</i>, so this <i>is</i> progress! We have eliminated all values of $x \le 3$. Before we began, $x$ could have been anywhere on the real number line; now we know that it can only be in the open interval to the right of $3$.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqVdy6aYUtTvYGw0Fr9aQaH9E2nlPTkOG22bUS93q28ZxkLbKeJqndQrDtIcc8QQRxS96boQR3zEvqvP4Uwe3Fuap1exLWAKL1lmKdPuR8HgGFvQPXyMCPAYeyia8aT9D8BkUXuiNK0acdLthn4UJfm8A1XpEoK5lWlNCmetDWmwWZ6Jafu9atKg1yig/s1210/x%20greater%20than%203.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="144" data-original-width="1210" height="77" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqVdy6aYUtTvYGw0Fr9aQaH9E2nlPTkOG22bUS93q28ZxkLbKeJqndQrDtIcc8QQRxS96boQR3zEvqvP4Uwe3Fuap1exLWAKL1lmKdPuR8HgGFvQPXyMCPAYeyia8aT9D8BkUXuiNK0acdLthn4UJfm8A1XpEoK5lWlNCmetDWmwWZ6Jafu9atKg1yig/w640-h77/x%20greater%20than%203.PNG" width="640" /></a></div><p>The fact that $x>3$ means that $x$ is "<i>definitely</i> more than <i>precisely</i> $3$" is, I think, sometimes not clear to students. They see inequalities as approximate because one way to think about them is that they capture uncertainty and tell us 'what $x$ might be'. This language of probability seems unfortunate here. If the solving of equations has been introduced to students through "I'm thinking of a number", and the student has to use the equation (like a 'clue') to figure out what the number is, then this may be problematic when we move to inequalities. The student has zero probability of being able to determine the teacher's secret number if the clue is 'just an inequality'.</p><p>Perhaps a better way to talk about this is in terms of <i>solution set</i>: all the values of $x$ that satisfy the equation or inequality. This way, we don't envisage that there is a single 'right answer', and we just unfortunately don't have enough information to determine it, since our single piece of information happens to be an inequality, which is 'imprecise' or 'vague'. Instead, we see our task as wanting to describe <i>all the possible values of</i> $x$ <i>that are consistent with the given information</i>. When we solve equations, that often turns out to be just one or two. With an inequality, we want to capture precisely those values that satisfy it. So $x>3$ is not saying that "$x$ is some number greater than 3, but we unfortunately don't know which number". Instead, we're saying that "the solution set is <i>all</i> of the numbers greater than 3 <i>and no others</i>".</p><p>I think this is the way I would deal with a problem I've sometimes seen, where a student writes something like $x>2$ and claims that this is correct. "No," you say. "The answer is that $x$ is greater than 3." And the student says, "Well, if the mystery number we're looking for is greater than 3, then it's certainly going to be greater than 2, so I'm right!" They think you can't mark them wrong for making a true statement about this 'mystery number'. Your answer may have pinned the number down slightly more tightly, by ruling out the numbers between 2 and 3, but $x>2$ is right too (in a way in which something like $x<2$ wouldn't be) (Note 2)!</p><p>The point is that we're not seeking <i>a single mystery number,</i> and trying to guess what it might be, but a solution set of <i>all the possible numbers</i>. The student's solution set $x>2$ contains a whole load of numbers that are less than or equal to 3, and these are not just unnecessary but <i>impossible</i>, so the student's solution set is the wrong one.</p><p>If we want to avoid these difficulties, then there is certainly more to solving linear inequalities than just "Solve it like an equation, but put the inequality sign instead of the equals sign!" But I think the idea of treating an unknown as a 'mystery number' perhaps has its problems when it comes to solving inequalities. We don't just want any old interval that definitely contains a certain mystery number; we want an interval that <i>doesn't contain</i> any numbers which the given inequality <i>rules out</i>. The language of <i>solution set</i> seems to make this much easier to talk about.</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. Have you encountered students having these kinds of questions/confusions?</p><p>2. How do you explain to students what is going on when they are solving inequalities?</p><h3 style="text-align: left;">Notes</h3><p>1. Interestingly, in practice, no matter what you do, it doesn't seem possible to create a significantly biased coin (Gelman & Nolan, 2002). (Of course, a double-headed coin would do the trick, though!)</p><p>2. This reminds me of a staffroom discussion about whether a student should receive most of the marks for obtaining a solution like $x<3$ to an inequality question to which the correct answer was $x>3$: "At least they got the right number; they just had the inequality sign the wrong way round" versus "They could hardly have been more wrong - the only possible answer that could have been <i>less</i> correct than this would have been $x \le 3$"!</p><h3 style="text-align: left;"><b>Reference</b></h3><p>Gelman, A., & Nolan, D. (2002). You can load a die, but you can't bias a coin. <i>The American Statistician, 56</i>(4), 308-311. <a href="https://doi.org/10.1198/000313002605">https://doi.org/10.1198/000313002605</a> ($)</p><p><br /></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-89531221343155747322023-02-16T07:00:00.034+00:002023-02-16T07:00:00.153+00:00Don't forget the units?<p><i>Sometimes, the units (e.g., cm) that come with a quantity can really help to make sense of what's going on. But do we always need units?</i></p><p>Recently, I was with some teachers who were arguing about a question like this:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>What is the area of this rectangle?</i></p></blockquote><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicWmYVip_JXAOmK39pyGoLjrk39YLFyZ-gzhAOUC3kETzwkTmoXug-4Lijdb2DXTvrPGvg0NtdrUJ-9JumfdmMQx5QOTd-XvtDJfWJ3rTUkA889VDjF4DS4SC8DFUPdUT74x9NBv1cABxswIZdnpsvjx9s-n6TJ1UH0w7h1WVC5cBzQ2Ce_Uklr2DeGA/s516/Fig%201.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="287" data-original-width="516" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicWmYVip_JXAOmK39pyGoLjrk39YLFyZ-gzhAOUC3kETzwkTmoXug-4Lijdb2DXTvrPGvg0NtdrUJ-9JumfdmMQx5QOTd-XvtDJfWJ3rTUkA889VDjF4DS4SC8DFUPdUT74x9NBv1cABxswIZdnpsvjx9s-n6TJ1UH0w7h1WVC5cBzQ2Ce_Uklr2DeGA/s320/Fig%201.png" width="320" /></a></div><p>Some of the teachers were complaining that the question was ‘wrong’, because the question writer had apparently ‘forgotten the units’! This was seen as ironic, because we are always telling students, “Don’t forget to include the units”, and yet here was a situation where this error had apparently been made in the writing of the question.</p><p>“You can’t have an area of 8,” someone said – “it has to be 8 <i>somethings</i>, like 8 centimetres squared.” The whole question was completely unspecified – what on earth is a “4 by 2” rectangle – imagine going into a shop to buy a carpet that is “4 by 2” – without some units it is completely useless!</p><p>I didn’t agree. I am quite happy to have a line segment of length 4 or a rectangle of area of 8. In pure mathematics, these things are dimensionless numbers. When we calculate the area enclosed between the curve $y=x^2$ and the curve $y=x(2-x)$, the answer is $\frac{1}{3}$. It doesn’t have any units, even though it <i>really is</i> an area. It isn’t $\frac{1}{3}$ of anything in particular, although I suppose you could say that it is $\frac{1}{3}$ of a unit square, if you wanted to. It would certainly be absurd to write the answer as $\frac{1}{3}$ cm$^2$ unless you were in some applied context in which you’d stated that the scales along the $x$ and $y$ axes were marked off in centimetres. The same applies if we use a formula like $$\int_a^b \sqrt{1+\left( \frac{dy}{dx} \right)^2} dx$$ to calculate the length along a curve: the answer has no units.</p><p>Confusion about units with things like this leads to students thinking that they need to write the words ‘square units’ after a definite integral. Similar confusions sometimes lead students to want to write the word ‘radians’ after $$\int_0^1 \frac{1}{1+x^2} dx=\tan^{-1} 1=\frac{\pi}{4},$$ or possibly even writing $$\int_0^1 \frac{1}{1+x^2} dx=45°,$$ which makes no sense at all! (How would you respond to the question: "When you do a definite trigonometric integral, should you give the answer in radians or degrees?")</p><p>The debate around units seems to be one where both sides think that the other side is demonstrating some kind of dangerous misconception. Contexts are very important, as are the applications of mathematics, but I am not convinced that everything is always made clearer by setting it in context. The abstract concept of area can be used to solve real-world problems, like painting walls and laying carpets, but there is also just the abstract notion of area, which is measured in dimensionless numbers. If you are happy that you can have a number like 8, all on its own, which isn't a measure of anything in any particular unit, then it ought to be OK to have a length of 8 or an area of 8 too.</p><p>A similar issue arises when people object to tasks like:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>Write down 5 positive integers with a mean of 7.</i></p><p><i>Write down 5 positive integers with a mean of 7 and a median of 4. </i></p></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>etc.</i></p></blockquote><p>Sometimes, teachers object that when you are calculating a mean of a real-life dataset the data points are very unlikely to be nice, neat positive integers. And would it really make sense to calculate summary statistics from data sets containing so few numbers? How meaningful is a median when there are just 5 data values altogether? People who object in this kind of way would be much happier if all of the data points had a couple of decimal places, and ideally would like us to have 500 data points, rather than 5, and handle them in a modern fashion using technology.</p><p>This all seems very valuable to me, and I am all for students grappling with realistic, messy datasets, with all the opportunities they present for data cleaning, examining outliers and using descriptive statistics to get a handle on what’s going on. In such a task, there is a purpose - something you want to find out from the data - and the focus becomes less on the nitty gritty of adding up and dividing and more on asking meaningful questions and using the mathematics to figure out meaningful answers (i.e., mathematical modelling).</p><p>But I don’t see that kind of work as an <i>alternative</i> to tasks like the ones above. The arithmetic and geometric means (as well as the harmonic mean and other kinds of mean) are all essentially (and certainly were originally) <i>pure mathematics concepts</i>. Certainly, they have important applications to statistics, and elsewhere, but if you are dealing with the <a href="https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means" target="_blank">AM-GM inequality</a>, for instance, there is no reason to think that the quantities being averaged must constitute some kind of 'realistic data set'.</p><p>There are times when context really helps students to get a sense of the underlying mathematics, but there are also times when context can get in the way. It seems likely to me that learners might get a better understanding of what the mean ‘means’ by using - at least at first - simple, easy to apprehend numbers. Tasks, for instance, in which you add an additional small integer value to a small, simple data set, and notice if the mean increases or decreases, or combine two small, simple data sets of different sizes, and explore what happens to the mean, seem very valuable to me. It can be through this kind of work that learners build a sense that the mean is the 'equal shares' value that 'balances' all the values in the set. In such tasks, it would be impossible to notice anything amid the noise of vast quantities of awkward numbers. Later on, of course, when <i>applying</i> the concept of the mean to the real world, we can bring those insights to bear on larger, more realistic data sets, but having messiness from Day 1, as the default, seems undesirable to me. I think there is no reason to feel guilty about asking learners to find the mean of a few small positive integers.</p><p>This is not to say that it's always wrong to 'begin with complexity'. Often that can be motivating and lead to powerful mathematics. But paring back the complexity at times, so as to see the mathematical structure, can also be really insightful and can support powerful generalisations. When we say that a 2 by 4 rectangle has area 8, we are effectively making a general statement that subsumes all of these:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p>A 2 cm by 4 cm rectangle has an area of 8 cm$^2$. </p><p>A 2 km by 4 km rectangle has an area of 8 km$^2$. </p><p>A 2 mile by 4 mile rectangle has an area of 8 mile$^2$. </p><p>etc.</p></blockquote><p>For me, to say "a 2 by 4 rectangle has area 8" isn't wrong, even though it doesn't apply directly to any one specific <i>real-world</i> area.</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. Do you feel that 'an area of 8' is wrong? Why / why not?</p><p>2. When do you think that contexts are helpful and when do they get in the way?</p><p><br /></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com2tag:blogger.com,1999:blog-2036014053389751696.post-24519993347393789392023-02-02T07:00:00.006+00:002023-02-02T09:05:47.044+00:00Non-expository video clips<p><i>How can video clips be used effectively in the teaching of mathematics? And I don't mean clips of someone explaining something...</i></p><p>If I do a <i>Google</i> search for "maths videos", I discover a tonne (258m hits) of short clips of mostly people explaining various bits of mathematics - with varying degrees of clarity and accuracy. I'm sure that some of these may have their uses, but they're not what I'm interested in in this blogpost. I assume that clips like those are rarely used in the classroom, if you have a live teacher who can do the explaining themselves in a more interactive fashion.</p><p>What I'm focused on in this blogpost are what I call 'non-expository' video clips. These are not trying to tell you something or explain something mathematical. They might not even be created with mathematics education in mind, although sometimes they are. But they are intriguing and engaging in their own right, and have obvious potential for mathematical discussions - whether it's to introduce a new concept or to apply some recently-taught ideas. They are just a minute or two at the most in length.</p><p>Lots of these I first found via <i>Twitter</i>, and I'd like to acknowledge whoever it might have been (now long forgotten) who forwarded them to me.</p><p>You don't need sound for any of these.</p><p>Here's the first one:</p><h4 style="text-align: left;">Cookie cutter</h4>
<blockquote class="twitter-tweet"><p dir="ltr" lang="en">Cookie cutter making machine <a href="https://t.co/XyGCn51N0V">pic.twitter.com/XyGCn51N0V</a></p>— Tool Of The Day (@toolotheday) <a href="https://twitter.com/toolotheday/status/1077263110958972929?ref_src=twsrc%5Etfw">December 24, 2018</a></blockquote> <script async="" charset="utf-8" src="https://platform.twitter.com/widgets.js"></script>
<p>I think you could use this clip with pretty much any age of learner. You could just play the clip and let discussion emerge, or you could ask:</p><p><i>What do you notice? What do you wonder?</i></p><p>Getting students to describe as precisely as they can in words what they have seen can be helpful in getting them talking about it, and mathematical terms might find their way into what they say quite naturally.</p><p>If the discussion doesn't take a mathematical turn by itself, you can always ask:</p><p><i>Where is the mathematics here?</i></p><p>or</p><p><i>What do you think is mathematical about this?</i></p><p>For me, this particular clip triggers thoughts about area and perimeter. If you wanted to be more directive, you could ask explicitly:</p><p><i>What does this have to do with perimeter?</i></p><p>There is still lots of room for different comments to be made, even with this level of direction. Someone could start by saying what they understand by the word 'perimeter'. Someone else might say that the perimeter is getting smaller, and then someone else might disagree with that and say that the perimeter is constant, but the <i>area enclosed</i> is getting smaller.</p><p>Another way to use the clip would be to introduce the idea of a cookie cutter first, so that everyone knows what one is, bearing in mind that not all children may have had experiences of home baking.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit5Ry6mvrY6PPpTwrNJStGXjFddUF7gRAyNv-sQWBh0q338RFQS9YEKqqTEMZP1Yv1G-WBkGztxk70Bg3fKlOGXDJRiWF5HpfTPlbenD0_fCI-2ai9Tu27zFI8F_JfstxGXkIVkGwW591tzkV2zMdN-uOssrSRJwc6Hu3MWbkjiEfFH1FSHLQ-679rVg/s5456/cookie_cutter_cookie_cutters.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3632" data-original-width="5456" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit5Ry6mvrY6PPpTwrNJStGXjFddUF7gRAyNv-sQWBh0q338RFQS9YEKqqTEMZP1Yv1G-WBkGztxk70Bg3fKlOGXDJRiWF5HpfTPlbenD0_fCI-2ai9Tu27zFI8F_JfstxGXkIVkGwW591tzkV2zMdN-uOssrSRJwc6Hu3MWbkjiEfFH1FSHLQ-679rVg/w400-h266/cookie_cutter_cookie_cutters.jpg" width="400" /></a></div><p style="text-align: left;">Or you could begin one step back from that, by showing a picture like this:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6_JNdC03hCAt6xc7GUXe47zRZIS1rGw-FRIMMs6GX6La2aTL-VAeXeIxncKvIQ-Z19caayUUNcwDzH9U3RQ8AuGqiTnhaVaxzqOVCFx9NMVRk0T6C3uThT3MNeVwql3j09GfYyMOz8yUU1kZSNJoeqQUlW0H5ZIYMLdYV50MLhhigVO0IH8h-WFJZew/s1371/gingerbread_gingerbread_men_cookies.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1024" data-original-width="1371" height="299" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6_JNdC03hCAt6xc7GUXe47zRZIS1rGw-FRIMMs6GX6La2aTL-VAeXeIxncKvIQ-Z19caayUUNcwDzH9U3RQ8AuGqiTnhaVaxzqOVCFx9NMVRk0T6C3uThT3MNeVwql3j09GfYyMOz8yUU1kZSNJoeqQUlW0H5ZIYMLdYV50MLhhigVO0IH8h-WFJZew/w400-h299/gingerbread_gingerbread_men_cookies.jpg" width="400" /></a></div><p style="text-align: left;">You could ask what kitchen equipment would be needed to make these, and then how they think cookie cutters themselves are made. Students might have interesting ideas about that - and then you could ask, "Would you like to see a video clip of how they are made?"</p><h4 style="text-align: left;">Drawing a freehand circle</h4><p style="text-align: left;">Here's another example of a clip that never fails to engage students:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/eAVpT34nxZM" width="320" youtube-src-id="eAVpT34nxZM"></iframe></div><p style="text-align: left;">Students could come to the board and see if they can do it. If you have an electronic whiteboard, you could record snapshots of their attempts (they could sign their name in the middle of their circle, so that you can tell which one is which). If you have a traditional board, you would need to take a photograph of the board after they have walked away (so that they aren't in the shot), before the board is wiped. Each student gets only one go. Then you can print out some of the best ones (you could get six onto one sheet of A4 paper) and ask:</p><p style="text-align: left;"><i>How can we fairly decide which one is the best circle?</i></p><p style="text-align: left;">This can go in broadly two directions.</p><p style="text-align: left;">(i) Students think about what measurements they would need to make on the drawn circles. There is a lot of mathematics to consider, and this is not straightforward, because, for instance, <a href="https://en.wikipedia.org/wiki/Curve_of_constant_width" target="_blank">shapes with constant width</a> are not necessarily circular. They might suggest covering the drawn circle with accurate circles - maybe sandwiching it between the largest circle that will fit completely inside it and the smallest circle it will fit completely inside, and then finding the difference between the diameters of those two circles, a smaller value representing a 'more circular' circle.</p><p style="text-align: left;">(ii) Students treat it as a statistical project and ask everyone in the class to rank the 6 circles from best to worst. Then they have to think about what to do with these rankings to come up with an overall answer (there are several options here). If they take this approach, to avoid bias they might prefer to obscure the names written in the middle of the circles. Ideally, they might wish to use participants from a different class, who wouldn't have a chance of remembering who had drawn which circle.</p><p style="text-align: left;">Work based on drawing freehand circles could test hypotheses such as that it is easier to draw a large circle accurately than a small one, or that people improve at this the more they do it, or that doing it faster is better than doing it more slowly.</p><p style="text-align: left;">I think that lessons that use activities such as these can be very memorable. The idea that the perimeter could remain constant while the area changes might be referenced by saying, “Remember the cookie cutters?”</p><p>If you browse around <a href="https://www.youtube.com/">https://www.youtube.com/</a> or a site such as <a href="https://free-images.com/">https://free-images.com/</a> you can find all sorts of interesting images and videos that could support this kind of activity. Many years ago, I collected some together myself at <a href="http://www.mathematicalbeginnings.com/">http://www.mathematicalbeginnings.com/</a>.</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. Do you have examples of non-expository video clips that you often use?</p><p>2. Would you use clips like the ones mentioned here? Why / why not?</p><p>3. Which class(es) would you use them with, and how would you use them?</p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-58907944493287296822023-01-19T07:00:00.016+00:002023-01-19T07:00:00.176+00:00Is zero really a number?<p><i>Zero is a strange number - learners sometimes even doubt if it is a number... </i></p><p>In my articles in <i><a href="https://www.m-a.org.uk/mathematics-in-school" target="_blank">Mathematics in School</a></i>, I often address 'Questions pupils ask' (I even have <a href="https://members.m-a.org.uk/Shop/product/1114" target="_blank">a book of that name</a>, Foster, 2017). In this blogpost, I'm going to address a question that came from a learner and that relates to zero and units:</p><p><i>“Is there any point writing ‘metres’ after zero, as it will just be equal to zero?”</i></p><p>In other words, is ‘zero metres’ just exactly the same as the number ‘zero’, <i>without</i> any units? (“Zero m is just zero!”)</p><p>Teachers often stress the idea that ‘$1$ metre’ is <i>not at all</i> the same as the number $1$. We know that $1$ is a dimensionless number, whereas $1$ m is a length. So, we would never dream of writing:</p><p style="text-align: center;">$1$ m $= 1$, </p><p>as this would be a dimensional catastrophe – as bad as saying something like ‘$1$ m $= 1$ kg’.</p><p>But is zero a different matter? If pupils think of ‘$1$ m’ as ‘$1$, multiplied by a metre’, as indeed it looks symbolically, then ‘$0$ m’ is ‘$0$, multiplied by a metre’, which is surely just zero, since zero multiplied by <i>anything</i> is just zero. Sometimes, when students are simplifying algebraic expressions (e.g., collecting like terms), they might simplify something like $$8a - 3m + a + m - 2a + 2m$$ by writing $7a + 0m$, but the teacher would probably say that there is no need to write the $+$ $0m$. Is it any more relevant to mention that "we haven't got any $m$'s" as it is to mention the absence of other quantities ($7a + 0m + 0r + 0mr + 0m^3$, etc.). In a particular context it might be worth being explicit about the zero $m$s (I sometimes find this useful when solving simultaneous equations by elimination, for instance, so as to keep everything nicely lined up in columns), but in general we wouldn't regard it as worth mentioning, so we would simplify to just $7a$. So, this feeling that $0m$ is just 'nothing' perhaps suggests that ‘zero metres’ should also be simplified to ‘zero’; i.e., nothing at all. Having no metres just means that you have nothing at all. Indeed, perhaps, on all the different dimensional scales of different quantities, the zeros coincide:</p><p style="text-align: center;">$0$ m $= 0$ kg $= 0$ °C $= 0$.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHr-xwm4Tgvt8851iiMW16gEyzFBeMJx1oBb-5NSVBAuqwv9Wz4pH0y2LgHB3W2KdfxsV8AWE6V1LeJAL9bAij51vvjL4-J0KA69bqP5H6Fva1ZN_p6WsH5Y9Q6TYn4tv_ilFoDwc32eOyKQwR5GrnhSRWj7RpZYT598n820lVOq3PdJc2ESheYx2onQ/s4127/Image.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1929" data-original-width="4127" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHr-xwm4Tgvt8851iiMW16gEyzFBeMJx1oBb-5NSVBAuqwv9Wz4pH0y2LgHB3W2KdfxsV8AWE6V1LeJAL9bAij51vvjL4-J0KA69bqP5H6Fva1ZN_p6WsH5Y9Q6TYn4tv_ilFoDwc32eOyKQwR5GrnhSRWj7RpZYT598n820lVOq3PdJc2ESheYx2onQ/w640-h300/Image.png" width="640" /></a></div><p style="text-align: center;"><br /></p><p>Clearly, it it not necessary to know any conversion factors to know that zero in <i>any</i> length unit, say, will be zero in any other length unit:</p><p style="text-align: center;">$0$ cm $= 0$ m $= 0$ inches $= 0$ furlongs $= ... $</p><p>But can it make any sense to write two zero measures on <i>different</i> dimensional scales as though they are equal, like $0$ m $= 0$ kg? This <i>really</i> looks wrong.</p><p>If it is right, does this mean that if a question says, “Give the units in your answer” that that is a subtle, unintended clue that the answer <i>won’t be zero</i>? Should a student <i>not</i> be penalised in an exam for omitting the units for a question where the answer is zero, such as this one?</p><p><i>The temperature on Tuesday is 2°C.<br />On Wednesday, it is 3°C warmer than it was on Tuesday. <br />On Thursday, it is 8°C colder than it was on Wednesday. <br />On Friday, it is 3°C warmer than it was on Thursday. <br />What temperature is it on Friday? <br />Give the units in your answer. </i></p><p>Surely not, as here the units are <i>very</i> necessary, since an answer of ‘$0$’ for temperature could be $0$ °F or $0$ K, since the zeroes of temperature certainly <i>don’t</i> coincide with one another, let alone with the zero for kilograms. Maybe ‘all zeroes are equal’ only applies to quantities with a ‘true zero’, as opposed to those with ‘arbitrary zeroes’, like temperature.</p><p>But can this be right? Zero oranges and zero apples do not necessarily represent the same state of affairs: just because I’ve run out of oranges, it doesn’t necessarily follow that I must have run out of apples too.</p><p>One possible response is to say that, with measurement, 'zero never really means zero'. So, a statement like '$0$ m' really means 'zero metres, to some degree of accuracy', so this represents not <i>a point</i> but <i>an</i> <i>interval</i> on the metres number line, such as $-0.5 \leq $ length $ \lt 0.5$. (Although can length be negative? Possibly, say if it's a difference between two other lengths.) And now clearly $-0.5 \leq $ length $ \lt 0.5$ is different from, say, $-0.5 \leq $ mass $ \lt 0.5$, so the problem goes away. But in pure mathematics we <i>can</i> have an exact zero that is not rounded to any degree of accuracy.</p><p>Questions like this can make zero seem like ‘not a number’, or at least not like any other number, in ways that students may find disturbing. I remember a Year 10 (age 14-15) student remaining behind after a mathematics lesson to ask me a question she didn't want others to hear. I assumed it would be something personal, but it turned out she was embarrassed to ask ("This is probably a really silly question", etc.) the question: "I have always wondered, but is zero actually a number?"</p><p>It might be tempting to dismiss such questions. Of course zero is a number - it's on the number line. What else would we put half way between $1$ and $-1$? Would we want the number line to have a tiny, infinitesimally small gap 'at zero'? And if zero isn't a number, what else would it be? But there are instances where zero does indeed seem to be in a class of itself. One example is that it's neither positive nor negative. There are three kinds of real number: positive ones, negative ones, and then a class all of its own for the single number zero. I sometimes catch myself saying that the topic of 'directed numbers' refers to positive and negative numbers, or I might even call the topic 'positive and negative numbers', forgetting to say 'and zero', which is 'central' to the whole thing. It is interesting to contrast the question "Is zero a number?" with the (possibly related) question "Is infinity a number?", which I suspect different mathematics teachers would answer in different ways.</p><p>A question like the one I've discussed in this blogpost may feel very abstract, and why should we worry about such an unusual question? I am not claiming that lots of students are asking this particular question all the time. But uncertainty over things like this perhaps contributes to students' feelings that mathematics doesn't make any sense. Sometimes, I suspect, students who are confused or stuck in mathematics, and who we might regard as 'having difficulties', are in that position because they have thought <i>further</i> (rather than less) than their peers.</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. Do you think that $0$ cm is 'the same thing' as $0$ kg? Would you dare to put an equals sign between them? Why/ why not?</p><p>2. Would you mark a zero answer wrong for not having the units?</p><p>3. What other issues do you see students having with zero?</p><h3 style="text-align: left;">References</h3><p>Foster, C. (2017). <i><a href="https://members.m-a.org.uk/Shop/product/1114" target="_blank">Questions Pupils Ask</a></i>. Mathematical Association.</p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com3tag:blogger.com,1999:blog-2036014053389751696.post-20636977030546241782023-01-05T07:00:00.033+00:002023-01-05T07:00:00.166+00:00Proportionality<p><i>If 'proportional' or 'multiplicative' thinking/reasoning is <b>the</b> central idea in age 11-14 mathematics, then how might we do a better job of making approaches to this consistent across the curriculum, so that students develop a deeper understanding?</i></p><p>At <a href="https://www.lboro.ac.uk/departments/maths-education/" target="_blank">Loughborough University</a>, with colleagues Tom Francome and Chris Shore, we are currently working on putting together <a href="https://www.lboro.ac.uk/services/lumen/curriculum/" target="_blank">a complete set of free, editable teaching resources for mathematics at Key Stage </a>3 (ages 11-14), and we are particularly trying to do so in a 'coherent' fashion, meaning that everything is connected together into a story that makes sense (Note 1; Foster, Francome, Hewitt, & Shore, 2021). We hope it will be ready to release later in 2023.</p><p>In this blogpost, I'm going to reflect on some of the thinking that has gone in the 'Multiplicative Relationships' Unit in Year 7 (ages 11-12). This Unit isn't yet finished, but we have a structure and a bit of the detail, which I wanted to share here.</p><p>It seems to me that 'proportionality' is <i>the</i> central idea in 11-14 mathematics that so much else is related to, and so we make a big deal within the 'story of the curriculum' of $y=mx$. Moving too quickly on to $y=mx+c$ just muddies the main point, so we save the '$+c$' for later on (Foster, 2022) (Note 2). Initially, we want students to become very comfortable working with $y=mx$. Think of all the things that come under the umbrella of $y=mx$ and can be viewed as instances of this:</p><p></p><ul><li>straight-line graphs through the origin</li><li>ratio and proportion</li><li>similar triangles</li><li>gradient</li><li>multiplication and division as inverses</li><li>speed, density and other 'rates'</li><li>rearranging formulae like $V=IR$ and $F=ma$</li><li>the basis for trigonometry (see Foster, 2021)</li></ul><p>Often each of these content areas comes with a different notation, and this is something that we want to address in the name of 'coherence'. For example, we might write $y=mx$ for line graphs but $y=kx$ for 'proportionality', and students may miss that the 'constant of proportionality' $k$ is precisely the gradient of the corresponding $x$-$y$ graph. There is much to gain by seeing all multipliers and rates as gradients, so we try to use consistent notation to highlight this.</p><p>In addition to the different notations across topics, multiple different representations are commonly used, even just within the topic of ratio and proportion, such as ratio tables, double number lines, etc. Perhaps because everyone agrees that proportional reasoning is hard, there is a temptation to throw everything at it, piling lots of different ideas on students, hoping that something will make sense and stick. Instead, in our curriculum design work, we have tried to avoid this, and instead choose one powerful approach and then use it consistently and probe into it deeply.</p><p>Because $x$-$y$ graphs are not just a representation but part of the content of the curriculum (unlike more 'optional' representations, like ratio tables, that some teachers use and others might not), we focus throughout the <a href="https://www.lboro.ac.uk/services/lumen/curriculum/" target="_blank">LUMEN Curriculum</a> on <i>number lines</i> and <i>Cartesian graphs</i> (which we see as two number lines coming together at right angles, intersecting at the origin). So, before we tackle 'proportionality' as such, we spend a lot of time making sense of multiplication through the family of graphs of $y=mx$.</p><p>We begin by taking two identical number lines and stretching one of them and (eventually) rotating it by 90°.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkobMmimEfiz3y2yPanE0uqTYJyevQbNMkSxIQ29IeSyzi6biPPAGVd_CanjLp7IFOBygfTAjFPUMsImEm9VVIM2qj99_Ked5hUj5ft8_3eDolgvIJ0S1xIQUugsDwc3hNO_x2W9v1pMgcIX0Pdt8bGrkrhuzSm_9cjFXxqgksozBGWEqFIbtbRc-rXQ/s2849/Fig%201.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2849" data-original-width="2333" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkobMmimEfiz3y2yPanE0uqTYJyevQbNMkSxIQ29IeSyzi6biPPAGVd_CanjLp7IFOBygfTAjFPUMsImEm9VVIM2qj99_Ked5hUj5ft8_3eDolgvIJ0S1xIQUugsDwc3hNO_x2W9v1pMgcIX0Pdt8bGrkrhuzSm_9cjFXxqgksozBGWEqFIbtbRc-rXQ/w524-h640/Fig%201.png" width="524" /></a></div><p>This develops into the idea of a rule, linking two number lines via points in the Cartesian plane.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBqREvGYKUWBI2JUztquvnht1U8PxY6KeNO8BKhNX_0DOOMYI1RwLjzKnIBHPUYWUUfZtVUI_fgnhsIU4rsbbJYlq40pjdIbhpD_AKREwnvxIL8BLFk-uxG1-0WJ4Kr6Z-t3QHOZMrUPqKoDC0tv7PGoFqu6mHMxfvwjS3XU75ppaMH6mZEUbfy7mF5Q/s2220/Fig%2010.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2220" data-original-width="1941" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBqREvGYKUWBI2JUztquvnht1U8PxY6KeNO8BKhNX_0DOOMYI1RwLjzKnIBHPUYWUUfZtVUI_fgnhsIU4rsbbJYlq40pjdIbhpD_AKREwnvxIL8BLFk-uxG1-0WJ4Kr6Z-t3QHOZMrUPqKoDC0tv7PGoFqu6mHMxfvwjS3XU75ppaMH6mZEUbfy7mF5Q/w350-h400/Fig%2010.png" width="350" /></a></div><br /><p>We go on to stress <i>multipliers,</i> like the gradient, $m$, as the key number that takes you, by multiplication, from any (non-zero) number to any other number:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji1ewZpSW7DpNf1CR5BC-_HWNYn67TnZGsRTu6iu2TCSH0NV2al7T2i7Tcs8jXb5JpKYgD9MAzg8cQ595luvyRd4QaU0Keov2o5uMtuKkb9d6jp7DTQwQpa2uJ-pWMdz78bE6ZhxdfyCPqGJ1pjJN3YgLoimxs5TW_vy0Wj_NjtdOYRqfSRdJ-Fp80Xg/s1124/Fig%202.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="624" data-original-width="1124" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji1ewZpSW7DpNf1CR5BC-_HWNYn67TnZGsRTu6iu2TCSH0NV2al7T2i7Tcs8jXb5JpKYgD9MAzg8cQ595luvyRd4QaU0Keov2o5uMtuKkb9d6jp7DTQwQpa2uJ-pWMdz78bE6ZhxdfyCPqGJ1pjJN3YgLoimxs5TW_vy0Wj_NjtdOYRqfSRdJ-Fp80Xg/w320-h178/Fig%202.png" width="320" /></a></div><p>We use lots of examples like this to practise finding multipliers and missing numbers:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdCeu2fhpPHnd1GriHxlN0kefl_OE6ZJdMnlDN9gzeLpM6CMeeHEYlXfr82zTy_ZhmkjP7yULBviXkM-XPKnzRrIdVeKkwnx6U3UgJnyusWWtZKc2QAWtXVLZddKmjx_u27EZa5P-qvY1S8uXlr54zclhK1g2MD5HVlE1tjd9v2CNOyI5A6b9WRiAV2A/s920/Fig%203.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="837" data-original-width="920" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdCeu2fhpPHnd1GriHxlN0kefl_OE6ZJdMnlDN9gzeLpM6CMeeHEYlXfr82zTy_ZhmkjP7yULBviXkM-XPKnzRrIdVeKkwnx6U3UgJnyusWWtZKc2QAWtXVLZddKmjx_u27EZa5P-qvY1S8uXlr54zclhK1g2MD5HVlE1tjd9v2CNOyI5A6b9WRiAV2A/w320-h291/Fig%203.png" width="320" /></a></div><p>We then tackle a mixture of proportion problems with 'nice' numbers as well as ones with 'hard' numbers. Below, we begin with a situation that uses 'nice' numbers. First we raise and discuss the 'additive' error:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSA1eYPWzMkrrPpRfom1v75cwcCITA5xOlXOoropmLp_iO6upbvtpwPzfttBLJ1V7toQl6vy8YjlKKft0IhfIw3gGhvc3z7z8v9AXQgkgq9F-bI-T-apTUuLOqg6ga8ix0qSuJRGG_o0aNDjhUI3kqcf0u9rnfBr8a_soRQWfgz2WnqBXqn8bJ-uzvoQ/s1583/Fig%204.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="995" data-original-width="1583" height="402" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSA1eYPWzMkrrPpRfom1v75cwcCITA5xOlXOoropmLp_iO6upbvtpwPzfttBLJ1V7toQl6vy8YjlKKft0IhfIw3gGhvc3z7z8v9AXQgkgq9F-bI-T-apTUuLOqg6ga8ix0qSuJRGG_o0aNDjhUI3kqcf0u9rnfBr8a_soRQWfgz2WnqBXqn8bJ-uzvoQ/w640-h402/Fig%204.png" width="640" /></a></div><p>Then, we work multiplicatively, first 'between variables' (i.e., from one variable to the other), using the same arrow notation for multipliers as we used earlier:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzqkasQN4FTrNKaWPMMy5naiwvWHlyr0pvZD9qsBcN2L9yiISUAl055D9L9zNVrga60a7rqM8ZfTyDUyOfSFr12_igVZsycKmZli_YrmABGe2xhdLCBKofZdB8WEYJTwgIFdGoWnBmktNvTPTpmg_M7blZXMIePl72GzRcYh9pXN2XNuJ-9JHNHyEfLg/s1858/Fig%205.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1858" data-original-width="1466" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzqkasQN4FTrNKaWPMMy5naiwvWHlyr0pvZD9qsBcN2L9yiISUAl055D9L9zNVrga60a7rqM8ZfTyDUyOfSFr12_igVZsycKmZli_YrmABGe2xhdLCBKofZdB8WEYJTwgIFdGoWnBmktNvTPTpmg_M7blZXMIePl72GzRcYh9pXN2XNuJ-9JHNHyEfLg/w504-h640/Fig%205.png" width="504" /></a></div><p>Then we find 'within variables' multipliers:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjasjhXNG5Kb8-ME9Gzv6MSIIEuotHXEiUKqN66v0TIY3ONkkVa9oYB1yd0UhrFbc3zc6K4Vs8adykILPB4m6oZ4bccKLYKKvTPG90WC44VSb4GEqynqiohgyTUEC1y-Kbb7Inj_G3Tk6fffpCmrmsigGwMUkjl8RXj6DQg1uq7lR5_a47P5Jp3RW9l1g/s1879/Fig%206.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1879" data-original-width="1433" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjasjhXNG5Kb8-ME9Gzv6MSIIEuotHXEiUKqN66v0TIY3ONkkVa9oYB1yd0UhrFbc3zc6K4Vs8adykILPB4m6oZ4bccKLYKKvTPG90WC44VSb4GEqynqiohgyTUEC1y-Kbb7Inj_G3Tk6fffpCmrmsigGwMUkjl8RXj6DQg1uq7lR5_a47P5Jp3RW9l1g/w488-h640/Fig%206.png" width="488" /></a></div><p>We contrast multipliers 'between variables' (which we call <b>rates</b>) with multipliers 'within variables', which we call <b>scale factors</b>. Scale factors are always dimensionless, whereas rates sometimes have units (e.g., here Rosie's multiplier was £/km). </p><p>Whether a rate or a scale factor is more convenient depends on the numbers; hence, tasks like this:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4pG8XszDTsmy-pe919WIDlLP3TtMWUNY1qxwnncedXwSIBqnzbci587Sn6iKJIkhu1rT6tRmfZFDVdUkdZNLi3XUIK-6-FOXCOgUN3dQQdX1v9Ol2aPQMZbSLGyUyJSEHKWeqkNbQbhZ5QnE7sVrTWackBKoGm6wfZ7laPEwyknfd7Xz6EaEZpraRpQ/s2033/Fig%207.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2033" data-original-width="1874" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4pG8XszDTsmy-pe919WIDlLP3TtMWUNY1qxwnncedXwSIBqnzbci587Sn6iKJIkhu1rT6tRmfZFDVdUkdZNLi3XUIK-6-FOXCOgUN3dQQdX1v9Ol2aPQMZbSLGyUyJSEHKWeqkNbQbhZ5QnE7sVrTWackBKoGm6wfZ7laPEwyknfd7Xz6EaEZpraRpQ/w590-h640/Fig%207.png" width="590" /></a></div><p>This eventually builds up to being able to use any of the elements from this kind of diagram:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwootBPQ0YMaDLQ8toOTkxrNlhSG8zdepLadWhXOMEAYMF7ux3P2LAXxlNth6eucB9dtqyhw4JStLBKjVRJSAJJNTNVRGqACtaCi2LW92DIlzV7iRETPTx9TSWm_JV3J3iGHuYYeJNxnaalI3AzNiZBSDqnHb0zK2uIxa1b-ox-H8RfXPcxhtyDunGLA/s1929/Fig%208.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1645" data-original-width="1929" height="546" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwootBPQ0YMaDLQ8toOTkxrNlhSG8zdepLadWhXOMEAYMF7ux3P2LAXxlNth6eucB9dtqyhw4JStLBKjVRJSAJJNTNVRGqACtaCi2LW92DIlzV7iRETPTx9TSWm_JV3J3iGHuYYeJNxnaalI3AzNiZBSDqnHb0zK2uIxa1b-ox-H8RfXPcxhtyDunGLA/w640-h546/Fig%208.png" width="640" /></a></div><p>(The diagram looks overwhelming with everything included. But in any situation you would only use 2 of these arrows at once, always the same colour as each other.)</p><p>This is all very much work in progress, and we'd be very glad of any thoughts or criticisms of what we're doing!</p><h3 style="text-align: left;">Questions to reflect on </h3><p>1. Do you agree about the centrality of proportionality in the lower secondary mathematics curriculum?</p><p>2. What do you like and dislike about the approach outlined here? </p><p>3. In what ways is it similar to or different from what you typically do?</p><h3 style="text-align: left;">Notes </h3><p>1. To find out more about the LUMEN Curriculum, go to <a href="https://www.lboro.ac.uk/services/lumen/curriculum/">https://www.lboro.ac.uk/services/lumen/curriculum/</a>. </p><p>2. And, when it comes, I think that $y=mx+c$ might be better encountered as $y-c=mx$. This way, rather than seeing $y=mx+c$ as a <i>non-example</i> of proportionality, we see it as <i>another example</i> of proportionality, but we just 'have the wrong origin'. So, the shift to $y-c$ as our variable, rather than $y$, makes it understandable as another instance of a proportional relationship. </p><h3 style="text-align: left;">References </h3><p>Foster, C. (2021). On hating formula triangles. <i>Mathematics in School, 50</i>(1), 31–32. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20On%20hating%20formula%20triangles.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20On%20hating%20formula%20triangles.pdf</a></p><p>Foster, C., Francome, T., Hewitt, D., & Shore, C. (2021). Principles for the design of a fully-resourced, coherent, research-informed school mathematics curriculum. <i>Journal of Curriculum Studies, 53</i>(5), 621–641. <a href="https://doi.org/10.1080/00220272.2021.1902569">https://doi.org/10.1080/00220272.2021.1902569</a></p><p>Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. <i>For the Learning of Mathematics, 42</i>(3), 21–27. <a href="https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf">https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf</a></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com1tag:blogger.com,1999:blog-2036014053389751696.post-75936557202974669512022-12-22T07:00:00.023+00:002022-12-22T07:00:00.151+00:00Mixing the dimensions in models of number<p><em>Not all commonly-used representations of number are necessarily equally helpful. We shouldn't just assume that anything 'visual' will automatically be helpful - some representations might even be counterproductive.</em></p><p>An emphasis on conceptual understanding in mathematics often seems to be linked with the use of visual representations. If you care about helping your students to make sense of mathematics – as opposed to just following standard, symbolic procedures – then you are likely to be keen on visual models. If a picture can paint a thousand words, then in mathematics a helpful visualisation might easily surpass multiple sentences of wordy explanation, whether written or oral, or many lines of mathematical symbols. It may also be that visualisation offers ways to support students who are more likely to be disadvantaged by traditional approaches to learning mathematics (Gates, 2018).</p><p>I very much like diagrammatic ways of showing mathematical properties and relationships (e.g., see Mattock, 2019, for a beautiful collection of examples), and some of my favourite books are the volumes of <em>Proofs without Words</em> (Nelson, 1993, 2000, 2016). However, I think a positive view of diagrams can sometimes lead us to be a little uncritical about cases where diagrams may be problematic. We should not just assume that anything ‘visual’ must be a good thing. In particular, if certain representations have the potential to create or embed particular misconceptions, so making understanding <i>harder</i> to achieve, then this is something we should worry about. The fact that students may say that they <em>like</em> certain visual representations is not enough, as it may not be possible for them to be aware at the time of possible problems coming down the line as a result of the representations that they are using. It relies on the teacher to look ahead and consider how future problems might be being set up by what is currently taking place.</p><p>I am a big fan of 1-dimensional, ‘linear’ models, such as number lines, and I include in this category any representation that has just one variable or dimension, even if it isn’t drawn in a straight line (see Foster, 2022). So, for me, a <i>circular</i> number line, like on a speedometer or clock, is still a kind of 1-dimensional number line, as is a <i>spiral</i> number line. So is a number <i>track</i>, such as the snaking squares on a snakes-and-ladders board, because all of these are still unidimensional representations (you can only go either forwards or backwards – ignoring the snakes and ladders themselves!). I see all of these as linear, even though they of course have to take up 2-dimensional space, otherwise we wouldn’t be able to see them (Note 1).</p><p>However, I think that I am coming round to the view that I am <em>not</em> a fan of <em>2-dimensional</em> representations of number, because they are inevitably <em>mixed-dimensional</em>, and I think this is quite problematic (Foster, 2022). To explain what I mean, consider ‘algebra tiles’, as embodied in diagrams (or physical or virtual manipulatives) like those shown in Figure 1 (Note 2).</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEha3lE_6YFZlhNrNIKQKEIE9vLMybtCkfuUjKHRm2mfbQVJz3l7VDm7XpHpFVPWB8NzrJj1MIJh5MjFHGbatlOeNDkKa6LFwiIYPUnt21zg03L9jt1PWNgb93xI66U9z-Q0r5DU2_8JREltdciSUMLw8bE3TaHJKFd0-NSbaC0k-E52zz0uhP0--KziJA=s800" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="173" data-original-width="800" height="138" src="https://blogger.googleusercontent.com/img/a/AVvXsEha3lE_6YFZlhNrNIKQKEIE9vLMybtCkfuUjKHRm2mfbQVJz3l7VDm7XpHpFVPWB8NzrJj1MIJh5MjFHGbatlOeNDkKa6LFwiIYPUnt21zg03L9jt1PWNgb93xI66U9z-Q0r5DU2_8JREltdciSUMLw8bE3TaHJKFd0-NSbaC0k-E52zz0uhP0--KziJA=w640-h138" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="text-align: left;"><br /><i>Figure 1. </i></span><span style="text-align: left;"><i>Algebra tiles representing (a) $3(x+2)\equiv3x+6$ and (b) $(3x+6)(x+1)\equiv3x^2+9x+6$</i></span></td></tr></tbody></table><div class="separator" style="clear: both; text-align: left;"><span><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span>In Figure 1a, the area 6 is represented by a blue 2 × 3 rectangle, and, in Figure 1b, we have another area of 6, this time represented by a blue 1 × 6 rectangle. This is fine, because it seems perfectly reasonable to say that both of these represent the same number 6 (Figure 2a), because they have the same area as each other. We could break up either one of them and fit it completely into the space occupied by the other. </span><span>However, my difficulty is that in Figure 1b we </span><em>also</em><span> have the number 6 represented by the (1-dimensional) purple </span><i>line segment</i><span> at the top right (see Figure 2b), meaning that the same number is represented, in the same diagram, by both a 1-dimensional line segment and a 2-dimensional area.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiAyInB5pAsmNZnIgHAzMaNcsL7zc8Lo-4ovkTDlHBnwzhmDkHlEw5ldH6kOMfvfd_AuanKBQyZNTrunV6C0d5N9azWq892nNwkdihCr7rIjFj60VwJ15UkiTCcnTJmZlRpYoUrMTYz7BGxrJRHzxJVs7KuF2SniqdWR98dcYO0BYTUOwhcVNQ8Bnx_lw=s400" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="198" data-original-width="400" height="198" src="https://blogger.googleusercontent.com/img/a/AVvXsEiAyInB5pAsmNZnIgHAzMaNcsL7zc8Lo-4ovkTDlHBnwzhmDkHlEw5ldH6kOMfvfd_AuanKBQyZNTrunV6C0d5N9azWq892nNwkdihCr7rIjFj60VwJ15UkiTCcnTJmZlRpYoUrMTYz7BGxrJRHzxJVs7KuF2SniqdWR98dcYO0BYTUOwhcVNQ8Bnx_lw=w400-h198" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-style: italic; text-align: left;"><br />Figure 2. (a) a reasonable equality; (b) an unreasonable equality</span><br style="font-style: italic; text-align: left;" /></td></tr></tbody></table><br /></div><div class="separator" style="clear: both; text-align: left;">I think this <em>mixed-dimensional</em> feature of area models of number is unfortunate, and becomes quite problematic the more you think about it. A rectangle and a line segment are not just different things (like two different rectangles) – they are different <em>kinds</em> of things. It’s OK if we sometimes represent a number by a 1-dimensional length, and other times by a 2-dimensional area – and we maybe sometimes represent it by all sorts of other things – but to do <em>both</em> of these simultaneously in the same diagram seems like asking for trouble. I am not sure how much relational understanding (Skemp, 1976) can be said to be going on if students have not noticed or thought about this. And this is not something that <em>occasionally</em> happens with representations like these, in certain awkward cases – it happens every time.</div><p>I am perfectly happy that there is no dimensional problem with writing an expression like $3x^2+9x+6$, because $x$ is a pure number, without any units, so this is just of the form 'number + number + number', which is equal (unsurprisingly) to 'a number'. But, as soon as you start to represent an expression like this using algebra tiles, it begins to look very much as though you have 'area + area + length', or maybe 'area + length + length'. In Figure 1a, $3x+6$ is 'area + area', but when, in Figure 1b, we want to multiply this expression further by $x+1$, we somehow have to shift our thinking down a dimension, and reconceive $3x+6$ as 'length + length', so that we can multiply it by another ‘length’ (the $x+1$), so as to obtain a quadratic expression, which is now represented as an area. Alternatively, we could retain $3x+6$ as area + area, and multiply it by length + length to give a <em>volume</em> in 3 dimensions, and, although this is tricky to sketch on paper, 3D models can be insightful. But then you are stuck if you want to go any further.</p><p>I think this problem is often overlooked, because algebra tiles are often used to multiply two linear expressions, like <em>$x+b$</em> and <em><em>$x+d$</em></em>. We are very pleased with the fact that the leading term in the expansion is <em><em>$x^2$</em></em>, and that ‘<em><em>$x$</em></em> squared’ is represented by ‘a square’. This seems great – we visualise an <em>algebraic</em> square by means of a <em>geometrical</em> square – what could be nicer? But, we overlook that the other terms in the expansion have less convenient interpretations – we have a $bx$ and a $dx$ rectangle, and a $bd$ rectangle, and it’s unclear why it’s appropriate for all three of these to be rectangles, given that things very like them were lengths at the beginning.</p><p>Even if you restrict ‘algebra’ tiles to numbers, and use them to work out things like $12\times46=(10+2)\times(40+6)=552$, I think you have exactly the same problem. The 2, say, in the original 12 looks visually like a completely different kind of a thing from the 2 in the 552, but they are both meant to be 2 ones. One of the most basic requirements of a good representation is that the same thing (e.g., 2) should be represented by the same thing (e.g., <i>either</i> a line segment of length 2 <i>or</i> a rectangle of area 2, but not both).</p><p>There is something nice about the dynamic in a classroom where students are fiddling around with physical algebra tiles, rearranging them and fitting them into rectangles and talking about what they are doing. It seems like just the sort of activity that should develop conceptual understanding. Students are actively manipulating representations that make the mathematical structure visible and figuring out what is possible and what is impossible. But, whether or not the jigsaws get completed, I worry that mixed-dimensional models like this have the potential to be more confusing than helpful.</p><p>I have expanded on the argument of this blogpost in Foster (2022).</p><h3 style="text-align: left;"><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. Do you use <i>mixed-dimensional</i> representations of number, like algebra tiles? If so, when and why?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. What do you think about the concerns I've expressed in this post?</span></p><h3 style="text-align: left;">Notes</h3><p>1. Manipulatives, like Cuisenaire rods or cubes, are harder to classify, because you can do lots of things with them, and not all of these things are 'linear'. For example, you can use them to make rectangles with an area of 12 square units. So I think whether they are ‘linear’ or not depends on what you do with them.</p><p>2. To generate diagrams like this conveniently, go to <a href="https://mathsbot.com/manipulatives/tiles">https://mathsbot.com/manipulatives/tiles</a>.</p><h3 style="text-align: left;">References</h3><p>Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. <i>For the Learning of Mathematics, 42</i>(3), 21–27. <a href="https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf">https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf</a></p><p>Gates, P. (2018). The importance of diagrams, graphics and other visual representations in STEM teaching. In R. Jorgensen, & K. Larkin (Eds), <em>STEM education in the Junior Secondary: The state of play</em> (pp. 169-196). Springer. <a href="https://www.researchgate.net/profile/Peter-Gates-4/publication/319086868_The_Importance_of_Diagrams_Graphics_and_Other_Visual_Representations_in_STEM_Teaching/links/5d094203299bf1f539cef6d3/The-Importance-of-Diagrams-Graphics-and-Other-Visual-Representations-in-STEM-Teaching.pdf">https://www.researchgate.net/profile/Peter-Gates-4/publication/319086868_The_Importance_of_Diagrams_Graphics_and_Other_Visual_Representations_in_STEM_Teaching/links/5d094203299bf1f539cef6d3/The-Importance-of-Diagrams-Graphics-and-Other-Visual-Representations-in-STEM-Teaching.pdf</a></p><p>Mattock, P. (2019). <em>Visible Maths: Using representations and structure to enhance mathematics teaching in schools</em>. Crown House Publishing Ltd.</p><p>Nelsen, R. B. (1993). <em>Proofs without words: Exercises in visual thinking</em>. The Mathematical Association of America.</p><p>Nelson, R. B. (2000). <em>Proofs without words II: More exercises in visual thinking</em>. Washington. The Mathematical Association of America.</p><p>Nelson, R. B. (2016). <i>Proofs without words III: Further exercises in visual thinking</i>. The Mathematical Association of America.</p><p><!--wp:paragraph-->
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<!--/wp:paragraph--></p><p>Skemp, R. (1976). Instrumental understanding and relational understanding. <em>Mathematics Teaching, 77</em>, 20-26. <a href="http://math.coe.uga.edu/olive/EMAT3500f08/instrumental-relational.pdf">http://math.coe.uga.edu/olive/EMAT3500f08/instrumental-relational.pdf</a></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com3tag:blogger.com,1999:blog-2036014053389751696.post-9389479984698867692022-12-08T07:00:00.006+00:002022-12-08T07:00:00.163+00:00Dividing into thirds<p><i>How accurately do things need to be</i><i> drawn</i><i> to evidence conceptual understanding? When are accurate drawings helpful and when are they unnecessary?</i></p><p>Suppose that you asked a child to divide a disc into thirds, and suppose they drew something like this:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhC9K9kSt2tpO4ZebigjbYi5REeNbEs1J3p0WMYTYypqC5rZYO81twHOtyNtyRu4fHEaWv4srNCR8MeZ3IGaJDCZ--2tQQ9Wgt-SwmSlqQL1TUmPGCDjFaY8mraztop9FWDP_OUOvZrhJwLwY1Yq_ngbYGPk77eysOcyCHHg0yLcS5MKT5lv9jAI9tVkQ=s1768" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1587" data-original-width="1768" height="179" src="https://blogger.googleusercontent.com/img/a/AVvXsEhC9K9kSt2tpO4ZebigjbYi5REeNbEs1J3p0WMYTYypqC5rZYO81twHOtyNtyRu4fHEaWv4srNCR8MeZ3IGaJDCZ--2tQQ9Wgt-SwmSlqQL1TUmPGCDjFaY8mraztop9FWDP_OUOvZrhJwLwY1Yq_ngbYGPk77eysOcyCHHg0yLcS5MKT5lv9jAI9tVkQ=w200-h179" width="200" /></a></div><p style="text-align: left;">How would you respond? Are they right? It is only a sketch, after all.</p><p style="text-align: left;">Now imagine an equally scruffy sketch like this:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhxJKTg1z58vN3lEtlcb4WirM9Tk86t3TsktvHlZBXAr-FSSqyMa2DDsk7Y-gOvrdL1tum5ZiHZ9MknjhUamY7pbwrWcf5GQJb-glbwh8dy1Oh00m08wce7K-N9sLLuzyd_QwNIY32vpeUDsAccBENoiWdlM0JGTR0qw5AKX7hjqe-3jZO7F_A-ykY5HA=s1783" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1783" data-original-width="1781" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEhxJKTg1z58vN3lEtlcb4WirM9Tk86t3TsktvHlZBXAr-FSSqyMa2DDsk7Y-gOvrdL1tum5ZiHZ9MknjhUamY7pbwrWcf5GQJb-glbwh8dy1Oh00m08wce7K-N9sLLuzyd_QwNIY32vpeUDsAccBENoiWdlM0JGTR0qw5AKX7hjqe-3jZO7F_A-ykY5HA=w200-h200" width="200" /></a></div><p>Full marks this time? I can imagine the second one being treated as more acceptable than the first, even if both were drawn equally accurately.</p><p>In fact, here, the first one of these is drawn <i>more</i> accurately into thirds than the second one is. But, might that just be 'luck', and not evidencing a clear understanding of what 'thirds' are? What are we trying to judge here? A small child may have a better eye than me, and be better at estimating equal areas accurately than I am. Is that the thing that matters?</p><p>Perhaps the teacher is expecting/hoping to see sectors drawn, and they view parallel lines suspiciously, as they worry that the child might be intending to indicate <i>evenly-spaced</i> lines, as shown below, which divide the vertical <i>diameter</i> into thirds, but would <i>not</i> divide the <i>area of the</i> <i>disc</i> into thirds. Intending to draw this might be counted as a 'misconception'.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgRftVlbR7c8cT1jFyfsKL-6Hr9KaA58c7HMa6FA_b39G9puf72Yixz7zLkDePX7RRxY4W_ZkcTzczmCOw3b7nK8KXaFNHYlfZ7YWFabBy1RzZy1_QfsAiKFGy0_Nd3F49f5zOSUNqEWsonLHOxVzi08b-gUwoS9Q1eJil3K226BaEhKBtjHekdfaABrA=s1446" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1446" data-original-width="1445" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEgRftVlbR7c8cT1jFyfsKL-6Hr9KaA58c7HMa6FA_b39G9puf72Yixz7zLkDePX7RRxY4W_ZkcTzczmCOw3b7nK8KXaFNHYlfZ7YWFabBy1RzZy1_QfsAiKFGy0_Nd3F49f5zOSUNqEWsonLHOxVzi08b-gUwoS9Q1eJil3K226BaEhKBtjHekdfaABrA=w200-h200" width="200" /></a></div><p>But, is it fair to the first child to assume that this is necessarily what they must be thinking? Without talking to them about their idea, it seems hasty to dismiss what they have drawn. But, they might struggle to express in words that the two lines are somewhat closer to each other than they are to the ends of the vertical diameter - and it would certainly be hard to say <i>how much</i> closer they ought to be (Note 1).</p><p>Dividing a circle into thirds precisely with parallel lines is tricky, and requires some calculation.</p><p>Suppose we have a unit circle below, centre $O$.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgOjJ6oj260fnSOBQKo0YxsZYkH1fC3G_I3KZVcNLpQxYoxjLRHgEqvxdmWy6JahjGvkwyLSlAbKvp53EwpSkSsOPTiG6pGpUyX8Q4u0LhEKAsqIREheY5c_89Qg5PiUZywdJJxE74JlJi_QuNgu07OXsV536-A65jlDIB9WpB3PZiXa3HD805xTygduQ=s1446" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1446" data-original-width="1445" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEgOjJ6oj260fnSOBQKo0YxsZYkH1fC3G_I3KZVcNLpQxYoxjLRHgEqvxdmWy6JahjGvkwyLSlAbKvp53EwpSkSsOPTiG6pGpUyX8Q4u0LhEKAsqIREheY5c_89Qg5PiUZywdJJxE74JlJi_QuNgu07OXsV536-A65jlDIB9WpB3PZiXa3HD805xTygduQ=w200-h200" width="200" /></a></div><p>We want to know the angle $\theta$ radian which makes the area of the blue shaded segment exactly $\frac{1}{3}$ of the area of the whole disc.</p><p>Now, </p><p>$$\text{area of blue segment}=\text{area of sector } – \text{ area of isosceles triangle}.$$</p><p>So,</p><p>$$\frac{\pi}{3}=\frac{1}{2}\times1^2\theta-\frac{1}{2}\times1^2\sin\theta,$$</p><p>giving</p><p>$$\frac{2\pi}{3}=\theta-\sin\theta.$$</p><p>This equation cannot be solved analytically, but we can get a numerical solution as accurately as we wish, and this turns out to be $\theta=2.605,325,...$ radian. Converting to degrees, this is $149.27^{\circ}$, correct to 2 decimal places. It is kind of neat that the required triangle is so close to a $15$-$15$-$150$ isosceles triangle.</p><p>Drawn accurately, it looks like this:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhSW70VFUywYFdmshfxm2NAdr-2PC1FK-WvKsbGEk3M20j41Ze6SfLuhIIvSxbOuUeqir1vNRSgiEbfR3fKCCMNU9RcUpuJegAlqteeEMysD7cvFftVCki2-2291UD8dMuev4z6ii52v4u62N1Zr3mfliNIwetJXNSFcl1L5yFBKOy79Cxz8qgCxACipw=s1639" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1627" data-original-width="1639" height="199" src="https://blogger.googleusercontent.com/img/a/AVvXsEhSW70VFUywYFdmshfxm2NAdr-2PC1FK-WvKsbGEk3M20j41Ze6SfLuhIIvSxbOuUeqir1vNRSgiEbfR3fKCCMNU9RcUpuJegAlqteeEMysD7cvFftVCki2-2291UD8dMuev4z6ii52v4u62N1Zr3mfliNIwetJXNSFcl1L5yFBKOy79Cxz8qgCxACipw=w200-h199" width="200" /></a></div><p>which might be a bit hard to distinguish from the equally-spaced <i>incorrect</i> version drawn above.</p><p style="text-align: left;">Here they are side by side:</p>
<table style="margin: auto; text-align: center;"><tbody>
<tr><td><img border="0" data-original-height="1446" data-original-width="1445" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEgc53pLCBQeZfL3uk0dSUccBH7Ei1piXNOGI2srlp6ceYywJRXZEOe_M_mWPnnA-S719KqtqZ2tZYKyG_bDelTMDeggFMMe4d_cNgQ9eEeMIIKRuEW1jQvWzqQgztdHKLS6aQ6Z20kWoX4MywMCrpGKd2As4GSYNZbsLhh0HuQY2nIiXfm5wmEouVmp4g=w200-h200" style="text-align: center;" width="200" /></td><td><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi951nmsVjtZSXFPnI961sisy56dl03DtqqYZoS4G1M1wT9fcuWX10yHQkdz9Jt-RX_rViArZ97EZLvEIhHgZJ_Dfs53i__bQI0YmL6rmv8PhjpIkjLCEsCsui7wiIBa56LcttsLggsdFkbl3QoXphXza2qWVBbULH_umZjhWU3BQKTX8_i_rbD-og2BQ=s1639" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="1627" data-original-width="1639" height="199" src="https://blogger.googleusercontent.com/img/a/AVvXsEi951nmsVjtZSXFPnI961sisy56dl03DtqqYZoS4G1M1wT9fcuWX10yHQkdz9Jt-RX_rViArZ97EZLvEIhHgZJ_Dfs53i__bQI0YmL6rmv8PhjpIkjLCEsCsui7wiIBa56LcttsLggsdFkbl3QoXphXza2qWVBbULH_umZjhWU3BQKTX8_i_rbD-og2BQ=w200-h199" width="200" /></a></td></tr></tbody></table>
<div style="text-align: center;"><span style="text-align: left;"><br /></span></div><div style="text-align: center;"><br /></div><div style="text-align: left;">With a bit of thought, it is clear that the vertical positions of these lines are just as good for <i>any </i>ellipse with a vertical major axis of this length, as others have noted (see <a href="https://www.forbes.com/sites/kevinknudson/2015/08/11/cutting-into-thirds-or-how-a-mathematician-spends-an-hour-figuring-out-how-to-divide-a-casserole/">https://www.forbes.com/sites/kevinknudson/2015/08/11/cutting-into-thirds-or-how-a-mathematician-spends-an-hour-figuring-out-how-to-divide-a-casserole/</a>):</div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgUupdfv4UVYeXwqPn_JsCbyWgCi84YC3xWIVbll0l-dc0E3rPkltqkniVDC8mhlg9ilti1LpySdxmUb5rMkZV-d5ZKycxJ7bdGdr-Y_0ey-L0pAPn1gnJ6fOtsTdgTWDiMerMmykEUNTqck_5AL7hfzxjYH5cSaJI_hy8shmpii5aOhYhGUbERHMD-3w=s1614" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1614" data-original-width="938" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEgUupdfv4UVYeXwqPn_JsCbyWgCi84YC3xWIVbll0l-dc0E3rPkltqkniVDC8mhlg9ilti1LpySdxmUb5rMkZV-d5ZKycxJ7bdGdr-Y_0ey-L0pAPn1gnJ6fOtsTdgTWDiMerMmykEUNTqck_5AL7hfzxjYH5cSaJI_hy8shmpii5aOhYhGUbERHMD-3w=w116-h200" width="116" /></a></div><p>This leaves me thinking that <i>no one</i> is going to be good at drawing these lines in the right position by eye. And I am not sure to what extent learning about fractions ought to be dependent on ability to estimate the relative sizes of areas bounded by arcs, like these. I don't think I'm particularly good at it (Note 2). But how should we get a sense of children's understandings of fractions when we look at their drawings and explanations? And should we be more cautious in our assumptions when we look at their drawings?</p><p>It isn't always clear what we intend to communicate when we make sketch drawings like these, and when the accurate details matter and when they don't. It certainly isn't easy to draw perfect circles freehand on a whiteboard (see Foster, 2015), and, even if you are using technology to display perfect circles, projectors can do funny, distorting things. I have seen a lesson on circle theorems in which none of the circles displayed was remotely circular. They looked fine on the teacher's computer screen (as they presumably did at home the night before, when planning the lesson), but, once projected, they were distinctly oval. How much does this matter? No circle in the real world can ever be completely perfect, so we always have to use our imagination. Every student in the lesson knew that the lesson was about 'circle theorems', not 'oval theorems', and that these images were <i>representing</i> circles. So was this therefore fine? Where the diagram is deficient, the viewer has to do some of the work to visualise it properly, and sometimes this can be helpful. But, in this case, given that it was an introductory lesson to circle theorems, I felt that the distorted figures were merely an extra burden on everyone's cognitive load. They seemed more likely to get in the way of the students' attempts to make sense of the relevant geometrical properties and relationships.</p><p>People find it much harder to judge the relative area of sectors than of rectangles (Burch & Weiskopf, 2014) - look at the same data below presented as a pie chart and then as a bar chart:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi6fv64yF6CwH402YurQ-V0yggxZB1bn_hGcZfBON1y3pqyQykI--Vju1iaU_iQaWTzOYpDLrWYLdoxxvM6Qy0AStqFEkI2PYtwPG2h4WPu-ovAgVjaa8W7eSg8yHyw_zDGe4VO3ltaxAt319ISddFoSIBWhxhldfGhiBnIKCoaJfwHFcFs0FFjV5GYKg=s8539" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2779" data-original-width="8539" height="208" src="https://blogger.googleusercontent.com/img/a/AVvXsEi6fv64yF6CwH402YurQ-V0yggxZB1bn_hGcZfBON1y3pqyQykI--Vju1iaU_iQaWTzOYpDLrWYLdoxxvM6Qy0AStqFEkI2PYtwPG2h4WPu-ovAgVjaa8W7eSg8yHyw_zDGe4VO3ltaxAt319ISddFoSIBWhxhldfGhiBnIKCoaJfwHFcFs0FFjV5GYKg=w640-h208" width="640" /></a></div><br /><p>Perhaps it is time we moved away from using circle diagrams for teaching fractions altogether (see Foster, 2022)?</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. How would you convince someone that the calculated position of the lines for a circle works just as well for any ellipse?</p><p>2. What do you see as the role of circle drawings like this for learning about fractions? Are other visuals (e.g. rectangles) preferable?</p><p>3. When is accuracy important and when isn't it?</p><h3>Notes</h3><p>1. I suppose you could also say that the division into sectors divides the <i>circle</i> (i.e., the circumference) into thirds, as well as divides the <i>disc</i> (i.e., the area) into thirds, whereas the parallel lines divide only the <i>area</i> into thirds.</p><p>2. Similar criticisms might be made about the value of training students to estimate angles in degrees with better accuracy. Tasks like this <a href="https://nrich.maths.org/1235">https://nrich.maths.org/1235</a> can be fun, but how important is this as a mathematical skill that is worth improving? I think that such tasks have value in gaining a sense of what 'one degree, ten degrees, etc.' look like, which seems as important as knowing what '1 centimetre' looks like.</p><h3 style="text-align: left;">References</h3><p>Burch, M., & Weiskopf, D. (2014). On the benefits and drawbacks of radial diagrams. In E. Huang (Ed.), <i>Handbook of human centric visualization</i> (pp. 429-451). Springer.</p><p>Foster, C. (2015). Exploiting unexpected situations in the mathematics classroom. <i>International Journal of Science and Mathematics Education, 13</i>(5), 1065–1088. <a href="https://doi.org/10.1007/s10763-014-9515-3">https://doi.org/10.1007/s10763-014-9515-3</a></p><p>Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. <i>For the Learning of Mathematics, 42</i>(3), 21–27. <a href="https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf">https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf</a></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com1tag:blogger.com,1999:blog-2036014053389751696.post-75140775187058939892022-11-24T07:00:00.012+00:002022-11-24T08:53:08.550+00:00Fractions as factors<p><i>Can a factor be a fraction?</i></p><p>People sometimes agonise over whether a fraction such as $\frac{2}{3}$ can be called ‘a factor’ of another number, such as 6 (Foster, 2022). If factors are defined as “numbers that divide exactly into another number” (BBC Bitesize, n.d.), then, since $6÷\frac{2}{3}=9$, and $9$ is an integer, shouldn’t $\frac{2}{3}$ be regarded as a factor of $6$?</p><p>Perhaps we decide that we want to be able to say that the number $6$ has exactly $4$ factors ($1, 2, 3$ and $6$), and so we don’t allow numbers like $–2$ or $\frac{2}{3}$ to be factors. If so, we could use a tighter definition of factor, such as: <i>A factor is a positive integer that divides into another number a positive integer number of times</i> (or, equivalently, we could say ‘without any remainder’). However, in many instances we might want to think of factors more broadly than this. For example, when factorising $x^2±5x+6$, it might be helpful to think of the $6$ as having four possible <i>factor pairs</i> ($\{1, 6\}, \{2, 3\}, \{–1, –6\}$ and $\{–2, –3\}$). Similarly, when using the <i>factor theorem</i>, we typically treat negative numbers as ‘factors’. In other contexts, it might seem natural to regard a number like $\frac{2}{3}$ as being a factor of $\frac{4}{3}$, since it ‘goes into it’ twice, or even $x$ as a factor of $x^2$, since it goes into it $x$ times, regardless of whether or not $x$ might be an integer. We might even pull out an irrational number, such as $\pi$, from an expression like $2\pi r-\pi l$ to give $\pi(2r-l)$, and call this 'factorising', and refer to $\pi$ here as a ‘common factor’, although it is certainly not an integer, and is not even rational (Foster, Francome, Hewitt, & Shore, 2022).</p><p>‘Factor’ seems to be one of those words that is used differently in different contexts, even within school mathematics, and I think it isn’t really possible to settle on a fixed definition which will always apply (see Foster, Francome, Hewitt, & Shore, 2022, for a similar discussion about the word ‘fraction’). Perhaps the best approach to awkward issues like these is to acknowledge them and explore them. Turn the issue into a task: <i>What would happen if we allowed fractions to be factors?</i> Perhaps we call them ‘fraction factors’.</p><h4 style="text-align: left;">Exploring 'fraction factors'</h4><p>Students often think of fractions as ‘numbers less than 1’, and they may initially think that <i>any</i> fraction would be a fraction factor of <i>any</i> integer, but of course this isn’t right. Although $\frac{2}{3}$ would be a fraction factor of $6$, it <i>wouldn’t</i> be a fraction factor of $5$, since $5÷\frac{2}{3}=\frac{15}{2}$, or $7.5$, which is not an integer. All <i>unit</i> fractions ($1/n$, where $n$ is an integer $\neq 0$) would be fraction factors of <i>every</i> integer, since they are by definition fraction factors of $1$, and $1$ is a factor of every integer. But when would a <i>non-unit</i> fraction be a fraction factor of an integer? Could we ask for <i>all</i> the fraction factors of $6$? Clearly not, because this list would include all of the unit fractions, and there are infinitely many of them.</p><p>There are many opportunities here for students to form conjectures and to find counterexamples – and, in each case, to try to find the <i>simplest</i> counterexample they can. It can also be helpful to look at the question the other way round, and ask what integers a fraction like $\frac{5}{12}$, say, would be a fraction factor of.</p><p>The conclusion is quite simple, but perhaps not that easy for students to arrive at without quite a bit of useful exploration. A fraction $\frac{p}{q}$, with $p,q \neq 0$, in its lowest terms, will divide an integer $m$ if and only if $m÷\frac{p}{q}$ is an integer. This is equivalent to saying that $\frac{mq}{p}$ must be an integer. Since $p$ and $q$ are co-prime, $\frac{mq}{p}$ will be an integer if and only if $p$ is a factor of $m$. So, the fraction factors of $6$ are fractions that, when simplified, have the (positive integer) factors of $6$ as their numerators:</p><p style="text-align: center;">$$\frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \frac{6}{n}$$ for all integer $n \neq 0$.</p><p>Perhaps this seems obvious, but I think it can be quite unintuitive that, say, $\frac{2}{17}$ is a fraction factor of $6$, but $\frac{4}{3}$ isn’t.</p><p>A less algebraic – and perhaps clearer – way to appreciate why the numerator matters, but the denominator doesn’t, is to realise that $\frac{1}{q}$, as a unit fraction, will always be a fraction factor of <i>any</i> integer, regardless of what $q$ is, because $q$ of them will always fit into every $1$. For the same reason, $\frac{p}{q}$ will necessarily be a fraction factor of $p$, because $q$ of them will fit exactly into the integer $p$. So, $\frac{p}{q}$ will be a fraction factor of any integer of which $p$ is a factor. </p><h3 style="text-align: left;">Things to reflect on</h3><p>1. Do you agree that it isn't possible to have a single definition of 'factor' that applies across all of school mathematics? Why / why not?</p><p>2. If you agree, which other technical mathematical terms do you think may be problematic in this kind of fashion? (See Foster et al., (2022) for a discussion of 'fraction' in this regard.)</p><h3 style="text-align: left;">References</h3><p>BBC Bitesize (n.d.). What are factors? <a href="https://www.bbc.co.uk/bitesize/topics/zfq7hyc/articles/zp6wfcw">https://www.bbc.co.uk/bitesize/topics/zfq7hyc/articles/zp6wfcw</a></p><p>Foster, C. (2022, October 13). How open should a question be? [Blog post]. <a href="https://blog.foster77.co.uk/2022/10/how-open-should-question-be.html">https://blog.foster77.co.uk/2022/10/how-open-should-question-be.html</a></p><p>Foster, C., Francome, T., Hewitt, D., & Shore, C. (2022). What is a fraction? <i>Mathematics in School, 51</i>(5), 25–27. <a href="https://www.foster77.co.uk/Foster%20et%20al.,%20Mathematics%20in%20School,%20What%20is%20a%20fraction.pdf">https://www.foster77.co.uk/Foster%20et%20al.,%20Mathematics%20in%20School,%20What%20is%20a%20fraction.pdf</a></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-73420362283184222332022-11-10T07:00:00.005+00:002022-11-10T07:00:00.146+00:00Is area more difficult than volume?<p><i>I have a tendency to assume that concepts get more difficult as the number of dimensions increases. Length is pretty straightforward, surely (how long is a piece of string?). Area is a bit harder, because we are in 2 dimensions now, and volume is even harder, because that's 3 dimensions. However, I'm not sure that this is right or that it makes much sense to think of building up in this way. After all, a point is zero-dimensional, and that is certainly not a simple thing to get your head around at all!</i></p><p>I think it is much easier to get an intuitive sense of volume than it is of area. From a very young age, children build with blocks and pour sand and water into containers, so they are engaging with 3-dimensional concepts such as volume right from babyhood. We are 3-dimensional beings and live in a 3-dimensional world, so we really ought to feel at home working with a concept like volume. By contrast, I think that area and length may be more inherently difficult concepts conceptually, as we never see these things in their true reality. A 1-dimensional horizontal line segment is represented on paper by a very thin rectangle, because it has to have a little bit of vertical height, otherwise it would be invisible. Indeed, this 'rectangle' is really a very shallow cuboid of ink, sitting on top of the sheet of paper, so it's an approximate cuboid rather than a 'line'. Everything we see that is intended as an approximation to or representation of something 1-d or 2-d is really forced to be actually 3-d just because that's the kind of world we live in.</p><p>I was thinking about this recently when considering how to design some lessons on area. What does it mean for two shapes to have <i>the same area</i>? By contrast, equal volumes is fairly easy to grasp. Two hollow shapes have the same capacity (or volume of space/air inside them) if you could fill them up using the same quantity of liquid (Note 1). If they are solid objects, then they have the same volume as each other if they <i>displace</i> the same quantity of a liquid that they are submerged within. For children who have played with floating and sinking objects in the bath, this is familiar territory. You can easily be sure whether two objects have the same volume or whether one has greater volume than the other, and which way round it is. It might not be easy to estimate volumes at a glance in practice - I am always surprised that an ordinary drinks can contains 330 cm$^3$ (it never looks big enough to contain that many centimetre cubes) - but it's clear what 'greater volume' means and how we could, in principle, find out (Note 2).</p><p>This is all much harder with area. Two shapes have the same area if you can cut up one of them and fit the pieces exactly inside the other one, with no gaps and no overlaps. This is not easy at all. What if the shapes have awkward edges? Can a disc of radius $1$ be 'cut up and fitted exactly inside' a square with side length $\sqrt{\pi}$? You would have to make infinitely many cuts to do it: can you be sure that there would definitely be no tiny gaps or overlaps? To a young child, there is nothing exceptional about a circle - they see circles all the time - so this should surely not be a 'hard' example to think about. Similarly, we expect children to accept things like the fact that shearing a rectangle parallel to one of its sides doesn't change the area. Maybe we push over a stack of paper to illustrate this. But is it really so clear that the area is definitely not changing?</p><p>I was thinking about lessons I've seen in which the teacher aimed for more intuitive understandings of area. It is normal to choose a small square 'unit' and try to cover a given shape with a finite number of these units. So children might place small plastic squares over a drawing of a shape to see how many are needed to cover it completely without any gaps. Sometimes the answer is an integer (e.g., for a carefully-chosen rectangle), or it might be a non-integer, but still exact, if half-squares, say, can be carefully totalled. Sometimes, it may involve estimation (e.g., tracing a hand or foot onto squared paper and approximating the total surface area). Occasionally units other than a square are used, and so long as the units tessellate we can often still tile many shapes with them. But I think all of this is much harder than pouring liquids into and out of various vessels.</p><p>What about more informal approaches to area? I saw a lesson in which area was informally understood as 'the amount of ink you need to colour in the shape', whereas perimeter was 'the amount of ink you need to draw around the edge of the shape'. Students had felt-tip pens and were illustrating area and perimeter of some shapes by colouring them in (Note 3). Of course, there is a lot of imprecision here, but perhaps the act of colouring in helps to reinforce the nature of the concept we are focused on. Time can be brought in as a proxy for area, so the shape with larger area is the one that takes longer to colour in (colouring races to determine which shape has greater area). There are some assumptions here about things like the same pen being used, students with equal 'colouring speed', and so on. And, of course, features of the shape that might mean that some shapes of area 10 cm$^2$, say, are quicker to colour than others - compare these two, for example:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcqicHMkhDSQ2YkEPWDLxqNvyY95EjYmNtD72tzgN64vZkUXlpxu-wnYFw5MN0U_O2pHdOS5I2FpmeQXEfrVbhPIWdjUSZPEcDkGrl9-Bg7eQsZbwMZbbyW3bw7WXukfGlL5BbPnEyHGAefkkIEUOSbImfWvqYi9OyYrzuHUvq1cmOQCC9LmwCnoouvQ/s1009/equal%20area.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1009" data-original-width="892" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcqicHMkhDSQ2YkEPWDLxqNvyY95EjYmNtD72tzgN64vZkUXlpxu-wnYFw5MN0U_O2pHdOS5I2FpmeQXEfrVbhPIWdjUSZPEcDkGrl9-Bg7eQsZbwMZbbyW3bw7WXukfGlL5BbPnEyHGAefkkIEUOSbImfWvqYi9OyYrzuHUvq1cmOQCC9LmwCnoouvQ/s320/equal%20area.PNG" width="283" /></a></div><p>Other possibilities for building helpful intuitions include things like cutting out shapes in thick card (with uniform thickness) and weighing them against each other to determine which has larger area. Or cutting them out of pastry and baking them and seeing which weighs more or takes longer to eat. All of this kind of work is a long way from the more common emphasis on quickly getting to the calculations. If you ask a child, "What is area?" they may be quite likely to say, "Base times height". Admittedly, "What is area?" is a hard question for anyone to answer. But stating a formula for calculating the area of one specific type of shape (i.e., a parallelogram) is not the same as having a good fundamental sense of what area is all about.</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p>1. Do you agree that area is more difficult than volume? If not, why not?</p><p>2. How important do you think it is to develop an intuitive understanding of concepts like area?</p><p>3. Would you use any of the approaches mentioned here? Why / why not?</p><h3 style="text-align: left;">Notes</h3><p>1. We'll assume throughout that the shapes are made of something like plastic, which doesn't absorb the liquid or get deformed by it! There are all kinds of other physical assumptions in play, such as the assumed incompressibility of liquids, and so on.</p><p>2. For reflections on some of the ambiguities around volume and surface area, see Foster (2011).</p><p>3. Colouring in is not always the most educational of activities in school, but in this case it seemed that it might be.</p><h3 style="text-align: left;">Reference</h3><p>Foster, C. (2011). Productive ambiguity in the learning of mathematics. <i>For the Learning of Mathematics, 31</i>(2), 3–7. <a href="https://www.foster77.co.uk/Foster,%20For%20The%20Learning%20of%20Mathematics,%20Productive%20Ambiguity%20in%20the%20Learning%20of%20Mathematics.pdf">https://www.foster77.co.uk/Foster,%20For%20The%20Learning%20of%20Mathematics,%20Productive%20Ambiguity%20in%20the%20Learning%20of%20Mathematics.pdf</a></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com1tag:blogger.com,1999:blog-2036014053389751696.post-21809416841211515362022-10-27T07:00:00.019+01:002022-10-27T07:00:00.164+01:00Butterfly effects when adapting tasks<p><i>Sometimes a superficially small tweak to a task - changing just one little thing - can dramatically alter it, and mean that a lot more thinking - or very different thinking - is needed. In many cases, students don't need to be taught any additional facts or methods beyond what they have already learned - they know everything necessary to be able to figure out what difference the change makes.</i></p><p>The details matter in mathematics task design and adaptation. Here are just a few examples of small ways in which tasks can be adapted to make them more challenging and so that they may provoke deeper thought. For more on ways to adapt mathematics tasks, see Prestage and Perks (2013). The tasks below all take the form "You know this, but what about <i>this</i>?"</p><h4 style="text-align: left;">1. Change the base</h4><p>It's sometimes said that <i>You don't understand long division until you can do it in any base</i>.</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>You know how to work out $252 \div 6$ in base $10$, but can you do it in base $16$?</i></p><p><i>In which other bases does $252 \div 6$ give an <b>integer</b> answer? Why?</i></p></blockquote><p>Doing arithmetic in different bases used to be much more common in UK mathematics teaching, and it is still quite prevalent in schools in many countries. I find that people often assume that it's much more complicated than it is. I once heard a staffroom conversation that went something like this:</p><p><i>A: I think we're just too obsessed with base 10 and we don't take opportunities to work in other bases.<br />B: Yes, I guess it's just a convenience thing - it's so handy, for example, to be able to multiply by 10 by just placing a zero on the end.<br />A: Do you know how to multiply by 7 in base 7?<br />B: Er, no, not off the top of my head, but I should think it's very complicated. That's my point about why we always resort to base 10.</i></p><p>Exploring <i>factors</i> of numbers in different bases can be a good way to see why person <i>B</i> is wrong about this!</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>You know the factors of $12$ in base $10$, but what are the factors of $12$ in base $16$?</i></p><p><i>In which bases is $12$ prime? Why?</i></p></blockquote><p>By changing something, away from the familiarity of base 10, we can 'make the familiar strange' and see things in new ways. It is such an apparently small change, but there are many tasks in which a change of base can be insightful (see Foster, 2007). For example,</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>You know that, in base 10, $\frac{1}{2}$ is a terminating decimal and $\frac{1}{3}$ is a recurring decimal. Which fractions terminate and recur in other bases? </i><i>Is there a base in which </i><i>$\frac{1}{2}$ is a </i><i>recurring</i><i> decimal and $\frac{1}{3}$ is a terminating decimal? Or in which they are both recurring or both terminating? W</i><i>hat is </i><i>$\frac{1}{2}$</i><i> in base 16? What about in base 15? What about in other bases? </i></p></blockquote><h4 style="text-align: left;">2. Introduce modulus signs</h4><p>The definition of the <i>modulus function</i> is very simple. You could use the ideas below when teaching students about the modulus function for the first time. But you could also use them with <i>any</i> students - even quite young ones - who don’t ‘need’ to learn about the modulus function. You would be using the modulus function as a way to get them to think more deeply about things they do 'need' to think about.</p><p>The modulus function $\lvert x \rvert$ is defined as:</p><p>$$\lvert x \rvert=\left\{ \begin{array}{@{}ll@{}} x, & \text{if}\ x \ge 0 \\ -x, & \text{if}\ x \lt 0 \\ \end{array}\right.$$</p><p>But the idea is much simpler and more accessible than this formal notation might suggest. The value of $\lvert x \rvert$ is the absolute 'size' of $x$, regardless of its sign. So $\lvert 18 \rvert = 18$, but $\lvert -18 \rvert = 18$ also. This means that $\lvert x \rvert$ is never negative (Note 1).</p><p>That is all the formal teaching that is needed. The task is then is to try some things that students can already do, but with a modulus function thrown into the mix - a superficially small change, but with a 'butterfly' effect.</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>You know what the graph of $y=\sin x$ looks like. What about $y=\lvert \sin x\rvert $? Or $y=\sin \lvert x \rvert$?</i></p><p><i>You know what the graph of $y=x+1$ looks like. What about $y=\lvert x \rvert+1$ or $y=\lvert x+1 \rvert$?</i></p><p><i>You know what the graph of $y=(x+1)^2$ looks like. What about $y=\lvert x+1 \rvert^2$? Or </i><i>$y=( \lvert x \rvert + 1)^2$?</i></p></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>You know how to solve $3x-2=10$. </i><i>What about $\lvert 3x-2 \rvert =10$?</i></p><p><i>You know how to solve $3x-2=2x+1$. </i><i>What about </i><i>$\lvert 3x-2 \rvert = \lvert 2x-1 \rvert$?</i></p><p><i>You know how to solve $3x-2<2x+1$. </i><i>What about $\lvert 3x-2 \rvert < \lvert 2x-1 \rvert$?</i></p><p><i>You know how to find the integral $\int (x^2-1) dx$. What about </i><i>$\int \lvert x^2-1 \rvert dx$?</i></p></blockquote><p>Introducing modulus signs is a little like introducing $\pm$ signs (see Foster, 2012) in that you are making a superficially small change that has the effect of requiring students to think quite hard.</p><p>Finally, for modulus functions, take a look at this graph:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiidOEFMbRBEdUvLPKxY4lwK6JXy3V5BpV4oI9pYoDtYhR5Ru2a99ST8AxLG2Zx-PycTyni1-TsdT75gpV3x_4X7LDPr7_K3AZnbfHwIDRcbYTn_kC18s2nvJG7YPfy6sMmYP5233lTgR8IUixLWvsK28863m_GZvHZl4_8zJPyV2HRHOhmlOcHBwkkdw=s7333" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="4111" data-original-width="7333" height="358" src="https://blogger.googleusercontent.com/img/a/AVvXsEiidOEFMbRBEdUvLPKxY4lwK6JXy3V5BpV4oI9pYoDtYhR5Ru2a99ST8AxLG2Zx-PycTyni1-TsdT75gpV3x_4X7LDPr7_K3AZnbfHwIDRcbYTn_kC18s2nvJG7YPfy6sMmYP5233lTgR8IUixLWvsK28863m_GZvHZl4_8zJPyV2HRHOhmlOcHBwkkdw=w640-h358" width="640" /></a></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>What combination of modulus functions could produce this graph? </i>(See Note 2 for the answer.)</p></blockquote><h4 style="text-align: left;">3. Go non-linear</h4><p>Older students may know, or have a sense of, what happens if you change a function $f(x)$ into, say, $f(-x)$, or $f(mx)$, where $m$ is a constant. But what happens to well known graphs of $y=f(x)$ if we change:</p><p></p><ul style="text-align: left;"><li>$x \mapsto \frac{1}{x}$; e.g., $\sin {(\frac{1}{x})}$, $e^{\frac{1}{x}}$?</li><li>$x \mapsto x^2$; e.g., $\sin {x^2}$, $e^{x^2}$?</li></ul><div>This is something that older students may find interesting to explore. It is worth delaying going to graph-drawing software until after you have thought hard about it, and have definite conjectures to check out. Turning to graph-drawing software too soon risks circumventing the thinking, rather than supporting it.</div><div><br /></div><h4>4. Look for in-between values and concepts</h4><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><p><i>You probably know what the graph of $y=x^n$ looks like for positive integer $n$ and for $n=-1$. But what about for other negative integer $n$? And what about for <b>non-integer</b> $n$, both positive and negative (Note 2)?</i></p><p><i>You probably know about 'highest common factors' (or 'greatest common divisors'). What about '<b>second</b>-highest common factors'? When do/don't they exist? What can you work out about them?</i></p><p><i>You probably know about 'common multiples' as the numbers where sequences like $3n$ and $8n$ intersect. But where do sequences like $3n+1$ and $8n-2$ intersect?</i></p></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>You probably know how to construct a perpendicular bisector? How would you construct a perpendicular trisector?</i> </p></blockquote><h4 style="text-align: left;">Conclusion</h4><p><i>Variation theory</i> currently receives a lot of attention, and can be powerful for helping students attend to what is the same and what is different in a sequence of tasks. In this spirit, little changes <i>beyond</i> what students <i>need</i> to know, and into unknown terrain, can be interesting extension tasks and valuable tasks for everyone to deepen their knowledge and understanding.</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. What other butterfly examples do you have, where 'changing one little thing' has a big effect?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. When would you use tasks like this?</span></p><h3>Notes</h3><p>1. Of course, the notation $\lvert x \rvert$ is also used where $x$ (perhaps written as $\textbf{x}$) is a vector, or where $x$ is a complex number, or to indicate the determinant of a matrix (in which case $\lvert x \rvert$ <i>could</i> be negative). But in this post I am only thinking about $x$ as a <i>real number</i> ($x \in \mathbb{R}$).</p><p>2. The graph is $y= \lvert x-2 \rvert + \lvert x+2 \rvert - \lvert x-1 \rvert$. Students could invent tasks like this for each other.</p><p>3. If you use software to draw $y = x^n$, and set up a slider to vary the value of $n$, you may notice a 'flickering' behaviour for negative values of $x$. It is interesting to think about why this happens. If $n$ is rational and equal to $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and $\frac{p}{q}$ is in its lowest terms, then the domain of $x$ is:</p><p></p><ul><li>all of the real numbers, if $q$ is odd, but</li><li>all of the <i>non-negative</i> real numbers, if $q$ is even. </li></ul><p></p><p><span style="font-family: inherit;"></span></p><p>This means that, when $n$ is real, but not rational, the domain is all the <i>non-negative</i> real numbers. So, the graph flickers as you drag the slider because, as $n$ varies, the function keeps switching between being defined and undefined (see also Dobbs, 2017).</p><h3 style="text-align: left;">References</h3><p>Dobbs, D. E. (2017). Why the $n$th-root function is not a rational function. <i>International Journal of Mathematical Education in Science and Technology, 48</i>(7), 1120-1132. <a href="https://doi.org/10.1080/0020739X.2017.1319980">https://doi.org/10.1080/0020739X.2017.1319980</a> ($\$$)</p><p>Foster, C. (2007). Twenty–one forever! <i>Journal of Recreational Mathematics, 36</i>(3), 194–195. <a href="https://www.foster77.co.uk/Foster,%20Journal%20of%20Recreational%20Mathematics,%20Twenty-One%20Forever.pdf">https://www.foster77.co.uk/Foster,%20Journal%20of%20Recreational%20Mathematics,%20Twenty-One%20Forever.pdf</a></p><p>Foster, C. (2012). Plus–minus graphs. <i>Mathematics in School, 41</i>(2), 32–33. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Plus-Minus%20Graphs.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Plus-Minus%20Graphs.pdf</a></p><p>Prestage, S., & Perks, P. (2013). <i>Adapting and extending secondary mathematics activities: New tasks for old</i>. David Fulton Publishers.</p><p><br /></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com1tag:blogger.com,1999:blog-2036014053389751696.post-25596255129973685512022-10-13T07:00:00.001+01:002022-10-13T07:00:00.167+01:00How open should a question be?<p><i>People often say or imply that when teaching mathematics 'open questions' are simply better than 'closed questions'. Instead of asking students a closed question like “What is $3 \times 4$?”, which has only one right answer, it is supposed to be better to ask them to “Tell me some products that make 12”, because this more open question has multiple possible correct answers. The more possible correct answers there are, the more ways there are for students to be right, and that feels more positive and inclusive. Open questions promote students’ creativity and individuality. But is it as simple as 'the more open the better'? Should we be sorry whenever we use a closed question?</i></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjTz0qq6OaSLN3doH3qN-a7UholJgv1zCqbPVMYZGBAgBEZZnSdtvTONCY0skchFjW0CMMeVKCrs03ogdqIHGvH2yTr3UQT1CufVbGFiuUkzYm4tLoLROZE264aGcEgXx-x9ArWndhBCHV5JmQR5efWum1taNPQ2gBMmPBfh3lI-wq_fA-DqdTNIO-o0g=s600" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="349" data-original-width="600" height="186" src="https://blogger.googleusercontent.com/img/a/AVvXsEjTz0qq6OaSLN3doH3qN-a7UholJgv1zCqbPVMYZGBAgBEZZnSdtvTONCY0skchFjW0CMMeVKCrs03ogdqIHGvH2yTr3UQT1CufVbGFiuUkzYm4tLoLROZE264aGcEgXx-x9ArWndhBCHV5JmQR5efWum1taNPQ2gBMmPBfh3lI-wq_fA-DqdTNIO-o0g=s320" width="320" /></a></div><p></p><p>Any question can always be made more open, but is that always an improvement? In the “Tell me some products that make 12” question above, why specify ‘products’? Instead, we could say “Tell me some <i>calculations</i> that make 12”, and that would be even more open. Or we could go even further and say “The answer is 12, what could the question be?” (see Ball & Ball, 2011, p. 20, for the '42' version of this). Perhaps the <i>ultimate</i> open question would be something like “Make up a question and an answer of your own choosing.” Is that the best possible question of all?</p><p>It seems to me that these questions can’t simply be getting better and better the more open we make them. Whether these questions are good or bad must surely depend on the teacher’s purposes in asking them. Let’s think about these questions in turn and how they might be more or less useful for different purposes.</p><h4 style="text-align: left;">1. Tell me some <i>products</i> that make 12.</h4><p>This could be a nice way to start thinking about factor pairs. One student says “$3 \times 4$”; another says “$6 \times 2$”. Perhaps, after a slightly longer pause, another student says “$1 \times 12$”. Now what do you do? If you continue to wait, then, after perhaps an even longer pause, you may be offered things like “$4 \times 3$” or “$24 \times \frac{1}{2}$” or even “$(-6) \times (-2)$” or perhaps (possibly?) “$n \times \frac{12}{n}$”. Is any of that helpful? I think it depends on your didactical purposes. Answers like these could lead to drawing rectangles of equal area (12) but different semi-perimeters, and explorations into <i>equable rectangles</i> (see Foster, 2021), and perhaps even aligning all these rectangles on a grid, with a common vertex at (0, 0), to create points at the diagonally-opposite vertex that lie on the curve $y=\frac{12}{x}$.</p><p>But, if you are defining factors as <i>positive integers</i> (so 3 is a factor of 12, but $\frac{2}{3}$ isn't), then most of these pairs are going to end up being <i>non-examples</i> of that, and we have drifted from the original purpose of thinking about factor pairs. Non-examples can be really important for nailing down what something is, but if you want to think about factors then we need to be working on the positive integers. So, although opening things up can feel accepting and positive, it can also be confusing for students and distracting from your intentions. If you want to focus on factors, maybe there is nothing wrong with being upfront about this, and saying what you mean by ‘factor’. Then you can ask, “Tell me a factor of 12”, and then ask for another one, and another one, and you still have multiple possible right answers - but this time there is a finite number of them. This means that, additionally, you can work on trying to justify <i>how we can be sure that we’ve got them all</i>. Being <i>less</i> inclusive about possible allowed numbers shuts down <i>some</i> possibilities (we can no longer have $-6$ or $\frac{1}{2}$) but opens up others: Now we can ask "Could there be any more? How can you be sure?" So, 'less open' doesn't necessarily mean fewer possibilities - it's just different.</p><h4 style="text-align: left;">2. Tell me some <i>calculations</i> that make 12.</h4><p>With this question, asking for ‘calculations’ enormously broadens out the possibilities that students could offer. Is that a good thing or a bad thing? If your learners are lower primary, they might be highly likely to offer additions and subtractions, and this could be a useful way in to number bonds to 12. Even with secondary-age students, simple responses like $10 + 2$ could lead you to explore the partitions of 12; i.e., <i>How many ways are there of making 12 from summing positive integers?</i> There are lots of interesting things to explore here. The total of 12 can be made by summing <i>pairs</i> of positive integers in 6 ways ($1+11$, $2+10$, $3+9$, $4+8$, $5+7$ and $6+6$) and by summing <i>three</i> positive integers at a time in 12 ways ($1+1+10$, $1+2+9$, ...). But how does this pattern continue - and what happens with other numbers than 12? (See Foster, 2011, pp. 110-112, for the details.)</p><p>But, if this is where you want to be heading, it might be preferable to ask a more <i>closed</i> question like "How can you express 12 as a sum of positive integers?" Otherwise, you might end up with things like $\frac{120}{10}$ or $2 \times 2 \times 3$ or $1+2 \times 4 + 3$ or $\sqrt{144}$, which could take you in all sorts of other directions (e.g., decimals, fractions, prime factorisation, priority of operations, powers). I have often seen teachers ask very open questions, even though they have something quite specific in mind. Whatever the students respond with, they praise it and write it up on the board - but actions speak louder than the board pen, and the students are watching to see what the teacher is <i>eventually</i> going to do. No matter how much the teacher says, "These are all great ideas ...", the students know that they should <i>ignore everything before the 'but'</i>, and so "... <i>but</i> today we're going to focus on..." is the point at which the teacher appears to come clean about what they <i>really</i> wanted right from the beginning. The person who said $10 + 2$ was 'right' and everyone else was 'nice try'.</p><p>So, I'm not convinced that open questions that lead to lots of possibilities are really all that inclusive if the teacher's response to all of the ideas except one is just to <i>write it on the board and ignore it</i>. But what is the alternative if you are using hyper-open questions? Allowing different students to work on totally different '12'-related problems splits the class into groups doing mathematics that may be as unrelated to each other as it is to the work being done by the students in the classroom down the corridor. And it can be hard to ensure that everyone has a suitably accessible and worthwhile challenge that will lead to learning something useful. The possibilities for sharing and building on each other's contributions later are quite limited if all that the different tasks have in common is that they have something to do with the number 12.</p><p>So, for me, prompt #2, although undeniably more 'open' than prompt #1, is more likely to <i>close down</i> the possibilities for purposeful sense-making of mathematics. Whereas adding constraints to #2, such as "It must include a cube root", or, for sixth formers, "It must include a logarithm", or "It must include an imaginary number", might make this more interesting and purposeful, depending on the teacher's intentions. Provided that there aren't too many, constraints often lead to creativity, whereas total freedom may not inspire anything.</p><h4 style="text-align: left;">3. The answer is 12, what could the question be?</h4><p>Now we have broadened out beyond even ‘calculations’. Students might say “What is the order of rotational symmetry of a dodecagon?” or “What is the maximum total you can get with two ordinary dice?” or “How many millimetres are there in 1.2 cm?” This might feel like a nice creative task for getting students to reflect on different ways in which 12 can appear in mathematics, but you could end up with quite a few less-mathematical responses, or low-level mathematical responses, like “What number bus goes to the sports centre?” or “How old is Amit’s sister?” or "What are the last two digits of the school phone number?" or "How many days of Christmas are there in the song?" (see <a href="https://en.wikipedia.org/wiki/12_(number)">https://en.wikipedia.org/wiki/12_(number)</a> for all things 'twelve'). By the time we reach a question as open as #3, we have really abdicated all responsibility for setting the direction of travel of the lesson, and ‘anything goes’ now. I guess I could maybe imagine asking this question as a lesson 'filler', or as a revision task: "Make up a question on each topic we've studied this term - and make all the answers '12'." But I don't think I could use this task to introduce anything new or focused on any particular content area, as it is too hard to predict what might happen with it. But maybe others are more expert at doing this than I am?</p><h4 style="text-align: left;">More 'mathematical' can mean more <i>closed</i></h4><p>Tony Gardiner (n.d., p. 31) has written that ‘While there are exceptions, it is generally true to say that good problems in school mathematics are almost never open-ended!" He advocates open-middled problems, where there is a closed start and end but multiple possible routes through. I think that teachers worry about using closed questions because they think they are too straightforward. But it is a big mistake to think that closed questions are likely to be simple or trivial. Most of the famous hard unsolved problems in mathematics are closed - even binary yes/no (e.g., Is the Riemann hypothesis true?) (see Foster, 2015).</p><p>Often we can make an open question more challenging by <i>closing</i> it, and that's usually a highly mathematical thing to do. The question we considered above, "Tell me a factor of 12", is open, but this should naturally lead to the desire to find them <i>all</i> by closing the question to: "What are the factors of 12?" This is now closed, because there is only one right answer (i.e., 1, 2, 3, 4, 6, 12). In a sense, giving just 5 of these is not $\frac{5}{6}$ correct, but wrong. This is not 'multiple answers' any more, because the question asks for <i>all</i> of them. It is as closed as the question "How many factors does 12 have?" (Answer: 6). Similarly, the closed question "Solve $x^2=9$" has one right answer, consisting of the two solutions $x=-3$ or $3$, whereas the open question "Find any solution to $x^2=9$" has two possible right answers, $x=-3$ or $x=3$. This may seem like splitting hairs, but the closed question is far more powerful - and mathematical - because we not only find <i>all</i> of the solutions but we know that there <i>can't be any more</i>, so we are finished and have complete knowledge of the situation (see Foster, 2022).</p><p>Thinking in this way, many of the open questions used in school can be viewed as merely poorly-specified or incomplete closed questions. As an example, consider the question: "I'm thinking of two numbers. Their sum is 12. What are the numbers?" We could respond "Impossible - insufficient information!", or we could give a few example pairs (which are effectively of the form "If one number is this, then the other number would be that"). Or we could plot <i>all</i> possible example pairs on the line $x+y=12$, which feels like as complete an answer as is possible. But is it that the question is open or merely that the (single) answer to this closed question happens to be an infinite solution set? (Note 1)</p><p>A more clear-cut example might be: "A number rounded to 1 decimal place is 3.1. What might the number have been?" This is open, and, again, in this case, there are infinitely many possible answers. The corresponding closed question would be "Which numbers round to 3.1 to 1 decimal place?", and here the answer would be all $x$ in the interval $3.05 \le x < 3.15$. There's a sense in which this is 'the complete answer'. But, I think it doesn't quite capture everything that might emerge from the open question. For example, someone might respond to the open version by saying $\pi$, and, although that is of course included in the range specified in the closed version, I might not have explicitly considered $\pi$ (or even the possibility of any irrational numbers) when writing down the 'complete' answer. The double inequality expresses the whole range of possibilities without necessarily examining it in detail.</p><p>To me there is certainly an important difference between a closed question like "Which numbers round to 3.1 to 1 decimal place?" and a closed question like "What is 3.08 rounded to 1 decimal place?", but the difference isn't about open/closed, as, for me, both are equally closed. The difference here is that the first question is an <i>inverse question</i>, and inverse questions are always harder and more interesting than direct questions (e.g., factorising versus multiplying, square rooting versus squaring, integrating versus differentiating).</p><p>Open questions are often essentially 'Give me an example of...', example-generation questions (see Watson & Mason, 2006), and I think these are extremely valuable when used to complement closed 'Characterise all the possibilities' kinds of questions. We need both. But making open questions more closed can be just as valuable mathematically and didactically as making closed questions more open.</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. What do you think about the value of 'open' questions? When do you use them or avoid them? Why?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. When is it good to be more open? When is it good to be more closed?</span></p><h3 style="text-align: left;">Note</h3><p style="text-align: left;">1. I guess one way to treat this as open would be to offer multiple ways of specifying the solution set (e.g., $2x+2y=24$, $y=12-x$, $x+y-12=0$, $y=-x+12$, $\frac{x+y}{2}=6$, etc.)</p><h3 style="text-align: left;">References</h3><p>Ball, B. & Ball, D. (2011). <i>Rich task maths 1</i>. Association of Teachers of Mathematics. </p><p>Foster, C. (2011). <i>Resources for teaching mathematics 11–14</i>. London: Continuum.</p><p>Foster, C. (2015). Fitting shapes inside shapes: Closed but provocative questions. <i>Mathematics in School, 44</i>(2), 12–14. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20In%20School,%20Fitting%20shapes%20inside%20shapes.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20In%20School,%20Fitting%20shapes%20inside%20shapes.pdf</a></p><p>Foster, C. (2021). Area and perimeter. <i>Teach Secondary, 10</i>(6), 13. <a href="https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Area%20and%20perimeter.pdf">https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Area%20and%20perimeter.pdf</a></p><p>Foster, C. (2022). Starting with completing the square. <i>Mathematics in School</i>.</p><p>Gardiner, T. (n.d.). <i>Beyond the soup kitchen: Thoughts on revising the Mathematics “Strategies/Frameworks” for England</i>. <a href="https://www.cimt.org.uk/journal/gardiner.pdf">https://www.cimt.org.uk/journal/gardiner.pdf</a></p><p>Watson, A., & Mason, J. (2006). <i>Mathematics as a constructive activity: Learners generating examples</i>. Routledge.</p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-44662820740018014312022-09-29T07:00:00.011+01:002022-09-29T07:00:00.178+01:00Tasks that can't fail<p><i>Do you have tasks that you use in lessons that "can't fail"? Favourite tasks that you've used many times and that always seem to go well? What features of these tasks seem to make this happen?</i></p><p>When mentoring trainee teachers in school, I found that it could be useful to have tasks to share with them to try out as some of their first experiences working with students (Foster, 2021). I think there's a place for tasks which "always seem to go well" and never seem to fail. Of course, we should 'never say never'. But rather than asking the trainee teacher to teach whatever happens to come next in the scheme of learning, for their first experiences I would usually give them something which requires minimal introduction and explanation and which students are likely to engage in with some enthusiasm, and will lead to productive conversations and the potential for a nice whole-class discussion. (These kinds of tasks can also be useful for 'interview lessons', when going for a job in another school.)</p><p>Sometimes trainee teachers, as one of their first experiences, are asked to do "the last 10 minutes" of a lesson. But I think this is very difficult. Even with lots of experience, it can be hard to gauge how long something will last, and the skill of drawing things out or speeding them up is really hard. Games like mathematical bingo are particularly difficult for lesson endings, I find, because it's hard to predict when someone will win and the game will finish, and if there's just 1 minute of the lesson left is there time to play another round? Also, students may be tired and not always at their best at the end of a lesson - especially if the lesson is followed by break or lunch or home, or if something fractious happened during the lesson. And the trainee teacher can be in the undesirable position of starting something off that goes well and then having to stop everyone part way through so as not to crash through the bell.</p><p>So, I prefer to ask the beginning trainee teacher to do the <i>beginning</i> of a lesson, and take as long or as short a time as they wish - and then I will pick up from wherever they leave off. If their section ends up taking 5 minutes, that's fine. If it ends up taking 20 minutes, that's also fine. Obviously, timekeeping is an important part of teaching, but I think it's helpful to focus on one thing at a time, so I try to remove time pressure in the first early experiences, so that the teacher can focus on what they are doing and what the students are doing, and experience the freedom of 'going with something' that takes off. So I want to take some things away, that they <i>don't</i> need to worry about until later. Above all, I want the teacher's first experience teaching a class to be a positive one that doesn't put them off. I want them to feel that teaching is something they can enjoy and that they want to do and can get better at doing.</p><p>So, what kinds of tasks can be effective for this? Over the years, I've developed a list of "can't fail" tasks for these kinds of purposes. Here are two:</p><h4 style="text-align: left;">1. Four fours</h4><p>I probably don't need to say much about this task, as it is such a classic:</p><p><i>What numbers can you make from four 4s?</i></p><p>(For more details, see Foster, 2020.) Sometimes teachers use this task in order to work on specific topics, such as priority of operations, but it can also be a great task for general numeracy and developing careful, systematic thinking. I think 'Four fours' can work with pretty much any age/stage of class. It requires very little set-up at the start and it is easy to get students sharing what they've come up with. I've used it with all ages in school and also with teachers.</p><h4 style="text-align: left;">2. Possibility tables</h4><p>Collaborative 'group work' is often perceived as an extremely difficult thing to do well, and perhaps something for trainee teachers to delay attempting until they have more experience (Foster, 2022). I think this is not necessarily the case, and pairwork, in particular, can be highly effective in situations in which the students have a clear joint task, which they both need to contribute to. One way to encourage this is through the resources that you provide. I remember one day realising in a flash of insight - as though it were a huge revelation - that the photocopying cost of one A3 piece of paper is (about) the same as that of two A4 pieces of paper. So, for the same cost, I could give every pair of students one A3 sheet, rather than give every student one A4 sheet. One A3 sheet and one pencil between two students can, if the task is well designed, almost 'force' effective pair work.</p><p>One "can't fail" task that seems to do this is what I call <i>Possibility tables</i> (Foster, 2015). These are a bit like what John Mason calls <i>structured variation grids</i> (see <a href="http://mcs.open.ac.uk/jhm3/SVGrids/SVGridsMainPage.html#What_is_an_SVGrid">http://mcs.open.ac.uk/jhm3/SVGrids/SVGridsMainPage.html#What_is_an_SVGrid</a>). For example, the possibility table below (<a href="https://www.foster77.co.uk/Symmetry%20Combinations%20TASK%20SHEET.pdf" target="_blank">pdf version</a>) varies order of rotational symmetry across the top and the number of lines of symmetry down the side. The task is for students to write the name, or draw a sketch, of any figure that could go in each cell of the table. If a cell is impossible to fill, they should indicate so, and try to explain why. (The notion that some combinations may be impossible is 'on the table' from the start.)<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6PSzwfsYOLOuZRCCwetIAMhocrBlgpV3uCmuclkAdVPqc7wLZ__UIwVMMHYer3W4zO5iZPDNvGvLnaFrjTuKb-RlTgJu-M77-1vpVJIi5hTab2-XAAMKzWoxiHe89PpCNtKIyJ6b5R5HUvLBEqpmaBF8m4DR8i7BkSUE7FWX4I4X8cTY5FMLie3DPnw/s3508/Symmetry%20Combinations%20TASK%20SHEET.png" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="2480" data-original-width="3508" height="452" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6PSzwfsYOLOuZRCCwetIAMhocrBlgpV3uCmuclkAdVPqc7wLZ__UIwVMMHYer3W4zO5iZPDNvGvLnaFrjTuKb-RlTgJu-M77-1vpVJIi5hTab2-XAAMKzWoxiHe89PpCNtKIyJ6b5R5HUvLBEqpmaBF8m4DR8i7BkSUE7FWX4I4X8cTY5FMLie3DPnw/w640-h452/Symmetry%20Combinations%20TASK%20SHEET.png" width="640" /></a></p><p>Again, with a task like this, there is very little set-up at the start from the teacher. A quick check or reminder about the definitions of 'order of rotational symmetry' and 'line of symmetry' is all that is required, and, if some students are still a little unsure of these, they will have ample opportunity to clarify that through the task.</p><p>The dynamic of <i>one large sheet of paper and one shared pencil </i>(pencil is preferable to pen, because then ideas are easily erased and replaced) seems to 'force' discussion. (A separate blank sheet of rough paper may also be useful, for trying out ideas.) If <i>both</i> students have a pencil, they can end up working relatively independently at opposite ends of the sheet, which is probably not what I would desire, whereas <i>one</i> shared pencil seems to lead to conversations about what is possible and what should be put where. But if students are very proficient at collaborative work then a pencil each might be OK (as in the drawing below). Often a 'wrong' figure can be viewed as 'right; just in the wrong place', and we can just move it to a different cell, and this sometimes means that we end up with multiple figures in some cells, which is fine. We can ask: <i>Why do some cells seem to be easier to find figures for than others?</i></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvBHEwa8-OEraHYXQ9MZMYUMV9fMBP5YKHxeGIRZE8h2bz3J6NksvEpou1onow7IyeHMUAkMAeLmUzT0ECpgD6Hmdd5zqYLFld0klzYXJ7TF2pKWIwyF6smU45Tg3jDUohpfY3mf8LKYnbytyw4PwsJsne7LdjTlF9unJI3sByFUixzAf4s09gKTRnhg/s7634/students%20doing%20possibility%20table.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="5894" data-original-width="7634" height="494" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvBHEwa8-OEraHYXQ9MZMYUMV9fMBP5YKHxeGIRZE8h2bz3J6NksvEpou1onow7IyeHMUAkMAeLmUzT0ECpgD6Hmdd5zqYLFld0klzYXJ7TF2pKWIwyF6smU45Tg3jDUohpfY3mf8LKYnbytyw4PwsJsne7LdjTlF9unJI3sByFUixzAf4s09gKTRnhg/w640-h494/students%20doing%20possibility%20table.png" width="640" /></a></div><p>You could say 'shape' instead of 'figure'. If students can't think of 'mathematical' shapes to try, you could suggest capital letters of the alphabet. One strategy is to work through the cells systematically, trying to think of shapes that would fit. Another is to first think of shapes and then decide where they go.</p><p>I find that lots tends to emerge from this task. Like 'Four fours', I've used it with students from primary age up to sixth form, and with trainee and experienced teachers as participants. It is possible to use university-level mathematics to reason about which symmetry combinations are definitely impossible. And there are many other pairs of variables that can generate other possibility tables.</p><p>These are two tasks I would generally be confident to give to a beginning teacher in the expectation that they would be likely to have a positive experience.</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. What do you make of these tasks?</p><p>2. What "can't fail" tasks do you use?</p><h3 style="text-align: left;">References</h3><p>Foster, C. (2015). Symmetry combinations. Teach Secondary, 4(7), 43–45. <a href="https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Symmetry%20Combinations.pdf">https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Symmetry%20Combinations.pdf</a></p><p>Foster, C. (2020). Revisiting 'Four 4s'. <i>Mathematics in School, 49</i>(3), 22–23. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Revisiting%20Four%204s.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Revisiting%20Four%204s.pdf</a></p><p>Foster, C. (2021). First things first. <i>Teach Secondary, 10</i>(6), 82–83. <a href="https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20First%20things%20first.pdf">https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20First%20things%20first.pdf</a></p><p><span>Foster, C. (2022). The trouble with groupwork. <i>Teach Secondary, 11</i>(5), 70–71. <a href="https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20The%20trouble%20with%20groupwork.pdf">https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20The%20trouble%20with%20groupwork.pdf</a></span></p><p><br /></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-50291316066185001172022-09-15T07:00:00.004+01:002022-09-15T07:00:00.155+01:00Always simplify your answer<p><i>Einstein is supposed to have said that “Everything should be made as simple as possible, but no simpler”. Mathematics questions often say 'Simplify your answer', or, if not explicitly stated, then this is often assumed, but is it a 'simple' matter to say what 'Simplify' actually means?</i></p><p>A student was calculating the radius of a circle with unit area. They wrote</p><p>$$\pi r^2=1$$</p><p>$$r^2=\frac{1}{\pi}$$</p><p>$$r=\frac{1}{\sqrt{\pi}}$$</p><p>$$r=\frac{\sqrt{\pi}}{\pi}$$</p><p>When challenged about the final step, they said that they were 'rationalising the denominator'. The teacher said, "You mean '<i>ir</i>rationalising' the denominator?", since $\pi$ is irrational. But the attempt at humour was not really right, because the denominator was irrational <i>before and after</i> this step. However, I have some sympathy with what the student was doing, presumably by analogy with things like</p><p>$$\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}.$$</p><p>Dividing by a rational number, like $3$, is much 'nicer' than dividing by an irrational number, like $\sqrt{3}$, and so rationalising denominators feels like a good thing to do, and comes under the heading of 'simplifying your answer'. But with $\sqrt{\pi}$, of course, that is different, because $3$ is rational, whereas $\pi$ is not. But $\pi$ is typographically <i>almost</i> a numeral, and we may sometimes think of it in that way, and, in cases like this, the fact that $\pi$ happens to be irrational feels separate from the square-rooting issue. Somehow, $\frac{\sqrt{\pi}}{\pi}$ does kind of look nicer than $\frac{1}{\sqrt{\pi}}$; perhaps $\sqrt{\pi}$ seems <i>even more irrational</i> than $\pi$? In fact, although it is irrational, we tend to think of $\pi$ as a beautiful, elegant number, whereas a decimal approximation, like $3.14$, although rational, does not seem anywhere near so nice. And I suppose $\sqrt{\pi}$ seems uglier than $\pi$, although $\sqrt{\pi}$ does turn up in some interesting places (e.g., <a href="https://en.wikipedia.org/wiki/Normal_distribution" target="_blank">the Gaussian distribution</a>).</p><p>This got me thinking about how confusing it can be for students to appreciate what counts as 'simplified', and there is some mathematical aesthetics here along with some perhaps rather arbitrary inconsistencies. Students may first meet the idea that there are multiple ways of representing the same thing when they encounter equivalent fractions. There, writing a fraction in its 'simplest form', or 'lowest terms', means reducing it to the smallest possible integers. There is something intuitive about 'simple' and 'small integers' being the same thing.</p><p>But things soon become more complicated (see Foster, 2021). Everyone would agree that $2x$ is simpler than $3x-8x+7x$, say, but is $2(x+1)$ simpler than $2x+2$? Simplifying algebra sometimes seems to mean writing in the most condensed form, "using the least possible amount of ink", but of course $\frac{\sqrt{3}}{3}$ uses <i>more</i> ink, and more/larger numbers, than $\frac{1}{\sqrt{3}}$, since $3>1$. We would probably prefer to write $-1+x$ as $x-1$, and this uses slightly less ink (we save a '$+$' sign, Note 1), but we would not <i>always</i> do this. If we were writing complex numbers in 'real-part, imaginary-part' form, we might prefer $-1+i$ to $i-1$, especially if we are combining (adding, say) several complex numbers, and don't want to mix up the real and imaginary parts.</p><p>Similarly, if solving a set of three simultaneous equations in three unknowns, we might prefer to write something like $-x+0y+2z$, so as to keep the unknowns aligned and in order, rather than 'simplifying' this to $2z-x$. Is $x^{-1}+y^{-1}+z^{-1}$ simpler or less simple than its equivalent form, $\frac{x+y+z}{xyz}$? I think it depends on the context. There are lots of situations in which we seem to prefer using <i>more</i> ink. And we would certainly rather write an exact number like $e^{\pi}-\pi$, rather than a very good approximation to this, $20$, which is unarguably 'simpler' and certainly uses less ink (Note 2).</p><p>Conversion of units provides another possible example. If you were calculating $1 \text{ cm} + 1.54 \text{ cm}$, to obtain $2.54 \text{ cm}$, would you regard it as 'simplifying' to convert this to $1 \text{ inch}$? What if you happened to end up with an answer of $7.62 \text{ cm}$ or $3.81 \text{ cm}$? Would you spot that they were 'simple' multiples of an inch, and, if so, would you convert to inches? I suppose it would depend on the context, but I don't think I would do this unless there was a good reason.</p><p>Ambiguity over 'simplification' continues as the mathematics becomes more complicated. Differentiating $\sin^2 x$ to obtain $2\sin x \cos x$, should the student 'simplify' this to $\sin 2x$? What if they were instead given $\frac{1}{2}\sin^2 x$ to differentiate, and so obtained $\sin x \cos x$, this time <i>without</i> the factor of $2$. Any pressure to go to $\frac{1}{2}\sin 2x$ feels less here. If, instead of using the chain rule on $\sin^2 x$, they had used trigonometric identities to convert to $\frac{1}{2}(1-\cos(2x))$, then they would 'instantly' obtain $\sin 2x$ as the derivative. But, otherwise, I would not expect students to switch $2\sin x \cos x$ into $\sin 2x$. But am I being inconsistent over identities? If they obtained an answer of $\sin^2x+\cos^2x$, then I certainly <i>would </i>expect them to simplify this to $1$!</p><p>I think it's pretty difficult to explain what exactly we mean by 'Simplify', and to specify what counts as simplified and what doesn't. When I devise trigonometric identity questions, with the instruction 'Simplify', I try to ensure that there is an equivalent form to the expression that I provide that is <i>uncontroversially</i> by far the shortest and 'simplest'; otherwise, it is hard to say that the question has a right answer. But how do I judge the student who arrives at an equivalent expression to that, if all the statements, including the starting one, are equivalent. Agonising over things like this reminds me of the method Paul Halmos (1985) recounted being taught by one of his students for how to answer any trigonometric identities question:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;">If you're told to prove that some expression A is equal to a different-looking B, you put A at the top left corner of the page, B at the bottom right, and, using correct but trivial substitutions, keep changing them, working from both ends to the middle. When they meet, stop. If the identity you were given is a true one (it always is), everything on the page is true. To be sure, somewhere near the middle of the page there is a gigantic step, probably as big as the original problem, but very few paper graders will ever find it, or, if they find it, dare to mark you down for it - it is, after all, true! (Halmos, 1985, p. 25).</p></blockquote><p>I think that the idea that every expression has a unique, 'most simplified' form is not really right - and finding this magic form (and knowing when you've got it) is certainly a hard thing to communicate to students. Perhaps we need to be open about the fact that the simplest, most elegant way to leave an answer is to some extent a matter of judgment.</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. How do you explain to your students what is required for 'simplified' answers?</p><p>2. Can you think of other examples of ambiguous or confusing situations involving simplification?</p><h3 style="text-align: left;">Notes</h3><p>1. I suppose if we wanted to be super-picky about this, we could argue about whether the '$-$' in '$-1$' might be written as a smaller line, like '$\text{-}$', than the '$-$' in '$1-x$'.</p><p>2. No one knows 'why' $e^{\pi}$ (<i><a href="https://en.wikipedia.org/wiki/Gelfond%27s_constant" target="_blank">Gelfond's constant</a>)</i> is so close to $\pi + 20$. Maybe it doesn't really make sense to ask for 'explanations' of things like this (see <a href="https://en.wikipedia.org/wiki/Mathematical_coincidence">https://en.wikipedia.org/wiki/Mathematical_coincidence</a>).</p><h3 style="text-align: left;">References</h3><p>Foster, C. (2021). Questions pupils ask: What are 'like terms'? <i>Mathematics in School, 50</i>(4), 20–21. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20What%20are%20'like%20terms'.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20What%20are%20'like%20terms'.pdf</a></p><p>Halmos, P. R. (1985). <i>I want to be a mathematician: An automathography</i>. Springer Science & Business Media.</p><p><br /></p><p><br /></p><p> </p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com2tag:blogger.com,1999:blog-2036014053389751696.post-25543968551656675962022-09-01T07:00:00.015+01:002022-09-01T08:17:34.188+01:00Interactive introductions<p><i>How do you introduce a new mathematical topic or concept? Do you give students a task to do, or do you start by explaining everything?</i></p><p>I think most teachers do a mixture of these things, depending on the topic and the class, and sometimes they orchestrate something that is kind of in between - what I call an <i>interactive introduction</i>. This is highly teacher-led, but aims to be more like a conversation and discussion than a monologue. This doesn't mean that it it is a 'free for all', in which anyone can just say anything that occurs to them. Nor is the teacher merely relying on one or two students happening to know what they wish to teach and telling everyone else. Realistically, only a few of the students will get to contribute orally to any particular <i>interactive introduction</i>. But, when an <i>interactive introduction</i> works well, all of the students will be equally able to 'participate' by engaging in the thinking process. They follow the thinking of the discussion, which is carefully planned <i>not to depend on any knowledge which the teacher hasn't yet taught</i>. And the teacher plans the <i>interactive introduction </i>to involve moments of puzzlement and surprise. The students are not left to figure out the content for themselves, but nor are they presented with it on a plate, all tidied up and complete. The teacher leads them to ask and answer the relevant questions.</p><p>It is easy to write a paragraph like that one, having my cake and eating it, and making it all sound so good. But how about some examples? Over time, many teachers have developed really nice ways to introduce topics, but I am not sure that these typically get shared so much. Teachers often share 'resources', which usually means either worksheets oriented towards the students - tasks for the students to do - or <i>PowerPoint</i> presentations for the teacher, that generally explain content and provide examples and exercises. Neither of these is quite what I'm talking about when I say an <i>interactive introduction.</i></p><p>So, I'm going to share a few examples of how I have introduced certain common topics. I'm not making any claims for greatness here, and I'm sharing them as <i>Word</i> files so that you can cut and edit as you wish, if you find anything there that you want to use/develop/improve. I've kept each one to 1 side of paper, but hopefully there's enough here for you to see what I'm trying to do. Certainly, any kind of 'scripted' lesson has to be 'made your own' before you can authentically use it - I wouldn't envisage reading out any of this word for word, but instead attempting to capture the overall idea and adapting it to your own style and purposes. I've chosen the specific mathematical examples used in them quite carefully - certainly much more carefully than I could have done if you'd asked me on the spur of the moment to get up and explain something, unprepared. I think the particular examples might be the most valuable part of these <i>interactive introductions</i>, but please see what you think. I'd be very happy for lots of criticism of them in the Comments below. If you hate them, that's fine!</p><p>So here are:</p><p><b><i>1. A first lesson on 'standard form'. (I discussed this one in <a href="http://www.mrbartonmaths.com/blog/research-in-action-16-writing-a-maths-curriculum-with-colin-foster/">my most recent podcast with Craig Barton</a>.) (<a href="https://www.foster77.co.uk/Standard%20Form%20Interactive%20Introduction.docx" target="_blank">Word</a> and <a href="https://www.foster77.co.uk/Standard%20Form%20Interactive%20Introduction.pdf" target="_blank">pdf</a> formats)</i></b></p><p><b><i>2. A first lesson on 'enlargement'. (<a href="https://www.foster77.co.uk/Enlargements%20Interactive%20Introduction.docx" target="_blank">Word</a> and <a href="https://www.foster77.co.uk/Enlargements%20Interactive%20Introduction.pdf" target="_blank">pdf</a> formats and the associated <a href="https://www.foster77.co.uk/Enlarging%20the%20head%20teacher.pptx" target="_blank">PowerPoint</a> file.)</i></b></p><p><b><i>3. A first lesson on 'circles and $\pi$'. </i></b><b><i>(<a href="https://www.foster77.co.uk/Circles%20and%20pi%20Interactive%20Introduction.docx" target="_blank">Word</a> and <a href="https://www.foster77.co.uk/Circles%20and%20pi%20Interactive%20Introduction.pdf" target="_blank">pdf</a> formats and the associated <a href="https://www.foster77.co.uk/Running%20around%20a%20circle.pptx" target="_blank">PowerPoint</a> file.)</i></b></p><p>And, finally, as a bit of a further experiment, I've also had a go at making <b style="font-style: italic;"><a href="https://youtu.be/IapWK-C-Uyk" target="_blank">a video of me introducing the idea of complex numbers</a> (<a href="https://www.foster77.co.uk/Introducing%20i%20(sheet).docx" target="_blank">Word</a> and <a href="https://www.foster77.co.uk/Introducing%20i%20(sheet).pdf" target="_blank">pdf</a> </b><b style="font-style: italic;">formats </b><i style="font-weight: bold;">of the sheets).</i><i style="font-weight: bold;"> </i>This is the sort of thing I would do with a sixth-form class in which I could assume that the students were familiar with the quadratic formula but have had no formal teaching about $i$. (You might also wish to see the related article, Foster, 2018.) Of course, in real life it wouldn't be a monologue like this, and would be 'interactive' to some degree. (And apologies for the sound quality on this recording - it turned out that the microphone wasn't plugged in, so it was recording through my laptop, but I didn't want to bother re-recording it!)</p><p>So, this is a shorter blogpost than usual, in order to give you time to look at the materials I've linked to.</p><p>Now over to you - comments, criticisms and improvements, please...</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. What are your thoughts on the idea of 'interactive introductions'?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. What comments do you have on any of these specific examples?</span></p><h3>Note</h3><p><span style="white-space: pre;">1. </span>You can listen to the episode here: <a href="http://www.mrbartonmaths.com/blog/research-in-action-16-writing-a-maths-curriculum-with-colin-foster/">http://www.mrbartonmaths.com/blog/research-in-action-16-writing-a-maths-curriculum-with-colin-foster/</a></p><h3 style="text-align: left;">Reference</h3><p>Foster, C. (2018). Questions pupils ask: Is i irrational? <i>Mathematics in School, 47</i>(1), 31–33. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Is%20i%20irrational.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Is%20i%20irrational.pdf</a></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com1tag:blogger.com,1999:blog-2036014053389751696.post-69612678447620408692022-08-18T07:00:00.001+01:002022-08-18T07:00:00.148+01:00The Differentiator<p><i>I have been very inspired by Leslie Dietiker's way of thinking about mathematics lessons as 'stories' (see Dietiker, 2015). In this post, I'm thinking more about how actual stories might be used within mathematics lessons (see <a href="https://www.mathsthroughstories.org/">https://www.mathsthroughstories.org/</a>). I don't mean historical anecdotes about famous mathematicians (Gauss summing the positive integers, Galois fighting a duel, etc.). I am thinking of completely fictitious stories that get at some mathematical concept or idea. </i><i>So, yes, I'm in 'holiday mode', but I'm hoping that this can be more than 'a bit of fun'. I'm going to give a calculus example suitable for ages 16-18, so as to counter the idea that stories are just for little children. But I'm hoping that you might be able to adapt the idea for any topic or age. I think something like this could be quite memorable but you would probably only want to do it occasionally...</i></p><p>In function-land, where all the elementary functions live, the polynomials were terrified. The <i>Differentiator</i> was stalking the land, striking fear into the hearts of all well-behaved functions. Poor $x^3$ had already been attacked, and, now as $3x^2$, had the scars to show it. Poor little $x$ was scared out of his mind – unlike $x^3$, he knew he had only <i>two</i> chances with The Differentiator before he would be reduced to nothing. But, it was really the <i>constants</i> who were most afraid - everyone knew that The Differentiator had had $\pi$ for lunch yesterday and not a crumb remained.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgFr16MQslolGoJVzPbpkUhO3CP9uiVrfeYGeG9xIFrG9N2wxiGVYnhhQlbcZry6BP2XrWvZqpTJUj8-ImBeQoeVokWDLJbKELbcqy8HjKJF3_Xlex-hwQeu6F1HIRvbpo-MXYMFpvxstj_c2-nyFrVq3o-0hjSoqIhCWEJHnfgOvETr0LiSDhxcwFnYw=s702" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="702" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgFr16MQslolGoJVzPbpkUhO3CP9uiVrfeYGeG9xIFrG9N2wxiGVYnhhQlbcZry6BP2XrWvZqpTJUj8-ImBeQoeVokWDLJbKELbcqy8HjKJF3_Xlex-hwQeu6F1HIRvbpo-MXYMFpvxstj_c2-nyFrVq3o-0hjSoqIhCWEJHnfgOvETr0LiSDhxcwFnYw=s320" width="274" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>The Differentiator</i> having $\pi$ for lunch<br /></td></tr></tbody></table><p>Everyone was looking to $x^{100}$ to help, and she was putting on as brave a face as she could. But even she knew that, sooner or later, her time would be up, and there was nothing anyone could do about it. The polynomials’ days were literally numbered.</p><p>The hopes of the entire community were pinned on their hero, $e^x$. They knew that $e^x$ could laugh in the face of The Differentiator: “Do your worst!” $e^x$ would say, and then stand back, completely unperturbed, as The Differentiator went into action. It was as though $e^x$ were inoculated against the terrors of The Differentiator.</p><p>Some of the polynomials, such as $x^2$, had decided to take refuge by hiding behind $e^x$, with some success:</p><p>$$\frac{d}{dx} \left( e^{x}x^2 \right) =e^x(x^2+2x).$$</p><p>or by climbing up onto $e$ for protection:</p><p>$$\frac{d}{dx} \left( e^{x^2} \right)=2xe^{x^2}.$$</p><p>But then, one fateful day, as The Differentiator was out prowling around, The Differentiator finally met their match, in the shape of $e^{2x}$. Unaware of what they were facing, The Differentiator attacked:</p><p>$$\frac{d}{dx} \left(e^{2x} \right) = 2e^{2x}.$$</p><p>Unbelievably, $e^{2x}$ immediately grew to twice the size! The Differentiator hit again:</p><p>$$\frac{d}{dx} \left(2e^{2x} \right) = 4e^{2x}.$$</p><p>And again, and again. But, the more times $e^{2x}$ was attacked, the stronger it became. Sixteen times its original height, $16e^{2x}$ stared down at The Differentiator, who – now defeated – turned on their heels and fled, and was never seen in function-land again.</p><p style="text-align: center;">***</p><p>Stories like this don't need to be lengthy - this one's a bit too long, I think. You could challenge students to make up their own - maybe a sequel, in which <i>The Integrator</i> comes to the rescue?</p><p>I have used this kind of story to lead in to thinking about 'differentiation-proof' functions, like $e^x$. At A-level, students meet the first-order differential equation:</p><p>$$\frac{dy}{dx}=y$$</p><p>and, with it, the idea of a <i>differentiation-proof function</i>.</p><p><i>“I’m thinking of a function, and when I differentiate it I get exactly the same function back. What might my function be?”</i></p><p>Starting with this prompt, students may suggest the <i>zero function,</i> $y=0$, and this is a trivial case of the general solution. They might think that $y=e^x$ is the only possibility for a general solution, but, of course, we need an arbitrary constant, and any constant multiple of this will also work, so the general solution is $y=Ae^x$, where $A$ is any constant, and the $A = 0$ case gives the zero function.</p><p>The general solution can be derived by separating the variables:</p><p>$$\frac{dy}{dx}=y$$</p><p>$$\int \frac{1}{y}\frac{dy}{dx}dx=\int 1dx$$</p><p>$$ \ln \lvert y \rvert=x+c$$</p><p>$$y=e^{x+c}=Ae^x,$$</p><p>where $c$ and $A$ are constants.</p><p>We can differentiate $y=Ae^x$ as many times as we like, and it never changes, so it might seem that this is ‘the’ solution to the general equation $\frac{d^n y}{dx^n}=y$, where $n$ is <i>any</i> positive integer.</p><p>But, in fact it is only <i>a </i>solution, not <i>the</i> solution, because an $n$th-order differentiatial equation ought to have a general solution containing $n$ arbitrary constants, and $y=Ae^x$ contains only one arbitrary constant. Students of Further Mathematics may come across the second-order differential equation $\frac{d^2 y}{dx^2}=y$, which has general solution $y=Ae^x+Be^{-x}$, with <i>two</i> arbitrary constants, $A$ and $B$, this time. So, here is the general solution to finding a function which, when differentiated twice, returns to the original function. And note that this is not necessarily (unless $B= 0$) identical to the original function after just <i>one</i> differentiation.</p><p>It is natural for students to wonder about how this pattern might continue for the <i>third</i>-order differential equation $\frac{d^3 y}{dx^3}=y$. It is clear that $y=Ae^x$ will be one part of this, since we have seen that this satisfies <i>any</i> differential equation of the form $\frac{d^n y}{dx^n}=y$, where $n$ is a positive integer, since $Ae^x$ is differentiation-proof. But, now that we have a <i>third</i>-order differential equation, we should be expecting <i>two more</i> arbitrary constants, and how can we find them?</p><p>The term $Be^{-x}$ term has the wrong parity, since differentiating this three times gives $-Be^{-x}$, rather than $Be^{-x}$, and it seems like we have hit a brick wall. However, since terms involving $e^{kx}$ have served us very well so far, it may seem natural to use $y=e^{kx}$ as a trial solution in $\frac{d^n y}{dx^n}=y$. This gives us</p><p>$$k^n e^{kx}=e^{kx}$$ </p><p>Since $e^{kx}$ is never zero, we require that $k^n=1$, so the $k$s that we need are the $n$<i>th roots of unity</i>.</p><p>When $n = 1$, $k = 1$, and we have $y=Ae^x$, as we found.</p><p>When $n = 2$, $k = \pm 1$, and we have $y=Ae^x+Be^{-x}$, as we also found.</p><p>Now that $n = 3$, we still have $k = 1$, but we also have two <i>complex</i> roots, $k = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$, so our general solution should be</p><p>$$y=Ae^x+Be^{\left( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \right) x} + Ce^{\left( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \right) x},$$</p><p>which we can write as</p><p>$$y=Ae^x+B'e^{-\frac{x}{2}}\cos \left( \frac{\sqrt{3}x}{2} \right) + C'e^{-\frac{x}{2}}\sin \left( \frac{\sqrt{3}x}{2} \right),$$</p><p>or even, if we wish, as</p><p>$$y=Ae^x+B'\exp{\left( e^{ \frac{2 \pi i}{3} } x \right)}+C'\exp{\left( e^{- \frac{2 \pi i}{3} } x \right)}.$$</p><p>The $n=4$ case is much neater:</p><p>$$y=Ae^x+B''e^{-x}+C''e^{ix}+D''e^{-ix}$$</p><p>And we have explored the general terrain of 'differentiation-proof' functions!</p><h3 style="text-align: left;">Question to reflect on</h3><p>1. What stories might you use in mathematics lessons to stimulate some worthwhile mathematical thinking?</p><h3 style="text-align: left;">Reference</h3><p>Dietiker, L. (2015). Mathematical story: A metaphor for mathematics curriculum. <i>Educational Studies in Mathematics, 90</i>(3), 285-302. <a href="https://doi.org/10.1007/s10649-015-9627-x">https://doi.org/10.1007/s10649-015-9627-x</a></p><p><br /></p><p><br /></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-36743136998217896462022-08-04T07:00:00.010+01:002022-08-04T07:00:00.159+01:00Misremembering Goldbach’s Conjecture<p><i>It's the holiday, so a shorter, lighter blogpost today, and only one reflection question. I hope you are having a good break!</i></p><p>When I went on <a href="http://mrbartonmaths.com/podcast/">Craig Barton’s podcast</a> for the first time (Note 1), he asked me (as he asks all his guests) to recount a ‘favourite failure’ - a situation where things didn't go to plan. I had so many to choose from that I had plenty of ideas leftover after the episode, so I thought I’d relate another one here… </p><p>This is about the time when I misremembered <i>Goldbach’s Conjecture</i>, which states:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>Every even integer greater than 2 is the sum of two primes.</i></p></blockquote><p>Unfortunately, for some reason, I misremembered it as:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>Every integer greater than 2 is the sum of two primes.</i></p></blockquote><p>If I had taken a moment to reflect on this, I would have realised that this obviously couldn’t be right, but it was one of those situations where I was distracted or ill or something (I can’t remember the specifics of my excuses!). And so I noticed nothing and carried on...</p><p>I wanted my Year 8 class (age 12-13) to work on something a bit exploratory and to understand the notion of a ‘counterexample’ – and also get in a bit of incidental practice on recognising prime numbers, which we had just been working on. So, this seemed like a good way to address all of that.</p><p>So, I told the class that Goldbach’s Conjecture was one of the best-known unsolved problems in all of mathematics, and I explained what a counterexample was. No one knows how to prove that Goldbach’s Conjecture is true, but, if it is false, all it needs is one counterexample to demonstrate that. A single counterexample can do a great deal of work!</p><p>The students seemed interested in this:</p><p><i>“Would we be famous if we found a counterexample?”<br />“Sure!”</i></p><p>There was immediately a bit of confusion about the number 3, which should have alerted me to the fact that something was wrong. Some pupils had written $3 = 1 + 2$, but others were – correctly – saying that 1 is not a prime number, in which case 3 would be a counterexample. I knew that 1 did used to be considered as a prime number (see Foster, 2016), so I thought perhaps this was just a historical glitch, so I decided that we would say “every integer <i>greater than 3</i>”, rather than 2, to avoid that problem.</p><p>And so the students began work. Of course, I knew very well that they would not find a counterexample, since all numbers at least as far as $4 \times 10^{18}$ have been checked (Note 1). If ever a teacher knew ‘the right answer’, I knew that the right answer here was that there would be no counterexample today!</p><p>The students began work on their own or in pairs, writing (at least, those working more systematically!) things like this:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;">$4 = 2 + 2$<br />$5 = 2 + 3$<br />$6 = 3 + 3$<br />$7 = 2 + 5$<br />$8 = 3 + 5$<br />$9 = 2 + 7$<br />$10 = 5 + 5$<br />$11 = …$</div></blockquote><p>I walked around casually observing what was going on, my mind drifting a little, perhaps. I engaged in some discussion with students about who Goldbach was, why prime numbers matter, and so on, in quite a relaxed way. This was basically routine practice with primes in a more interesting context (a kind of <i><a href="http://www.mathematicaletudes.com/">mathematical etude</a></i>, see Foster, 2018).</p><p>I gradually became aware that I could hear the number 11 being muttered quite a bit.<a href="https://blogger.googleusercontent.com/img/a/AVvXsEjrw2oLVc1nKUaMtnoUCtPT2C5UBjd75grDG9-N9yMxN2OUMlw7dtTeZUB631oyq5HOsT0FpREz_Fqu8X6lad8CB4VliHBYsWWQnxa-pxQnfiCa9HTFMfXpiAgxBwgmjBJEYdREBUGGfQwoG0uN4rPAHR8EErtKTXu4VgaUeNV0vIDRGdCFq8J6xWcalA=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="380" data-original-width="600" height="254" src="https://blogger.googleusercontent.com/img/a/AVvXsEjrw2oLVc1nKUaMtnoUCtPT2C5UBjd75grDG9-N9yMxN2OUMlw7dtTeZUB631oyq5HOsT0FpREz_Fqu8X6lad8CB4VliHBYsWWQnxa-pxQnfiCa9HTFMfXpiAgxBwgmjBJEYdREBUGGfQwoG0uN4rPAHR8EErtKTXu4VgaUeNV0vIDRGdCFq8J6xWcalA=w400-h254" width="400" /></a></p><p>Then a few people started to say that they had found a counterexample, and it was 11. I decided that this would be a good opportunity to stop everyone and highlight the importance of ‘being systematic’. There's 'being systematic' in the sense of choosing your numbers according to some pattern, rather than haphazardly, but there's also 'being systematic' when you check each number. It’s easy to think you have found a number which can’t be made by summing two primes, and it may just be because you haven’t thoroughly checked all the possibilities. You haven’t managed to <i>find</i> a pair of primes that sum to 11, but that doesn’t mean that there <i>isn’t one</i>. The only way to be sure is to be systematic and check all the possibilities in such a way that you can be sure that you haven’t missed any. “Go back and check – be systematic – make sure you haven’t missed a possibility!” All good advice, to be sure.</p><p>I vividly remember the moment that one student came to the board and said, essentially:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>Look, the only possibilities for 11 are:</i></p></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><i>1 and 10, but neither is prime <br />2 and 9, but 9 isn’t prime <br />3 and 8, but 8 isn’t prime <br />4 and 7, but 4 isn’t prime <br />5 and 6, but 6 isn’t prime <br />So, 11 is a counterexample. </i></div></blockquote><p>Ordinarily, I would have been very pleased with such a proof by exhaustion. But, I remember staring at the board thinking, “What am I missing?” Even if we included 1 as prime, it would have to go with 10, which had certainly never been prime in anyone’s book!</p><p>As I tried to figure out what was going on, the class became more excited at my puzzlement:<br /><i>“We’ve done it! We’ve solved this big maths problem – and it wasn’t even that hard!”, “Are we going to be on the news, sir?”, “Maybe no one ever bothered to check 11 because they assumed someone else had already checked it? Sometimes it’s the easy things that get missed!”, etc. </i></p><p>Obviously, if there were a counterexample, it was going to be considerably higher than 11. So, feeling desperate, I now Googled “Goldbach’s Conjecture”: “Every even integer greater than 2 is the sum of two primes.” ‘<i>Even, even, even</i>!’ (Having computers connected to the internet in every classroom has to be one of the great pedagogical advances of recent decades.)</p><p>Of course, with hindsight it is very clear that the only way to make an odd number by summing two integers is if one of the integers is odd and the other one is even. And the only even prime is 2. So, the only way my version of Goldbach’s Conjecture could be true is if every odd number were 2 greater than a prime. This is equivalent to saying that every odd number is prime, and although it is true that (almost, with the exception of 2), every prime number is odd, the reverse is not the case. This is why we had the problem with 3, because 1, which is $3-2$, is not prime. But then 5, 7 and 9 all have primes 2 less than them, so everything seems fine for a while, but then 11 doesn’t, because 9 is not prime, and that’s why it had appeared to be a counterexample. So, at least there was something mildly interesting to understand in relation to my mistake. Obviously, a counterexample to one conjecture is not necessarily a counterexample to a different conjecture.</p><p>There was understandably limited enthusiasm now for going back and checking for counterexamples to the <i>real</i> Goldbach Conjecture. It felt like the moment had passed, and perhaps the objectives of understanding what a counterexample is and gaining facility with primes had been accomplished more or less anyway.</p><p>I reflected afterwards on the strange feeling of seeing the student's apparently flawless proof and yet not believing it – the feeling that ‘there must be something wrong even though I can’t see what it is’. However rational we might aspire to be about mathematics, we are influenced by more than merely logical arguments. I was quite sure that the students must have made a mistake and omitted a possibility, and I was reluctant to believe even the very simple mathematics of their proof until I had appreciated the wrong assumption that <i>I</i> had begun the whole lesson with.</p><h3><span style="font-family: inherit;">Question to reflect on</span></h3><p>1. Do you have any 'armchair responses' (AssocTeachersMaths, 2020; Foster, 2019) to my ‘favourite failure’ or to any of your own?</p><h3 style="text-align: left;">Notes</h3><p><span style="white-space: pre;">1. </span>You can listen to the episode here: <a href="http://www.mrbartonmaths.com/blog/colin-foster-mathematical-etudes-confidence-and-questioning/">http://www.mrbartonmaths.com/blog/colin-foster-mathematical-etudes-confidence-and-questioning/</a> </p><p>2.<span style="white-space: pre;"> </span>See <a href="http://sweet.ua.pt/tos/goldbach.html">http://sweet.ua.pt/tos/goldbach.html</a></p><h3 style="text-align: left;">References</h3><p>AssocTeachersMaths. (2020, July 13). <i>Armchair Responses to Classroom Events - with Colin Foster</i> [Video]. YouTube. <a href="https://youtu.be/L0ovhillL0c">https://youtu.be/L0ovhillL0c</a></p><p>Foster, C. (2016). Questions pupils ask: Why isn’t 1 a prime number? <i>Mathematics in School, 45</i>(3), 12–13. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20isn't%201%20a%20prime%20number.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20isn't%201%20a%20prime%20number.pdf</a></p><p>Foster, C. (2018). Developing mathematical fluency: Comparing exercises and rich tasks. <i>Educational Studies in Mathematics, 97</i>(2), 121–141. <a href="https://doi.org/10.1007/s10649-017-9788-x">https://doi.org/10.1007/s10649-017-9788-x</a></p><p>Foster, C. (2019). Armchair responses. <i>Mathematics in School, 48</i>(3), 26–27. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Armchair%20responses.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Armchair%20responses.pdf</a></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-73613603861273278562022-07-21T07:00:00.014+01:002022-07-21T07:00:00.150+01:00Making rounding interesting<p><i>Are there any 'boring' topics in mathematics? Understandably, mathematics teachers tend to be kind of professionally committed to the idea that all mathematics topics are interesting. If even the teacher doesn’t find the topic interesting, then what hope is there for the students? And yet, perhaps, if we are completely honest about it, we find some topics a bit harder to be enthusiastic about. For me, ‘rounding’ is that kind of a topic. But can it be made interesting?</i></p><p>I wonder if rounding is any mathematics teacher’s favourite topic? Somehow I doubt it, although perhaps, following this post, lots of people will write in the comments that it is theirs, which would be interesting! Even if it perhaps isn’t the most exciting topic, it’s certainly one that contributes to success in high-stakes assessments. Students will be repeatedly penalised throughout their examination paper if they don’t correctly round their answers to the specified degree of accuracy, so it’s certainly something that needs teaching. Boring but important?</p><p>When I suggest that rounding is not a very intrinsically interesting topic, I am not talking about ‘estimation’. That is certainly something that can be extremely interesting and engaging. I really like beginning with some scenario, such as a jar of sweets, and asking students for their off-the-top-of-their-head guesstimates of how many there are, and then coming up with a variety of different ways to improve on this (Note 1). Ideally this leads to the notion that quicker, rougher estimates are not necessarily ‘worse’ than more accurate ones, and choosing the optimal level of accuracy depends on the <i>purpose</i> for which you need the estimate and how much time you have available and how much effort seems worthwhile. Level of accuracy needs to be fit for purpose. A good way to promote this is to ask questions like:</p><p></p><ul style="text-align: left;"><li>Which weighs more - a cat or 10,000 paperclips?</li><li>Which mathematics teacher in our school do you think is closest to being a billion seconds old?<br /><i>[Apologies, but I don't know where I first came across these examples - please say in the comments if there is someone I should acknowledge for these.]</i></li></ul><p></p><p>In these, it is clear that your estimates only need to be accurate enough to answer the question. There is no point obtaining more accuracy than you need to do that.</p><p>But here I’m not thinking about contextual estimates like that but the more abstract kind of questions that you see in textbooks and on examinations, like:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;">Estimate the answer to $0.278 \times 73.4-\sqrt{48.3}$.</p></blockquote><p>These questions that ask for ‘an estimate’ but don’t specify how accurate it should be are a bit nonsensical, it seems to me. You could always answer <i>any</i> question like this with ‘zero’, and the only hard part would be working out what degree of accuracy this was to (which the question never asks for). For example, the answer to this calculation turns out to be 13, to the nearest integer, so this would be 0 to the nearest 100, 0 to the nearest 1000, or (to be on the safe side) 0 to the nearest billion! Any point on the number line is always ‘close’ to zero if you zoom out far enough. So, you will technically never go wrong with a question like this by answering ‘0 to the nearest trillion’ – although of course mark schemes would not reward you for that!</p><p>More seriously, the usual approach that is taught within school mathematics is to round each individual number to 1 significant figure, with the possible exception of when you are about to find a root, where you might fiddle it to the nearest convenient number instead. So, in this example, although 48.3 would round to 50 to 1 significant figure, or 48 to 2 significant figures, we might instead choose to round it to 49, because $\sqrt{49}=7$. Doing that, we would get something we should be able to do easily in our head: $0.3×70-\sqrt{49}=21-7=14$.</p><p>The issue of how good our estimate might be (and therefore what it might be good <i>for</i>) is not really addressed at this level, and students would be expected simply to leave their answer as 14, without any idea how close this is likely to be to the true answer, or even whether it is an underestimate or an overestimate. But is this $14±1$ or $14±1000$ or what? This is really a bit strange, as, in any real situation, a lot of the value in estimating is in getting <i>bounds</i>. We may not care exactly what the answer is, but it is usually important to know that it is definitely between some number and some other number. Simply throwing back an answer like ‘14’, which we know is almost certainly not exactly right, without having any idea <i>how</i> wrong it is, doesn’t seem very useful. Usually, we are estimating a number in order to enable us to make some real-life decision – how much paint to buy, or how many coaches to order – and those all require us to commit to some actual quantity. So, really, I don’t want to know ‘roughly 14’ – I want to know ‘definitely between 10 and 20’ or definitely between 10 and 15. So perhaps we should teach it this way? (Note 2)</p><p>Then, we can consider that how much effort it is worth going to in order to get a more accurate estimate depends on how narrow I want my bounds to be. It’s foolish to act as though more accuracy is simply an absolute good. (Sitting down and calculating more and more digits of $\pi$ forever would not serve any useful purpose.) Sometimes, when peer marking, students will be told to give themselves more marks if their answer is closer to the ‘true’ answer, but I think this reinforces an unhelpful view of estimation. It also encourages students to 'cheat', by first calculating the exact answer, rounding this answer, and then constructing some fake argument for how they legitimately got it. If more accuracy were always better, we would always use a calculator or computer and get the answer to as high a degree of accuracy as we possibly could. But, with estimation, the sensible thing to do is to spend your effort according to how <i>useful</i> any extra accuracy would be in the particular context that applies. These seem to me the important issues in estimation, and they go largely unaddressed in the lessons on estimation that I see.</p><h4 style="text-align: left;">Exploring rounding</h4><p>One way to make the topic of ‘rounding’ a bit more interesting is to begin to explore some of these issues. For example, in the calculation above, since (almost) all of the numbers were rounded to 1 significant figure, it might seem sensible to give the <i>answer</i> to 1 significant figure, which would be 10, suggesting that this means $10±5$. In this case, the exact answer to the original calculation (13.45537…) is also 10, to 1 significant figure, which is good, and there seems to be an assumption within school mathematics that almost has the status of a theorem:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>Claim 1: </i>If you round each number in a calculation to 1 significant figure, then the answer will also be correct to 1 significant figure.</p></blockquote><p>However, there is no reason at all why this should be true, and you might like to consider what the simplest counterexample is that you can find. When can you be more confident using this heuristic, and when should you be more cautious?</p><p style="text-align: center;">***</p><p>A simple counterexample would be $3.5+3.5$, which comes to 8 if you round each of the 3.5s to 4 (to 1 significant figure) before you add them, but 8 is <i>not</i> the correct answer to 1 significant figure, because of course it should be 7.</p><p>A slightly more complicated scenario that might be worth exploring with students involves rounding two numbers in a <i>subtraction</i>, so we could begin with a question like this:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;">Estimate the answer to $14.2-12.9$.</p></blockquote><p>(You might ask, "Why estimate something so simple, and not just calculate?", and the point of this is not to be a realistic rounding scenario, but a simplified situation to help us see what is actually going on with rounding.)</p><p>So, here's another claim:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>Claim 2: </i>If we round each number to the nearest integer, then the answer will be correct to the nearest integer.</p></blockquote><p>Is this claim always, sometimes or never true?</p><p>Students will need a bit of time to figure out what the claim even means. Using the notation $[x]$ to mean “round $x$ to the nearest integer”, we could write:</p><p>$[14.2]-[12.9] = 14-13 = 1$</p><p>And $[14.2-12.9] = [1.3] = 1$.</p><p>So that checks out in this case.</p><p>So, in this notation, the question is:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;">When is $[a-b] = [a]-[b]$?</p></blockquote><p>I think this is a potentially interesting task, where there is plenty to think about, but it also generates some repetitive but somehow acceptable routine practice. (I call such tasks <i><a href="http://www.mathematicaletudes.com/">mathematical etudes</a></i> – see Foster, 2018). You might like to try it yourself before reading on.</p><p style="text-align: center;">***</p><div>Running through this for all possibilities of $14.x-12.y$, where $x$ and $y$ are single digits between 1 and 9, we find this situation: </div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiLIk0CQMGzso4jbcV1UOaZhWZMkCsPmDKfBquykAvNRd4IwzcD_eRfYlV7Uo1QT36GpSAUabvBkYcfXR8PJBGDF20uqg6I-mruhS5neC7gU5ip45KxsjgHdCetwFFGFYKz7feusSBFwlcRXXcfOalJJ9bS--48Mu8JGLNmJ1Q0j8Xw8lV2iwC6RLa95g=s600" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="228" data-original-width="600" height="245" src="https://blogger.googleusercontent.com/img/a/AVvXsEiLIk0CQMGzso4jbcV1UOaZhWZMkCsPmDKfBquykAvNRd4IwzcD_eRfYlV7Uo1QT36GpSAUabvBkYcfXR8PJBGDF20uqg6I-mruhS5neC7gU5ip45KxsjgHdCetwFFGFYKz7feusSBFwlcRXXcfOalJJ9bS--48Mu8JGLNmJ1Q0j8Xw8lV2iwC6RLa95g=w640-h245" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 1. $[14.x-12.y]$ and $[14.x]-[12.y]$ compared, where $x$ and $y$ are single digits between 1 and 9.<br />Ticks indicate where equality holds.</td></tr></tbody></table><p>So, a counterexample to Claim 2 would be any of the <i>empty</i> cells in this table; for example, $[14.7-12.3] = [2.4] = 2$ but $[14.7]-[12.3] = 15-12 = 3$.</p><p>We might prefer to write this on one line as: </p><p>$$3 = 15-12 = [14.7]-[12.3] ≠ [14.7-2.3] = [2.4] = 2$$</p><p>Applying some deduction, we might say:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;">1. If both numbers in the subtraction are rounded up, or both are rounded down, we should get a tick. These are the ticks in the green squares in the table above.</p></blockquote><p>We might also be tempted to say:</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;">2. If one number is rounded up and the other number is rounded down, we should <i>not</i> get a tick.</p></blockquote><p>As you can see from Table 1, this is false, as can be seen by the ticks in <i>some</i> of the white squares. Why does this happen? For example, in $14.3-12.8$, the minuend rounds down and the subtrahend rounds up, and we get no tick, since the difference, 1.5, rounds up to 2, whereas $14-13 = 1$. However, this doesn’t always happen. For example, in $14.2-12.8$, as before, the minuend rounds down and the subtrahend rounds up, giving $14-13 = 1$, but this time the difference is only 1.4, which rounds <i>down</i> to 1, so we <i>do</i> get a tick.</p><p>There is lots to explore here, and the idea of comparing the result from applying some function before and after some composition - i.e., $f(x \pm y) \stackrel{?}{=} f(x) \pm f(y)$ - is a highly mathematical question.</p><p>When I look back at mathematics tasks that I have designed over the years, I now notice that they often cluster around certain ‘favourite’ topics. Without meaning to, I have unintentionally avoided certain topics – perhaps those that, like ‘rounding’, seem intrinsically less interesting – and cherrypicked other topics to design tasks for. At Loughborough, <a href="https://www.lboro.ac.uk/services/lumen/curriculum/">we are currently working on designing a complete set of teaching materials for Year 7-9</a>, so we are now working in the same kind of situation as teachers – we can’t miss anything out! And this has led me to thinking about how to address some of those potentially ‘less interesting’ topics, which is proving fun.</p><h3 style="text-align: left;"><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. Are there mathematics topics that you personally find less interesting to teach? Which ones? Why?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. What tasks can make these topics more interesting for you and for your learners?</span></p><p style="text-align: left;"><span style="font-family: inherit;">3. For rounding, what other tasks can you devise? </span>Is $[a+b] \stackrel{?}{=} [a]+[b]$ an easier or harder problem? What about $[ab] \stackrel{?}{=} [a][b]$?</p><h3 style="text-align: left;">Notes</h3><p>1. Dan Meyer is the expert at designing tasks like this; e.g., <a href="https://blog.mrmeyer.com/2009/what-i-would-do-with-this-pocket-change/">https://blog.mrmeyer.com/2009/what-i-would-do-with-this-pocket-change/</a>; <a href="https://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/">https://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/</a>; <a href="https://www.101qs.com/70-water-tank-filling">https://www.101qs.com/70-water-tank-filling </a></p><p>2. I think the various versions of the “approximately-equal sign” $≈$ are not really our friends here, because a statement like $13≈10$ doesn’t really have a precise meaning.</p><h3 style="text-align: left;">References</h3><p>Eastaway, R. (2021). <i>Maths on the back of an envelope: Clever ways to (roughly) calculate anything</i>. HarperCollins.</p><p>Weinstein, L., & Adam, J. A. (2009). <i>Guesstimation</i>. Princeton University Press.</p><p>Weinstein, L. (2012). <i>Guesstimation 2.0</i>. Princeton University Press.</p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-4005989727753790922022-07-07T07:00:00.012+01:002022-07-07T07:00:00.144+01:00A football on the roof<p><i>I am always on the lookout for 'real-life' mathematics that is of potential relevance and interest to students but where the mathematics isn't trivial and the context isn't contrived. Too often the scenario is of potential interest but the mathematics is spurious, and doesn't really offer anything in the actual situation that couldn't be done more easily without mathematics. It is not easy to find good examples, but I think this is one that might provide some opportunities to work on topics such as similar triangles and ratio.</i></p><p>Some students lost a football on a flat roof and wanted to know whether the ball had rolled off and fallen down behind the back of the building (i.e., gone forever) or whether it was worth climbing up to retrieve it (Note 1). There weren't any tall buildings nearby that they could access to get a good view of the roof. What they needed to know was how far back from the building they needed to stand so as to be sure that <i>if</i> the ball was there they would be able to see it.</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: left;"><i>"Can you see it?"<br /></i><i>"No, but I just need to go back a bit further."<br /></i><i>"If it was there, you'd be able to see it by now."<br /></i><i>"I'm not sure."</i></div></blockquote><p>In particular, going as far back from the building as possible, given the constraints of the <i>surrounding</i> buildings, did the fact that they couldn't see the football mean that it was <i>definitely not there</i>, or could it be that it was just not visible over the edge of the building (Figure 1)?</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjbwHM5T2-5hrvBHBwPOP5Tfa3LsRL_91uClfW3JJa4S99t8DXyFcM6R--Zc-CNi5CSBWAK3pXksakIfn1F8Fqz06SEZ_oSt2EuhGqB6SUnUoR-6NxEIdK_zBOpZPyE4ZOnDOzDYfQvZETyOHtgMTMskkIeySQhgHySq9n_gi2anSo2j52Vtyh5TfkK5g=s600" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="250" data-original-width="600" src="https://blogger.googleusercontent.com/img/a/AVvXsEjbwHM5T2-5hrvBHBwPOP5Tfa3LsRL_91uClfW3JJa4S99t8DXyFcM6R--Zc-CNi5CSBWAK3pXksakIfn1F8Fqz06SEZ_oSt2EuhGqB6SUnUoR-6NxEIdK_zBOpZPyE4ZOnDOzDYfQvZETyOHtgMTMskkIeySQhgHySq9n_gi2anSo2j52Vtyh5TfkK5g=s16000" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. A football on the roof</td></tr></tbody></table><p>This problem is reminiscent of lessons in which students determine the height of a tree near the school using a clinometer, but it feels somehow different. There is not usually any good reason for wanting to know the height of a tree, and it is usually hard to find any way to decide afterwards whether the students' estimates are reasonably accurate or not. In this case, there is a clear 'need to know' and, ultimately, when the site manager brings a ladder, the students would discover if they were right or wrong, so it feels as though something is at stake.</p><p>A good way to start would be to decide on simplifying assumptions that it seems sensible to make; i.e., things that we might sensibly choose to ignore. For example:</p><p></p><ul style="text-align: left;"><li>assume that the ground and all roofs are perfectly horizontal</li><li>assume that the roof in question is free from any debris</li><li>assume that the building has height 4 m and goes back 6 m</li><li>assume that the football is perfectly spherical, with diameter 22 cm</li><li>assume (worst-case scenario) that the ball is right at the back of the roof, against the brick wall</li></ul><p></p><p>Students may suggest more outlandish things, such as assuming that light travels in straight lines or that the curvature of the earth is negligible, and I would include these as well if they raised them.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhvjhsmd2MIDEUazgNe7JM9ws-A_WwIQaragURWm7qYAgEypdIfgpzco3ywecwttuWZZm3DVk5XJy01XQxJ57ue33GgN_1EwI7_q_n2VxFOJPSxo1GZWxw7vDCkks-3zy6ga4sPe0Hjj_bFRNszesjouj5_yKALLbr5zMSG1p29lnOkVzQYV-JXCheqsg=s800" style="margin-left: auto; margin-right: auto; text-align: center;"><img border="0" data-original-height="349" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEhvjhsmd2MIDEUazgNe7JM9ws-A_WwIQaragURWm7qYAgEypdIfgpzco3ywecwttuWZZm3DVk5XJy01XQxJ57ue33GgN_1EwI7_q_n2VxFOJPSxo1GZWxw7vDCkks-3zy6ga4sPe0Hjj_bFRNszesjouj5_yKALLbr5zMSG1p29lnOkVzQYV-JXCheqsg=s16000" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Careful analysis (diagram not drawn to scale)</td></tr></tbody></table><p>Let's start with a careful analysis, which uses trigonometry and is 'a sledgehammer to crack a nut' for this simple scenario. This is <i>not</i> the approach that I would envisage students taking.</p><p>Lots of the work has been done in the diagram (Figure 2), and our units are metre throughout.</p><p>We have</p><p>$$\tan \theta = \frac{r}{l}$$</p><p>$$\tan 2\theta = \frac{h}{d}$$</p><p>Using the identity</p><p>$$\tan 2\theta \equiv \frac{2 \tan \theta}{1-\tan^2 \theta}$$</p><p>we obtain</p><p>$$\frac{h}{d}= \frac{2 \left( \frac{r}{l} \right) }{1- \left( \frac{r}{l} \right)^2},$$</p><p>giving</p><p>$$d= \frac{h(l^2-r^2)}{2rl}.$$</p><p>We can now substitute in some reasonable values:</p><p></p><ul style="text-align: left;"><li>$h=4-1.8=2.2$; the height of the building subtract the maximum eye height of the student when standing on tip toes or jumping,</li><li>$l=6-0.22=5.78$; the depth of the shed subtract the diameter of the football, and</li><li>$r=0.11$.</li></ul><p></p>This gives $d=57.8$, so the student would <i>just</i> be able to see the top of the football from about 58 metre back from the shed.<br /><p>But the trigonometry here is overkill for the nature of this problem and the accuracy required, so it would be much quicker and more reasonable to use the <i>simplified</i> diagram shown in Figure 3.</p><p style="text-align: left;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEij69bdbGh98yxl3UCPykGlGmij-1pLWOo-Ta2c4ZoRUHCgonLsialdowRiJ7-r8qDx8CPr7KUbE6yOjZ3xBFh3zflcGky44KfKE27KEjAHXZ6GcDPrN7ubg0yw2WsGnrZj37JDLLgpRY3YxqnStD5egvbEQswj259Tds5GS6JFpSX8T2psTQHg2-M_VQ=s800" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="348" data-original-width="800" src="https://blogger.googleusercontent.com/img/a/AVvXsEij69bdbGh98yxl3UCPykGlGmij-1pLWOo-Ta2c4ZoRUHCgonLsialdowRiJ7-r8qDx8CPr7KUbE6yOjZ3xBFh3zflcGky44KfKE27KEjAHXZ6GcDPrN7ubg0yw2WsGnrZj37JDLLgpRY3YxqnStD5egvbEQswj259Tds5GS6JFpSX8T2psTQHg2-M_VQ=s16000" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 3. Rougher analysis (diagram not drawn to scale)</td></tr></tbody></table><p></p><p style="text-align: left;">For this rougher analysis, we don't need to use $\tan$ explicitly and can just equate corresponding ratios in similar triangles.</p><p style="text-align: left;">So,<br /></p><p style="text-align: left;">$$\frac{d}{h}=\frac{l}{2r},$$</p><div><div style="text-align: left;"><p style="text-align: left;">giving that</p><p style="text-align: left;">$$d=\frac{hl}{2r},$$</p><p style="text-align: left;">so, with the same values as above, this again gives $d=57.8$, correct to 1 decimal place, and the same conclusion that the student would <i>just</i> be able to see the top of the football from about 58 metre back from the shed.</p><p style="text-align: left;">In the situation where $l \gg r$, we can see that in our first equation</p><p style="text-align: left;">$$d= \frac{h(l^2-r^2)}{2rl}$$</p><p style="text-align: left;">the bracket $(l^2-r^2)$ will, to a good approximation, reduce to $l^2$, giving</p><p style="text-align: left;">$$d \approx \frac{hl^2}{2rl}=\frac{hl}{2r},$$</p><p style="text-align: left;">as before. So, all routes lead to an answer of about 58 metre.</p><p style="text-align: left;">But, what if the playground extends only, say, 40 metre before meeting another building? Would it be worth the students going inside and fetching a chair to stand on? Would that make enough difference to be worth the trouble?</p><p style="text-align: left;">The beauty of having derived a formula is that a question like this can be answered instantly by substitution. All that changes here is that $h$ reduces from 2.2 to, say, 1.7.</p><p style="text-align: left;">So,</p><p style="text-align: left;">$$d=\frac{hl}{2r}=\frac{1.7 \times 5.78}{2 \times 0.11}=44.7,$$</p><p style="text-align: left;">and so this would <i>not</i> quite be enough to work within the available space, since $44.7 > 40$. Rearranging the equation to give</p><p style="text-align: left;">$$h=\frac{2rd}{l}$$</p><p style="text-align: left;">reveals that, unless you can find a stool of height at least $2.2 -1.52 = 0.68$ metre, then there is no point bothering.</p><p style="text-align: left;">The mathematics here is not profound, but the result is not guessable without it. I think we need more tasks like this, where a little bit of mathematics (not pages and pages) tells you something practically useful that you couldn't have guesstimated accurately enough without it.</p><h3 style="text-align: left;"><span style="font-family: inherit;">Questions to reflect on</span></h3><p>1. Would your students find a task like this credibly realistic and engaging? How might you improve it?</p><p>2. What 'real-life' tasks do you use that are both non-trivial mathematically and non-embarrassing in terms of correspondence with reality?</p><h3 style="text-align: left;">Note</h3><p>1. Disclaimer: Nothing in this post should be taken to endorse climbing onto roofs to retrieve footballs!</p><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEirbaaJugduY-3x1gJubKY_xOlTfLMpEjyLojeuPumTAJJa6VPo5kE4o8eydPdpCEPF6z_hR6FFe0UaQfcrGJug7s576wjTddS_g_M2PbN6JP_iX6PM_kaZX4RwDkMoXxqycjdOljBk3eCr7IO3nzvDOszlKGwID20PDaafk7bDV2OSgFFbZpM4rAOjsg=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="298" data-original-width="600" src="https://blogger.googleusercontent.com/img/a/AVvXsEirbaaJugduY-3x1gJubKY_xOlTfLMpEjyLojeuPumTAJJa6VPo5kE4o8eydPdpCEPF6z_hR6FFe0UaQfcrGJug7s576wjTddS_g_M2PbN6JP_iX6PM_kaZX4RwDkMoXxqycjdOljBk3eCr7IO3nzvDOszlKGwID20PDaafk7bDV2OSgFFbZpM4rAOjsg=s16000" /></a></p></div></div>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-43693964077700927862022-06-23T07:00:00.010+01:002022-06-27T09:55:55.647+01:00Lines of not-very-good fit<p><i>Does anyone teach lines of best fit 'properly' in lower secondary school? I think whenever I’ve seen this concept taught, or taught it myself, it’s always been a bit wrong.</i></p><p>Typically, students are given a scatterplot, or they draw one themselves, and are asked to draw a straight line on top of it, by eye, but the instructions for how they are supposed to draw this line can be a bit vague. Maybe the teacher says something like, “The 'line of best fit' goes roughly through the middle of all the scatter points on a graph.” (BBC Bitesize: <a href="https://www.bbc.co.uk/bitesize/guides/zrg4jxs/revision/9">https://www.bbc.co.uk/bitesize/guides/zrg4jxs/revision/9</a>) I guess this is kind of right, but I think that any student hearing this is bound to misinterpret what this is supposed to mean.</p><p>Suppose you give students the $x$-$y$ scatterplot below (Note 1), and ask them to draw the best straight line they can that takes account of all these points. </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj_JB8KsO7W4dSyD0NgVua1Eai78ZbyG6AlTMf6XjDOgFMMkRLfY-rdIyzexjNAmB9BrtGEfXvMhVdmwYv3UTvuQZrkQzqZjp6ooAyEgJe2myHYNvsHrukpGY2QGzzO7ORkHNO3vsgtmDtzjfvArlPrZArFQT66V9W1JP31bqJaL7PT1rWljm1QcyScAg=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEj_JB8KsO7W4dSyD0NgVua1Eai78ZbyG6AlTMf6XjDOgFMMkRLfY-rdIyzexjNAmB9BrtGEfXvMhVdmwYv3UTvuQZrkQzqZjp6ooAyEgJe2myHYNvsHrukpGY2QGzzO7ORkHNO3vsgtmDtzjfvArlPrZArFQT66V9W1JP31bqJaL7PT1rWljm1QcyScAg=s320" width="320" /></a></div><p>Of course, they could draw something like this, which “goes roughly through the middle of all the scatter points” (10 points on either side).</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgMOVkQHcMHE9RjswSyqd2etr5PeAuRtobw1H_jvJphbdE-dClModmJ9yS58NB-X7KshYISBtTdUCY1oeYV-84OhDtJMlkdMuatxHhV3wSw8UkooD61Eg0Ac3auFSYRUv40cZxaOVYntsEff4oMHSVv5UY0lLJ4av2abMXB6J_ASW-OF9F5mWnyRmIR_Q=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgMOVkQHcMHE9RjswSyqd2etr5PeAuRtobw1H_jvJphbdE-dClModmJ9yS58NB-X7KshYISBtTdUCY1oeYV-84OhDtJMlkdMuatxHhV3wSw8UkooD61Eg0Ac3auFSYRUv40cZxaOVYntsEff4oMHSVv5UY0lLJ4av2abMXB6J_ASW-OF9F5mWnyRmIR_Q=s320" width="320" /></a></div><p>But, unless they are trying to be awkward, they will probably be much more likely to draw something like this.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiW59NhfOgtmuB9xnzy8-ES4qGLV5smo8hL_V1QkQAg_4ISFWTdWIclVh26p3G8z4hJCtA_4hvmQa1QdS7Fa6Knl-QltsZHv8_kuo_ccRxfx3sPmxA9RVCw1O_3icQWSAh5Kit7ZJEHfH60I9ZhIEk3bDFm4qMIKJSKvHNkh6J8IvNvs0uWu-QXUJzDqg=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEiW59NhfOgtmuB9xnzy8-ES4qGLV5smo8hL_V1QkQAg_4ISFWTdWIclVh26p3G8z4hJCtA_4hvmQa1QdS7Fa6Knl-QltsZHv8_kuo_ccRxfx3sPmxA9RVCw1O_3icQWSAh5Kit7ZJEHfH60I9ZhIEk3bDFm4qMIKJSKvHNkh6J8IvNvs0uWu-QXUJzDqg=s320" width="320" /></a></div><p>It 'goes through the middle' and is the kind of thing that the teacher is wanting (Note 2).</p><p>But, if you then display an <i>accurate</i> trend line, say using <i>Excel</i> (in black below), then it will be a bit off from what the students have drawn.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjVnw5dqrfAzlyQGbGqaNPHF8RrWSF0G5ZhVrO2zZkQl6iJ7afIRF_nMUYseOleVRnoi6jFSMNlqkjS0PPnN7I_vSOCCl6HbjxlMGJ3tYyniCIOYYgnG_DkDDq3UwBu5pvY-X8P4Dhkn9sNwT6p8JzQYuItv5FJID9HtZ-oDYpWFPj7ooamGbznb0IONQ=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjVnw5dqrfAzlyQGbGqaNPHF8RrWSF0G5ZhVrO2zZkQl6iJ7afIRF_nMUYseOleVRnoi6jFSMNlqkjS0PPnN7I_vSOCCl6HbjxlMGJ3tYyniCIOYYgnG_DkDDq3UwBu5pvY-X8P4Dhkn9sNwT6p8JzQYuItv5FJID9HtZ-oDYpWFPj7ooamGbznb0IONQ=s320" width="320" /></a></div><p style="text-align: left;">Here they are together, so you can see the difference:</p><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhaa7f6sRJbqFFMqqBFLLWT4jaO-or_gyfxgOeS6XBN_8vS7-SRpe7ZrIV3KY2ZNy2wTFAMPH1f1iDm2MZlTzl8PVNplaFiYJ6utSt0RWKgKoD837hVjVm2sgL56havFe2VMlJ7cqxG5SQ6UCdcEqYcE2B1sARFPTYd8TCtu4SXiomTbFgx24VyMdubKA=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhaa7f6sRJbqFFMqqBFLLWT4jaO-or_gyfxgOeS6XBN_8vS7-SRpe7ZrIV3KY2ZNy2wTFAMPH1f1iDm2MZlTzl8PVNplaFiYJ6utSt0RWKgKoD837hVjVm2sgL56havFe2VMlJ7cqxG5SQ6UCdcEqYcE2B1sARFPTYd8TCtu4SXiomTbFgx24VyMdubKA=s320" width="320" /></a></div><p>It is easy to put this discrepancy down to human error. The computer draws the <i>best possible</i> line, and the line we draw by eye is bound to be not quite right. Students might over-attend to a few prominent outliers, rather than <i>really</i> base their line on where the overall mass of the points is located. So there is nothing to worry about.</p><p>But there is more than random error going on here. I claim that the students are not even <i>trying</i> to draw the line that the computer is drawing. For example, if we switch the variables around (interchange the axes), presumably this would/should make no difference at all to the line that the students are trying to draw, relative to the positions of the points – it should just be a reflection of their line in the diagonal $y=x$. But the computer will give you a <i>completely different</i> regression line, because the regression line of $y$ on $x$ is in general quite different from the regression line of $x$ on $y$ - and sometimes dramatically so. The regression line of $x$ on $y$ is shown in blue below, on top of the black regression line of $y$ on $x$.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgU0ZrO-pgTm9hMW6MYmrJUwo_wW2l5-rvKDcwIxODlU3ihuopM5nnaW3EMoo3Lwj_jn5EoqWnlavTS96h99bV6PyCzMb_Liuz_-iX_Pk2hMcYR7P5T1SSITuUI7QajL6AfCzUwjzv1XAGwlBGCCe61VZ9ekjenFdPUMkzdd_fKP7-rnr-IV8HuR2GKSw=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgU0ZrO-pgTm9hMW6MYmrJUwo_wW2l5-rvKDcwIxODlU3ihuopM5nnaW3EMoo3Lwj_jn5EoqWnlavTS96h99bV6PyCzMb_Liuz_-iX_Pk2hMcYR7P5T1SSITuUI7QajL6AfCzUwjzv1XAGwlBGCCe61VZ9ekjenFdPUMkzdd_fKP7-rnr-IV8HuR2GKSw=s320" width="320" /></a></div>The black line minimises the sum of the squares of the <i>vertical</i> distances of the points from the line, whereas the blue line minimises sum of the squares of the <i>horizontal</i> distances of the points from the line. We should not expect the resulting lines to be the same. The black line gives the best linear prediction of the $y$ value, given the $x$ value; the blue line gives the best linear prediction of the $x$ value, given the $y$ value. The two lines answer two different questions.<br /><p style="text-align: left;">And neither of these questions is likely to be what the students are thinking about. The line the students are likely to be aiming for is the <i>principal axis</i> of the data. If we draw an ellipse around our data points, what the students are presumably trying to do is essentially find the major axis of this ellipse.</p><p style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjhCJxhoRmcfFDFHRHsQONeK1up7xzze0Bayoy15DGNk3_E7g71QQwZemQVaswReeOZJOZTXCjFmYfWVOmp9IZuO1-iuq7n3F3hLnlEVRuG0UsT1129pvEqGq96EVREjyoaFnFg4_3ItMl9-TfM8rsp1HWR4gr4KWWUH_nhNT4m8LZkVWhYT5UPNiVZCw=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjhCJxhoRmcfFDFHRHsQONeK1up7xzze0Bayoy15DGNk3_E7g71QQwZemQVaswReeOZJOZTXCjFmYfWVOmp9IZuO1-iuq7n3F3hLnlEVRuG0UsT1129pvEqGq96EVREjyoaFnFg4_3ItMl9-TfM8rsp1HWR4gr4KWWUH_nhNT4m8LZkVWhYT5UPNiVZCw=s320" width="320" /><br /></a><br /></p><p style="clear: both; text-align: left;">If we compare the principal axis (in red below) with the correct regression line of $y$ on $x$ (in black), we can see that they are not the same.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhZYSkcApY_ztoqd-Zc1ibjRE7CI_-B0ZYE4SLI47CjfLGvMdxo4i9FScgYAjEgjQhPmCNgPJj5j9D0TIVAh2PhIl_fcHvb1OJ8A5PlYnJkhdsdZBDo0omX783LELqltVFYBHjh_jDIzKHJhSeQVzZWmgGd8cuTDEwv_IpBMl3rKquVUeNUld3oDCxisQ=s708" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="708" data-original-width="708" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhZYSkcApY_ztoqd-Zc1ibjRE7CI_-B0ZYE4SLI47CjfLGvMdxo4i9FScgYAjEgjQhPmCNgPJj5j9D0TIVAh2PhIl_fcHvb1OJ8A5PlYnJkhdsdZBDo0omX783LELqltVFYBHjh_jDIzKHJhSeQVzZWmgGd8cuTDEwv_IpBMl3rKquVUeNUld3oDCxisQ=s320" width="320" /></a></div>If you consider thin, vertical slices of the ellipse, the black line approximately bisects these, and is close to the mean $y$-value of the points that are near to that value of $x$. Relative to this, the red line underestimates $y$ for low values of $x$ and overestimates them for high values of $x$.<br /><p>In school mathematics, lines of best fit are used to predict one variable from the other, so it's really regression lines that we need, not principal axes. (And, indeed, really we should use a different line to predict $y$ from $x$ [part (b) of the typical exam question, in which part (a) is to draw the line of best fit] from the line we use to predict $x$ from $y$ [part (c) of the typical exam question].) Even when the regression line and the principal axis happen to be close to each other, conceptually they are quite different. The principal axis minimises the sum of the squares of the (perpendicular) distances from the points to the line, whereas the ordinary-least-squares regression line minimises the sum of the squared <i>vertical</i> distances from the points to the line. It can be interesting to devise scenarios where these two lines are very similar and very different.</p><p>From a school teaching point of view, does this matter, or is it unnecessary quibbling? I have found that this discussion comes up sometimes when students complain that the line of best fit that the computer is producing ‘looks wrong’, especially when there are lots of points, and the correlation is fairly strong. They think they can draw a better one, and are puzzled why the computer is clearly giving them 'wrong' lines. The problem here is that the students have been misled about which line they should be aiming for, and Gelman and Nolan (2017, Chapter 4) have a nice approach to addressing this.</p><p>Maybe it is a relatively small point to worry about, but surely it would be a bit of a problem if students drawing something <i>closer</i> to the black line above were being penalised or criticised over those drawing something more like the red line.</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. How concerned are you about this distinction between regression lines and principal axes?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. What, realistically, might be done to address this in school-level mathematics?</span></p><p style="text-align: left;"><span style="font-family: inherit;">3. Are there other examples in school mathematics where it is usual to teach things 'a bit wrong'?</span></p><h3 style="text-align: left;">Notes</h3><p>1. The data used in this blogpost is available at: <a href="https://www.foster77.co.uk/Data%20for%20line%20of%20best%20fit%20blogpost.csv">https://www.foster77.co.uk/Data%20for%20line%20of%20best%20fit%20blogpost.csv</a></p><p>2. Students sometimes have a strong tendency to avoid going directly <i>through</i> any of the points. They have been told that they are not meant to 'join up' the points, and, as if to prove this, they try to keep away from any actual points altogether. Similarly, they may feel that it would be wrong to allow the line to pass directly through the origin, so they act as though the origin must be avoided at all costs.</p><h3 style="text-align: left;">Reference</h3><p>Gelman, A., & Nolan, D. (2017). <i>Teaching statistics: A bag of tricks</i>. Oxford University Press.</p><p><br /></p><p><br /></p></div>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com1tag:blogger.com,1999:blog-2036014053389751696.post-80156436801460734802022-06-09T07:00:00.000+01:002022-06-09T07:00:00.158+01:00Motivation for measurement<p><i>Optical illusions are almost universally intriguing. Young children can completely get them, but they can fascinate adults too. There is something captivating about being tricked by your eyes. And I think they can provide a great opportunity for motivating some geometry.</i></p><p>Topics in mathematics that involve accurate measurement can sometimes feel a bit<i> un</i>mathematical - more science than mathematics. For example, why is 'scale drawing' a topic in mathematics? Is this just a hangover from the days when 'technical drawing' was a marketable skill that was taught in schools? Converting scales is a useful application of ratio, but what is the mathematical purpose of making accurate drawings? Loci and construction are important topics for understanding concepts in geometry, and the central idea that compass constructions are 'exact in principle' seems to me to be important. But should it matter whether students can execute a perfect circle with their compasses or draw an angle of $35^\circ \pm 1^\circ$ using a protractor? Arguably, making neat constructions may depend more on the quality of the student's instruments (such as how well-tightened the screw on their compasses is) and on basic dexterity than on any mathematically-relevant skill. The beauty of mathematics is the ability to carry out exact calculations that mean that a correct mathematical sketch <i>not</i> drawn to scale is generally as useful as, or more useful than, an accurate scale drawing. For example, in an astronomical scenario (e.g., calculating the distance to the sun) we can sketch a 1 by 20,000 right-angled triangle, and this is much more helpful than trying to draw this to scale! In mathematics, we develop ways to calculate so that we don't <i>need</i> to make accurate drawings, so perhaps the main purpose of scale drawing is to show students how slow and tiresome things would be without mathematics (a Dan Meyer 'headache-aspirin' situation, see Meyer, 2015)?</p><p>Nevertheless, there are times when we need students to measure lengths and angles, and it is great when we can find purposeful ways to practise these skills (see Andrews, 2002). I think finding scenarios where there is a real (i.e., uncontrived) <i>need</i> to measure can be quite difficult, but optical illusions can be really helpful for this - and are fun in their own right.</p><p>There is a good list of many kinds of optical illusions at <a href="https://en.wikipedia.org/wiki/List_of_optical_illusions">https://en.wikipedia.org/wiki/List_of_optical_illusions</a>, and this includes things that are extremely weird, such as <i><b>Ames room</b></i> (<a href="https://en.wikipedia.org/wiki/Ames_room">https://en.wikipedia.org/wiki/Ames_room</a>). These might be fun to look at and talk about. However, I focus below on some examples of optical illusions that could have obvious, immediate relevance in motivating some primary or secondary school geometry and measurement.</p><p>The <i><b>Ebbinghaus illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Ebbinghaus_illusion">https://en.wikipedia.org/wiki/Ebbinghaus_illusion</a>) is a nice one. The orange discs below are equal in size, but don't look it.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhE7t-dBzReUe4rLDebCP0fMW8LXKEjIg0srtuLXuZZT7dqkaEDa_BvnD7IgRjHyVZtNEZYXEuDOXr0vdV2aEuwzFu-xZHehsfko4IFM2GGNchLmZ0umMvOHmHeWEPlOhgaeswyXovoKliIE5Sr9Erfdjr3RhuZITm5nF6CQyvVTWnme8ZEQ1R8WJ4S-w=s650" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="400" data-original-width="650" height="246" src="https://blogger.googleusercontent.com/img/a/AVvXsEhE7t-dBzReUe4rLDebCP0fMW8LXKEjIg0srtuLXuZZT7dqkaEDa_BvnD7IgRjHyVZtNEZYXEuDOXr0vdV2aEuwzFu-xZHehsfko4IFM2GGNchLmZ0umMvOHmHeWEPlOhgaeswyXovoKliIE5Sr9Erfdjr3RhuZITm5nF6CQyvVTWnme8ZEQ1R8WJ4S-w=w400-h246" width="400" /></a></div><p>This is just crying out for some measurement with a ruler. Can the diameters <i>really</i> be equal? But the centres of the circles are not marked, so how could you be sure you were accurately measuring the <i>diameter</i>, and not some other chord?</p><p>The <i><b>Delboeuf illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Delboeuf_illusion">https://en.wikipedia.org/wiki/Delboeuf_illusion</a>) is similar. The black discs are in fact equal, but the right-hand one looks larger:</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi_dEfThuY7hzxB0b-Np_iqf-GIqbAlxKnK373S7ghDlEjPjyKAI5apyD0r-Q3pg6BzjdXpBUaow9XbGQP2JuNvbRKpHwZ4RDVDi7FP41nB502DId6kw0qowHolZPv74cyVJRmVcMUg4gsVVOWg3HDfUz-GUTJWpIU-xYX5SpfcDlTsImppnaOkiGhuvA=s920" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="550" data-original-width="920" height="239" src="https://blogger.googleusercontent.com/img/a/AVvXsEi_dEfThuY7hzxB0b-Np_iqf-GIqbAlxKnK373S7ghDlEjPjyKAI5apyD0r-Q3pg6BzjdXpBUaow9XbGQP2JuNvbRKpHwZ4RDVDi7FP41nB502DId6kw0qowHolZPv74cyVJRmVcMUg4gsVVOWg3HDfUz-GUTJWpIU-xYX5SpfcDlTsImppnaOkiGhuvA=w400-h239" width="400" /></a></div><p>The <i><b>Moon illusion</b></i> is a nice variation on this (<a href="https://en.wikipedia.org/wiki/Moon_illusion">https://en.wikipedia.org/wiki/Moon_illusion</a>).</p><p style="text-align: left;">Without being prompted to do so, when presented with these illusions, students reach for their rulers. And so, of course, if you want every student to do the measuring, then you need to provide the images on paper, as displaying them on the screen allows only one student to do it on behalf of everyone else.</p><p style="text-align: left;">Students could also attempt to create <i>their own</i> drawings, some of which are illusory (something looks bigger but isn't) and some not (something looks bigger and <i>is</i>), and see if other students can decide which are which by eye - followed by measuring to check.</p><div><p></p><p>The <i><b>Müller-Lyer illusion</b></i> (<a href="https://en.wikipedia.org/wiki/M%C3%BCller-Lyer_illusion">https://en.wikipedia.org/wiki/M%C3%BCller-Lyer_illusion</a>) provides motivation for measuring the lengths of line segments. The two horizontal portions below are equal in length, but don't look it!</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjTXGNRHfy0pXC5agmjx_kQ0ovLWR7P7yMuPo6bTc-ED8z82KLvIL6UgexE9FocCbb2oX6ibejSlr4PkqedumPRl3scWgWe3p4Y7cfLnda8c4Edyu_4cQomba0LefL9znNG8d8vt83JwYMVhkzf4tklpedEElEIxAFR5GlL8VhdJaWzBwExbF5eJ2iv7w=s842" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="169" data-original-width="842" height="128" src="https://blogger.googleusercontent.com/img/a/AVvXsEjTXGNRHfy0pXC5agmjx_kQ0ovLWR7P7yMuPo6bTc-ED8z82KLvIL6UgexE9FocCbb2oX6ibejSlr4PkqedumPRl3scWgWe3p4Y7cfLnda8c4Edyu_4cQomba0LefL9znNG8d8vt83JwYMVhkzf4tklpedEElEIxAFR5GlL8VhdJaWzBwExbF5eJ2iv7w=w640-h128" width="640" /></a></div><p>Similar opportunities are provided by the <b style="font-style: italic;">Ponzo illusion</b> (<a href="https://en.wikipedia.org/wiki/Ponzo_illusion">https://en.wikipedia.org/wiki/Ponzo_illusion</a>),</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjz3VXY5dHZ04lLX28NjYyVPP__5zLlA98ARUuL4Sd-m-zgdsxiLlkHIQVD6FbFZhhGF9VgufYvxWrrn0yWWYBaJnernAH_lcg70KqqF1fnS96Q_vbTxjZjTlS4UZLJM-MohScIuU0g5FJm1Wk6HAeGcBUDm3UojcQGYBWog9TxR5i_xXAH-LWWcR11Mw=s225" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="166" data-original-width="225" height="472" src="https://blogger.googleusercontent.com/img/a/AVvXsEjz3VXY5dHZ04lLX28NjYyVPP__5zLlA98ARUuL4Sd-m-zgdsxiLlkHIQVD6FbFZhhGF9VgufYvxWrrn0yWWYBaJnernAH_lcg70KqqF1fnS96Q_vbTxjZjTlS4UZLJM-MohScIuU0g5FJm1Wk6HAeGcBUDm3UojcQGYBWog9TxR5i_xXAH-LWWcR11Mw=w640-h472" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="text-align: left;">Tony Philips, National Aeronautics and Space Adm., Public domain, via Wikimedia Commons</span></td></tr></tbody></table><div style="text-align: left;"><br /></div><div style="text-align: left;">the <i><b>Sander illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Sander_illusion">https://en.wikipedia.org/wiki/Sander_illusion</a>), where the two purple diagonals below are actually <i>equal</i> in length,</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhIMDwcVhp29fpGpXk47g8GS6iIXbnBADtYpxEHo1P_YwIBaC9O8T7CXskFrRQ8bNMUL_zMMSp54l7kxHWEY7Dq98wRL5tjRA2wlTz8IaErXFfq4crSLqOmJ8mjSc8dSP0oyFRgrd63N6d8o8NUw1KVK19YaN8duBhWpizjDTUg9XFGRhYwT698SfMq2g=s448" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="160" data-original-width="448" height="228" src="https://blogger.googleusercontent.com/img/a/AVvXsEhIMDwcVhp29fpGpXk47g8GS6iIXbnBADtYpxEHo1P_YwIBaC9O8T7CXskFrRQ8bNMUL_zMMSp54l7kxHWEY7Dq98wRL5tjRA2wlTz8IaErXFfq4crSLqOmJ8mjSc8dSP0oyFRgrd63N6d8o8NUw1KVK19YaN8duBhWpizjDTUg9XFGRhYwT698SfMq2g=w640-h228" width="640" /></a></div><div style="text-align: left;">and the <i><b>vertical-horizontal illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Vertical%E2%80%93horizontal_illusion">https://en.wikipedia.org/wiki/Vertical%E2%80%93horizontal_illusion</a>),</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjVPqCfYeD0WuZpP2L8FQqRMLOWn9fnoQwoslMTjHU7p-LkwCATKLD8_qxKAWgNAR2FNUkmor2mBMUOKEklf1S7JomqazPTOv6aKLJJlawGTm2LTMcIF77jcJ-pKqSfsmvSfzFW_XEhB22kqaBLAk8lhwXI4zb-9lNFlHtMQNJqBogvdfGyz5Xsz2yN4Q=s517" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="454" data-original-width="517" height="176" src="https://blogger.googleusercontent.com/img/a/AVvXsEjVPqCfYeD0WuZpP2L8FQqRMLOWn9fnoQwoslMTjHU7p-LkwCATKLD8_qxKAWgNAR2FNUkmor2mBMUOKEklf1S7JomqazPTOv6aKLJJlawGTm2LTMcIF77jcJ-pKqSfsmvSfzFW_XEhB22kqaBLAk8lhwXI4zb-9lNFlHtMQNJqBogvdfGyz5Xsz2yN4Q=w200-h176" width="200" /></a></div>where the vertical line segment looks longer, but isn't.</div><div><p style="text-align: left;">Asking students to attempt to draw a square by eye, using a straight edge (i.e., not a scaled ruler), can be revealing. Then the students measure everyone's and do some statistics to see whether, among the various drawings produced, 'squares' that are tall/narrow are more prevalent than ones which are short/wide. (The easiest way of keeping track of the orientation of each piece of paper is to have the students write their name at the top.) There are similar opportunities for statistical analysis in devising a way to decide how to judge the quality of people's freehand circles (see Foster, 2015, and Bryant & Sangwin, 2011).</p><p>The <i><b>café wall illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Caf%C3%A9_wall_illusion">https://en.wikipedia.org/wiki/Caf%C3%A9_wall_illusion</a>) is a bit more sophisticated, and this can be a good opportunity to encourage students to use precise language. What exactly do they mean by 'wavy', 'wonky' or 'not straight'? Do they mean <i>sloping straight</i> lines or <i>curves</i>? "Say what you see" can be a really a useful prompt to use with these figures, and you can follow up with requests for greater clarity.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhUDeFc9uPbqLfCgCK7bkuZUqmBeD65mUIOdJYmOcGCgeNT64MtNYmYdwELLUTXlkJZm_IttlxcGR2w3-8K6o680eFx1GD94cx66ct4QYy9DLLX2usc6Rd3Om4IWO4Z9rLYqZNTUolSZLwwhGFpj02k1Xe4gjujPNPKfrJFnMivT6h4pQkj8euq7Fewzg=s840" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="540" data-original-width="840" height="258" src="https://blogger.googleusercontent.com/img/a/AVvXsEhUDeFc9uPbqLfCgCK7bkuZUqmBeD65mUIOdJYmOcGCgeNT64MtNYmYdwELLUTXlkJZm_IttlxcGR2w3-8K6o680eFx1GD94cx66ct4QYy9DLLX2usc6Rd3Om4IWO4Z9rLYqZNTUolSZLwwhGFpj02k1Xe4gjujPNPKfrJFnMivT6h4pQkj8euq7Fewzg=w400-h258" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="text-align: left;">Fibonacci, CC BY-SA 3.0 <a href="http://creativecommons.org/licenses/by-sa/3.0/">http://creativecommons.org/licenses/by-sa/3.0/</a>, via Wikimedia Commons</span></td></tr></tbody></table><br />The <i><b>Zöllner illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Z%C3%B6llner_illusion">https://en.wikipedia.org/wiki/Z%C3%B6llner_illusion</a>) provides another opportunity for students to check parallelness,<div><br /></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj0H1idQJp_NhqRdkG2ILgixy8F44UDmrPH3sBjPzQFQB0SbEvzpGK9qrjOYTbaxSC61NKcs7-wFRMKUtEPq1pad1q9_LseZufM5OZMNx64QLXmdEFRmNKPiI-6To91ZLADM9M-amw7JWmdRpSeb5JEGBaQevG4rStHWVPNTfm9p_Dvh9bn_xRze4Iyiw=s1280" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="930" data-original-width="1280" height="233" src="https://blogger.googleusercontent.com/img/a/AVvXsEj0H1idQJp_NhqRdkG2ILgixy8F44UDmrPH3sBjPzQFQB0SbEvzpGK9qrjOYTbaxSC61NKcs7-wFRMKUtEPq1pad1q9_LseZufM5OZMNx64QLXmdEFRmNKPiI-6To91ZLADM9M-amw7JWmdRpSeb5JEGBaQevG4rStHWVPNTfm9p_Dvh9bn_xRze4Iyiw=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="text-align: left;">Fibonacci, CC BY-SA 3.0 </span><a href="http://creativecommons.org/licenses/by-sa/3.0/" style="text-align: left;">http://creativecommons.org/licenses/by-sa/3.0/</a><span style="text-align: left;">, via Wikimedia Commons<br /></span></td></tr></tbody></table><div><p>and the <i><b>Hering illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Hering_illusion">https://en.wikipedia.org/wiki/Hering_illusion</a>) is another example:</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgBbJN7JDqU1Y7_R48rsI_fVVs1lL-yv14zvv0Dn0lzcIXK6Z3Yo4H58coB-Q-0oEWjUNtSVGCuYGk50ixGRCVODZP-y9DqlQQed0unUL4ZMLSewR30PVLMWCxIRvw6_YkPuHXMwCFMeJHmjziTeao1PQ2MPrpiREReDnjxfDjLf-LQB60902sIlo-9lg=s512" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="512" data-original-width="320" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgBbJN7JDqU1Y7_R48rsI_fVVs1lL-yv14zvv0Dn0lzcIXK6Z3Yo4H58coB-Q-0oEWjUNtSVGCuYGk50ixGRCVODZP-y9DqlQQed0unUL4ZMLSewR30PVLMWCxIRvw6_YkPuHXMwCFMeJHmjziTeao1PQ2MPrpiREReDnjxfDjLf-LQB60902sIlo-9lg=s320" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="text-align: left;">Fibonacci, CC BY-SA 3.0 </span><a href="http://creativecommons.org/licenses/by-sa/3.0/" style="text-align: left;">http://creativecommons.org/licenses/by-sa/3.0/</a><span style="text-align: left;">, via Wikimedia Commons<br /></span></td></tr></tbody></table><p>For working with polygons, the <i><b>Ehrenstein illusion</b></i> (<a href="https://en.wikipedia.org/wiki/Ehrenstein_illusion">https://en.wikipedia.org/wiki/Ehrenstein_illusion</a>) is useful. What do we need to check to see if the shape really is a square? Is it enough just to measure the lengths of the four sides? Is it enough just to check that the angles are all right angles (and how many do we need to measure to do this?)? (The <i><b>Orbison illusion</b></i> - <a href="https://en.wikipedia.org/wiki/Orbison_illusion">https://en.wikipedia.org/wiki/Orbison_illusion</a> - provides similar opportunities.)</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhRpCcQBxfEIhnKagXkvBx8aIH4lM-Mdyy1LNy89WDsLIGvVozglxach2RfGHS0A09yc3xJaPypllOwxuzYwu1l6lLEseHHJ3tLHW20ugNBmKszunRvy5fBLHO2s8hyyiIecQrZbCk9LJEX3mfzmFEVfp0VwSsuqMFjPhUFDWg5cEi0_kycbAJ7Xq2hBg=s300" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="300" height="300" src="https://blogger.googleusercontent.com/img/a/AVvXsEhRpCcQBxfEIhnKagXkvBx8aIH4lM-Mdyy1LNy89WDsLIGvVozglxach2RfGHS0A09yc3xJaPypllOwxuzYwu1l6lLEseHHJ3tLHW20ugNBmKszunRvy5fBLHO2s8hyyiIecQrZbCk9LJEX3mfzmFEVfp0VwSsuqMFjPhUFDWg5cEi0_kycbAJ7Xq2hBg" width="300" /></a></div><p>Often, students address measurement objectives by spending lesson time measuring arbitrary line lengths or angles on a sheet, merely to improve their skill at measurement. Optical illusions can provide a rich context for doing similar work, where there is a motivation to discover whether, say, two lengths really are the same or not. I find that students will measure much more accurately, and with considerably more enthusiasm, when it has some purpose behind it, and I would call tasks like these <i><b>mathematical etudes</b></i> (<a href="http://www.mathematicaletudes.com/">http://www.mathematicaletudes.com/</a>) for measurement.</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. Do you find these optical illusions engaging? Would your students?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. How could you use these ideas to promote a <i>need</i> for measurement with one of your classes?</span></p><p style="text-align: left;"><span style="font-family: inherit;">3. What other tasks make measurement a meaningful mathematical activity?</span></p><h3 style="text-align: left;">References</h3><p>Andrews, P. (2002). Angle measurement: An opportunity for equity. <i>Mathematics in School, 31</i>(5), 16–18. <a href="https://nrich.maths.org/content/id/2855/AngleMeasurement.pdf">https://nrich.maths.org/content/id/2855/AngleMeasurement.pdf</a></p><p>Bryant, J., & Sangwin, C. (2011). <i>How round is your circle? </i>Princeton University Press.</p><p>Foster, C. (2015). Exploiting unexpected situations in the mathematics classroom. <i>International Journal of Science and Mathematics Education, 13</i>(5), 1065–1088. <a href="https://doi.org/10.1007/s10763-014-9515-3">https://doi.org/10.1007/s10763-014-9515-3</a></p><p>Meyer, D. (2015, June 17). If math is the aspirin, then how do you create the headache? [Blog post]. <a href="https://blog.mrmeyer.com/2015/if-math-is-the-aspirin-then-how-do-you-create-the-headache/">https://blog.mrmeyer.com/2015/if-math-is-the-aspirin-then-how-do-you-create-the-headache/</a></p></div></div>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com0tag:blogger.com,1999:blog-2036014053389751696.post-79802273752285369162022-05-26T07:00:00.001+01:002022-05-26T07:00:00.152+01:00Are two cars better than one?<p><i>So many ideas in probability are really quite unintuitive. How can we help learners make better sense of how simple probabilities combine, without relying on arbitrary rules and mysterious formulae?</i></p><p>I recently heard someone talking about whether they should get a second car for their household (Note 1). They were doubtful that it was a good idea. In addition to the cost and environmental impact, they said, “If you have two cars, there’s twice as much chance that something will go wrong with one of them.” This seemed self-evidently true, and everybody nodded sadly, and the conversation moved on.</p><p>I began thinking about how I might respond mathematically. I knew what they meant, and their statement might indeed be approximately true, but it couldn’t be <i>exactly</i> true. Even if all that you know about probabilities is that they are capped at 1 (i.e., 100% is the highest possible probability), it is clear that you cannot just go around doubling probabilities. Doubling any probability greater than 0.5 will give you a total probability greater than 1, which is impossible. And, even if you have super-reliable cars with a very small probability $p$ of failing, you would only need to ensure that you buy more than $\frac{1}{p}$ of them for the total probability to exceed 1.</p><p>So, why is such a plausible-sounding statement not right? And under what assumptions might the statement be <i>approximately</i> true?</p><p>Suppose that both cars have the same probably $p$ of ‘something going wrong with them’ in a certain time interval. Would these be independent events? If both cars are parked outside the same house, then they are likely to be subject to similar weather conditions and other factors, so it seems unlikely that failure of one would be completely unrelated to failure of the other. But let’s ignore this and suppose that the two events <i>are</i> independent, and also that the two cars are equally likely to fail. This would mean that the probability of <i>both</i> cars failing would be $p^2$. And this means that when we double $p$ we are <i>overcounting</i> by $p^2$ (see Figure 1), because we are counting the same situation of ‘something goes wrong’ when car A fails and counting it <i>again</i> when car B fails, for those occasions when they <i>both</i> fail. In the extreme case, where you had two completely useless cars, you would have a 100% chance of not being able to drive anywhere, but not a 200% chance! Now, if $p$ is very small, then $p^2$ will be <i>very very</i> small, and so we can perhaps ignore the overlap region. But, if $p$ is <i>not</i> very small, then $p^2$ will be <i>non</i>-negligible. This means that the correct probability for either (or both) cars failing is $2p-p^2$. </p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi5Mi8ot3JiOPeY9nfs8ldKLCHCwV7xGp9fHTlOh5V7PaoYBKipqaFFHljzffV4VGSWbO4HAd3dOCfaqodoEZCxAfCJFEWpa6116reDIP8ySWwrWSM3HDLR8vW77UJLSa7I5LrnKF0R0JNjSbNf_sQ-0BDFX9B0tISXJQ-aw-VSZUwXVoNZTbSLjIgalw=s600" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="378" data-original-width="600" height="202" src="https://blogger.googleusercontent.com/img/a/AVvXsEi5Mi8ot3JiOPeY9nfs8ldKLCHCwV7xGp9fHTlOh5V7PaoYBKipqaFFHljzffV4VGSWbO4HAd3dOCfaqodoEZCxAfCJFEWpa6116reDIP8ySWwrWSM3HDLR8vW77UJLSa7I5LrnKF0R0JNjSbNf_sQ-0BDFX9B0tISXJQ-aw-VSZUwXVoNZTbSLjIgalw=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. $p(A∪B)$</td></tr></tbody></table><p>Looking at the expression $2p-p^2$, we might wonder whether we can be absolutely sure that it is <i>always</i> less than 1 for all values of $p$. Is the $p^2$ definitely always sufficiently large to bring back $2p$ to below 1 whenever $p>0.5$? One way to see this is by completing the square, to obtain $1-(1-p)^2$, meaning that $2p-p^2$ is equal to '$1-$something that is never negative'. Figure 2 shows the graph $y=2p-p^2=p(2-p)$, and the curve has its maximum value of 1 at $p=1$, and so it never exceeds 1 for any value of $p$ (Note 2).</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi59nGOezj11m-Pd1mhsLPUQIOPprIxHc8Ka4BuUZZMD4RUJhcYt0npgizVxwYhMRxOskAYjF3lWOdxqdSW0-slDjuPS2tQUAtr9P64bqFAeKPXebXP625pZ8S1JKctEU5aH-KnJz5A_uhdaUKdsR75MM22SVq0t-E6d0Xw2FtjLzDZ1Zpnu9DIhufhCQ=s2258" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="2154" data-original-width="2258" height="381" src="https://blogger.googleusercontent.com/img/a/AVvXsEi59nGOezj11m-Pd1mhsLPUQIOPprIxHc8Ka4BuUZZMD4RUJhcYt0npgizVxwYhMRxOskAYjF3lWOdxqdSW0-slDjuPS2tQUAtr9P64bqFAeKPXebXP625pZ8S1JKctEU5aH-KnJz5A_uhdaUKdsR75MM22SVq0t-E6d0Xw2FtjLzDZ1Zpnu9DIhufhCQ=w400-h381" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. $y=2p$ and $y=2p-p^2$</td></tr></tbody></table><p>We can also see in Figure 2 that the line $y=2p$ is indeed a good approximation to the curve for small values of $p$. So, just doubling the probability <i>is</i> a reasonable approximation if you are considering very reliable cars.</p><p>Using the algebra, this reasoning is quite straightforward for anyone comfortable with quadratics and elementary probability. In probability, we can only add the probability of events A and B if they are <i>mutually exclusive</i> (i.e., $p(A∩B)=0$), so that the ‘Venn diagram identity’ $p(A∪B)\equiv p(A)+p(B)-p(A∩B)$ reduces to $p(A∪B)=p(A)+p(B)$. In other cases, we have to subtract the intersection, so as not to double count it.</p><p>I was happy with all of this, but I wanted to say something that didn't sound technical or rely on set theory or even Venn diagrams. Could I say in words why doubling was not quite right? I found it hard to come up with a good way to explain to my friend why the possibility of <i>both</i> cars going wrong was relevant to their statement, and why this indeed made their statement technically wrong, even if you were willing to make assumptions about things like independence and so on. If I had said that having two cars that might go wrong is not <i>quite</i> twice as bad as having one car that might go wrong, because, at least some times, <i>both</i> cars will <i>simultaneously</i> go wrong, I think they would be quite surprised! The commonsense response is that there is no consolation in having <i>both</i> cars fail on the same day - that is the worst possible nightmare, and indeed one of the reasons for contemplating having a second car was to try to be sure that they would always have one working car! They would probably respond that “When I said 'either-or', I was including 'both'!”, which misses the point. Yes, we want to <i>include</i> the chance of both cars failing - the point is that we want to include that possibility <i>only once,</i> not twice (Figure 3)! It is still true that $p^2<p$, and possibly dramatically so ($p^2 \ll p$), so the chance of having at least one working car has indeed risen from $1-p$ to $1-p^2$. The point is that if last week Car A failed, say, on Monday, Thursday and Saturday, whereas Car B failed on Tuesday, Saturday and Sunday, our daily frequency of car trouble would have been $\frac{5}{7}$ and not $\frac{6}{7}$, because we don't double count Saturday just because it was a double-failure day.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiSksFdcfwfXfbwlc6c0AY96ory-aaSMIPL1BRF3avJQK4U1n1S4nYDpmIQBilUbBEh8fJ50FLn7JMqd2hKZqO6Lo7aCLGDy9HSXOuVqbnxoAR1Zr2fFFSxTr_5w2G_Fi8jyCkwV4McNYmks9yRZl7biqmgJKSq6ivRrTSLtPr2Ll_eZOQ87oHinfFkGA=s1899" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1197" data-original-width="1899" height="202" src="https://blogger.googleusercontent.com/img/a/AVvXsEiSksFdcfwfXfbwlc6c0AY96ory-aaSMIPL1BRF3avJQK4U1n1S4nYDpmIQBilUbBEh8fJ50FLn7JMqd2hKZqO6Lo7aCLGDy9HSXOuVqbnxoAR1Zr2fFFSxTr_5w2G_Fi8jyCkwV4McNYmks9yRZl7biqmgJKSq6ivRrTSLtPr2Ll_eZOQ87oHinfFkGA=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="text-align: left;">Figure 3. $(A∪B)-(A∩B)$</span></td></tr></tbody></table><p>Perhaps this is part of a broader theme in mathematics of situations in which things can’t be simplistically added up. Other examples could include vectors that are not in the same direction, numerators of fractions that have different denominators, and dimensionally-incompatible quantities, such as distance and time (Foster, 2019). However, there is something particular about probability for me. I enjoy probability very much, but I think it's the area of mathematics in which I'm most likely to struggle to truly make sense or to be able to explain concepts clearly to others without using technical language and symbols (see Foster, 2021). Rarely when I'm doing a probability calculation do I have a rough ballpark estimate of the answer I should be getting, and, if I obtain an answer like $\frac{127}{351}$, it would be scarcely worth my while converting it to a decimal to see if it looked a 'reasonable' size, as I would have no idea how to tell reasonable from unreasonable. Do other people share this sense?</p><h3 style="text-align: left;"><span style="font-family: inherit;">Questions to reflect on</span></h3><p>1. Do you have a better way of explaining why doubling is invalid here?</p><p>2. Do you have other examples of 'similar' situations to this?</p><p>3. Do you share my sense that probability is often 'harder to explain' than other areas of mathematics?</p><h3 style="text-align: left;">Notes</h3><p>1. With apologies for the highly 'middle-class' nature of this 'first-world problem'!</p><p>2. The expression $1-(1-p)^2$ can also be obtained intuitively by saying that the required probability is the <i>complement</i> of the probability that both cars are working properly. Since the probability that either car is working properly is $1-p$, the probability that both (assumed independent) work properly is $(1-p)^2$, and so the probability that this is <i>not</i> the case must be $1-(1-p)^2$.</p><h3 style="text-align: left;">References</h3><p>Foster, C. (2019). Questions pupils ask: Why can’t it be distance plus time? <i>Mathematics in School, 48</i>(1), 15–17. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20can't%20it%20be%20distance%20plus%20time.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20can't%20it%20be%20distance%20plus%20time.pdf</a></p><p>Foster, C. (2021). In a spin. <i>Teach Secondary, 10</i>(1), 11. <a href="https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20In%20a%20spin.pdf">https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20In%20a%20spin.pdf</a></p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com3tag:blogger.com,1999:blog-2036014053389751696.post-91530182447769431932022-05-12T07:00:00.022+01:002022-05-12T09:58:25.748+01:00 Learning times tables efficiently<p><i>Times tables can be a controversial subject. Can we help students to learn their tables in ways that promote conceptual understanding? This is my take on teaching times tables. I imagine there will be some strong opinions...</i></p><p>For many children, learning the times tables feels like a huge mountain to climb. And for those who have tried and feel that they have failed, going back and trying again fills them with dread. Perhaps all seems to go well in the beginning, with the 2s, 5s and 10s, say, but before long we reach the 6s, 7s and 8s, and it feels like every new fact that is mastered displaces an old fact that then becomes lost. As more and more facts are covered, the potential for muddling them up increases (e.g., $7 \times 8$: Is it $54$, or maybe $48$?), until the student really doesn’t have much idea which things they know and which they don’t. In the worst-case scenario, the only thing the child <i>really</i> trusts is skip-counting up from zero every time. And with skip-counting you only have to make one mistake for all of your remaining numbers to be wrong.</p><p>Teachers are highly strategic in the order in which they teach the tables: often 2s, 5s, 10s, 4s, … etc. But the effect of this is that the ‘hard stuff’ (6s, 7s, etc.) is delayed, so that when it arrives it can feel overwhelming and as though it is coming at learners far too quickly. I am not sure that learning one table after another like this – however carefully planned the sequence – is ideal (Note 1).</p><p>Here is a different way, that tries to build up from the <i>multiplicative</i> connections between the facts and deliberately avoids any addition/subtraction/skip-counting approaches, so as to build on the <i>multiplicative</i> structure of the tables and work more in harmony with that.</p><p>At first sight, there are 144 facts to learn: </p><p style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjrULvcai6dIVvfD3neDUZdaAzEQOpUPMcKTriDUxTeAmD5oYmKJ2TjDLZPi6uCbv2iD3dHufiGFhMsosAy_30x28rXGNuabDTkDJrLDPhYvA1UfuAYPz5sDdKSkXNtpWplMEr_dLDyvUVFljhUTbk_16QYJ_EB5MqMXVM08rKgKSBvMa_MOpmhpOFpeA=s602" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="602" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjrULvcai6dIVvfD3neDUZdaAzEQOpUPMcKTriDUxTeAmD5oYmKJ2TjDLZPi6uCbv2iD3dHufiGFhMsosAy_30x28rXGNuabDTkDJrLDPhYvA1UfuAYPz5sDdKSkXNtpWplMEr_dLDyvUVFljhUTbk_16QYJ_EB5MqMXVM08rKgKSBvMa_MOpmhpOFpeA=s320" width="319" /></a><br /></p><p style="text-align: left;">But, of course, this is highly deceptive, and it is nowhere near as bad as this. Because of the commutativity of multiplication ($a \times b \equiv b \times a$, see Foster, 2022), we can immediately delete nearly half of these facts: </p><p style="text-align: left;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEicRUyanrZY5zNNnqYWhmPt1Q8z2X04uN1c46xYcExlbLHsjRdAx4kUDwfpJcofO1NoLoPTbEDhtF851rP05HPMjGRnBJ7Aj4lagZZKny5j2zNa1h8muDnb1PREe-kNaKgDGG6-_0bw12ptK6MZSDyFBkynxEoxC-fPaYgLry2bm6NZgw7nsVAdB8GbkA=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="600" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEicRUyanrZY5zNNnqYWhmPt1Q8z2X04uN1c46xYcExlbLHsjRdAx4kUDwfpJcofO1NoLoPTbEDhtF851rP05HPMjGRnBJ7Aj4lagZZKny5j2zNa1h8muDnb1PREe-kNaKgDGG6-_0bw12ptK6MZSDyFBkynxEoxC-fPaYgLry2bm6NZgw7nsVAdB8GbkA=s320" width="320" /></a></p><p> Everyone knows their 1-times table (which is almost as easy as the 0-times table, Note 2), so we can grey those out. And I will assume that the 2s (the even numbers), 5s, 10s and 11s (at least up to $9 \times 11$) are also known, or easily learned, and so I’ve marked those in green below:</p><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgdzOo6yBCwTNxCTL65QjApvoFnAsAdSPQ1fOs7hZikqjgAsFrX_jSxYnftozYfxOZ9S-xcTSfIDpAjux0Cosn9yi247Otdh2yVHwfFoDv3LKGdN_JNRyrj-mLccSBdZezlA3lbBowcbL7F-FTncyU3RPbBYnbvRnF1QX--jIGJ9qk2JX40MBoffYKYDw=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="600" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgdzOo6yBCwTNxCTL65QjApvoFnAsAdSPQ1fOs7hZikqjgAsFrX_jSxYnftozYfxOZ9S-xcTSfIDpAjux0Cosn9yi247Otdh2yVHwfFoDv3LKGdN_JNRyrj-mLccSBdZezlA3lbBowcbL7F-FTncyU3RPbBYnbvRnF1QX--jIGJ9qk2JX40MBoffYKYDw=s320" width="320" /></a></p><p>So, from the original 144, this now leaves just 30 which need some teaching. And these are the tougher ones. Because of how the picture looks at this point, the best way to tackle this, I think, is <i>not</i> to go table by table (Note 3), but to exploit the structure a bit more strategically. In particular, we want to begin with the <i>highest-leverage</i> multiplication facts – the ones that help most with getting others. When students arrive, say, at secondary school and clearly ‘do not know their tables’, it is basically these 30 that are the problem. Convincing them that their difficulty is not a functionally infinite number of unknown facts but a relatively small number can be helpful. (It really is not like having to memorise a telephone directory!) And starting with the ones most likely to be of immediate help seems to make sense.</p><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><p style="text-align: left;"><i>Big claim: </i>The most useful of these remaining products to know are the eight squares in red below:</p></blockquote><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjTNtV2vy2zMrD0I_l7mG75jjIi8BxlJJXYxRl0nX7yerPJQYrnH1QL_OBmNm1rlqfT-C1_IXfs9c9UknxRTPkWSy8KX9Qnnu7qorcJIHfN44X3uMHKzr63OwHDcGZ9Ca6urcLqCxSrhadkAjtCrnlxW9JlIJGjyfJVBJAkqK7REDXRAi4stL5mQcok_w=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="597" data-original-width="600" height="318" src="https://blogger.googleusercontent.com/img/a/AVvXsEjTNtV2vy2zMrD0I_l7mG75jjIi8BxlJJXYxRl0nX7yerPJQYrnH1QL_OBmNm1rlqfT-C1_IXfs9c9UknxRTPkWSy8KX9Qnnu7qorcJIHfN44X3uMHKzr63OwHDcGZ9Ca6urcLqCxSrhadkAjtCrnlxW9JlIJGjyfJVBJAkqK7REDXRAi4stL5mQcok_w=s320" width="320" /></a></p><p>In desperate circumstances, where students have repeatedly tried without success to master tables, I have been known to (reluctantly) settle for just knowing the squares. The beauty of the squares is that they march diagonally through the table, and so they really take you deep in amongst all the difficult facts. If you know the squares, the difficult products you <i>don’t</i> know are often only a step away.</p><p>For example, if you know that $8 \times 8 = 64$, then $7 \times 8$ must be $64-8 = 56$.<br />Or, if you know that $7 \times 7 = 49$, then $7 \times 8$ must be $49+7 = 56$.</p><p>So, the squares are really <i>high-leverage</i> facts to know, and I wouldn’t do anything else on the multiplication facts until the student knows these 8 squares. However, I am not really advocating pushing things like $7 \times 8 = 8 \times 8-8$, because students find this reasoning hard (Do I subtract 8 or 7?), and it breaks with the <i>multiplicative</i> theme.</p><p>So, instead, I would build differently from the squares:</p><p>$6 \times 3$ is half of $6 \times 6$ or double $3 \times 3$<br />$4 \times 8$ is half of $8 \times 8$ or double $4 \times 4$<br />$6 \times 12$ is half of $12 \times 12$ or double $6 \times 6$</p><p>This is really powerful. Mental doubling and halving may need some work, but that is very important anyway, so I am happy to be dependent on that (see Francome, 2020).</p><p>So, now we have three more facts, in orange below:</p><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhA9JJCNdAMwLqvwWAc18__wV0A2_H-93v77UxVE94gZvWDtEMkzdNCUr45aEj1rsFxtzA2aagqY_NxDxIP4hUpqR0ZYVWjs2GgRPx9u5n2c9FzqvnhoqnjB76osbYLF_A7oI_wzoxNpZRPruUuKsLEwJ6MqPelqbx8WfDz1-gxTxDf5StEGQJDzdcuBg=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="599" data-original-width="600" height="319" src="https://blogger.googleusercontent.com/img/a/AVvXsEhA9JJCNdAMwLqvwWAc18__wV0A2_H-93v77UxVE94gZvWDtEMkzdNCUr45aEj1rsFxtzA2aagqY_NxDxIP4hUpqR0ZYVWjs2GgRPx9u5n2c9FzqvnhoqnjB76osbYLF_A7oI_wzoxNpZRPruUuKsLEwJ6MqPelqbx8WfDz1-gxTxDf5StEGQJDzdcuBg=s320" width="320" /></a></p><p>Knowing that $6 \times 6 = 36$ is the single most powerful fact in the entire tables square, so long as you are able to mentally break down the 6s into 2s and 3s. Students who haven’t had much practice doing this ‘prime decomposition’ find it initially difficult, but this is at the heart of how multiplication works, so is an important awareness, and, with practice, it allows students to see why all the 36s in the table are equal (there are no 'coincidences' in the multiplication table):</p><p>$4 \times 9 = (2 \times 2) \times (3 \times 3) = (2 \times 3) \times (2 \times 3) = 6 \times 6 = 36$<br />So, $8 \times 9 = 2 \times 4 \times 9 = 2 \times 36 = 72$ and<br />$3 \times 12 = 6 \times 6$ (double the 3, halve the 12) $= 36$</p><p>These are in gold below:</p><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj8YFOQh8hpCcipBhw7Te3RIi7nLq2Gtw3kV0GXx7-xFD9J8-AkdS8wjmy8h3beXMjmRXKjvdm1G4-CRtNWp-oqX3eeVoTOxEB1DjV3ppZ7hWJAumcFIvGcSIKT6vmYM9sOWwDcaqINCZ1YGIQpjcwdMGl2ocCzhUT0d948DX7PSHrvzuyA1lpHAlqd2g=s602" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="602" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEj8YFOQh8hpCcipBhw7Te3RIi7nLq2Gtw3kV0GXx7-xFD9J8-AkdS8wjmy8h3beXMjmRXKjvdm1G4-CRtNWp-oqX3eeVoTOxEB1DjV3ppZ7hWJAumcFIvGcSIKT6vmYM9sOWwDcaqINCZ1YGIQpjcwdMGl2ocCzhUT0d948DX7PSHrvzuyA1lpHAlqd2g=s320" width="319" /></a></p><p>Next, I would do 12, 24, 48 and 96. If you learn that $3 \times 4 = 12$ (which most students will know), then $3 \times 8$ (double the 4), $6 \times 4$ (double the 3), $12 \times 4$ (double the 3 <i>twice</i>), $6 \times 8$ (double the 3 <i>and</i> double the 4) and $12 \times 8$ (double <i>three times</i>) all come along without too much trouble if students are fluent doublers - and only the the last one of these involves any 'carrying' when doubling.</p><p>This means that when students are stuck on $6 \times 8$, the prompt would <i>not</i> be to count up in 6s or work from the nearest multiple of 6 or 8 they can think of (e.g., $6 \times 10$). It would be: <i>Do you know $ \textit 3 \times \textit 4$? </i>(Both numbers are doubled, so the answer must be 12 <i>double-doubled, </i>which can be done easily mentally, without any 'carrying'.)</p><p>These six are in blue below, so, by this point, we have dealt with 20 of the tricky ones and there are just 10 left.</p><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi3osu9-LNuul9sEgRz1S1N3LSbQkNK2TsOMh_abpZ6xQD7Wsaf1rRvu41uMOO4wlf9WntOs77czvxbJrM8FsaWwpOm2CyLQyprY_Kw7F_nwDBRBMQ54nqXx5XVaHWURCh64pUPmDOqozdnfZkmHHRdOkFpVdS6gTBkJCxsRcMB96BM7M0KYt7o1ZVJJA=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="599" data-original-width="600" height="319" src="https://blogger.googleusercontent.com/img/a/AVvXsEi3osu9-LNuul9sEgRz1S1N3LSbQkNK2TsOMh_abpZ6xQD7Wsaf1rRvu41uMOO4wlf9WntOs77czvxbJrM8FsaWwpOm2CyLQyprY_Kw7F_nwDBRBMQ54nqXx5XVaHWURCh64pUPmDOqozdnfZkmHHRdOkFpVdS6gTBkJCxsRcMB96BM7M0KYt7o1ZVJJA=s320" width="320" /></a></p><p>The remaining ones are all ‘hard’, and we need to take time and care over these. I think I would spend 50% of my total energies on these 10.</p><p>There is 21, 42, 84 and 63 (in purple below), which all come from $3 \times 7$, which therefore needs to be learned. Then, given $3 \times 7$, we can do $6 \times 7$ (double), $12 \times 7$ (double twice) and $9 \times 7$ (triple). (None of these scalings is hard to do quickly mentally, as none involves any 'carrying'.)</p><p>Then there is 28 and 56 (in yellow below), where $8 \times 7$ is just double $4 \times 7$ (which is just double $2 \times 7$).</p><p>And then we have 27, 54 and 108 (in pink below), which come from $3 \times 9$, which needs to be learned (perhaps as $3^3$). We have $6 \times 9$ (double) and $12 \times 9$ (double twice).</p><p>Which just leaves 132 to remember (or know as $11^2 + 11$, which is possibly easier than double $6 \times 11$). I think this is probably the least connected of all of the multiplication facts, and so perhaps the hardest to remember.</p><p><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhHTpbfirgKfKDywsT1oQ47PXBRHDqjB3Thmm-9qKEKoF63NBHIXOet_5P6r3bMMElpJ_VwikV2FogpW-n5gYRN4YRwHFiapN5crJPyMPpdhSNkiWyVNKPz-ervFZ-lk8XovFs9p4dAXkR6fDzzJOd7XYx8oFPYKnsyygpvoTwt60yvAFwtdxhr1ITM3Q=s600" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="600" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhHTpbfirgKfKDywsT1oQ47PXBRHDqjB3Thmm-9qKEKoF63NBHIXOet_5P6r3bMMElpJ_VwikV2FogpW-n5gYRN4YRwHFiapN5crJPyMPpdhSNkiWyVNKPz-ervFZ-lk8XovFs9p4dAXkR6fDzzJOd7XYx8oFPYKnsyygpvoTwt60yvAFwtdxhr1ITM3Q=s320" width="320" /></a></p><p>So, in conclusion, this means that the only ones that potentially 'need' memorising are these 12:<br /></p><div style="text-align: center;">$3^2, 4^2, 6^2, 7^2, 8^2, 9^2, 11^2, 12^2, 3 \times 4, 3 \times 7, 3 \times 9$ and $11 \times 12.$</div>And, if you have the 3-times table, then that reduces this list to just the other 7 squares and $11 \times 12$, which really feels manageable. It does show the power you get from knowing the squares (Note 4).<p></p><p>I think the key to supporting all of this is in the kind of prompts that you provide when a student is stuck. Rather than asking them to figure it out from ‘anything relevant that you know’, or waiting patiently while they skip-count up from zero, with this approach you have a clear plan for how they might be getting from known things to unknown. With practice, figuring out something like $7 \times 8$ by saying ‘double $7 \times 4$, which is 28, so that's 56’ can be extremely quick, and the more you do this the more you are incidentally practising doubling. (And this is one of the trickiest ones, because doubling 28 involves a mental 'carry'.) Of course, nothing will be as fast as ‘just knowing’, but, where that has repeatedly failed for a student, then this kind of approach may help. And I would teach it to everyone for the sake of understanding the multiplicative connections (Note 5). I certainly prefer to spend energy on this than on those one-off mnemonics, like ‘5-6-7-8’ for $56 = 7 \times 8$, which are flukes that don't generalise.</p><p>Here is my attempt at a (rather messy) summary of where everything comes from (<a href="https://www.foster77.co.uk/Foster%20Times%20Tables%20Summary.pdf" target="_blank">pdf version</a>):</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQey38f2BeieQcCNieqmq5N3d_13QadwXEC0yrl52GAWkkriaF8sCS9cFANBruDxVVv2U2vXICti6XRj3IsH20_u5aXUgHz4OMOhG4KG602nzVKMioZ_sJkZR7BM-KyS2GErOgzGaRgnZxnLMJNr8JRLjyrDO8IOUNBuooVL6bKiPJTb4Y9990H8EJcg/s5100/Tables%20summary.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="5090" data-original-width="5100" height="399" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQey38f2BeieQcCNieqmq5N3d_13QadwXEC0yrl52GAWkkriaF8sCS9cFANBruDxVVv2U2vXICti6XRj3IsH20_u5aXUgHz4OMOhG4KG602nzVKMioZ_sJkZR7BM-KyS2GErOgzGaRgnZxnLMJNr8JRLjyrDO8IOUNBuooVL6bKiPJTb4Y9990H8EJcg/w400-h399/Tables%20summary.png" width="400" /></a></div><p style="text-align: left;">In conclusion, I am not suggesting that any of this is easy, especially with students who have experienced repeated failure with tables or have developed ‘tables anxiety’. There is no quick, easy fix. And I’m not saying that I think this approach is definitely the best (e.g., I don’t make anything much of the 9s, <a href="https://cdn.oxfordowl.co.uk/2013/08/13/10/05/56/722/9_x_Table_Trick.pdf">which can be fairly easy to learn</a>). But, I think that if you work through in this order you at least get the highest-leverage facts (e.g., the squares) before the lower-leverage ones (e.g., $11 \times 12$). However, if you have a better order - or entire approach - please put it in the comments below!</p><h3 style="text-align: left;">Questions to reflect on</h3><p>1. What are your best strategies for teaching the multiplication tables? Do you work differently with older learners who have previously been unsuccessful learning their multiplication tables? </p><p>2. What are the pros and cons of the different approaches you have tried? </p><p>3. What do you think of the scheme I have outlined above? Please respond in the comments if you can improve on it.</p><h3 style="text-align: left;">Notes</h3><p style="text-align: left;">1. For some great tables tasks that focus on conceptual understanding, see Faux (2018). See also the Position Statement on 'The Teaching and Learning of Multiplication Bonds' from the Joint ATM/MA Primary Group: <a href="https://www.m-a.org.uk/resources/1multiplicationbondsATMMA.pdf">https://www.m-a.org.uk/resources/1multiplicationbondsATMMA.pdf</a></p><p style="text-align: left;">2. Of course, except for the $1 \times n = 1 + n$ error, which seems to be particularly common with $1 \times 1 = 2$.</p><p style="text-align: left;">3. In the time of the Numeracy Strategies in the UK, everyone seemed to be chanting up and down in multiples on 'counting sticks', but I worry that that doesn't always help learners to remember which numbers belong in which tables. Once you move on to a new table, you trample all over numbers that have been learned in previous tables, with different but similar numbers appearing, and this interference makes it highly muddling for many students. It also feels 'additive' rather than 'multiplicative'.</p><p style="text-align: left;">4. Of course, some of the squares in this list could be derived from others in the list (e.g., $6^2$ is double-double $3^2$), but I tend to think that they are all important enough to know in their own right. But, if you disagree, then you could further reduce the list of base facts to just these nine: <span style="text-align: center;">$3^2, 4^2, 7^2, 9^2, 11^2, 3 \times 4, 3 \times 7, 3 \times 9$ and $11 \times 12$, and get everything from just them.</span></p><p style="text-align: left;"><span style="text-align: center;">5. The other advantage of 'just knowing' the tables, rather than working them out (even very quickly) is, of course, that you can work <i>backwards</i>, and when you see, say, 56, you immediately think 7s and 2s ($2^3 \times 7$). I think the kind of approach I've outlined here, focused on scaling up, rather than repeated addition, potentially helps with this, because, when you see 56, you are more likely to think 'double 28', and that can take you back to $14=2 \times 7$ and, via $4 \times 7$, to $8 \times 7$, so all the 'reverse doubling' helps to make visible the multiplicative structure that is there. Whereas thinking that 56 may be 7 or 8 more or less than some other half-remembered number doesn't do much for you.</span></p><h3 style="text-align: left;">References </h3><p>Faux, G. (2018). <i>Tables together</i>. Association of Teachers of Mathematics. <a href="https://www.atm.org.uk/shop/act114pk/Tables-Together-e-book/DNL170">https://www.atm.org.uk/shop/act114pk/Tables-Together-e-book/DNL170</a></p><p>Foster, C. (2022). Getting multiplication the right way round. <i>Mathematics in School, 51</i>(2), 16–17. <a href="https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Getting%20multiplication%20the%20right%20way%20round.pdf">https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Getting%20multiplication%20the%20right%20way%20round.pdf</a></p><p>Francome, T. (2020). Random chants: Generating a lot from a little using Excel. <i>Mathematics Teaching, 274</i>, 28-30.</p><p><br /></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com8tag:blogger.com,1999:blog-2036014053389751696.post-15026939423534288202022-04-28T07:00:00.001+01:002022-04-28T07:00:00.221+01:00Tangible contexts for mathematics<p><i>Do contexts help students to understand mathematics or do they just make it harder for them to untangle the mathematics from all the extraneous information? I think the answer is yes – both of these happen on different occasions. So, what is it that gives some contexts the potential to be powerfully illuminating?</i></p><p>I think the answer is not ‘relevance’ to a student’s personal interests. Relevance might be motivating, possibly, but it doesn’t necessarily make the context more illuminating of the mathematics. That way lies a ‘learning styles’ kind of fallacy, that every student needs a different context that is just right for them, and the magical right context will somehow make everything clear to them. I don’t think that’s right. And anyway, students often seem more switched on by contexts that take them <i>out</i> of their existing worlds (e.g., spaceships, dinosaurs, unicorns) than those which merely reference things they are already familiar/bored with. So I don't think matching personal interests is the most helpful approach. I think it's more likely that generally most students are helped by the <i>same</i> illuminating, well-chosen contexts, and not really so much by others. </p><h4 style="text-align: left;">Ratio and multiplicative/proportional reasoning</h4><p>Let’s take ratio or proportional/multiplicative reasoning as an example. This is widely acknowledged to be a (or possibly ‘the’) central concept in lower secondary mathematics. And something that many students really have a weak grasp of. If you wanted a concrete context to help students make sense of this area, what would you pick? If you opened a textbook at the ‘ratio’ chapter, what contexts would you expect to find?</p><p>Of course, ratio can be <i>applied</i> to all sorts of contexts, and it is important to do this and let students see how ratio can be relevant and important in a wide range of areas. That is fine. But what I am thinking about here is contexts that are deliberately used to try to develop students’ understanding of what ratio <i>is</i> and how it operates.</p><p>The problem for me with, for example, money as context is that if the ratio of money spent by, say, Usha and Sam is 3:1, and the ratio of money spent by Dave and Priya is <i>also</i> 3:1, it is quite hard to capture in words (or in pretty much any other way) what specifically it is about Usha/Sam and Dave/Priya that is <i>the same</i>, given that these ratios are <i>the same</i>. The ‘same ratio’ is a highly abstract concept here. So, although I think that money might at some point be a worthwhile context for using ratio, I don’t think it’s helpful for understanding what ratio <i>is</i>. My test is that I need to be able to complete the sentence: <i>“When the ratios are the same, the _____ is the same”</i> with something highly tangible and familiar (not mathematical) going into the blank space. For this reason, I think that most <i>discrete</i> ratios (money, different coloured beads on a string, different kinds of animals on a farm, boys and girls in a class, etc.) are not so useful.</p><h4 style="text-align: left;">Tangible context: paint</h4><p>Instead, I think the ratios of <i>continuous</i> quantities are much more useful to begin with, and, in particular, my go-tos are always drinks (Foster, 2007) and paint. The fact that most students probably never mix their own drinks, and even professional decorators rarely mix pots of paint together to make new colours (and when students mix their paint in art, this would be by eye) is irrelevant. The point of the context is not that it’s something students do every day, or even ever. The point is that it’s <i>easy to imagine </i>(what <a href="https://rme.org.uk/">Realistic Mathematics Education</a> calls 'realistic', and which means something closer to 'realisable').</p><p>The reason that I think these contexts are useful is that:<br /></p><ul style="text-align: left;"><li>“When the ratios of red paint to white paint, say, are the same, the paint is the <i>same colour</i>.” and</li><li>“When the ratios of orange juice to lemonade, say, are the same, the drinks <i>taste the same</i>.”</li></ul>And everyone knows what these things mean. This means that you can have a discussion about various hypothetical mixtures of red and white paint, or fizzy orange, and you can initially completely avoid the word ratio and any 'rules' about when 'ratios' are or aren’t equal. You can just ask: “Would they be the same colour?” or "Would they taste the same?", and everyone knows what you mean and can engage in the thinking that you want them to do.<p></p><p>With paint, I find that having the two colours as red and white is particularly useful, because you then have the word ‘pink’ available, in addition to talking about ‘redness’ and ‘darker/lighter’. This all helps the discussion to focus initially on the mathematical thinking, rather than terminology. Once students appreciate that 2:3 and 20:30 and 1:1.5 and 4:6 are all ‘the same colour’, then it is natural to try to capture this ‘sameness’, and we can use a word like ‘ratio’ to do so. But doing it the other way round, beginning by stating that 'We say that' 2:3 and 20:30 and 1:1.5 and 4:6 are all ‘the same ratio’ invites students to ask, “What do you mean?” And that puts the teacher in the position of having to do the justifying, whereas really you want the students to be doing this, based on something that they have already gained a sense of.</p><h4>Tangible context: fizzy orange</h4><p>For the same reason, making fizzy orange using orange juice and lemonade can be another really illuminating context (and you could possibly even do this one for real in the classroom, Foster, 2007). Lemonade is better than water, I think, not just because the mixture tastes better, but because then you can ask, “Which mixture will be <i>fizzier</i>?” as well as “Which mixture will be <i>more orangey</i>?” Really tangible contexts like these do a lot of the work for you. Every child knows that adding more orange juice won’t <i>necessarily</i> make the mixture taste more orangey, if you are also adding more lemonade.</p><p>I would often begin a discussion of this scenario by suggesting a few possible mixtures of orange juice and lemonade (as in the table below), and asking students which mixtures would taste the same, and which would taste different. For any ones that they think would taste different, I would ask them which would taste more orangey, and I find that that sometimes causes them to change their minds. You often get to a situation where they think one mixture would taste more organgey, but also more fizzy, and so that causes them to go back and think again.<a href="https://blogger.googleusercontent.com/img/a/AVvXsEibNWlvO9vNHVE7JWUrV6J7JHu1-0EpMZCBjK3JVVI6n65-SnJYicmmZe1DAdrFQiNei1QdBsAEL6G3-h1JVaucW5_u78i4f4TMaKAZbIBvRi-UYGesYsPHCBcEXFCllwSERKMwViObbUlplz4C_LJKrKql8lhhyyqNbkGpmFY8lL9woYL5y6yfXdISGw=s800" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="118" data-original-width="800" height="94" src="https://blogger.googleusercontent.com/img/a/AVvXsEibNWlvO9vNHVE7JWUrV6J7JHu1-0EpMZCBjK3JVVI6n65-SnJYicmmZe1DAdrFQiNei1QdBsAEL6G3-h1JVaucW5_u78i4f4TMaKAZbIBvRi-UYGesYsPHCBcEXFCllwSERKMwViObbUlplz4C_LJKrKql8lhhyyqNbkGpmFY8lL9woYL5y6yfXdISGw=w640-h94" width="640" /></a>As the discussion progresses, further possible mixtures are usually suggested by the students, and I would add these to the list. The point is to avoid telling students whether they are right or wrong, but to draw on their common sense and life experience to let them figure it out. They know everything they need to know to do this. This then forms a really good basis for more formal teaching of ratio.</p><p>For example:</p><i>Teacher: Would any of these mixtures taste the same? Are there any you’re sure would taste different?</i><br /><i>Student 1: D and E would taste different.</i><br /><i>Teacher: Why do you say that?</i><br /><i>Student 1: D would taste stronger than E because there’s less lemonade in it.</i><br /><i>Teacher: But D and E have the same amount of orange, don’t they, so shouldn’t they be equally orangey?</i><br /><i>Student 2: No, because the orange is spread out in more lemonade in E.</i><br /><i>Teacher: Can someone else explain what S2 is saying?</i><br /><i>...</i><br /><i>Teacher: Would any of these mixtures taste the same as each other?</i><br /><i>Student 3: A and B would taste the same.</i><br /><i>Teacher: Why do you say that?</i><br /><i>Student 3: Because they both have 1 more </i><i>lemonade</i><i> than orange.</i><br /><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"></blockquote><i>Teacher: Are there any other mixtures with 1 more litre of </i><i>lemonade</i><i> than orange?<br /></i><i>Student 4: Mixtures C and D.</i><br /><i>Teacher: So, would mixtures A, B, C and D all taste the same?</i><br /><div style="text-align: left;"><i>Students: Yes.</i></div><p>It’s likely at this point that some student will raise some doubt, perhaps relating to C being ‘nearly fifty-fifty’. Multiplicative language or thinking tends to appear around this point, if it hasn't already, which can then develop into getting the students to order A, B, C, D and E by ‘orangeyness’.</p><p>If this doesn't happen, then the teacher can be more proactive:</p><i>Teacher: Suppose I took two containers of Mixture A. How many litres would there be in each?</i><br /><i>Student 3: 5 litres.</i><br /><div style="text-align: left;"><i>Teacher: What would happen if I mixed them together?</i></div><p>Every student will appreciate that mixing identical mixtures will lead to twice as much mixture, but that it will taste <i>exactly</i> the same. So this gives us Mixture E. And students will have already agreed that Mixture E must be <i>less</i> orangey than Mixture D, so this provides the nudge for everyone to think more deeply. Mixtures A and D can't taste the same if mixtures A and E taste the same and mixtures D and E don't! The idea that mixing 'identically-tasting mixtures' (still avoiding the use of the word ‘ratio’) will lead to a new mixture with exactly the same taste is highly intuitive, and nobody will ever doubt this. And that kind of knowledge is all that is needed to develop all the necessary ideas of ratio through this kind of discussion.</p><h4>Tangible context: chromatography</h4><p>Finally, I think a really helpful science context is chromatography and <i>retardation factor</i> ($R_f$) values (Note 2). There could be potential for some cross-curricular practical work with chromatography paper and water-soluble marker pens. Different inks dotted along a pencil line at the bottom of a sheet of chromatography paper will move at different rates as the solvent soaks up the sheet (Figure 1). Each component will travel at a fixed fraction of the speed of the solvent, and the $R_f$ value of each is defined as</p><p>$$R_f = \frac {\text{distance travelled by the substance}} {\text{distance travelled by the solvent}}$$</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg7j0yGelh9TDrBh7EL1U_LpbF5HgEYDgdR07RcJZCkq5nPST4mh4HxR-i4u3nmUXLVzvExY6TI1NfRDYTKAcSUbfDwNfm9CsXRRCmN4eSqKYs3Vwctbrrn_S-y1OWM1dXsqIX1Oo40mTCbegO9--ZYCMEPr3KpR4vX6uCP5MEuYjYsKJvFESZghPs0LQ=s775" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="775" data-original-width="600" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEg7j0yGelh9TDrBh7EL1U_LpbF5HgEYDgdR07RcJZCkq5nPST4mh4HxR-i4u3nmUXLVzvExY6TI1NfRDYTKAcSUbfDwNfm9CsXRRCmN4eSqKYs3Vwctbrrn_S-y1OWM1dXsqIX1Oo40mTCbegO9--ZYCMEPr3KpR4vX6uCP5MEuYjYsKJvFESZghPs0LQ=w310-h400" width="310" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. Calculating the $R_f$ for the red substance.</td></tr></tbody></table><p>This seems to me like a perfect, dynamic scenario for understanding ratio, because molecules of a substance are highly obliging, and obey the rules perfectly (unlike, say, two runners in a race, running at different speeds, who need to negotiate bends and are likely to get tired at different rates). Here, when the ratio is the same, the height above the baseline on the chromatogram is the same (and the substance is likely to be the same). I would be keen to hear from anyone who has used this context as a way to explore ratio with students.</p><h3><span style="font-family: inherit;">Questions to reflect on</span></h3><p style="text-align: left;"><span style="font-family: inherit;">1. What examples of illuminating contexts do you use - for ratio, or for other topics? What is so good about them?</span></p><p style="text-align: left;"><span style="font-family: inherit;">2. When do you feel that contexts <i>do</i> and <i>do</i> <i>not</i> work well? Why?</span></p><h3 style="text-align: left;">Notes</h3><p>1. For a free lesson plan based on the fizzy orange idea, see <a href="https://www.map.mathshell.org/lessons.php?collection=8&unit=6230">https://www.map.mathshell.org/lessons.php?collection=8&unit=6230</a></p><p>2. People's recent familiarity with Covid lateral flow tests may also make this easier to grasp.</p><h3 style="text-align: left;">Reference</h3><p>Foster, C. (2007, May 24). Make maths sparkle. <i>SecEd</i>, 12. <a href="https://doi.org/10.12968/sece.2007.5.902">https://doi.org/10.12968/sece.2007.5.902</a>. Available at <a href="https://www.foster77.co.uk/Foster,%20SecEd,%20Make%20Maths%20Sparkle.pdf">https://www.foster77.co.uk/Foster,%20SecEd,%20Make%20Maths%20Sparkle.pdf</a></p>Colin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.com2