tag:blogger.com,1999:blog-2036014053389751696.comments2024-03-21T14:25:37.650+00:00Colin Foster's Mathematics Education BlogColin Fosterhttp://www.blogger.com/profile/12463017049484632672noreply@blogger.comBlogger36125tag:blogger.com,1999:blog-2036014053389751696.post-89801180843923609662024-03-13T10:00:04.985+00:002024-03-13T10:00:04.985+00:00I totally get the struggle with times tables; it&#...I totally get the struggle with times tables; it's like tackling a tricky mountain. This new approach sounds like a game-changer, connecting the dots in a way that feels natural and might just make those tables stick for good!Lisandra Tamezhttps://www.car.co.uk/noreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-2498188154871247522024-03-03T12:18:59.476+00:002024-03-03T12:18:59.476+00:00"Sometimes, I suspect, students who are confu..."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 further (rather than less) than their peers." Further than their teachers, in many cases. For one, distances aren't mathematical entities; 1m is outside of mathematics, so this conversation doesn't really belong in a mathematics class. The mathematical part of this problem might be to determine how many multiples (a number) of some standard quantity (in units such as metres, litres, etc.) measure a given quantity (length, volume, mass, etc.). That so-called 'mathematics teachers' can't make this distinction reflects on the levels of mathematics understanding within the profession. The argument that 0m is 0 therefore even more meaningless than comparing apples and oranges. The other issue is confounding measure (cardinality) and order. Positions (order) can be relative in ways that sizes can't. If 0 is used to represent/label some point (your origin), you're using numbers to refer to positions relative to the origin with respect to some other point (which defines your unit). Negative numbers make perfect sense in the context of positions, but negative lengths don't make sense. Temperature conversions involve both two distinct origins and two distinct units (centigrade and fahrenheit), that's why they shouldn't both be labelled 0; they may be labelled 0C or 0F to indicate their respective origins. The label 2C represents a point two multiples of the C-unit in a certain direction, just as -2F represents the point for which the origin, 0F, is two F-units to the right of that point. All too often we blame students for the failings of their teachers.Victornoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-70208942517337502652023-12-26T12:08:45.876+00:002023-12-26T12:08:45.876+00:00Question , can problems be solved at a higher simp...Question , can problems be solved at a higher simple level or am I going mad?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-64946512909184156092023-05-15T09:07:00.259+01:002023-05-15T09:07:00.259+01:00Really interesting. With your alternate vertical ...Really interesting. With your alternate vertical lines, and visually linking equality with inequality, I think there is also the potential to link the horizontal lines - an equal sign becomes an inequality by the two lines getting closer together at one end and therefore further apart at the other - no need for the crocodile, but still exploring the visualAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-54814535841917981432023-03-30T12:53:19.439+01:002023-03-30T12:53:19.439+01:00Splendid! Not long ago there was an outbreak of wh...Splendid! Not long ago there was an outbreak of whinging on social media about a GCSE problem that featured a tree-diagram question set "back-to-front": "we can't do this because we haven't been shown how to do it". To get top grades (and to impress at interview, to say nothing of general educational value), students need to be able to do questions they haven't see before, and so teachers need to empower their learners for problem-solving. This advice seems to be just what is needed - thanks!Owen Tollernoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-75835058863352797052023-03-29T16:32:26.436+01:002023-03-29T16:32:26.436+01:00Really interesting thoughts, Colin. I wonder to wh...Really interesting thoughts, Colin. I wonder to what degree relating inequality signs to magnitudes hinders when we get to negative numbers, though. Is thinking of 3<4 in terms of 'smaller' and 'bigger' problematic when we get to -4<-3, when -4 is a 'bigger' negative than -3?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-34649522889620418832023-03-20T13:02:14.090+00:002023-03-20T13:02:14.090+00:00Thanks as ever for an interesting and thoughtful b...Thanks as ever for an interesting and thoughtful blog. I would say it is okay when it makes things more fun (with obvious caveats to avoid racism or engendering unhelpful beliefs for life). I think its main use is when introducing ideas - I remember learning the game of chess from a book where pieces talked to one another, which I liked. <br /><br />Informal terms: to be controversial (ignorant?) I actually think we should use the terms "top of a fraction" and "bottom of a fraction" - I think they are just as precise, and IMHO clearer. I think informal terms can be helpful to aid understanding, but there is a big potential danger of then not recognising standard terms (with "top" and "bottom" above I these should be the standard terms) or being confused by having two terms for the same thing (since a very reasonable assumption is that if they were the same, there wouldn't be two terms!)<br /><br />Using own terms I think is helpful for a sense of ownership, and particularly good in say exploring a problem where the student spots some property, and it's really helpful to give it a name. In the debrief, I think it is helpful for the teacher to map these to any standard terms/concepts e.g. "parity", "odd function". Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-50221722958641733082023-02-16T15:44:08.578+00:002023-02-16T15:44:08.578+00:00This is my recent twitter thread. An engineer says...This is my recent twitter thread. An engineer says "the speed of a particle that travels d meters in t seconds is d/t meters per seconds". d and t are numbers, so you can put them into a calculator to work out d/t, and you don't have to tell the calculator what the units or dimensions are. In this usage, "d m" is a length, whilst "d" is a number.<br />OTOH, a mathematician, at least at University level, will talk about a distance d and a time t and a speed d/t. So here "d" is a distance, it has a dimension but no units. A distance can be divided by a time, but not on a calculator. The relationship can be illustrated by saying "this distance" (indicating two points) and "this time" (snapping fingers twice), and "this speed" (sweeping a finger from one point to the other in the indicated time). Understanding this equation does not require a system of units. So "d" can either mean a number (of meters) or a physical length. I detect some lack of clarity in school mathematics about which approach is being used, and I don't think that the distinction is taught either at school or university. The engineer will say the mathematician has forgotten to put in the units, but not so. The engineer's "d" is a completely different class of entity than the mathematician's.<br />Sio normally I'd say a numeric length needs a unit. But then we talk about unitless lengths and areas in the Cartesian (or Argand) plane, so here we are talking about the geometry of our abstract number system, and perhaps the words "length" and "area" are being misapplied in this contextJim Simonsnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-42703329352699180552023-02-16T13:21:06.682+00:002023-02-16T13:21:06.682+00:00I agree totally with all of this. I would add that...I agree totally with all of this. I would add that when I taught standard deviation I would always get the class to do one example (no more) - usually {4, 6, 7, 8, 10}, or {4, 6, 7, 7, 8, 10} if you insist on using (n - 1) - wholly without a calculator. I believe that "getting their hands dirty" (just the once) gave them an understanding of the formula and its meaning that would be lacking if all their calculations were simply "substituting into a formula" or using calculator software. <br /> What I would really like to know is how we can set public exams that test things like mean and SD realistically, i.e. requiring candidates to use computer software to analyse large and messy data. Surely we are at the stage when taking examinations online need not be restricted to candidates with learning difficulties?Owen Tollernoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-19434167930479105082023-02-05T00:19:20.367+00:002023-02-05T00:19:20.367+00:00I think this is really clever! By prioritising the...I think this is really clever! By prioritising the multiplicative links between the products you are moving away from the short term goal of memorisation and recall (which is important for fluency), and prioritising multiplicative thinking (which is also important for fluency, and a host of other mathematical things). It always surprises me how many children finding doubling and halving difficult, though, so I suppose one would have to be sure that this was a useable strategy for those children - and if not immediately useable, to dedicate enough time to developing doubling and halving fluency to make it useable for them. I have always done something similar (but not as clever, or as interlinked) by linking the 2's, 4's and 8's, as well as the 3's, 6's and 12's using doubling and halving, but still in the sequence of skip counting (starting from 1 or 2), rather than starting from the square numbers (which is immediately more powerful mathematically - and I'm feeling a little silly that I didn't see that before!). The only addition/modification I am pondering would be to begin with your proposed sequence, and then also point out the easily memorisable form of the 9's as we go, and also to think of 12's as the sum of 10's and 2's (but that is heading into additive-land, and multiplicative-land is a much more useful place to find oneself). Thanks for sharing this - I will be adopting all of it. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-34054070246099557522023-01-05T19:51:36.131+00:002023-01-05T19:51:36.131+00:00At a possibly earlier level:
Fractions: 2/5 = 20/5...At a possibly earlier level:<br />Fractions: 2/5 = 20/50 = 10/25 etc.<br />Percentages of: 10% of 360 is 36, 20% ... 72. 5% ... 18, etc.<br />Distance-Speed-Time calculations: 30 miles in 60 minutes, 3 miles in 6 minutes, 12 miles in 24 minutes etc.<br />Ratio: 2:3, 20:30, 10:15, etc. are all the same.<br />Pie Charts: 120 children = 360deg., 1 child = 3deg., 48 children = 144deg. etc.<br />Scale Drawing: 1 cm is1 km, 15 cm is 15 km, etc.<br /><br />Some of these maybe a bit out-of date!Wil Ransomenoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-8569074776632364802023-01-02T12:28:37.543+00:002023-01-02T12:28:37.543+00:00I have been thinking the same thing about algebra ...I have been thinking the same thing about algebra tiles, I have found it particularly problematic when trying to use them to model completing the square, they 'work', I think, only when dealing with positive terms (x^2 + 2x + 10) but for expressions like x^2 - 4x -10 you need to have tiles representing positive and negative areas and lengths. This type of representation seems more complicated than just learning the transformation from the perfect square. Anna Pickovernoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-42965287144441339352022-12-23T10:47:51.738+00:002022-12-23T10:47:51.738+00:00Not fractional powers but negative powers!
Linear...Not fractional powers but negative powers! <br />Linear models are also very important.Christine Lenghausnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-67139309283397570382022-12-23T10:02:49.744+00:002022-12-23T10:02:49.744+00:00There are many ways to view this. I see it as the ...There are many ways to view this. I see it as the generalisation for any base. So base 10, x^2 would be the hundreds column, x the tens or if it was base 2, x^2 would be the 4 column and x would be 2. The first (most right) column, before fractional powers, is ones.<br />If we had students that wanted to go deeper on the maths we could explain the different dimensions. For most of the students I teach the area model has been a game changer in terms of their understanding and success.Christine Lenghausnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-3134022596167573382022-12-11T02:39:01.772+00:002022-12-11T02:39:01.772+00:00I have used the scenario of two caves (used maskin...I have used the scenario of two caves (used masking tape to show on the carpet as upside down u) - one wide/short the other narrow/tall and two different paper "leaves" to measure (I cut a gum tree leaf and maple leaf). The sts named the cavemen and we set about to measure who had the biggest cave. Using thin gum leaf to measure across wide entrance and compare to maple leaf for narrow height. Not good way to compare. We had a discussion on what would be a good and fair shape to measure with. Squares are good!ðŸ’¥Christine Lenghausnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-32356107965308042212022-12-10T01:43:19.624+00:002022-12-10T01:43:19.624+00:00I very rarely draw circles for anything to do with...I very rarely draw circles for anything to do with fractions. <br />Using a clock analogy, I don't expect students to easily say 0.1 of an hour is 6 minutes (try dividing a circle into fifths or tenths). But every lawyer and accountant knows how to bill in 6 min increments!Christine Lenghausnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-46787865528255470322022-10-27T12:56:16.593+01:002022-10-27T12:56:16.593+01:00This is great. On the modulus function, I love to...This is great. On the modulus function, I love to explore the alternative definition âˆš(x^2), which links nicely to the norm of a vector. Of course it is the norm of a 1-dimensional vector. Generally 1-dimensional vectors is a great topic for making connections.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-71511244060128222982022-10-25T11:36:58.347+01:002022-10-25T11:36:58.347+01:00i think you are missing the whole point, he is sug...i think you are missing the whole point, he is suggesting logical thinking rather than rote learning...Attardnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-39727528810209948802022-10-23T08:03:03.317+01:002022-10-23T08:03:03.317+01:00With coordinate geometry, a lot of students want t...With coordinate geometry, a lot of students want to change gradients which are improper fractions to mixed numbers (to â€˜simplifyâ€™), but itâ€™s better to be able to see â€˜riseâ€™ and â€˜runâ€™, also when rearranging, itâ€™s usually better to have y=-2x+4 than y=4-2x, or even y=2(2-x). The word simplify is indeed a tricky one! Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-83514191047822220482022-09-16T20:39:31.358+01:002022-09-16T20:39:31.358+01:00Good stuff here. I teach my students two things a...Good stuff here. I teach my students two things about simplification. Mid-calculation, simplification is about making the next step as simple as possible, so think about what the next step will be before deciding what simplifications to make, beyond the incontestable ones. At the end of an answer, what counts as simplest is an aesthetic judgement, not a mathematical one, and people will differ. So don't worry about it, but present the answer the way you like it.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-37774967581089708952022-09-01T10:25:15.506+01:002022-09-01T10:25:15.506+01:00I think these are excellent, engaging and fun. I w...I think these are excellent, engaging and fun. I work as a tutor and for many students I have taught, their main problem is not mastering a technique, but understanding that it relates to something they might actually care about! The standard form example seems strongest to me in this context. The complex introduction is slightly weaker in this context, in that it seems quite reasonable the x^2 + 3 has no solutions, in the same way we say 0x = 3 has no solutions - however hopefully with an A-level class the possibility it has solutions is exciting. Bombelli's "wild thought" at solving the cubic x^3=15x+4 in the 1500s (I saw it mentioned p5 in Tristan Needham's Visual Complex Analysis) where pretending root -1 exists suddenly allows you to find the real root 4, for me seems a stronger motivation. However this needs balancing against it being more complicated, though students can check (2 +- i)^3 gives (2 +- 11i) and confirm Bombelli's thought wasn't too wild. (Also, I haven't commented on any previous blog, but I have read them, and found them really helpful and thoughtful. Thanks for writing them.) Francis Huntnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-51201368260903327602022-06-23T18:18:48.967+01:002022-06-23T18:18:48.967+01:00Another fascinating post. To me this shows that ca...Another fascinating post. To me this shows that calculated lines of best fit are several bridges too far for GCSE. Everything depends on the modelling assumptions used, and not only are these quite sophisticated in themselves but the whole concept of modelling assumptions is probably too advanced for GCSE. This is a typical example of the way in which Excel makes it very easy to draw poor or confusing charts. GCSE students aren't in a position (or at least don't have time) to discuss the question, "what do you mean by a line of best fit?"<br /> It might have been nice to mention the fact that, in your diagram showing the ellipse and black line, the black line goes through the points where the vertical tangents to the ellipse meet the ellipse. (This of course follows from your comment about "thin vertical slices".) The line of x on y goes through the corresponding points for the horizontal tangents. But then we start having to worry about controlled variables or bivariate data.<br /> I'd also suggest that this raises the question of whether a least-squares line of best fit is necessarily the best to use. Certainly it gives a high priority to minimising large distances of points from the line. Is this what we want? Are not the points far from the line the ones that should have less weight? Should we not want to give more emphasis to the points close to the line? Do we not just use the least-squares line because it has the simplest algebraic properties? <br /> I think we should just suppress any idea that there is a best-fit line that can be calculated, until we have the tools to discuss the issue properly.Owen Tollernoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-78180652917658082852022-06-07T22:51:18.920+01:002022-06-07T22:51:18.920+01:00I spent three consecutive years working with the w...I spent three consecutive years working with the weakest mathematicians in Y5. All left the year group with rapid recall of multiplication facts. Here's how... First, we used a little deck of flashcards per pair of kids to learn the 2s and 5s rapidly. (Kids quiz their partner. Wrong answers go on a pile to be re-asked.) Once these were solid (and not before), we created a new set with 3s and 4s. Any flashcards that are consistently answered correctly by both partners get removed from the deck.) You slowly add sets of facts (but only new facts so, for example, by the time you get to the 7s - the last set - you only need to include 7x7). <br /><br />The key things are (a) this needs to be done every day for five minutes, (b) encourage quick guessing and answer showing, and (c) put wrong answers on the re-ask pile, so kids are seeing these ones again. <br /><br />Oh, and I don't bother with 11s and 12s. Beyond school accountability purposes, there is little purpose to them.<br /><br />No wizardry needed. Just a set of flashcards per pair and lots of zero-stakes little-and-often practice. That's it.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-56892646709136106572022-06-01T10:00:55.559+01:002022-06-01T10:00:55.559+01:00I am sure KO is right - the expected cost per year...I am sure KO is right - the expected cost per year of repairs doubles (assuming identical cars!). I think expectations work more intuitively than probabilities, mainly because the expected value of the two RVs is the sum of the two expectations, even if the RVs are not independent.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-2036014053389751696.post-58041643862706296602022-05-26T07:15:20.460+01:002022-05-26T07:15:20.460+01:00Very nice - I wonder if the sentiment/intuition o...Very nice - I wonder if the sentiment/intuition of 'twice as much chance to go wrong' is really about expected values rather than probability - so really it's expressing that 2 cars may have twice as many total problems to deal with (still assuming the first car doesn't get used less etc.). KOhttps://www.blogger.com/profile/02891326341862553976noreply@blogger.com