# Colin Foster's Mathematics Education Blog

## 30 March 2023

### Teaching specific tactics for problem solving

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. How can we help students not just to 'do problem solving' but actually learn to get better at it?

#### What I mean by problem solving

The term problem solving 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 non-routine task - 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).

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 routine 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 what to do, as that's clear from the outset.

The alternative scenario is a tunnel which you cannot 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 starting, 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 non-routine task, or problem.

It's important to realise that routine tasks are not necessarily quick and easy - 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).

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?

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 assumed I was good at chess because I was good at mathematics.

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!

#### Teaching the rules but not the strategies

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 $180^\circ$, and then we give students increasingly challenging problems to solve that depend 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 everything 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.

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 The Mathematical Gazette in 1922. It involves what is sometimes referred to as the 80-80-20 triangle (see Figure 1). The task is to show that $x = 30$.

 $ABC$ is an isosceles triangle.$B = C = 80^\circ$.$CF$ at $30^\circ$ to $AC$ cuts $AB$ in $F$.$BE$ at $20^\circ$ to $AB$ cuts $AC$ in $E$.Prove $BEF = 30^\circ$.Figure 1. Edward Langley’s original problem (Langley, 1922, p. 173).

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 “The World’s Hardest Easy Geometry Problem” – 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!

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 @Cshearer41, 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!

I think the missing ingredient is problem-solving tactics (finer grained than 'strategies'). Of course, content knowledge of geometry (e.g., the angle sum of a plane triangle is $180^\circ$) is essential. It is necessary but not sufficient. 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 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 topic-specific strategies (Foster, under review). In the case of Langley’s problem, the key strategy turns out to be to add an auxiliary line 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!).

My recent Economic and Social Research Council (ESRC) project, Exploring socially-distributed professional knowledge for coherent curriculum design, 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 LUMEN Curriculum resources, 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 all students become powerful problem solvers.

#### Concluding thoughts

Writing these 26 blogposts over the course of my year as President of the Mathematical Association 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 @colinfoster77 and through my website https://www.foster77.co.uk/, and I hope to see many of you at the Joint Conference of Mathematics Subject Associations 2023 next week!

### Questions to reflect on

1. Do you agree about the value of teaching problem-solving tactics, in addition to 'content'? Why / why not?

2. Where in your teaching/curriculum do students encounter strategies such as 'Draw in an auxiliary line'?

3. How might you plan to teach other problem-solving tactics explicitly?

### Notes

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).

2. I use 'task' to refer to anything mathematical a student is asked to do: it could be written, oral or practical.

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 repdigit 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.

4. Langley was the founding editor of The Mathematical Gazette. A curious fact is that he apparently had a blackberry named after him – not a lot of people can say that!

5. One possible solution is given in the diagrams below: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 $30^\circ$.

### References

BAGs to Learn Podcast by Ben Gordon (2021, December 2). Episode 4 – Colin Foster – Problem Solving in the mathematics curriculum [Audio podcast]. https://anchor.fm/ben-gordon83/episodes/Episode-4---Colin-Foster---Problem-Solving-in-the-mathematics-curriculum-e1b5ic3

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. Journal of Mathematics Teacher Education. Advance online publication. https://doi.org/10.1007/s10857-021-09511-6

Chen, Y. (2019). 103.39 A lemma to solve Langley’s problem. The Mathematical Gazette, 103(558), 521-524. https://doi.org/10.1017/mag.2019.121

Fischer, B., Margulies, S., & Mosenfelder, D. (1982). Bobby Fischer teaches chess. Bantam Books.

Foster, C. (2019). The fundamental problem with teaching problem solving. Mathematics Teaching, 265, 8–10. https://www.atm.org.uk/write/MediaUploads/Journals/MT265/MT26503.pdf

Foster, C. (2021). Problem solving and prior knowledge. Mathematics in School, 50(4), 6–8. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Problem%20solving%20and%20prior%20knowledge.pdf

Foster, C. (2023). Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics. The Curriculum Journal. Advance online publication. https://doi.org/10.1002/curj.213

Langley, E. M. (1922). Problem 644. The Mathematical Gazette, 11(160), 173. https://doi.org/10.2307/3604747

Leikin, R. (2001). Dividable triangles—what are they? The Mathematics Teacher, 94(5), 392-398. https://doi.org/10.5951/MT.94.5.0392

Quadling, D. A. (1978). Last words on adventitious angles. The Mathematical Gazette, 174-183. https://doi.org/10.2307/3616686

Rike, T. (2002). An intriguing geometry problem. Berkeley Math Circle, 1-4.

Schoenfeld, A. H. (1985). Mathematical problem solving. Elsevier.

Shearer, C. & Agg, K. (2019). Geometry puzzles in felt tip: A compilation of puzzles from 2018. Independent.

## 16 March 2023

### Crocodiles and inequality signs

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?

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 $4>3$.

Figure 1. Will a hungry crocodile eat a mouse or an elephant?

Now, framed in this way, clearly this is a bit silly. Even very young children are likely to wonder:

What if the crocodile isn't hungry?

What if the smaller animal is tastier, more nutritious, easier to catch, or less likely to attack than the larger animal?

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 zoomorphism of an inequality symbol the thin end of a $<$-shaped wedge that we would just be much better off without?

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?

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.

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:

LHS $=$ RHS

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 $90^\circ$:

LHS $\lvert \rvert$ RHS

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.

I suppose there could be the danger with this of thinking that $\lvert \rvert$ 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

$\lvert -11 \rvert=11,$

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.

But the really nice thing about using $\lvert \rvert$ for equality would be that we could use very natural symbols for greater than and less than:

$3 < 4$ becomes $3$ |$\rvert \; 4$

$4 > 3$ becomes $4 \; \lvert$|$\; 3$

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.

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?

After all, writing $3$ |$\rvert \; 4$ is all very well, but how would you express $3 \le 4$, and how would you handle double (or more) inequalities, like  $3 < 4 < 5$? I think it would get quite messy and awkward. In our usual notation, I do 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 three less-than symbols.

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 $\LaTeX$ that 'bar' and 'hat' are precisely the words I need to type to produce them, so knowing these names is actually quite useful.

### Questions to reflect on

1. When is anthropomorphising or zoomorphising mathematical symbols OK and when is it not?

2. When are 'informal' names for symbols OK and when should they be avoided?

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?

### Reference

Wege, T. E., Batchelor, S., Inglis, M., Mistry, H., & Schlimm, D. (2020). Iconicity in mathematical notation: Commutativity and symmetry. Journal of Numerical Cognition, 6(3), 378-392. https://doi.org/10.5964/jnc.v6i3.314

## 02 March 2023

### Are probabilities and inequalities approximate?

If mathematics is about being certain and precise, then how can probability be part of mathematics, because probability is about not being sure?

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”.

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 real coin, undergoing any real 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 all 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.

Alternatively, the approximate aspect that the student was thinking about might have been the uncertainty of the outcome on any single coin flip. 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 any given flip 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.

Although $\frac{1}{2}$ is exactly in the middle of the probability scale that runs from $0$ to $1$, in a sense it represents maximum uncertainty, since if the probability were to take any other value 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.

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).

#### Inequalities

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'.

When we solve an equation like $2x+5=11$, we obtain an exact solution, $x=3$. We find that $x$ takes this one specific value, and no other, and that is that. But, when we solve an inequality like $2x+5>11$, we obtain a solution expressed as another inequality, $x>3$, and this may seem to students to be expressing some uncertainty, 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 still don't know what $x$ is!" a student might complain.

'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 all the possible answers. Or, with simultaneous equations, where the values of both unknowns need to be found before someone can claim to have solved it. But here there are infinitely many possible answers, so we seem to know very little indeed about what the value of $x$ is!

However, infinitely many possible value have also been ruled out, so this is 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 $3$.

The fact that $x>3$ means that $x$ is "definitely more than precisely $3$" 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'.

Perhaps a better way to talk about this is in terms of solution set: 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 all the possible values of $x$ that are consistent with the given information. 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 all of the numbers greater than 3 and no others".

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)!

The point is that we're not seeking a single mystery number, and trying to guess what it might be, but a solution set of all the possible numbers. 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 impossible, so the student's solution set is the wrong one.

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 doesn't contain any numbers which the given inequality rules out. The language of solution set seems to make this much easier to talk about.

### Questions to reflect on

1. Have you encountered students having these kinds of questions/confusions?

2. How do you explain to students what is going on when they are solving inequalities?

### Notes

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!)

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 less correct than this would have been $x \le 3$"!

### Reference

Gelman, A., & Nolan, D. (2002). You can load a die, but you can't bias a coin. The American Statistician, 56(4), 308-311. https://doi.org/10.1198/000313002605 (\$)

### Teaching specific tactics for problem solving

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 ...