Tasks that can't fail

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?

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.)

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.

So, I prefer to ask the beginning trainee teacher to do the beginning 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 don't 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.

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:

1. Four fours

I probably don't need to say much about this task, as it is such a classic:

What numbers can you make from four 4s?

(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.

2. Possibility tables

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.

One "can't fail" task that seems to do this is what I call Possibility tables (Foster, 2015). These are a bit like what John Mason calls structured variation grids (see http://mcs.open.ac.uk/jhm3/SVGrids/SVGridsMainPage.html#What_is_an_SVGrid). For example, the possibility table below (pdf version) 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.)

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.

The dynamic of one large sheet of paper and one shared pencil (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 both 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 one 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: Why do some cells seem to be easier to find figures for than others?

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.

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.

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.

Questions to reflect on

1. What do you make of these tasks?

2. What "can't fail" tasks do you use?

References

Foster, C. (2015). Symmetry combinations. Teach Secondary, 4(7), 43–45. https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Symmetry%20Combinations.pdf

Foster, C. (2020). Revisiting 'Four 4s'. Mathematics in School, 49(3), 22–23. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Revisiting%20Four%204s.pdf

Foster, C. (2021). First things first. Teach Secondary, 10(6), 82–83. https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20First%20things%20first.pdf

Foster, C. (2022). The trouble with groupwork. Teach Secondary, 11(5), 70–71. https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20The%20trouble%20with%20groupwork.pdf

Always simplify your answer

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?

A student was calculating the radius of a circle with unit area. They wrote

$$\pi r^2=1$$

$$r^2=\frac{1}{\pi}$$

$$r=\frac{1}{\sqrt{\pi}}$$

$$r=\frac{\sqrt{\pi}}{\pi}$$

When challenged about the final step, they said that they were 'rationalising the denominator'. The teacher said, "You mean 'irrationalising' the denominator?", since $\pi$ is irrational. But the attempt at humour was not really right, because the denominator was irrational before and after this step. However, I have some sympathy with what the student was doing, presumably by analogy with things like

$$\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}.$$

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 almost 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 even more irrational 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., the Gaussian distribution).

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.

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 more 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 always 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.

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 more 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).

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.

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 without 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 would expect them to simplify this to $1$!

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 uncontroversially 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:

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).

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.

Questions to reflect on

1. How do you explain to your students what is required for 'simplified' answers?

2. Can you think of other examples of ambiguous or confusing situations involving simplification?

Notes

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 '$1-x$'.

2. No one knows 'why' $e^{\pi}$ (Gelfond's constant) is so close to $\pi + 20$. Maybe it doesn't really make sense to ask for 'explanations' of things like this (see https://en.wikipedia.org/wiki/Mathematical_coincidence).

References

Foster, C. (2021). Questions pupils ask: What are 'like terms'? Mathematics in School, 50(4), 20–21. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20What%20are%20'like%20terms'.pdf

Halmos, P. R. (1985). I want to be a mathematician: An automathography. Springer Science & Business Media.

 

Interactive introductions

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 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 interactive introduction. 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 interactive introduction. But, when an interactive introduction 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 not to depend on any knowledge which the teacher hasn't yet taught. And the teacher plans the interactive introduction 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.

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 PowerPoint 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 interactive introduction.

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 Word 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 interactive introductions, 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!

So here are:

1. A first lesson on 'standard form'. (I discussed this one in my most recent podcast with Craig Barton.) (Word and pdf formats)

2. A first lesson on 'enlargement'. (Word and pdf formats and the associated PowerPoint file.)

3. A first lesson on 'circles and $\pi$'. (Word and pdf formats and the associated PowerPoint file.)

And, finally, as a bit of a further experiment, I've also had a go at making a video of me introducing the idea of complex numbers (Word and pdf formats of the sheets). 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!)

So, this is a shorter blogpost than usual, in order to give you time to look at the materials I've linked to.

Now over to you - comments, criticisms and improvements, please...

Questions to reflect on

1. What are your thoughts on the idea of 'interactive introductions'?

2. What comments do you have on any of these specific examples?

Note

1. You can listen to the episode here: http://www.mrbartonmaths.com/blog/research-in-action-16-writing-a-maths-curriculum-with-colin-foster/

Reference

Foster, C. (2018). Questions pupils ask: Is i irrational? Mathematics in School, 47(1), 31–33. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Is%20i%20irrational.pdf

The Differentiator

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 https://www.mathsthroughstories.org/). 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. 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...

In function-land, where all the elementary functions live, the polynomials were terrified. The Differentiator 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 two chances with The Differentiator before he would be reduced to nothing. But, it was really the constants who were most afraid - everyone knew that The Differentiator had had $\pi$ for lunch yesterday and not a crumb remained.

The Differentiator having $\pi$ for lunch

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.

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.

Some of the polynomials, such as $x^2$, had decided to take refuge by hiding behind $e^x$, with some success:

$$\frac{d}{dx} \left( e^{x}x^2 \right) =e^x(x^2+2x).$$

or by climbing up onto $e$ for protection:

$$\frac{d}{dx} \left( e^{x^2} \right)=2xe^{x^2}.$$

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:

$$\frac{d}{dx} \left(e^{2x} \right) = 2e^{2x}.$$

Unbelievably, $e^{2x}$ immediately grew to twice the size! The Differentiator hit again:

$$\frac{d}{dx} \left(2e^{2x} \right) = 4e^{2x}.$$

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.

***

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 The Integrator comes to the rescue?

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:

$$\frac{dy}{dx}=y$$

and, with it, the idea of a differentiation-proof function.

“I’m thinking of a function, and when I differentiate it I get exactly the same function back. What might my function be?”

Starting with this prompt, students may suggest the zero function, $y=0$, 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.

The general solution can be derived by separating the variables:

$$\frac{dy}{dx}=y$$

$$\int \frac{1}{y}\frac{dy}{dx}dx=\int 1dx$$

$$ \ln \lvert y \rvert=x+c$$

$$y=e^{x+c}=Ae^x,$$

where $c$ and $A$ are constants.

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 any positive integer.

But, in fact it is only a solution, not the 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 two 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 one differentiation.

It is natural for students to wonder about how this pattern might continue for the third-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 any 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 third-order differential equation, we should be expecting two more arbitrary constants, and how can we find them?

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

$$k^n e^{kx}=e^{kx}$$ 

Since $e^{kx}$ is never zero, we require that $k^n=1$, so the $k$s that we need are the $n$th roots of unity.

When $n = 1$, $k = 1$, and we have $y=Ae^x$, as we found.

When $n = 2$, $k = \pm 1$, and we have $y=Ae^x+Be^{-x}$, as we also found.

Now that $n = 3$, we still have $k = 1$, but we also have two complex roots, $k = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$, so our general solution should be

$$y=Ae^x+Be^{\left( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \right) x} + Ce^{\left( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \right) x},$$

which we can write as

$$y=Ae^x+B'e^{-\frac{x}{2}}\cos \left( \frac{\sqrt{3}x}{2} \right) + C'e^{-\frac{x}{2}}\sin \left( \frac{\sqrt{3}x}{2} \right),$$

or even, if we wish, as

$$y=Ae^x+B'\exp{\left( e^{ \frac{2 \pi i}{3} } x \right)}+C'\exp{\left( e^{- \frac{2 \pi i}{3} } x \right)}.$$

The $n=4$ case is much neater:

$$y=Ae^x+B''e^{-x}+C''e^{ix}+D''e^{-ix}$$

And we have explored the general terrain of 'differentiation-proof' functions!

Question to reflect on

1. What stories might you use in mathematics lessons to stimulate some worthwhile mathematical thinking?

Reference

Dietiker, L. (2015). Mathematical story: A metaphor for mathematics curriculum. Educational Studies in Mathematics, 90(3), 285-302. https://doi.org/10.1007/s10649-015-9627-x

Misremembering Goldbach’s Conjecture

It's the holiday, so a shorter, lighter blogpost today, and only one reflection question. I hope you are having a good break!

When I went on Craig Barton’s podcast 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… 

This is about the time when I misremembered Goldbach’s Conjecture, which states:

Every even integer greater than 2 is the sum of two primes.

Unfortunately, for some reason, I misremembered it as:

Every integer greater than 2 is the sum of two primes.

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...

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.

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!

The students seemed interested in this:

“Would we be famous if we found a counterexample?”
“Sure!”

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 greater than 3”, rather than 2, to avoid that problem.

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!

The students began work on their own or in pairs, writing (at least, those working more systematically!) things like this:

$4 = 2 + 2$
$5 = 2 + 3$
$6 = 3 + 3$
$7 = 2 + 5$
$8 = 3 + 5$
$9 = 2 + 7$
$10 = 5 + 5$
$11 = …$

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 mathematical etude, see Foster, 2018).

I gradually became aware that I could hear the number 11 being muttered quite a bit.

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 find a pair of primes that sum to 11, but that doesn’t mean that there isn’t one. 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.

I vividly remember the moment that one student came to the board and said, essentially:

Look, the only possibilities for 11 are:

1 and 10, but neither is prime 
2 and 9, but 9 isn’t prime 
3 and 8, but 8 isn’t prime 
4 and 7, but 4 isn’t prime 
5 and 6, but 6 isn’t prime 
So, 11 is a counterexample. 

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!

As I tried to figure out what was going on, the class became more excited at my puzzlement:
“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. 

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.” ‘Even, even, even!’ (Having computers connected to the internet in every classroom has to be one of the great pedagogical advances of recent decades.)

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.

There was understandably limited enthusiasm now for going back and checking for counterexamples to the real 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.

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 had begun the whole lesson with.

Question to reflect on

1. Do you have any 'armchair responses' (AssocTeachersMaths, 2020; Foster, 2019) to my ‘favourite failure’ or to any of your own?

Notes

1. You can listen to the episode here: http://www.mrbartonmaths.com/blog/colin-foster-mathematical-etudes-confidence-and-questioning/ 

2. See http://sweet.ua.pt/tos/goldbach.html

References

AssocTeachersMaths. (2020, July 13). Armchair Responses to Classroom Events - with Colin Foster [Video]. YouTube. https://youtu.be/L0ovhillL0c

Foster, C. (2016). Questions pupils ask: Why isn’t 1 a prime number? Mathematics in School, 45(3), 12–13. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20isn't%201%20a%20prime%20number.pdf

Foster, C. (2018). Developing mathematical fluency: Comparing exercises and rich tasks. Educational Studies in Mathematics, 97(2), 121–141. https://doi.org/10.1007/s10649-017-9788-x

Foster, C. (2019). Armchair responses. Mathematics in School, 48(3), 26–27. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Armchair%20responses.pdf

Making rounding interesting

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 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?

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 purpose 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:

• Which weighs more - a cat or 10,000 paperclips?
• Which mathematics teacher in our school do you think is closest to being a billion seconds old?
  [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.]

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.

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:

Estimate the answer to $0.278 \times 73.4-\sqrt{48.3}$.

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 any 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!

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: $0.3×70-\sqrt{49}=21-7=14$.

The issue of how good our estimate might be (and therefore what it might be good for) 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 bounds. 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 how 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)

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 useful 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.

Exploring rounding

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 answer 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:

Claim 1: If you round each number in a calculation to 1 significant figure, then the answer will also be correct to 1 significant figure.

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?

***

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 not the correct answer to 1 significant figure, because of course it should be 7.

A slightly more complicated scenario that might be worth exploring with students involves rounding two numbers in a subtraction, so we could begin with a question like this:

Estimate the answer to $14.2-12.9$.

(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.)

So, here's another claim:

Claim 2: If we round each number to the nearest integer, then the answer will be correct to the nearest integer.

Is this claim always, sometimes or never true?

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:

$[14.2]-[12.9] = 14-13 = 1$

And $[14.2-12.9] = [1.3] = 1$.

So that checks out in this case.

So, in this notation, the question is:

When is $[a-b] = [a]-[b]$?

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 mathematical etudes – see Foster, 2018). You might like to try it yourself before reading on.

***

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: 

Table 1. $[14.x-12.y]$ and $[14.x]-[12.y]$ compared, where $x$ and $y$ are single digits between 1 and 9.
Ticks indicate where equality holds.

So, a counterexample to Claim 2 would be any of the empty cells in this table; for example, $[14.7-12.3] = [2.4] = 2$ but $[14.7]-[12.3] = 15-12 = 3$.

We might prefer to write this on one line as: 

$$3 = 15-12 = [14.7]-[12.3] ≠ [14.7-2.3] = [2.4] = 2$$

Applying some deduction, we might say:

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.

We might also be tempted to say:

2. If one number is rounded up and the other number is rounded down, we should not get a tick.

As you can see from Table 1, this is false, as can be seen by the ticks in some 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 down to 1, so we do get a tick.

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.

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, we are currently working on designing a complete set of teaching materials for Year 7-9, 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.

Questions to reflect on

1. Are there mathematics topics that you personally find less interesting to teach? Which ones? Why?

2. What tasks can make these topics more interesting for you and for your learners?

3. For rounding, what other tasks can you devise? Is $[a+b] \stackrel{?}{=} [a]+[b]$ an easier or harder problem? What about $[ab] \stackrel{?}{=} [a][b]$?

Notes

1. Dan Meyer is the expert at designing tasks like this; e.g., https://blog.mrmeyer.com/2009/what-i-would-do-with-this-pocket-change/; https://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/; https://www.101qs.com/70-water-tank-filling 

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.

References

Eastaway, R. (2021). Maths on the back of an envelope: Clever ways to (roughly) calculate anything. HarperCollins.

Weinstein, L., & Adam, J. A. (2009). Guesstimation. Princeton University Press.

Weinstein, L. (2012). Guesstimation 2.0. Princeton University Press.

A football on the roof

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.

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 if the ball was there they would be able to see it.

"Can you see it?"
"No, but I just need to go back a bit further."
"If it was there, you'd be able to see it by now."
"I'm not sure."

In particular, going as far back from the building as possible, given the constraints of the surrounding buildings, did the fact that they couldn't see the football mean that it was definitely not there, or could it be that it was just not visible over the edge of the building (Figure 1)?

Figure 1. A football on the roof

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.

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:

• assume that the ground and all roofs are perfectly horizontal
• assume that the roof in question is free from any debris
• assume that the building has height 4 m and goes back 6 m
• assume that the football is perfectly spherical, with diameter 22 cm
• assume (worst-case scenario) that the ball is right at the back of the roof, against the brick wall

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.

Figure 2. Careful analysis (diagram not drawn to scale)

Let's start with a careful analysis, which uses trigonometry and is 'a sledgehammer to crack a nut' for this simple scenario. This is not the approach that I would envisage students taking.

Lots of the work has been done in the diagram (Figure 2), and our units are metre throughout.

We have

$$\tan \theta = \frac{r}{l}$$

$$\tan 2\theta = \frac{h}{d}$$

Using the identity

$$\tan 2\theta \equiv \frac{2 \tan \theta}{1-\tan^2 \theta}$$

we obtain

$$\frac{h}{d}= \frac{2 \left( \frac{r}{l} \right) }{1- \left( \frac{r}{l} \right)^2},$$

giving

$$d= \frac{h(l^2-r^2)}{2rl}.$$

We can now substitute in some reasonable values:

• $h=4-1.8=2.2$; the height of the building subtract the maximum eye height of the student when standing on tip toes or jumping,
• $l=6-0.22=5.78$; the depth of the shed subtract the diameter of the football, and
• $r=0.11$.

This gives $d=57.8$, so the student would just be able to see the top of the football from about 58 metre back from the shed.

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 simplified diagram shown in Figure 3.

Figure 3. Rougher analysis (diagram not drawn to scale)

For this rougher analysis, we don't need to use $\tan$ explicitly and can just equate corresponding ratios in similar triangles.

So,

$$\frac{d}{h}=\frac{l}{2r},$$

giving that

$$d=\frac{hl}{2r},$$

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 just be able to see the top of the football from about 58 metre back from the shed.

In the situation where $l  \gg r$, we can see that in our first equation

$$d= \frac{h(l^2-r^2)}{2rl}$$

the bracket $(l^2-r^2)$ will, to a good approximation, reduce to $l^2$, giving

$$d \approx \frac{hl^2}{2rl}=\frac{hl}{2r},$$

as before. So, all routes lead to an answer of about 58 metre.

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?

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.

So,

$$d=\frac{hl}{2r}=\frac{1.7 \times 5.78}{2 \times 0.11}=44.7,$$

and so this would not quite be enough to work within the available space, since $44.7 > 40$. Rearranging the equation to give

$$h=\frac{2rd}{l}$$

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.

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.

Questions to reflect on

1. Would your students find a task like this credibly realistic and engaging? How might you improve it?

2. What 'real-life' tasks do you use that are both non-trivial mathematically and non-embarrassing in terms of correspondence with reality?

Note

1. Disclaimer: Nothing in this post should be taken to endorse climbing onto roofs to retrieve footballs!

Lines of not-very-good fit

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.

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: https://www.bbc.co.uk/bitesize/guides/zrg4jxs/revision/9) 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.

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. 

Of course, they could draw something like this, which “goes roughly through the middle of all the scatter points” (10 points on either side).

But, unless they are trying to be awkward, they will probably be much more likely to draw something like this.

It 'goes through the middle' and is the kind of thing that the teacher is wanting (Note 2).

But, if you then display an accurate trend line, say using Excel (in black below), then it will be a bit off from what the students have drawn.

Here they are together, so you can see the difference:

It is easy to put this discrepancy down to human error. The computer draws the best possible 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 really base their line on where the overall mass of the points is located. So there is nothing to worry about.

But there is more than random error going on here. I claim that the students are not even trying 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 completely different 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 $x$.

The black line minimises the sum of the squares of the vertical distances of the points from the line, whereas the blue line minimises sum of the squares of the horizontal 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.

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 principal axis 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.

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.

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 $x$.

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 vertical distances from the points to the line. It can be interesting to devise scenarios where these two lines are very similar and very different.

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.

Maybe it is a relatively small point to worry about, but surely it would be a bit of a problem if students drawing something closer to the black line above were being penalised or criticised over those drawing something more like the red line.

Questions to reflect on

1. How concerned are you about this distinction between regression lines and principal axes?

2. What, realistically, might be done to address this in school-level mathematics?

3. Are there other examples in school mathematics where it is usual to teach things 'a bit wrong'?

Notes

1. The data used in this blogpost is available at: https://www.foster77.co.uk/Data%20for%20line%20of%20best%20fit%20blogpost.csv

2. Students sometimes have a strong tendency to avoid going directly through 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.

Reference

Gelman, A., & Nolan, D. (2017). Teaching statistics: A bag of tricks. Oxford University Press.

Motivation for measurement

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.

Topics in mathematics that involve accurate measurement can sometimes feel a bit unmathematical - 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 not 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 need 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)?

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) need to measure can be quite difficult, but optical illusions can be really helpful for this - and are fun in their own right.

There is a good list of many kinds of optical illusions at https://en.wikipedia.org/wiki/List_of_optical_illusions, and this includes things that are extremely weird, such as Ames room (https://en.wikipedia.org/wiki/Ames_room). 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.

The Ebbinghaus illusion (https://en.wikipedia.org/wiki/Ebbinghaus_illusion) is a nice one. The orange discs below are equal in size, but don't look it.

This is just crying out for some measurement with a ruler. Can the diameters really be equal? But the centres of the circles are not marked, so how could you be sure you were accurately measuring the diameter, and not some other chord?

The Delboeuf illusion (https://en.wikipedia.org/wiki/Delboeuf_illusion) is similar. The black discs are in fact equal, but the right-hand one looks larger:

The Moon illusion is a nice variation on this (https://en.wikipedia.org/wiki/Moon_illusion).

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.

Students could also attempt to create their own drawings, some of which are illusory (something looks bigger but isn't) and some not (something looks bigger and is), and see if other students can decide which are which by eye - followed by measuring to check.

The Müller-Lyer illusion (https://en.wikipedia.org/wiki/M%C3%BCller-Lyer_illusion) provides motivation for measuring the lengths of line segments. The two horizontal portions below are equal in length, but don't look it!

Similar opportunities are provided by the Ponzo illusion (https://en.wikipedia.org/wiki/Ponzo_illusion),

Tony Philips, National Aeronautics and Space Adm., Public domain, via Wikimedia Commons

the Sander illusion (https://en.wikipedia.org/wiki/Sander_illusion), where the two purple diagonals below are actually equal in length,

and the vertical-horizontal illusion (https://en.wikipedia.org/wiki/Vertical%E2%80%93horizontal_illusion),

where the vertical line segment looks longer, but isn't.

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).

The café wall illusion (https://en.wikipedia.org/wiki/Caf%C3%A9_wall_illusion) 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 sloping straight lines or curves? "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.

Fibonacci, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons

The Zöllner illusion (https://en.wikipedia.org/wiki/Z%C3%B6llner_illusion) provides another opportunity for students to check parallelness,

Fibonacci, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons

and the Hering illusion (https://en.wikipedia.org/wiki/Hering_illusion) is another example:

Fibonacci, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons

For working with polygons, the Ehrenstein illusion (https://en.wikipedia.org/wiki/Ehrenstein_illusion) 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 Orbison illusion - https://en.wikipedia.org/wiki/Orbison_illusion - provides similar opportunities.)

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 mathematical etudes (http://www.mathematicaletudes.com/) for measurement.

Questions to reflect on

1. Do you find these optical illusions engaging? Would your students?

2. How could you use these ideas to promote a need for measurement with one of your classes?

3. What other tasks make measurement a meaningful mathematical activity?

References

Andrews, P. (2002). Angle measurement: An opportunity for equity. Mathematics in School, 31(5), 16–18. https://nrich.maths.org/content/id/2855/AngleMeasurement.pdf

Bryant, J., & Sangwin, C. (2011). How round is your circle? Princeton University Press.

Foster, C. (2015). Exploiting unexpected situations in the mathematics classroom. International Journal of Science and Mathematics Education, 13(5), 1065–1088. https://doi.org/10.1007/s10763-014-9515-3

Meyer, D. (2015, June 17). If math is the aspirin, then how do you create the headache? [Blog post]. https://blog.mrmeyer.com/2015/if-math-is-the-aspirin-then-how-do-you-create-the-headache/