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