Mixing the dimensions in models of number

Not all commonly-used representations of number are necessarily equally helpful. We shouldn't just assume that anything 'visual' will automatically be helpful - some representations might even be counterproductive.

An emphasis on conceptual understanding in mathematics often seems to be linked with the use of visual representations. If you care about helping your students to make sense of mathematics – as opposed to just following standard, symbolic procedures – then you are likely to be keen on visual models. If a picture can paint a thousand words, then in mathematics a helpful visualisation might easily surpass multiple sentences of wordy explanation, whether written or oral, or many lines of mathematical symbols. It may also be that visualisation offers ways to support students who are more likely to be disadvantaged by traditional approaches to learning mathematics (Gates, 2018).

I very much like diagrammatic ways of showing mathematical properties and relationships (e.g., see Mattock, 2019, for a beautiful collection of examples), and some of my favourite books are the volumes of Proofs without Words (Nelson, 1993, 2000, 2016). However, I think a positive view of diagrams can sometimes lead us to be a little uncritical about cases where diagrams may be problematic. We should not just assume that anything ‘visual’ must be a good thing. In particular, if certain representations have the potential to create or embed particular misconceptions, so making understanding harder to achieve, then this is something we should worry about. The fact that students may say that they like certain visual representations is not enough, as it may not be possible for them to be aware at the time of possible problems coming down the line as a result of the representations that they are using. It relies on the teacher to look ahead and consider how future problems might be being set up by what is currently taking place.

I am a big fan of 1-dimensional, ‘linear’ models, such as number lines, and I include in this category any representation that has just one variable or dimension, even if it isn’t drawn in a straight line (see Foster, 2022). So, for me, a circular number line, like on a speedometer or clock, is still a kind of 1-dimensional number line, as is a spiral number line. So is a number track, such as the snaking squares on a snakes-and-ladders board, because all of these are still unidimensional representations (you can only go either forwards or backwards – ignoring the snakes and ladders themselves!). I see all of these as linear, even though they of course have to take up 2-dimensional space, otherwise we wouldn’t be able to see them (Note 1).

However, I think that I am coming round to the view that I am not a fan of 2-dimensional representations of number, because they are inevitably mixed-dimensional, and I think this is quite problematic (Foster, 2022). To explain what I mean, consider ‘algebra tiles’, as embodied in diagrams (or physical or virtual manipulatives) like those shown in Figure 1 (Note 2).

Figure 1. Algebra tiles representing (a) $3(x+2)\equiv3x+6$  and (b) $(3x+6)(x+1)\equiv3x^2+9x+6$

In Figure 1a, the area 6 is represented by a blue 2 × 3 rectangle, and, in Figure 1b, we have another area of 6, this time represented by a blue 1 × 6 rectangle. This is fine, because it seems perfectly reasonable to say that both of these represent the same number 6 (Figure 2a), because they have the same area as each other. We could break up either one of them and fit it completely into the space occupied by the other. However, my difficulty is that in Figure 1b we also have the number 6 represented by the (1-dimensional) purple line segment at the top right (see Figure 2b), meaning that the same number is represented, in the same diagram, by both a 1-dimensional line segment and a 2-dimensional area.

Figure 2. (a) a reasonable equality; (b) an unreasonable equality

I think this mixed-dimensional feature of area models of number is unfortunate, and becomes quite problematic the more you think about it. A rectangle and a line segment are not just different things (like two different rectangles) – they are different kinds of things. It’s OK if we sometimes represent a number by a 1-dimensional length, and other times by a 2-dimensional area – and we maybe sometimes represent it by all sorts of other things – but to do both of these simultaneously in the same diagram seems like asking for trouble. I am not sure how much relational understanding (Skemp, 1976) can be said to be going on if students have not noticed or thought about this. And this is not something that occasionally happens with representations like these, in certain awkward cases – it happens every time.

I am perfectly happy that there is no dimensional problem with writing an expression like $3x^2+9x+6$, because $x$ is a pure number, without any units, so this is just of the form 'number + number + number', which is equal (unsurprisingly) to 'a number'. But, as soon as you start to represent an expression like this using algebra tiles, it begins to look very much as though you have 'area + area + length', or maybe 'area + length + length'. In Figure 1a, $3x+6$ is 'area + area', but when, in Figure 1b, we want to multiply this expression further by $x+1$, we somehow have to shift our thinking down a dimension, and reconceive $3x+6$ as 'length + length', so that we can multiply it by another ‘length’ (the $x+1$), so as to obtain a quadratic expression, which is now represented as an area. Alternatively, we could retain $3x+6$ as area + area, and multiply it by length + length to give a volume in 3 dimensions, and, although this is tricky to sketch on paper, 3D models can be insightful. But then you are stuck if you want to go any further.

I think this problem is often overlooked, because algebra tiles are often used to multiply two linear expressions, like $x+b$ and $x+d$. We are very pleased with the fact that the leading term in the expansion is $x^2$, and that ‘$x$ squared’ is represented by ‘a square’. This seems great – we visualise an algebraic square by means of a geometrical square – what could be nicer? But, we overlook that the other terms in the expansion have less convenient interpretations – we have a $bx$ and a $dx$ rectangle, and a $bd$ rectangle, and it’s unclear why it’s appropriate for all three of these to be rectangles, given that things very like them were lengths at the beginning.

Even if you restrict ‘algebra’ tiles to numbers, and use them to work out things like $12\times46=(10+2)\times(40+6)=552$, I think you have exactly the same problem. The 2, say, in the original 12 looks visually like a completely different kind of a thing from the 2 in the 552, but they are both meant to be 2 ones. One of the most basic requirements of a good representation is that the same thing (e.g., 2) should be represented by the same thing (e.g., either a line segment of length 2 or a rectangle of area 2, but not both).

There is something nice about the dynamic in a classroom where students are fiddling around with physical algebra tiles, rearranging them and fitting them into rectangles and talking about what they are doing. It seems like just the sort of activity that should develop conceptual understanding. Students are actively manipulating representations that make the mathematical structure visible and figuring out what is possible and what is impossible. But, whether or not the jigsaws get completed, I worry that mixed-dimensional models like this have the potential to be more confusing than helpful.

I have expanded on the argument of this blogpost in Foster (2022).

Questions to reflect on

1. Do you use mixed-dimensional representations of number, like algebra tiles? If so, when and why?

2. What do you think about the concerns I've expressed in this post?

Notes

1. Manipulatives, like Cuisenaire rods or cubes, are harder to classify, because you can do lots of things with them, and not all of these things are 'linear'. For example, you can use them to make rectangles with an area of 12 square units. So I think whether they are ‘linear’ or not depends on what you do with them.

2. To generate diagrams like this conveniently, go to https://mathsbot.com/manipulatives/tiles.

References

Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. For the Learning of Mathematics, 42(3), 21–27. https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf

Gates, P. (2018). The importance of diagrams, graphics and other visual representations in STEM teaching. In R. Jorgensen, & K. Larkin (Eds), STEM education in the Junior Secondary: The state of play (pp. 169-196). Springer. https://www.researchgate.net/profile/Peter-Gates-4/publication/319086868_The_Importance_of_Diagrams_Graphics_and_Other_Visual_Representations_in_STEM_Teaching/links/5d094203299bf1f539cef6d3/The-Importance-of-Diagrams-Graphics-and-Other-Visual-Representations-in-STEM-Teaching.pdf

Mattock, P. (2019). Visible Maths: Using representations and structure to enhance mathematics teaching in schools. Crown House Publishing Ltd.

Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. The Mathematical Association of America.

Nelson, R. B. (2000). Proofs without words II: More exercises in visual thinking. Washington. The Mathematical Association of America.

Nelson, R. B. (2016). Proofs without words III: Further exercises in visual thinking. The Mathematical Association of America.

Skemp, R. (1976). Instrumental understanding and relational understanding. Mathematics Teaching, 77, 20-26. http://math.coe.uga.edu/olive/EMAT3500f08/instrumental-relational.pdf

Dividing into thirds

How accurately do things need to be drawn to evidence conceptual understanding? When are accurate drawings helpful and when are they unnecessary?

Suppose that you asked a child to divide a disc into thirds, and suppose they drew something like this:

How would you respond? Are they right? It is only a sketch, after all.

Now imagine an equally scruffy sketch like this:

Full marks this time? I can imagine the second one being treated as more acceptable than the first, even if both were drawn equally accurately.

In fact, here, the first one of these is drawn more accurately into thirds than the second one is. But, might that just be 'luck', and not evidencing a clear understanding of what 'thirds' are? What are we trying to judge here? A small child may have a better eye than me, and be better at estimating equal areas accurately than I am. Is that the thing that matters?

Perhaps the teacher is expecting/hoping to see sectors drawn, and they view parallel lines suspiciously, as they worry that the child might be intending to indicate evenly-spaced lines, as shown below, which divide the vertical diameter into thirds, but would not divide the area of the disc into thirds. Intending to draw this might be counted as a 'misconception'.

But, is it fair to the first child to assume that this is necessarily what they must be thinking? Without talking to them about their idea, it seems hasty to dismiss what they have drawn. But, they might struggle to express in words that the two lines are somewhat closer to each other than they are to the ends of the vertical diameter - and it would certainly be hard to say how much closer they ought to be (Note 1).

Dividing a circle into thirds precisely with parallel lines is tricky, and requires some calculation.

Suppose we have a unit circle below, centre $O$.

We want to know the angle $\theta$ radian which makes the area of the blue shaded segment exactly $\frac{1}{3}$ of the area of the whole disc.

Now, 

$$\text{area of blue segment}=\text{area of sector } – \text{ area of isosceles triangle}.$$

So,

$$\frac{\pi}{3}=\frac{1}{2}\times1^2\theta-\frac{1}{2}\times1^2\sin\theta,$$

giving

$$\frac{2\pi}{3}=\theta-\sin\theta.$$

This equation cannot be solved analytically, but we can get a numerical solution as accurately as we wish, and this turns out to be $\theta=2.605,325,...$ radian. Converting to degrees, this is $149.27^{\circ}$, correct to 2 decimal places. It is kind of neat that the required triangle is so close to a $15$-$15$-$150$ isosceles triangle.

Drawn accurately, it looks like this:

which might be a bit hard to distinguish from the equally-spaced incorrect version drawn above.

Here they are side by side:

With a bit of thought, it is clear that the vertical positions of these lines are just as good for any ellipse with a vertical major axis of this length, as others have noted (see https://www.forbes.com/sites/kevinknudson/2015/08/11/cutting-into-thirds-or-how-a-mathematician-spends-an-hour-figuring-out-how-to-divide-a-casserole/):

This leaves me thinking that no one is going to be good at drawing these lines in the right position by eye. And I am not sure to what extent learning about fractions ought to be dependent on ability to estimate the relative sizes of areas bounded by arcs, like these. I don't think I'm particularly good at it (Note 2). But how should we get a sense of children's understandings of fractions when we look at their drawings and explanations? And should we be more cautious in our assumptions when we look at their drawings?

It isn't always clear what we intend to communicate when we make sketch drawings like these, and when the accurate details matter and when they don't. It certainly isn't easy to draw perfect circles freehand on a whiteboard (see Foster, 2015), and, even if you are using technology to display perfect circles, projectors can do funny, distorting things. I have seen a lesson on circle theorems in which none of the circles displayed was remotely circular. They looked fine on the teacher's computer screen (as they presumably did at home the night before, when planning the lesson), but, once projected, they were distinctly oval. How much does this matter? No circle in the real world can ever be completely perfect, so we always have to use our imagination. Every student in the lesson knew that the lesson was about 'circle theorems', not 'oval theorems', and that these images were representing circles. So was this therefore fine? Where the diagram is deficient, the viewer has to do some of the work to visualise it properly, and sometimes this can be helpful. But, in this case, given that it was an introductory lesson to circle theorems, I felt that the distorted figures were merely an extra burden on everyone's cognitive load. They seemed more likely to get in the way of the students' attempts to make sense of the relevant geometrical properties and relationships.

People find it much harder to judge the relative area of sectors than of rectangles (Burch & Weiskopf, 2014) - look at the same data below presented as a pie chart and then as a bar chart:

Perhaps it is time we moved away from using circle diagrams for teaching fractions altogether (see Foster, 2022)?

Questions to reflect on

1. How would you convince someone that the calculated position of the lines for a circle works just as well for any ellipse?

2. What do you see as the role of circle drawings like this for learning about fractions? Are other visuals (e.g. rectangles) preferable?

3. When is accuracy important and when isn't it?

Notes

1. I suppose you could also say that the division into sectors divides the circle (i.e., the circumference) into thirds, as well as divides the disc (i.e., the area) into thirds, whereas the parallel lines divide only the area into thirds.

2. Similar criticisms might be made about the value of training students to estimate angles in degrees with better accuracy. Tasks like this https://nrich.maths.org/1235 can be fun, but how important is this as a mathematical skill that is worth improving? I think that such tasks have value in gaining a sense of what 'one degree, ten degrees, etc.' look like, which seems as important as knowing what '1 centimetre' looks like.

References

Burch, M., & Weiskopf, D. (2014). On the benefits and drawbacks of radial diagrams. In E. Huang (Ed.), Handbook of human centric visualization (pp. 429-451). Springer.

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

Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. For the Learning of Mathematics, 42(3), 21–27. https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf

Fractions as factors

Can a factor be a fraction?

People sometimes agonise over whether a fraction such as $\frac{2}{3}$ can be called ‘a factor’ of another number, such as 6 (Foster, 2022). If factors are defined as “numbers that divide exactly into another number” (BBC Bitesize, n.d.), then, since $6÷\frac{2}{3}=9$, and $9$ is an integer, shouldn’t $\frac{2}{3}$ be regarded as a factor of $6$?

Perhaps we decide that we want to be able to say that the number $6$ has exactly $4$ factors ($1, 2, 3$ and $6$), and so we don’t allow numbers like $–2$ or $\frac{2}{3}$ to be factors. If so, we could use a tighter definition of factor, such as: A factor is a positive integer that divides into another number a positive integer number of times (or, equivalently, we could say ‘without any remainder’). However, in many instances we might want to think of factors more broadly than this. For example, when factorising $x^2±5x+6$, it might be helpful to think of the $6$ as having four possible factor pairs ($\{1, 6\}, \{2, 3\}, \{–1, –6\}$ and $\{–2, –3\}$). Similarly, when using the factor theorem, we typically treat negative numbers as ‘factors’. In other contexts, it might seem natural to regard a number like $\frac{2}{3}$ as being a factor of $\frac{4}{3}$, since it ‘goes into it’ twice, or even $x$ as a factor of $x^2$, since it goes into it $x$ times, regardless of whether or not $x$ might be an integer. We might even pull out an irrational number, such as $\pi$, from an expression like $2\pi r-\pi l$ to give $\pi(2r-l)$, and call this 'factorising', and refer to $\pi$ here as a ‘common factor’, although it is certainly not an integer, and is not even rational (Foster, Francome, Hewitt, & Shore, 2022).

‘Factor’ seems to be one of those words that is used differently in different contexts, even within school mathematics, and I think it isn’t really possible to settle on a fixed definition which will always apply (see Foster, Francome, Hewitt, & Shore, 2022, for a similar discussion about the word ‘fraction’). Perhaps the best approach to awkward issues like these is to acknowledge them and explore them. Turn the issue into a task: What would happen if we allowed fractions to be factors? Perhaps we call them ‘fraction factors’.

Exploring 'fraction factors'

Students often think of fractions as ‘numbers less than 1’, and they may initially think that any fraction would be a fraction factor of any integer, but of course this isn’t right. Although $\frac{2}{3}$ would be a fraction factor of $6$, it wouldn’t be a fraction factor of $5$, since $5÷\frac{2}{3}=\frac{15}{2}$, or $7.5$, which is not an integer. All unit fractions ($1/n$, where $n$ is an integer $\neq 0$) would be fraction factors of every integer, since they are by definition fraction factors of $1$, and $1$ is a factor of every integer. But when would a non-unit fraction be a fraction factor of an integer? Could we ask for all the fraction factors of $6$? Clearly not, because this list would include all of the unit fractions, and there are infinitely many of them.

There are many opportunities here for students to form conjectures and to find counterexamples – and, in each case, to try to find the simplest counterexample they can. It can also be helpful to look at the question the other way round, and ask what integers a fraction like $\frac{5}{12}$, say, would be a fraction factor of.

The conclusion is quite simple, but perhaps not that easy for students to arrive at without quite a bit of useful exploration. A fraction $\frac{p}{q}$, with $p,q \neq 0$, in its lowest terms, will divide an integer $m$ if and only if $m÷\frac{p}{q}$ is an integer. This is equivalent to saying that $\frac{mq}{p}$ must be an integer. Since $p$ and $q$ are co-prime, $\frac{mq}{p}$ will be an integer if and only if $p$ is a factor of $m$. So, the fraction factors of $6$ are fractions that, when simplified, have the (positive integer) factors of $6$ as their numerators:

$$\frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \frac{6}{n}$$ for all integer $n \neq 0$.

Perhaps this seems obvious, but I think it can be quite unintuitive that, say, $\frac{2}{17}$ is a fraction factor of $6$, but $\frac{4}{3}$ isn’t.

A less algebraic – and perhaps clearer – way to appreciate why the numerator matters, but the denominator doesn’t, is to realise that $\frac{1}{q}$, as a unit fraction, will always be a fraction factor of any integer, regardless of what $q$ is, because $q$ of them will always fit into every $1$. For the same reason, $\frac{p}{q}$ will necessarily be a fraction factor of $p$, because $q$ of them will fit exactly into the integer $p$. So, $\frac{p}{q}$ will be a fraction factor of any integer of which $p$ is a factor. 

Things to reflect on

1. Do you agree that it isn't possible to have a single definition of 'factor' that applies across all of school mathematics? Why / why not?

2. If you agree, which other technical mathematical terms do you think may be problematic in this kind of fashion? (See Foster et al., (2022) for a discussion of 'fraction' in this regard.)

References

BBC Bitesize (n.d.). What are factors? https://www.bbc.co.uk/bitesize/topics/zfq7hyc/articles/zp6wfcw

Foster, C. (2022, October 13). How open should a question be? [Blog post]. https://blog.foster77.co.uk/2022/10/how-open-should-question-be.html

Foster, C., Francome, T., Hewitt, D., & Shore, C. (2022). What is a fraction? Mathematics in School, 51(5), 25–27. https://www.foster77.co.uk/Foster%20et%20al.,%20Mathematics%20in%20School,%20What%20is%20a%20fraction.pdf

Is area more difficult than volume?

I have a tendency to assume that concepts get more difficult as the number of dimensions increases. Length is pretty straightforward, surely (how long is a piece of string?). Area is a bit harder, because we are in 2 dimensions now, and volume is even harder, because that's 3 dimensions. However, I'm not sure that this is right or that it makes much sense to think of building up in this way. After all, a point is zero-dimensional, and that is certainly not a simple thing to get your head around at all!

I think it is much easier to get an intuitive sense of volume than it is of area. From a very young age, children build with blocks and pour sand and water into containers, so they are engaging with 3-dimensional concepts such as volume right from babyhood. We are 3-dimensional beings and live in a 3-dimensional world, so we really ought to feel at home working with a concept like volume. By contrast, I think that area and length may be more inherently difficult concepts conceptually, as we never see these things in their true reality. A 1-dimensional horizontal line segment is represented on paper by a very thin rectangle, because it has to have a little bit of vertical height, otherwise it would be invisible. Indeed, this 'rectangle' is really a very shallow cuboid of ink, sitting on top of the sheet of paper, so it's an approximate cuboid rather than a 'line'. Everything we see that is intended as an approximation to or representation of something 1-d or 2-d is really forced to be actually 3-d just because that's the kind of world we live in.

I was thinking about this recently when considering how to design some lessons on area. What does it mean for two shapes to have the same area? By contrast, equal volumes is fairly easy to grasp. Two hollow shapes have the same capacity (or volume of space/air inside them) if you could fill them up using the same quantity of liquid (Note 1). If they are solid objects, then they have the same volume as each other if they displace the same quantity of a liquid that they are submerged within. For children who have played with floating and sinking objects in the bath, this is familiar territory. You can easily be sure whether two objects have the same volume or whether one has greater volume than the other, and which way round it is. It might not be easy to estimate volumes at a glance in practice - I am always surprised that an ordinary drinks can contains 330 cm$^3$ (it never looks big enough to contain that many centimetre cubes) - but it's clear what 'greater volume' means and how we could, in principle, find out (Note 2).

This is all much harder with area. Two shapes have the same area if you can cut up one of them and fit the pieces exactly inside the other one, with no gaps and no overlaps. This is not easy at all. What if the shapes have awkward edges? Can a disc of radius $1$ be 'cut up and fitted exactly inside' a square with side length $\sqrt{\pi}$? You would have to make infinitely many cuts to do it: can you be sure that there would definitely be no tiny gaps or overlaps? To a young child, there is nothing exceptional about a circle - they see circles all the time - so this should surely not be a 'hard' example to think about. Similarly, we expect children to accept things like the fact that shearing a rectangle parallel to one of its sides doesn't change the area. Maybe we push over a stack of paper to illustrate this. But is it really so clear that the area is definitely not changing?

I was thinking about lessons I've seen in which the teacher aimed for more intuitive understandings of area. It is normal to choose a small square 'unit' and try to cover a given shape with a finite number of these units. So children might place small plastic squares over a drawing of a shape to see how many are needed to cover it completely without any gaps. Sometimes the answer is an integer (e.g., for a carefully-chosen rectangle), or it might be a non-integer, but still exact, if half-squares, say, can be carefully totalled. Sometimes, it may involve estimation (e.g., tracing a hand or foot onto squared paper and approximating the total surface area). Occasionally units other than a square are used, and so long as the units tessellate we can often still tile many shapes with them. But I think all of this is much harder than pouring liquids into and out of various vessels.

What about more informal approaches to area? I saw a lesson in which area was informally understood as 'the amount of ink you need to colour in the shape', whereas perimeter was 'the amount of ink you need to draw around the edge of the shape'. Students had felt-tip pens and were illustrating area and perimeter of some shapes by colouring them in (Note 3). Of course, there is a lot of imprecision here, but perhaps the act of colouring in helps to reinforce the nature of the concept we are focused on. Time can be brought in as a proxy for area, so the shape with larger area is the one that takes longer to colour in (colouring races to determine which shape has greater area). There are some assumptions here about things like the same pen being used, students with equal 'colouring speed', and so on. And, of course, features of the shape that might mean that some shapes of area 10 cm$^2$, say, are quicker to colour than others - compare these two, for example:

Other possibilities for building helpful intuitions include things like cutting out shapes in thick card (with uniform thickness) and weighing them against each other to determine which has larger area. Or cutting them out of pastry and baking them and seeing which weighs more or takes longer to eat. All of this kind of work is a long way from the more common emphasis on quickly getting to the calculations. If you ask a child, "What is area?" they may be quite likely to say, "Base times height". Admittedly, "What is area?" is a hard question for anyone to answer. But stating a formula for calculating the area of one specific type of shape (i.e., a parallelogram) is not the same as having a good fundamental sense of what area is all about.

Questions to reflect on

1. Do you agree that area is more difficult than volume? If not, why not?

2. How important do you think it is to develop an intuitive understanding of concepts like area?

3. Would you use any of the approaches mentioned here? Why / why not?

Notes

1. We'll assume throughout that the shapes are made of something like plastic, which doesn't absorb the liquid or get deformed by it! There are all kinds of other physical assumptions in play, such as the assumed incompressibility of liquids, and so on.

2. For reflections on some of the ambiguities around volume and surface area, see Foster (2011).

3. Colouring in is not always the most educational of activities in school, but in this case it seemed that it might be.

Reference

Foster, C. (2011). Productive ambiguity in the learning of mathematics. For the Learning of Mathematics, 31(2), 3–7. https://www.foster77.co.uk/Foster,%20For%20The%20Learning%20of%20Mathematics,%20Productive%20Ambiguity%20in%20the%20Learning%20of%20Mathematics.pdf

Butterfly effects when adapting tasks

Sometimes a superficially small tweak to a task - changing just one little thing - can dramatically alter it, and mean that a lot more thinking - or very different thinking - is needed. In many cases, students don't need to be taught any additional facts or methods beyond what they have already learned - they know everything necessary to be able to figure out what difference the change makes.

The details matter in mathematics task design and adaptation. Here are just a few examples of small ways in which tasks can be adapted to make them more challenging and so that they may provoke deeper thought. For more on ways to adapt mathematics tasks, see Prestage and Perks (2013). The tasks below all take the form "You know this, but what about this?"

1. Change the base

It's sometimes said that You don't understand long division until you can do it in any base.

You know how to work out $252 \div 6$ in base $10$, but can you do it in base $16$?

In which other bases does $252 \div 6$ give an integer answer? Why?

Doing arithmetic in different bases used to be much more common in UK mathematics teaching, and it is still quite prevalent in schools in many countries. I find that people often assume that it's much more complicated than it is. I once heard a staffroom conversation that went something like this:

A: I think we're just too obsessed with base 10 and we don't take opportunities to work in other bases.
B: Yes, I guess it's just a convenience thing - it's so handy, for example, to be able to multiply by 10 by just placing a zero on the end.
A: Do you know how to multiply by 7 in base 7?
B: Er, no, not off the top of my head, but I should think it's very complicated. That's my point about why we always resort to base 10.

Exploring factors of numbers in different bases can be a good way to see why person B is wrong about this!

You know the factors of $12$ in base $10$, but what are the factors of $12$ in base $16$?

In which bases is $12$ prime? Why?

By changing something, away from the familiarity of base 10, we can 'make the familiar strange' and see things in new ways. It is such an apparently small change, but there are many tasks in which a change of base can be insightful (see Foster, 2007). For example,

You know that, in base 10, $\frac{1}{2}$ is a terminating decimal and $\frac{1}{3}$ is a recurring decimal. Which fractions terminate and recur in other bases? Is there a base in which $\frac{1}{2}$ is a recurring decimal and $\frac{1}{3}$ is a terminating decimal? Or in which they are both recurring or both terminating? What is $\frac{1}{2}$ in base 16? What about in base 15? What about in other bases? 

2. Introduce modulus signs

The definition of the modulus function is very simple. You could use the ideas below when teaching students about the modulus function for the first time. But you could also use them with any students - even quite young ones - who don’t ‘need’ to learn about the modulus function. You would be using the modulus function as a way to get them to think more deeply about things they do 'need' to think about.

The modulus function $\lvert x \rvert$ is defined as:

$$\lvert x \rvert=\left\{ \begin{array}{@{}ll@{}} x, & \text{if}\ x \ge 0 \\ -x, & \text{if}\ x \lt 0 \\ \end{array}\right.$$

But the idea is much simpler and more accessible than this formal notation might suggest. The value of $\lvert x \rvert$ is the absolute 'size' of $x$, regardless of its sign. So $\lvert 18 \rvert = 18$, but $\lvert -18 \rvert = 18$ also. This means that $\lvert x \rvert$ is never negative (Note 1).

That is all the formal teaching that is needed. The task is then is to try some things that students can already do, but with a modulus function thrown into the mix - a superficially small change, but with a 'butterfly' effect.

You know what the graph of $y=\sin x$ looks like. What about $y=\lvert \sin x\rvert $? Or $y=\sin \lvert x \rvert$?

You know what the graph of $y=x+1$ looks like. What about $y=\lvert x \rvert+1$ or $y=\lvert x+1 \rvert$?

You know what the graph of $y=(x+1)^2$ looks like. What about $y=\lvert x+1 \rvert^2$? Or $y=( \lvert x \rvert + 1)^2$?

You know how to solve $3x-2=10$. What about $\lvert 3x-2 \rvert =10$?

You know how to solve $3x-2=2x+1$. What about $\lvert 3x-2 \rvert = \lvert 2x-1 \rvert$?

You know how to solve $3x-2<2x+1$. What about $\lvert 3x-2 \rvert < \lvert 2x-1 \rvert$?

You know how to find the integral $\int (x^2-1) dx$. What about $\int \lvert x^2-1 \rvert dx$?

Introducing modulus signs is a little like introducing $\pm$ signs (see Foster, 2012) in that you are making a superficially small change that has the effect of requiring students to think quite hard.

Finally, for modulus functions, take a look at this graph:

What combination of modulus functions could produce this graph? (See Note 2 for the answer.)

3. Go non-linear

Older students may know, or have a sense of, what happens if you change a function $f(x)$ into, say, $f(-x)$, or $f(mx)$, where $m$ is a constant. But what happens to well known graphs of $y=f(x)$ if we change:

• $x \mapsto \frac{1}{x}$; e.g., $\sin {(\frac{1}{x})}$, $e^{\frac{1}{x}}$?
• $x \mapsto x^2$; e.g., $\sin {x^2}$, $e^{x^2}$?

This is something that older students may find interesting to explore. It is worth delaying going to graph-drawing software until after you have thought hard about it, and have definite conjectures to check out. Turning to graph-drawing software too soon risks circumventing the thinking, rather than supporting it.

4. Look for in-between values and concepts

You probably know what the graph of $y=x^n$ looks like for positive integer $n$ and for $n=-1$. But what about for other negative integer $n$? And what about for non-integer $n$, both positive and negative (Note 2)?

You probably know about 'highest common factors' (or 'greatest common divisors'). What about 'second-highest common factors'? When do/don't they exist? What can you work out about them?

You probably know about 'common multiples' as the numbers where sequences like $3n$ and $8n$ intersect. But where do sequences like $3n+1$ and $8n-2$ intersect?

You probably know how to construct a perpendicular bisector? How would you construct a perpendicular trisector? 

Conclusion

Variation theory currently receives a lot of attention, and can be powerful for helping students attend to what is the same and what is different in a sequence of tasks. In this spirit, little changes beyond what students need to know, and into unknown terrain, can be interesting extension tasks and valuable tasks for everyone to deepen their knowledge and understanding.

Questions to reflect on

1. What other butterfly examples do you have, where 'changing one little thing' has a big effect?

2. When would you use tasks like this?

Notes

1. Of course, the notation $\lvert x \rvert$ is also used where $x$ (perhaps written as $\textbf{x}$) is a vector, or where $x$ is a complex number, or to indicate the determinant of a matrix (in which case $\lvert x \rvert$ could be negative). But in this post I am only thinking about $x$ as a real number ($x \in \mathbb{R}$).

2. The graph is $y= \lvert x-2 \rvert + \lvert x+2 \rvert - \lvert x-1 \rvert$. Students could invent tasks like this for each other.

3. If you use software to draw $y = x^n$, and set up a slider to vary the value of $n$, you may notice a 'flickering' behaviour for negative values of $x$. It is interesting to think about why this happens. If $n$ is rational and equal to $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and $\frac{p}{q}$ is in its lowest terms, then the domain of $x$ is:

• all of the real numbers, if $q$ is odd, but
• all of the non-negative real numbers, if $q$ is even. 

This means that, when $n$ is real, but not rational, the domain is all the non-negative real numbers. So, the graph flickers as you drag the slider because, as $n$ varies, the function keeps switching between being defined and undefined (see also Dobbs, 2017).

References

Dobbs, D. E. (2017). Why the $n$th-root function is not a rational function. International Journal of Mathematical Education in Science and Technology, 48(7), 1120-1132. https://doi.org/10.1080/0020739X.2017.1319980 ($\$$)

Foster, C. (2007). Twenty–one forever! Journal of Recreational Mathematics, 36(3), 194–195. https://www.foster77.co.uk/Foster,%20Journal%20of%20Recreational%20Mathematics,%20Twenty-One%20Forever.pdf

Foster, C. (2012). Plus–minus graphs. Mathematics in School, 41(2), 32–33. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Plus-Minus%20Graphs.pdf

Prestage, S., & Perks, P. (2013). Adapting and extending secondary mathematics activities: New tasks for old. David Fulton Publishers.

How open should a question be?

People often say or imply that when teaching mathematics 'open questions' are simply better than 'closed questions'. Instead of asking students a closed question like “What is $3 \times 4$?”, which has only one right answer, it is supposed to be better to ask them to “Tell me some products that make 12”, because this more open question has multiple possible correct answers. The more possible correct answers there are, the more ways there are for students to be right, and that feels more positive and inclusive. Open questions promote students’ creativity and individuality. But is it as simple as 'the more open the better'? Should we be sorry whenever we use a closed question?

Any question can always be made more open, but is that always an improvement? In the “Tell me some products that make 12” question above, why specify ‘products’? Instead, we could say “Tell me some calculations that make 12”, and that would be even more open. Or we could go even further and say “The answer is 12, what could the question be?” (see Ball & Ball, 2011, p. 20, for the '42' version of this). Perhaps the ultimate open question would be something like “Make up a question and an answer of your own choosing.” Is that the best possible question of all?

It seems to me that these questions can’t simply be getting better and better the more open we make them. Whether these questions are good or bad must surely depend on the teacher’s purposes in asking them. Let’s think about these questions in turn and how they might be more or less useful for different purposes.

1. Tell me some products that make 12.

This could be a nice way to start thinking about factor pairs. One student says “$3 \times 4$”; another says “$6 \times 2$”. Perhaps, after a slightly longer pause, another student says “$1 \times 12$”. Now what do you do? If you continue to wait, then, after perhaps an even longer pause, you may be offered things like “$4 \times 3$” or “$24 \times \frac{1}{2}$” or even “$(-6) \times (-2)$” or perhaps (possibly?) “$n \times \frac{12}{n}$”. Is any of that helpful? I think it depends on your didactical purposes. Answers like these could lead to drawing rectangles of equal area (12) but different semi-perimeters, and explorations into equable rectangles (see Foster, 2021), and perhaps even aligning all these rectangles on a grid, with a common vertex at (0, 0), to create points at the diagonally-opposite vertex that lie on the curve $y=\frac{12}{x}$.

But, if you are defining factors as positive integers (so 3 is a factor of 12, but $\frac{2}{3}$ isn't), then most of these pairs are going to end up being non-examples of that, and we have drifted from the original purpose of thinking about factor pairs. Non-examples can be really important for nailing down what something is, but if you want to think about factors then we need to be working on the positive integers. So, although opening things up can feel accepting and positive, it can also be confusing for students and distracting from your intentions. If you want to focus on factors, maybe there is nothing wrong with being upfront about this, and saying what you mean by ‘factor’. Then you can ask, “Tell me a factor of 12”, and then ask for another one, and another one, and you still have multiple possible right answers - but this time there is a finite number of them. This means that, additionally, you can work on trying to justify how we can be sure that we’ve got them all. Being less inclusive about possible allowed numbers shuts down some possibilities (we can no longer have $-6$ or $\frac{1}{2}$) but opens up others: Now we can ask "Could there be any more? How can you be sure?" So, 'less open' doesn't necessarily mean fewer possibilities - it's just different.

2. Tell me some calculations that make 12.

With this question, asking for ‘calculations’ enormously broadens out the possibilities that students could offer. Is that a good thing or a bad thing? If your learners are lower primary, they might be highly likely to offer additions and subtractions, and this could be a useful way in to number bonds to 12. Even with secondary-age students, simple responses like $10 + 2$ could lead you to explore the partitions of 12; i.e., How many ways are there of making 12 from summing positive integers? There are lots of interesting things to explore here. The total of 12 can be made by summing pairs of positive integers in 6 ways ($1+11$, $2+10$, $3+9$, $4+8$, $5+7$ and $6+6$) and by summing three positive integers at a time in 12 ways ($1+1+10$, $1+2+9$, ...). But how does this pattern continue - and what happens with other numbers than 12? (See Foster, 2011, pp. 110-112, for the details.)

But, if this is where you want to be heading, it might be preferable to ask a more closed question like "How can you express 12 as a sum of positive integers?" Otherwise, you might end up with things like $\frac{120}{10}$ or $2 \times 2 \times 3$ or $1+2 \times 4 + 3$ or $\sqrt{144}$, which could take you in all sorts of other directions (e.g., decimals, fractions, prime factorisation, priority of operations, powers). I have often seen teachers ask very open questions, even though they have something quite specific in mind. Whatever the students respond with, they praise it and write it up on the board - but actions speak louder than the board pen, and the students are watching to see what the teacher is eventually going to do. No matter how much the teacher says, "These are all great ideas ...", the students know that they should ignore everything before the 'but', and so "... but today we're going to focus on..." is the point at which the teacher appears to come clean about what they really wanted right from the beginning. The person who said $10 + 2$ was 'right' and everyone else was 'nice try'.

So, I'm not convinced that open questions that lead to lots of possibilities are really all that inclusive if the teacher's response to all of the ideas except one is just to write it on the board and ignore it. But what is the alternative if you are using hyper-open questions? Allowing different students to work on totally different '12'-related problems splits the class into groups doing mathematics that may be as unrelated to each other as it is to the work being done by the students in the classroom down the corridor. And it can be hard to ensure that everyone has a suitably accessible and worthwhile challenge that will lead to learning something useful. The possibilities for sharing and building on each other's contributions later are quite limited if all that the different tasks have in common is that they have something to do with the number 12.

So, for me, prompt #2, although undeniably more 'open' than prompt #1, is more likely to close down the possibilities for purposeful sense-making of mathematics. Whereas adding constraints to #2, such as "It must include a cube root", or, for sixth formers, "It must include a logarithm", or "It must include an imaginary number", might make this more interesting and purposeful, depending on the teacher's intentions. Provided that there aren't too many, constraints often lead to creativity, whereas total freedom may not inspire anything.

3. The answer is 12, what could the question be?

Now we have broadened out beyond even ‘calculations’. Students might say “What is the order of rotational symmetry of a dodecagon?” or “What is the maximum total you can get with two ordinary dice?” or “How many millimetres are there in 1.2 cm?” This might feel like a nice creative task for getting students to reflect on different ways in which 12 can appear in mathematics, but you could end up with quite a few less-mathematical responses, or low-level mathematical responses, like “What number bus goes to the sports centre?” or “How old is Amit’s sister?” or "What are the last two digits of the school phone number?" or "How many days of Christmas are there in the song?" (see https://en.wikipedia.org/wiki/12_(number) for all things 'twelve'). By the time we reach a question as open as #3, we have really abdicated all responsibility for setting the direction of travel of the lesson, and ‘anything goes’ now. I guess I could maybe imagine asking this question as a lesson 'filler', or as a revision task: "Make up a question on each topic we've studied this term - and make all the answers '12'." But I don't think I could use this task to introduce anything new or focused on any particular content area, as it is too hard to predict what might happen with it. But maybe others are more expert at doing this than I am?

More 'mathematical' can mean more closed

Tony Gardiner (n.d., p. 31) has written that ‘While there are exceptions, it is generally true to say that good problems in school mathematics are almost never open-ended!" He advocates open-middled problems, where there is a closed start and end but multiple possible routes through. I think that teachers worry about using closed questions because they think they are too straightforward. But it is a big mistake to think that closed questions are likely to be simple or trivial. Most of the famous hard unsolved problems in mathematics are closed - even binary yes/no (e.g., Is the Riemann hypothesis true?) (see Foster, 2015).

Often we can make an open question more challenging by closing it, and that's usually a highly mathematical thing to do. The question we considered above, "Tell me a factor of 12", is open, but this should naturally lead to the desire to find them all by closing the question to: "What are the factors of 12?" This is now closed, because there is only one right answer (i.e., 1, 2, 3, 4, 6, 12). In a sense, giving just 5 of these is not $\frac{5}{6}$ correct, but wrong. This is not 'multiple answers' any more, because the question asks for all of them. It is as closed as the question "How many factors does 12 have?" (Answer: 6). Similarly, the closed question "Solve $x^2=9$" has one right answer, consisting of the two solutions $x=-3$ or $3$, whereas the open question "Find any solution to $x^2=9$" has two possible right answers, $x=-3$ or $x=3$. This may seem like splitting hairs, but the closed question is far more powerful - and mathematical - because we not only find all of the solutions but we know that there can't be any more, so we are finished and have complete knowledge of the situation (see Foster, 2022).

Thinking in this way, many of the open questions used in school can be viewed as merely poorly-specified or incomplete closed questions. As an example, consider the question: "I'm thinking of two numbers. Their sum is 12. What are the numbers?" We could respond "Impossible - insufficient information!", or we could give a few example pairs (which are effectively of the form "If one number is this, then the other number would be that"). Or we could plot all possible example pairs on the line $x+y=12$, which feels like as complete an answer as is possible. But is it that the question is open or merely that the (single) answer to this closed question happens to be an infinite solution set? (Note 1)

A more clear-cut example might be: "A number rounded to 1 decimal place is 3.1. What might the number have been?" This is open, and, again, in this case, there are infinitely many possible answers. The corresponding closed question would be "Which numbers round to 3.1 to 1 decimal place?", and here the answer would be all $x$ in the interval $3.05 \le x < 3.15$. There's a sense in which this is 'the complete answer'. But, I think it doesn't quite capture everything that might emerge from the open question. For example, someone might respond to the open version by saying $\pi$, and, although that is of course included in the range specified in the closed version, I might not have explicitly considered $\pi$ (or even the possibility of any irrational numbers) when writing down the 'complete' answer. The double inequality expresses the whole range of possibilities without necessarily examining it in detail.

To me there is certainly an important difference between a closed question like "Which numbers round to 3.1 to 1 decimal place?" and a closed question like "What is 3.08 rounded to 1 decimal place?", but the difference isn't about open/closed, as, for me, both are equally closed. The difference here is that the first question is an inverse question, and inverse questions are always harder and more interesting than direct questions (e.g., factorising versus multiplying, square rooting versus squaring, integrating versus differentiating).

Open questions are often essentially 'Give me an example of...', example-generation questions (see Watson & Mason, 2006), and I think these are extremely valuable when used to complement closed 'Characterise all the possibilities' kinds of questions. We need both. But making open questions more closed can be just as valuable mathematically and didactically as making closed questions more open.

Questions to reflect on

1. What do you think about the value of 'open' questions? When do you use them or avoid them? Why?

2. When is it good to be more open? When is it good to be more closed?

Note

1. I guess one way to treat this as open would be to offer multiple ways of specifying the solution set (e.g., $2x+2y=24$, $y=12-x$, $x+y-12=0$, $y=-x+12$, $\frac{x+y}{2}=6$, etc.)

References

Ball, B. & Ball, D. (2011). Rich task maths 1. Association of Teachers of Mathematics. 

Foster, C. (2011). Resources for teaching mathematics 11–14. London: Continuum.

Foster, C. (2015). Fitting shapes inside shapes: Closed but provocative questions. Mathematics in School, 44(2), 12–14. https://www.foster77.co.uk/Foster,%20Mathematics%20In%20School,%20Fitting%20shapes%20inside%20shapes.pdf

Foster, C. (2021). Area and perimeter. Teach Secondary, 10(6), 13. https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Area%20and%20perimeter.pdf

Foster, C. (2022). Starting with completing the square. Mathematics in School.

Gardiner, T. (n.d.). Beyond the soup kitchen: Thoughts on revising the Mathematics “Strategies/Frameworks” for England. https://www.cimt.org.uk/journal/gardiner.pdf

Watson, A., & Mason, J. (2006). Mathematics as a constructive activity: Learners generating examples. Routledge.

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