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.