Opening up tasks: the ‘what-if-not’ technique

When was the last time you got caught up in the quest for the ‘perfect’ task? Scouring the internet for a rich activity to make all students think, about mathematical ideas, and in meaningful ways. Even for an experienced teacher, with a curated set of reliable sources¹ and the skill to judge whether a task aligns with their learning intention, this can be a huge time-sink. The challenge is even greater for novice teachers. Where to look? Which websites are trustworthy? Is the task a good fit for the topic? For the students? It’s understandable that we can find ourselves short of time and defaulting to the textbook. 

Textbooks are often maligned in mathematics education. They can be too helpful in their treatment of rich tasks, as Dan Meyer explains here and here (and be sure to dig into the comments on the blog post). And they frequently contain problems that are either unappetising or lacking that certain je ne sais quoi. But they are fertile ground for adaptation, and often the resource closest to hand.

For problems that provide too much support, Dan Meyer advocates that we be less helpful — ‘You can always add. You can’t subtract.’, as one of his workshop participants observed. For problems that are lacking in richness, John Mason encourages us to adopt techniques designed for the purpose of ‘’opening up tasks’ as they frequently make a task richer than its original version.’ (Mason 2006, p. 60).

The ‘what-if-not’ strategy is one such technique. First coined by Stephen Brown and Marion Walter, the technique gets extensive treatment in their book ‘The Art of Problem Solving’. For a more concise introduction to the technique, I prefer ‘Adapting and extending secondary mathematics activities: new tasks for old’ by Stephanie Prestage and Pat Perks (pp. 5-8).

Put simply, the ‘what-if-not’ strategy starts by taking a question and listing all its attributes. Once we have exhausted our brainstorming, we take one attribute from the list and repeatedly ask the question ‘What if it is not … [the attribute], but … [something else]?’ to generate a collection of different problems. Let’s see this in action with two examples.

Our first example

Problem: Solve for x:

Before launching into the technique, I strongly recommend trying the problem yourself. Doing the problem also helps sensitise us to things to notice, as John Mason would say. Note that the converse is also a very powerful technique — approaching a problem by first asking ‘What do I notice?’ can help us to enter and perhaps find a successful resolution.

We then generate a list of attributes:

• The lengths of two sides (the legs) of the triangle are known
• The length of the third side (the hypotenuse) is unknown
• The side lengths are integer
• One side length is 5m
• One side length is 12m
• The triangle is right angled
• Two of the interior angles of the triangle are unknown
• and so on

Next we take one of the attributes from the list and ask the question ‘What if not … [the attribute], but … [something else]? 

What if it is not side lengths of 5m and 12m but something else?

Problem: Solve for x:

Keeping the format of the problem constant and varying the side lengths strategically provides a range of triangles for students to gain practice on, whilst also providing opportunities to think mathematically by attending to mathematical structure. (Craig Barton terms this ‘Intelligent Practice’.)

What if it is not the legs of the triangle that are known but the hypotenuse?

Problem: What right-angled triangles can you find with an hypotenuse of 13m?

Giving the answer (‘reversing the question’) is a powerful technique! Applied here, the strategy also encourages practice in applying Pythagoras’ theorem and attending to mathematical structure. However, this time students are working with one triangle and making decisions about which side lengths to specify.

What if the hypotenuse is also known?

Problem: What can we say about whether this triangle is right-angled or not?

This opens up either an application of or an investigation into the converse of Pythagoras’ theorem, depending on whether students have met this concept yet.

What if it is not a right-angled triangle? 

Problem: What could be the length of the third side?

Removing a constraint suddenly opens up a infinite number of solutions and, in doing so, encourages exploration of the Pythagorean inequality theorem.

What if it is not a side length of 5m but something else?

Problem: Solve for x: 

A careful choice of a new side length may serve as an introduction to Pythagorean triples.

What if another internal angle of the triangle is known?

Problem: Solve for x:

Students who have encountered trigonometry now have a variety of ways in which to find the length of the hypotenuse (e.g. trigonometric ratios, sine rule, cosine rule, Pythagoras’ theorem), which may raise the complexity of the question as they now need to select and correctly apply an appropriate procedure.

None of these questions as Prestage and Perks note ‘is better than the other. Each serves a different purpose for the teacher and for the learner. Crucially, the teacher needs to know for what mathematical purpose any question is being given.’ Experimenting with the question in this way has given us ideas as to what concepts we might investigate in a unit of work on Pythagoras’ theorem.

Our second example

Problem: Which has the bigger area: a 3 metre square or an equilateral triangle in which each side is 4 metres long?

If you tried the problem² as I recommended earlier, you will have discovered that the solution to this one has a feature that may not be apparent on the surface. 

We then generate a list of attributes:

• There is a square
• There is an equilateral triangle
• There are two shapes mentioned
• The length of a side of the square is 3 metres
• The length of a side of the equilateral triangle is 4 metres
• The side lengths are integer
• The side length of the equilateral triangle is 1 metre more than the side length of the square
• The difference between the side lengths is 1 metre
• We need to compute areas of the shapes
• The goal is to find which has the bigger area
• The task is given in written form
• There is a four-sided regular polygon
• There is a three-sided regular polygon
• and so on.

When I chose this example, I wasn’t sure I’d generate much of a list, but one idea quickly sparks another.

Next we take one of the attributes from the list and ask the question ‘What if not … [the attribute], but … [something else]? This may lead to several interesting lines of inquiry to wonder about.

What if it is not a square with side lengths of 3 m and a triangle with side lengths of 4 m, but vice versa?

Problem: Which has the bigger area: a 4 metre square or an equilateral triangle in which each side is 3 metres long?

Now this is not a particularly interesting problem to me, as the square has longer side lengths than the triangle, and so I would expect the area of the square to be bigger. But it leads me to wonder …

What if it is not given in written form, but presented as a diagram drawn to scale?

Problem: Which has the bigger area?

We might now notice that triangle appears bigger than the square, and so are perhaps surprised when the square turns out to have a bigger area than the triangle. This might lead us to investigate an intuitive approach to understanding why this is so.

What if it is not 3m and 4m respectively? What adjustments can be made to the side lengths of each shape to maintain some interesting feature? This is a very open variation that I’m not yet sure what to do with.

Problem: Which has the bigger area?

What if it is not an equilateral triangle but a circle? 

Problem: Which has the bigger area: a 3 metre square or a circle in which the diameter is 4 metres?

Now I’m wondering if the square can be circumscribed by the circle (that is, the square fits snugly within the circle). That’s another rabbit hole. Riffing on the regular polygon theme …

What if it is not an equilateral triangle but a regular pentagon?

Problem: Which has the bigger area: a 3 metre square or a regular pentagon in which each side is 3 metres long?

This led me down a rabbit hole of calculating the areas for regular polygons in general. 

What if it is not area to be calculated but perimeter?

Problem: Which has the bigger perimeter: a 3 metre square or an equilateral triangle in which each side is 4 metres long?

While the calculation is more straightforward, it sparks a new direction to investigate about shapes with equal perimeters!

Each line of inquiry is now inviting us to generate a new list of attributes to notice and wonder about.

Summary

The ‘what-if-not’ technique puts mathematics at the heart. We’re not looking to change the presentation of the task, or the context in which it is set. We are closely attending to the mathematical features of the problem, and analysing what happens as we make very considered and localised changes. Sometimes the questions that are generated stray far from the original mathematical concept, but encourage us (and our students) to ‘develop a flexibility of thinking about questions and about mathematics which is useful in adapting and finding new questions’ (Prestage and Perks, p. 7). 

This technique also encourages a spirit of inquiry very much in line with how research in mathematics is done. As Walter and Brown (2005) say:

“Great advances in knowledge have taken place by people who have had the courage to look at a cluster of attributes and to ask, “What-If-Not?” As we mentioned earlier, perhaps the most famous such instance in mathematics involves the development of non-Euclidean geometry. Up through the 18th century, mathematicians had tried in vain to prove the parallel postulate as a theorem. It took 2000 years before mathematicians were prepared to even ask the question, “What if it were not the case that through a given external point there was exactly one line parallel to a given line? What if there were at least two? None? What would that do the structure geometry?”

Or another example, by asking ‘what if it is not a² + b² = c² (as in the Pythagorean equation) but aⁿ + bⁿ = cⁿ for any integer n greater than 2’, we arrive at Fermat’s Last Theorem (or Fermat’s Conjecture) which tantalised mathematicians for centuries. 

The algebraic topologist Peter Hilton said “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” The ‘what-if-not’ technique is central to developing mathematical thinking in us and in our students. The framework encourages us to generate and pursue new questions and, in doing so, to immerse ourselves in authentic mathematical practices.

References

Brown, S. I. (2005). The art of problem posing. (3rd ed.). Lawrence Erlbaum.

Mason, J., & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks. Tarquin. 

Prestage, S., & Perks, P. (2006). Adapting and Extending Secondary Mathematics Activities: New Tasks For Old. David Fulton Publishers. https

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[1] e.g. map.mathshell.org, nrich.maths.org, openmiddle.com, donsteward.blogspot.com, to name a few good ones.

[2] I found this problem in a blog post by Grant Wiggins that, because I disagreed with his categorisation about problems like these, ultimately sharpened my thinking around routine exercises and problematic tasks; read my summary here.