Imagine a world of maths where-
- teachers take their time on a topic,
- teachers can go in-depth with new concepts
- (not just skate the surface),
- … and breathe.
Recently, I’ve found myself going back and forth on the idea of creating change in maths education.
On the one hand, things are SO hard to shift (think: curriculum and assessment structures, school timetables, etc.).
But on the other hand, meaningful and sustainable change starts with just one step.
I believe that there are very real ways to bring this fairytale world, of slowing down and going deeper, to life.
In this article, I'm sharing one of those ways. It’s a practical strategy you can use to-
- Turn 1 problem into 5 more
- Help all learners look at the maths they’re learning differently
- (not just the fast finishers or ‘A-students’)
- Plan lessons for many topics and concepts,
- And, save you time.
How to turn 1 problem into 5 more
This strategy starts by taking a problem and asking yourself: “Where might this go next?”. With one question comes new opportunities for students to go far deeper and to explore fascinating uncharted mathematical territory.
Let's look at how this works with an example…
A very ordinary task
Say your original question is this fairly ordinary perimeter one. It's the sort of problem you'll find in a worksheet, question bank or textbook – and it gives us the perfect starting point.
So… Where might you go next?
Here are five strategies you can use to create variations on the original problem – and make it sparkle.
Variation 1: Start with the answer
By starting with the answer, students will naturally become more flexible in their thinking. They'll also use logic, justification and proof to make sense of the concept they're working with.
Example:
- What other rectangles can you find that have a perimeter of 44 m?
- If you're just using integer side lengths- how do you know when you've found them all?
Variation 2: Make the numbers harder
The moment we make the numbers we're dealing with harder, the problem levels up – and our thinking needs to change.
You can do this, for example, by changing the numbers to be FAR larger, or to be fractions, decimals, mixed numerals, or negative integers.
Example:
- What other rectangles can you find that have a perimeter of 44 m, and side lengths that are NOT whole numbers?
- What's your strategy for finding these?
Variation 3: Link to a related concept
The most powerful skill in mathematics is to be able to form new connections and find a relationship between different ideas.
When linking to a related concept, by asking students for their observations, they will slow down their thinking and make sense of what they've done.
Example:
- Out of all the possible rectangles that have a perimeter of 44 m, what's the largest area? smallest?
- What do you notice about the different areas and the side lengths?
Variation 4: Change one parameter
By changing one parameter (and one parameter only), students can use their existing strategies and thinking from one problem and apply it to the next.
If we change too much at once, then we minimise the chance that students will be able to form connections and see how the problems are related.
Example 1:
- What other SHAPES can you find that have a perimeter of 44 m?
- Can you find a shape with that perimeter and 3 sides? 5 sides? 6 sides? 10 sides? n sides?
Example 2:
- A shape is made up of three 7 m by 15 m rectangles. What might the perimeter of that shape be?
- Can you make the perimeter any smaller? larger?
Variation 5: Generalise
The moment students start to generalise – or find a rule – they are stepping back from the problem and considering, at a broader level, what it means.
In other words, they're abstracting out information and demonstrating an in-depth understanding of what's going on.
Example 1:
- A shape has a perimeter of 44 m and n sides of equal length. How long is each side?
Example 2:
- If a rectangle has a perimeter of P m, what might its side lengths be?
Why bother turning 1 problem into 5?
By turning one problem into five, we are inviting students to examine a concept from different angles. We're saying to them: “This is an idea that's worth your focus”.
And through that focus, comes new opportunities to form valuable and longer-lasting mathematical connections.
Creating the conditions for deeper learning doesn't mean you need to throw out your resources and overhaul your practice – but it does mean you can take steps, one at a time, towards meaningful and sustainable change.
Interesting methodology. Love how you took one problem and extended it other level.
Thanks Marty. I actually find it a fun creative exercise tweaking the ordinary problem & taking into new directions.
Love this idea. It’s practical, it starts where teachers already are, and it trains the habit (both for the teacher and student!) of not stopping at solving a problem as written.
I do something similar when I give kids puzzles — I ask them “How would you make this puzzle easier? How would you make it harder?”
To dramatize the idea of parametrizing a question, you could pull out a physical dial, and attach it to the question, and ask what happens if I turn this dial. Or you could show a slider control beneath the question, and ask what the two ends of the slider could be labeled.
To disrupt the idea that every math question has exactly one answer, I’ve started constructing sets of questions where one of the questions has no solution, and another has multiple solutions. So the question to students becomes — which question has no solutions or multiple solutions?
Thanks Scott.
Your point about ‘training the habit’ is spot on. The physical dial/slider idea is fantastic – and I can imagine that over time, students would start to internalise the visual of the slider/dial and no longer need the actual prop.
Reminds me of The art of Problem Posing by Brown and Walter. 👍
Indeed!
It’s a great book.
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