Being Nimble with Numbers

In this blog, I want to discuss one of the issues in Maths education: The focus on one method to solve problems of a particular type. The example I will use throughout is the unitary method: When you work out the cost per unit of an item.

The unitary method

The following illustration of the unitary method comes from a very good textbook (Lipson et. al., 2022): 


Students can apply this method to compare "cost per unit" when deciding which of the following to buy:

Hopefully, students realise that 100g would be a good unit here, rather than feel compelled to use 1g.

Using one method: The good and the bad

It is a good thing to practise a method that applies in as many contexts as the unitary method does. It gives the student a strategy and enables them to master it. It also gives the teacher direction on how to teach. I am all for explicit teaching and it is difficult to teach explicitly in the absence of a repeatable method.
The bad thing about emphasising one method is that it can take away a student's confidence to use alternative methods and removes the incentive for them to understand the context of a problem and then apply the strategy that makes the most sense in that context.

A different context

Take this example, where division, an operation that many students find difficult, would not be necessary at all:
Given that we need to buy 60 chocolates, I would want the following strategy to be available to the students:
Option A: We need 5 packs of 12. These will cost :
Option B: We need 3 packs of 20. These will cost:
Is this not easier than working out the cost per chocolate bar in each of the packs?

Let's also take the following example:
Here also, I would think it easiest to work out the cost per kg. Basically, I would be using the unitary method but with 1kg as the unit. That's one multiplication: Option B represents a cost of $6.80 per kg. Therefore, Option A represents better value.

Conclusion

Students need to be exposed to a variety of problems and allowed to experiment with and share different strategies for solving them. Standard methods are still going to be useful as one way of getting to an answer. What I am advocating in this post is simply this: Giving students some time to consider the problem, understand the context and then retrieve the right strategy from their toolbox. 

Reference

Jones, P., Lipson, K., Main, D., Tulloch, B., Humberstone, R., Karakoussis, P., & Staggard, K. (2022). General Mathematics VCE Units 1&2 (2nd ed.) [Review of General Mathematics VCE Units 1&2]. Cambridge University Press.

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