The very first blog in this series, One Method: Many Topics, looked at how pupils can sometimes use the same method to solve questions across numerous areas of mathematics. This is because even when topics appear different on the surface, their underlying principles may be similar. This post explores benefits of learning multiple methods for a single topic or a single question.
While I have always had some appreciation for teaching multiple methods, it was a trip to a handful of schools in Tokyo (with the IMPULS Project) that really opened my eyes up to how much can be learned from comparing methods – thanks to my school for sending me! For more about this, see the suggested reading at the bottom of the page.
Anyway, one of my pupils recently asked me:
“Sir, why do we learn more than one method to solve things? Surely that means we’ll need to learn twice as much stuff than we would if we just learned one method for each topic?”
This was a great question! I replied by briefly summing up a few reasons that she seemed happy with. Two key reasons were the following:
- The method that is the easiest for one question isn’t always the easiest!
- By comparing multiple methods, we can make connections and learn bigger things about the topic.
Each of these reasons will be explored in Part 1 and Part 2 of this post.
Part 1: The method that is the easiest for one question isn’t always the easiest!
Let’s consider calculating areas for compound shapes made from rectangles. For example:
One way to solve this could be to split the shape into smaller rectangles, find the area of each one and then add them together. However, another way to solve it could be to fill in the empty space, making a small rectangle within a larger one, then find the area of each and subtract.
Usually, I tend to find that pupils prefer the ‘split and add’ method over the ‘fill and subtract’ method. But let’s look at another shape. Which method would be the easiest with the one below?
The Splitters and Adders of the class would probably still argue that their preferred method is easier. But what if we nudge that empty space a tiny bit to the left?
Is splitting and adding still the most straight forward method to use? Why might students who choose to ‘fill and subtract’ have an easier time? Or how about if we had nudged it 1 cm downwards instead?
Finding the area of this using splitting and adding would involve four rectangles; filling and subtracting would involve two.
We can make a similar argument when pupils are learning to compare fractions. For example:
The most popular method for this tends to be to find equivalent fractions that have a common denominator and then compare.
But how about if we switch the numbers in those fractions round?
To apply the same method here, we’d have to use a common denominator of 165. An alternative could be to find fractions with a common numerator instead.
But arguably learning multiple methods is only half of the battle; the other half is learning how to judge when one method is easier than another. A key component to problem solving is making decisions about what methods to use. Therefore, after pupils learn a couple of different methods for a question type, they may find it helpful to practise this kind of decision making. This could be through an exercise that, rather than asking them to work out answers, simply asks them state which method they would choose. For example:
Part 2 of this post will discuss how comparing multiple methods can lead to bigger generalisations about the topic.
If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: ‘Many Methods: One Topic (Part 1)’
If you found this post interesting, then I would also recommend the following:
- This conference paper by Takahashi describes problem-solving lessons in Japan: Beyond Show and Tell: Neriage for Teaching Through Problem-Solving – Ideas from Japanese Problem-Solving Approaches for Teaching Mathematics
- The idea of isolating small, key skills for students to practise separately (e.g. deciding what method to used) is discussed by Barton in Chapter 8 of his book: How I Wish I’d Taught Maths
Previous blog posts:
This series, ‘Making Connections’:
‘What I Learned From Shanghai’ Series:
- Not As Complicated As It Looks
- A Shanghai Lesson on Adding Negative Numbers
- Do Sweat The Small Stuff
‘Planning Thought’ Series: