It’s no secret that students regularly struggle with fractions. It isn’t necessarily that the concept of a fraction itself is difficult. Most students, for example would understand what I meant if I asked for half, or a quarter, of a pizza.
What students do struggle with, though, is the operations we ask them to perform using fractions. I’m talking about addition, subtraction, multiplication, division, simplifying, etc. These are often taught using a series of rules, or ‘tricks’, that give students a sequence of steps to follow in order to solve a problem.
Why might we want to use ‘tricks’? Aside from sometimes being faster, one might argue that it allows students to perform operations that can otherwise be difficult to visualise or ‘think through’.
Take division as an example. We’re first introduced to the idea of division as ‘sharing’. If I have six coins and share them equally between two friends, hopefully we’re all agreed that both friends will have three coins.
But what about division by a half? Taking the same problem, I am now sharing six coins between half a friend. That sounds a little strange! Understandably, then, it can feel like fractions are easier to teach using tricks, allowing students to side-step the mathematical meaning.
But what might go wrong with trick-based learning? Well, let us consider my personal ‘favourite’ the Butterfly Method, described as a memorable way to add or subtract fractions (see attached resource for image). It follows the approach shown below, where you construct a butterfly-like shape around the fractions to be added (or subtracted).
Without going into too much detail, you are essentially asked to multiply across the diagonals (the wings), put the products either side of two antennae, and add a body consisting of the multiplication of the denominators (the ‘18’). Finally, add the numbers in the antennae for the numerator and use the number in the body for the denominator, simplifying as necessary.
I don’t know about you, but I’m struggling to find the mathematical learning from that!
Let’s say, though, that a particular student struggling with fractions was perfectly able to follow these steps. When it came to their exam, they applied the ‘Butterfly Method’, and got the right answer. This might sound okay in practice, but what needs to be considered is the long-term effect of teaching fractions this way; not just the ability to pass a single exam.
I’ve seen many students try, and fail, to re-create the ‘Butterfly method’ from memory, and not just for addition either. They try this for multiplication and division also, to which the method was never intended.
Frankly, I don’t blame them. It’s not easy to remember rules that you haven’t applied in a while. I’ve experienced this many times myself.
Take one example. A few years ago, I spent some time learning a sequence of steps that would allow me to solve a Rubik’s cube. I could take any regular Rubik’s cube, and solve it relatively efficiently.
Could I do the same now? Not a chance. I know this because I tried recently after stumbling across a Rubik’s cube at my workplace. I attempted to dig into my memory banks to recall the sequence of steps I’d learnt so well years before but, ultimately, failed.
What went wrong? Well that’s pretty simple… I’d forgotten how to solve a Rubik’s Cube. Moving on?
Not yet. The problem was that despite learning the steps needed to solve a Rubik’s cube, I hadn’t taken any time to understand the fundamental reasons why they allowed me to solve it. The individual steps themselves were, essentially, meaningless to me. I am now no more able to solve a Rubik’s cube than I had been when I picked one up for the first time.
This is exactly the problem with rule/trick-based learning. Without the ‘why’ you are relying purely on memory, which can be fragile at best! It is not a desirable outcome if, years later, students are thinking ‘I remember something about forming a Butterfly shape’ when confronted with fractions.
We also haven't touched on the efficiency of these tricks. Let's consider what I believe to be the ideal solution to the earlier problem, shown below (see attached resource for image). It uses ideas fundamental to working with fractions and also, in my opinion, is more elegant! At no point do we end up with a denominator of 18 that needs simplifying.
I’d also be perfectly happy with the solution below (see attached resource for image). Sure, the need to draw out a pizza shaped object might indicate some problems working with the numbers directly (potentially a problem for more awkward fractions) but it still shows me that the student is thinking about the problem correctly.
So what’s my overall message? Well, what I’m suggesting is that we start thinking about teaching for the long-term, which only works if we value student understanding more than ‘getting to the answer’.
That is not to say we need to ditch shortcuts, or tricks, entirely. What is crucial, however, is that we think about whether the shortcuts we are teaching benefit student learning in the long-term. I would argue the ‘Butterfly Method’ does not meet this criteria.