Everyone knows how to divide fractions, right? You just flip-and-multiply.
Unfortunately, although this will enable you to solve problems all day long, it is an example of a procedure performed ‘without understanding’ and so that is bad.
If you know that means the same as “two divided by nine” then I can demonstrate how it works. Of course, if you don’t know this then I will have to prove it to you first so that you have ‘understanding’. No, scratch that, I will have to create some rich task that enables you to come to that understanding for yourself or something. But let’s assume you get it. We can say
If so, it must also be true that
However, given that we can write this as
If so, it must also be true that
Continuing, we can see that
Given that we can rewrite as
If we wanted to, we could extend this logic to any pair of fractions by replacing the numbers with algebraic terms.
So, are you happy now? Do you understand? Are we all OK to go away and flip-and-multiply?
Perhaps not. After all, do you really understand? I mean, like really? Can you actually picture what is going on? No? OK then, here’s a picture:
By completing our fraction division this way, we can avoid flip-and-multiply and instead spend our time drawing lots of diagrams and colouring them in. Is this better? What if you tried to do this way?
Note a few important points. Firstly, flip-and-multiply is a very simple rule to apply. Once you have it in long-term memory alongside procedures for simplifying fractions (e.g. or ) you can tackle problems involving fraction division with relative ease.
In contrast, visually solving fraction division problems involves a large number of interacting elements and will therefore be high in cognitive load. Similarly, following the explanation of why flip-and-multiply works requires tracking through many steps and following the logic at each step. This imposes a high cognitive burden (which ironically will be reduced if you already have a lot of rules in your long-term memory – see the discussion here).
Imagine a school makes the decision that ‘understanding must come first’. In this school, flip-and-multiply is forbidden and students have to solve all such problems using diagrams like the one above. This will have two effects. Firstly, it will limit the number of problems that students may be asked to solve to those that can be feasibly represented visually. Secondly, it will impose a far higher cognitive burden on students. Many students who may have successfully deployed flip-and-multiply will struggle with the diagram approach.
This seems counterproductive and doctrinaire. Why not teach students the rule and allow them to use it? What harm would come of that? There is nothing to stop you explaining it and you could use both a diagram approach and the kind of rough proof I have given above in order to do this. Some students may follow it and some may not. Some may learn the rule and later come to appreciate how it works. In fact, that’s quite likely because the more they have in long-term memory, the easier it is to follow the proof. However, pretty much all students should be able to apply the rule in solving problems, provided they have the prerequisite skills.
If you insist that understanding must always come first then you are placing limits on students. By raising the cognitive burden, such an approach creates confusion and, for many students, will result in the exact opposite of the intended outcome.