Dividing a Fraction by a Fraction – A Japanese Approach (Part One)

As some of you may know, I’m currently in Japan learning about Japanese Lesson Study, with a focus on mathematics. Lesson study itself has nothing specific to do with mathematics, but is a particular way in which the Japanese conduct professional development for their staff. It has been utterly fascinating to watch and learn about, and I’ll be sure to write more about it soon enough. For now though a brief description will have to suffice: Lesson study involves a small group of teachers team planning a lesson over the course of a few weeks (the detail and thought behind the lesson is very impressive) with an agreed focus such as, for example, developing conceptual understanding, or meaningful group participation etc. The lesson is then observed by a (large) number of teachers (sometimes just school staff, sometimes much broader), and then immediately after the lesson a post lesson discussion is held where everyone discusses how to drive the quality of learning forwards. I’m sure that whole process raises some questions, which I’ll try and get to in another post. In this post I want to talk about the strategies I observed for one of the most difficult lessons to teach in school mathematics – the division of fractions by fractions, with a few of my own thoughts thrown in for good measure.

The lesson we observed was part two, with, I assume, at least another two parts to to follow on.

Part one (unseen by me) was effectively a discussion and sort of think tank in which students considered the following problem:

“3/4 of a bucket of paint covers 2/5 of a boat. How much of the boat will a full paint bucket cover?”

The lesson was taught to Grade 6 students (UK Grade 7). During that (unseen) lesson, students derived, with very little to no guidance from the teacher (i can attest a little to this based on the many other lessons i observed which follow a similar structure and strategy) that the way, or, *a* way, ┬áto approach the problem is to divide 2/5 by 3/4 – something they can’t actually do mathematically yet.

This in itself is a marked departure from UK / US strategies. I would never dream of utilising an entire lesson to enable students to effectively just think through a problem and derive what it’s actually asking you to do mathematically – not necessarily because I disagree with it or think it’s a waste of time, but just because I can’t picture it working with many of the classes I’ve taught in the UK due to likely behavioural and perseverance issues. ┬áRegardless, this is not intended as a “we should copy them!” post. I’m just interested in their approach.

One of the ways in which the problem was understood by students was by the use of a tool I’ve personally never used nor seen before, a ‘double number line’. I have no doubt that some of you will have, but this was a new one on me.

The double number line is used as a visual proportional reasoning tool, much like bar modelling or, my preferred proportional reasoning tool highlighted by the late Malcolm Swan in the UK, the four squares method (i’m sure it has a cleverer name somewhere):


The double number line has some advantages (and some disadvantages) over the squares method. Primarily, you can highlight two things at once a bit easier (see in a minute) whereas the squares would, I suppose, require two separate diagrams.

Anyway, the double number line is joint together at zero, and then each proportionately similar element is lined up accordingly. Imagine a proportions x/y table, then make it a diagram basically. So for our paint and boats problem, we’d have a diagram like this:

double number line

The area painted is indicated on the top row, and the paint used is on the bottom. The empty square is what we want to find out. This would not be given to students. Students would come up with this diagram themselves, although they would have used it fairly frequently in similar situations previously. I cannot emphasise enough how much the teacher does not interfere with thinking time, and how long it goes on for (!) we’re talking a very, very long time compared to how I’ve seen it done, or indeed how I’ve done it. A general rule followed is that any hint you give to students immediately narrows their thinking, and that’s bad. I find it hard to disagree, at least in principle. Japanese teachers don’t seem to have the balance here working well enough yet – and I certainly don’t either. Whilst I emphasise thinking time and deep discussion, I like the idea of making them struggle a lot more than I do, but trying to keep them motivated and behaving well is something you cannot take for granted in the UK. Although behaviour was never an issue in the lessons I saw, discussions were so drawn out sometimes that some students were clearly drifting off or losing enthusiasm. In fairness this was always picked up in the post lesson discussions. I suppose the message is that we all know this stuff is hard to do, but worth persistently trying to improve upon.

Anyway, moving on, some students began to see the problem more like this:


So students are thinking about how they get from 3/4 to 1 on the bottom line, in a way that’s easy to them. I should add they can divide and multiply fractions by integers comfortably. Here lies the advantage of this diagram over the four squares. We can show two relationships rather than just one. Incidentally the disadvantage of this method to me is that you can’t show the relationship between the 3/4 and the 2/5 very well.

They then apply the same strategy to the top line and divide 2/5 by 3 (which they can do) and multiply their answer by 4 (which they can also do). So at this point a valid, smart approach to the problem has been thought of, and the problem can be solved. Hurrah! However, there is so much emphasis on thinking through the problem rather than getting the answer, so one method is simply not the end of the matter.

Another student notices that if you divide 3/4 by 3/4 then the answer is 1, because anything divided by itself is always 1. And so they suggest that 2/5 could be divided by 3/4 to obtain the answer, regardless of the fact that they don’t know how to do that mathematically. Fascinating stuff, and I was genuinely surprised at the range of responses time after time, lesson after lesson, when allowing students the freedom to think things through deeply without the interjections of the teacher to willingly or subconsciously shape their thoughts in a particular way.

A third idea was that you could multiply 3/4 by its reciprocal (they know about reciprocals) to get 1, hence you could possibly multiply 2/5 by 4/3. Incidentally no connection was made between dividing by 3/4 and multiplying by 4/3.

All of this was done in lesson 1, and recapped to some extent in the lesson I observed, with the teacher aim of establishing that 2/5 divided by 3/4 was a valid mathematical process to perform to solve the boat problem – to set up lesson 2 to learn how to do it. I’ll get to that in my next post…!

If you want to learn more about Lesson Study as a tool for teacher professional development (I *really* recommend you do, it’s fascinating) then there’s a conference in Nottingham very soon dedicated to it. Here’s the link.