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

So as mentioned in part 1, the first section of the lesson I observed recapped interactively both the problem at hand, and the deduced fact that 2/5 divided by 3/4 was a valid approach to solving it.

In my mind, whilst there was undoubtedly a *lot* of time put into reaching this point, a really nice advantage of doing so is that the students are not only invested in the problem far more than they might have been otherwise, but they’re also seeing a kind of validity in the need for dividing a fraction by a fraction as a ‘tool’ to add to their maths toolkit. This aligns somewhat with Andrew Blair’s UK Inquiry Maths approach.

As mentioned in part 1, or at least alluded to, there is a much greater emphasis on manipulating questions to make them easier to tackle or understand. We saw it when describing how to get proportionately from 3/4 to 1, and the strategy played its part again in this lesson when students were given thinking time to try and tackle 2/5 divided by 3/4. The teacher essentially said something along the lines of ‘i’m going to give you some time to think about this and i want you to see what you can find out’ as his instruction, and then walked the room as students diligently worked away independently of one another trying to make a dent in the problem. This fascinates me for a couple of reasons. Firstly, it seems initially to ask students to solve something they don’t have the tools to solve, until you realise that actually they do, they just haven’t been instructed that they do. Nor do they know any quick algorithms – which is exactly the point – they get to study it mathematically, think, ponder, try out ideas, fail, try again, predict, confirm and so forth. They do what real mathematicians do, and some of them do it really well.

The most rewarding part for me is that all the ideas and approaches are collated and discussed on the board as a group, so no particular approach is preferred, or ignored. All are equally valid at this stage, and later students will debate which is/are most efficient and why.

Before we get to the different methods, I should add that all the lessons I observed used only blackboards / whiteboards, large pieces of paper (some pre-prepared with things for reference, or enlarged versions of diagrams, some would be written on live to copy down student ideas for all the class to see) and magnets to hold things in place. No technology whatsoever, apart from a visualiser in two lessons. This may seem at odds with the notion of Japan being technologically superior to the universe, however seeing it first hand I’m sold that interactive whiteboards are the devil (I was before if I’m honest, but this just made it brutally obvious).

Back to the lesson. After students were given a long time (maybe 15 minutes?) to generate ideas, the teacher began to display several fully formed solutions that students had created by themselves. Here’s the first:

work 1

The teacher asked the class to explain what the student was thinking. They had to get inside the head of the student without the student herself explaining anything. The teacher spent careful attention on where the 4 came from in the second line – why 4? What’s the overall strategy for doing that?

Students explained that the strategy was to make the divisor an integer, because that’s something they can work with and ‘do’. The fact that the divisor is a fraction is what’s stopping their progress, so their logical approach is ‘stop it being a fraction’.

The student was then asked to explain why they multiplied *both* fractions by 4, rather than just the second one. Their explanation was that they ‘used the property of division’, which after more probing referred to multiplying both numerator and denominator by the same amount to maintain balance. It struck me that perhaps then, the student saw the problem like this:


Or at least as a/b.

A second student’s work showed this:


Initially there was a mistake in the above working, but it was weeded out after the discussion about ‘the property of division’ in the first example. Again it was discussed in depth, and the class decided the strategy here was again to rewrite the problem so that we weren’t dividing by a fraction. In this case, both dividend and divisor are multiplied by the reciprocal of the divisor so that we’re dividing by 1 instead of a fraction.

The final method displayed was this:


Yet again, the strategic thinking is ‘stop the divisor being a fraction’, but this time we pick the lowest common multiple of both denominators so that we get rid of both fractions at once. Again a lot of emphasis is placed on the strategic choice of ’20’, where it came from, why it’s a good idea etc.

This all obviously took up  a lot of time. And no doubt we’d be criticised in the UK for not moving onto independent work where students have 10 questions to solve etc, but I couldn’t help but admire the depth of discussion, and the ingenuity of the students.

At this point students were asked to spot commonalities between all 3 methods, and again some great responses were given.

“They all get the same answer”

“They all do the division at the end, not the start”

“They all make the divisor into a whole number”

“8 appears on the left in two of them”

That last point was focused on by the teacher. They revisited all 3 methods and explicitly added any ‘missing steps’ (ie, “show your working” for us UK teachers), and out pops the key similarity the teacher wanted them to notice:


This was a subtle but excellent point in the lesson. Making every step more explicit helped guide the students to spot similarities, but also to understand the maths. In one example the numbers were the other way around (3×5) and this was also discussed to make sure people knew it was ‘ok’ even though it looked slightly different. A nice quick recap of the commutative property of multiplication was led by a student.

So with this commonality, students were intrigued as to why it always occurred, and reflected back upon the original problem (2/5 divided by 3/4). Here the teacher began to pull the lesson in the direction he wanted more than at any other point. He explicitly wrote the question alongside this commonality to make it more obvious to students what it was he wanted them to see. Sure enough, they spotted that the commonality was linked to the question in that it was multiplying the dividend by the reciprocal of the original divisor.

The lesson drew to a finish with the promise that in the next lesson they would look into whether that could be generalised – which inevitably would lead to another long discussion(!).

A few thoughts:

The ‘hints narrow reasoning’ thing has definitely struck a chord for me, although as mentioned previously, the balance is incredibly difficult and i doubt there’s a one size fits all approach to this.

The amount of time discussing and probing ideas is also a really hard balance to grasp, and I’m not convinced this teacher had it right, but it was certainly better than my own efforts.

Discussing commonalities between methods is a really powerful teaching/learning strategy that I don’t do.

I wonder how many students understood everything, and how many might just shrug it off and use the algorithm with no further thought. Checking who is ‘with us’ in the lesson and adapting is something I think the UK do really, really well, and thoroughly.

Planning the most difficult lessons in great depth over time with other experts is exactly the way it should be done – down to the finest details of what examples you use and what questions you’re going to ask / anticipate.



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,  to 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.  Regardless, 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.