Sorry we keep lying to you…

It’s well documented that many school students and adults alike are less than fond of mathematics. It tends to be a theme I discuss on here. I’ve singled out over-emphasis on speed of processes, misguided attempts at trying to convince students it’s entirely relevant to their daily lives, and of course, not explaining things,  as factors. These are all ways in which we, the teachers, are sometimes subconsciously influencing things – but it’s certainly not all our fault though. We live in a country where, typically, it’s a badge of honour to be crap at maths and still miraculously live a normal life. Our bloated curriculum and imbalanced subject hierarchy don’t exactly help either. I was reading a maths book for trainee teachers the other day and I came across a familiarly painful explanation of the column method of subtraction, stating that you ‘cannot subtract three from two, so you must borrow…” blah blah blah. It got me thinking – just how much do we lie to our poor little mushy brained students when they’re absorbing what they think is the truth? What are the consequences as the truth suddenly changes conveniently and we forget to tell them? Perhaps I should hashtag this as #fakemaths. I know the best hashtags. Nobody does hashtags better than me. I digress…

Below is a letter I have kindly drafted for my maths teachers over the years to send back to me. I’ll even pay for the postage.

Dear Ed,

By strange coincidence, we, your former maths teachers, have all ended up in a room together with only a pen and some paper and your name and address on a sticky label. We figured it meant something, so we started talking and realised we’ve all been guilty of a bad thing or three. We hereby apologise for the following truths we accidentally lied to you about (because reasons):

  1. Really sorry we told you that you can’t subtract a big number from a little number. We just meant don’t do it or the algorithm is harder to interpret.
  2. Whoops, you know the squares? They’re rectangles too. You could have had that mark.
  3. My method isn’t the preferred method, it’s just one of many, and efficiency is subjective.
  4. Showing all the steps really doesn’t mean you’re a better mathematician, it just means I can mark your work against the criteria I’ve been given.
  5. The even numbers in the book weren’t harder than the odd numbers, I just didn’t have anything else to give you to do.
  6. You don’t need to be good at maths to go to a supermarket.
  7. You don’t tend to use algebra explicitly in restaurants.
  8. We told you ‘because it works’ because we didnt know why it worked.
  9. Opinions (before we’ve told you facts) aren’t really wrong as such.
  10. Equals doesn’t mean ‘find the answer’
  11. Algebra is not the same as 2 apples + 1 apple = 3 apples
  12. We weren’t really adding a zero at all – but shhhh don’t tell anyone.
  13. You don’t “just have to memorise it”.
  14. There’s no such thing as ‘borrowing’ in maths – except for forgotten pencils and rubbers.
  15. Wrong answers aren’t stupid.
  16. Mathematicians struggle too.

Hope we didn’t accidentally destroy the best subject in the world, but we totally understand if you think we’re lying to you about that.

Signed blah.

 

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Problem Solving #3

p2a

“A square is constructed as shown. Point A is the midpoint of a side. Prove that triangle BCD is isosceles (and not equilateral)”

If you don’t want to be swayed by my thinking, solve it now rather than reading ahead.

 

 

I like proof questions because they give you a little comfort blanket – you know the answer before you start. I stared at the diagram a little while, explaining internally how each line is constructed as a kind of mental warm up I suppose. Then I thought to myself “I’ll try and find the side lengths”. I think I was swayed to pursue sides rather than angles because I identified this triangle fairly quickly and figured it’d probably be useful:

p2b

I gave it side lengths of 2 and 1 for easy numbers, and so the hypotenuse must be

\sqrt{5}

Then I focused for a little while on this red triangle:

p2c

I hoped it’d be similar, but couldn’t convince myself that it was by using side lengths. At this point I only know the hypotenuse is 2.

So I thought maybe I could with angles.

p2d

I don’t know the values for the red and green angles. I could work them out as I know the side lengths, but I don’t want to yet. Instead, I noted that the green and red angles sum to 90 degrees. Therefore, this angle must also be ‘red’:

p2e

And as angles in a triangle sum to 180, and we have a 90 degree angle at point C, then the third angle in the red triangle must be ‘green’, so it’s a similar triangle to the one I started with. Phew!

p2f

Why is that useful? Well, now I can calculate side DC using similarity.

\frac{DC}{2}=\frac{2}{\sqrt{5}}

DC = \frac{4\sqrt{5}}{5}

Well, I’ve proved that triangle BCD isn’t equilateral at least.

Now to find side BC…

I know that angle BDC must also be ‘red’:

p2g

I could just use the Cosine rule and find side BC to be 2 or the surd for CD (I assume it would come out at 2 by inspection), but then I’d have to actually work out those angles, and it was late and I needed to get to work early because I’ve been locked out of my work emails all weekend (stupid password change requirements are so inconvenient). I had one last look before bed and thought ‘maybe there’s a way to show that angle DCB is also ‘red’…

The next morning I ate some toast (this is a theme, maths and toast. I highly recommend it). and stared at this diagram:

p2h

I stared for a good long while, thinking of that angle next to the right angle. My mind got a bit confused with itself as one of my unwritten approaches in my head was ‘assume it IS red, then what else’, and I found myself in a loop of using something else that would be true if it WAS red, to show that it was red… that doesn’t work in maths sadly.

I even sketched this to see if it would jolt any thoughts out of my brain:

p2i

It didn’t. Not that I’m consciously aware of anyway. As I brushed the crumbs from my face and sipped my obligatory morning mug of Yorkshire Tea, an idea finally struck.

I focused again on the red triangle:

p2d

It’s a right-angled triangle, so I can circumscribe it. Would that help?

p2j

It was a better idea than no idea. A few more details can be deduced now:

p2k

So the top line is a tangent, and the midpoint on the side of the square is the centre of the circle…

p2l

And the dotted line is a radius, so now I have another pair of red and green angles… i feel like i’m making progress now, but I couldn’t quite make what felt like a final move to unlock the proof. So i drew a second circle as I lost hope that the first would be sufficient on its own (perhaps it is and I missed something):

p2n

Notice I’ve ditched that tangent, it’s not important. At last, from this second circle I can happily convince myself that everything works, and the original triangle BCD is isosceles:

I know the two green angles are green from the previous diagram, and i know the dots A, B, C, D & F all lie on the new circle because …

Triangle BDF forms a right triangle for the new circle drawn.

Triangle DBA forms a right triangle with the same height, so the midpoint of its hypotenuse is the same as for BDF – so it’s the same circumscribed circle (so A lies on the circle from BDF)

We know ECD is 90 degrees, so DCA is 90 degrees, so that can be circumscribed.

DA is the hypotenuse of DBA and DCA so all three right triangles have the same circle circumscribing them all. I’m now convinced C lies on both circles.

That’s important because it tells us that BFC is also circumscribed by the same circle, and so FCB is 90 degrees and so DCB is… RED!

I have a feeling I might not have needed both circles for my solution. But I’ve moved on.

Here is the solution approach provided with the problem:

sol

Problem Solving #2

Whilst eating some toast I felt the need to have a go at another geometry problem this morning:

ps2a

(not drawn to scale)

“If the radius of both circles is 2, and the area of both the equilateral triangle and rectangle are the same, find the dimensions of the rectangle”

I liked the diagram and the idea so I thought I’d have a go.

This question felt easier than the last one in that i have a direction to go in right from the start. I’m told the areas are the same, and I can use the diagonal of the rectangle to get a pythagorean formula, and a second formula from the triangle, then solve. So there’s less pondering at the initial stage:

ps2b

Using the red triangle, we get

x^2+y^2=4^2

The second formula looked a bit trickier to determine. I decided to go for this triangle:

ps2c

Which is a 30,60,90 triangle, and so has side ratios 1:2:\sqrt{3}

I mixed up which side matched which part of the ratio initially, but it didnt look right so I went back and corrected it fairly quickly.

\frac{a}{2}=\sqrt{3}\ so\ a=2\sqrt{3}

Now I have the side length, I can derive the area:

2\sqrt{3}*0.5*3=3\sqrt{3}

And so I have my second formula for the rectangle:

xy=3\sqrt{3}

squaring this gives :

x^2y^2=27, so\ x^2=\frac{27}{y^2}

Substitute into the other equation:

y^2+\frac{27}{y^2}=16

y^4-16y^2+27=0

Now I have a quartic, which I’m not happy about. Fortunately as I only have a power of 4 and 2 to deal with, I can substitute with say, z:

z = y^2

z^2-16z+27=0

z=8\pm\sqrt{37}

y^2=8\pm\sqrt{37}

This is getting yucky. At this point I square rooted everything and started messing with

xy=3\sqrt{3}

Which got even messier to the point that I stopped and had a rethink. I went back to

y^2=8\pm\sqrt{37}

And it dawned on me that if I use

x^2=16-(8\pm\sqrt{37})

then I can see by observation that when:

y^2=8+\sqrt{37}

x^2=8-\sqrt{37}

and when

y^2=8-\sqrt{37}

x^2=8+\sqrt{37}

Hence the rectangle has dimensions \sqrt{8+\sqrt{37}} ,\ \sqrt{8-\sqrt{37}}

A nice twist at the end, but the answer was ugly, and made me sad.

Problem Solving

Someone tweeted me this puzzler a couple of days ago (I forget who it was, apologies):

puz

“If the square has side 2cm, what is the radius of the circle?”

I had a stab at it today and thought I’d share my process, which may well be awful, but as Bob Hoskins famously didn’t quite say, “it’s good to share”

My first thought was “it’s a circle, I need a radius, where can I get one”. So I drew in a few lines:

puz2

I know that where the dotted diagonal lines cross is the centre of the circle, and I drew in the bold black line because I know I can calculate it, which might come in handy.

I thought I better have a visual reminder of what lengths are equal, in the hope it might prompt a new thought:

puz3

Then I just stared at it for a while.

I couldn’t visualise how calculating the black diagonal line would help me, even though I could do it. So I drew in another line as I thought about it:

puz4

Which helped me spot the blue arrow thing which kicked in a thought about circle theorems. At this point, note I didn’t have a direction I was going in, I was just trying stuff out. Testing the waters to see where it’s shallow. I thought about the circle theorem thing, and eventually decided against it, and ditched the idea. However, looking at the diagram with the two angles on it made me realise the direction I wanted to take:

puz6

I want to work with this triangle, because it has the radius (twice, bonus) as its side length, and I can use the length that I know I can calculate (the longest side of the green triangle, which was mentioned in the first diagram). So now all I need is the angles inside the green triangle and off I go. And to get those, I can use the big red bottom left triangle:

puz7

So now I have some ugly trig to work through:

puz8

[/insert elevator music whilst ugly trig is being worked through]

and out pops r = 1.25 cm.

The nice answer makes me suspect there’s a far nicer approach to this. I’ll ponder it some more.

*Update: A few people have sent me their approaches, which are predictably simpler than mine. Interestingly, some are asking why I ‘chose to solve it in this complicated way’. I didn’t choose to! This is where my first thoughts took me. Anyway, here are some alternative approaches:

DIpXZC4VoAAVGuz

And from the comments section:

“If you let the height of the square be r + h, where r is the radius of the circle and the width of the square to be 2w (w = 1), you can form a triangle with sides w and h and hypotenuse r. From here do a little Pythagoras and the answer of 1.25 drops out quite quickly. “

“thaMographe” Review

I’m not sure where I’ve been for the last five years or so, but I’ve recently discovered a cool tool called the ‘thaMographe’, invented by Thierry Delattre. It looks like this:

le-thamographe-1-seul-instrument-de-math-au-lieu-de-4

It’s a small tool and the inventor claims it can conveniently replace a compass / set square / protractor / ruler. An all in one tool a bit like a swiss army knife but for maths.

Anyone who has taught constructions at school will be familiar with the typical frustrations that come about regardless of how well you plan for them: super loose compasses, students jabbing each other, compasses with the pencil fastener missing, students getting cross because their compass slips as they’re drawing a circle, or they wobble it as they try to figure out how to manoevre their hand 360 degrees around their other hand. It’s a tough unit of work, if only because of the frustrating elements of the equipment we use. And so I was pretty excited to find what could be a solution to all this – not to mention a possible money saver for schools.

I’ve been using one for three days now and I’m really impressed with how easy it is to use. The video below demonstrates the basics:

You’ll notice that a key feature of the tool is the central line, which enables quick and easy drawing of shapes and angles, without lifting the pencil from the paper. Pretty innovative stuff!

*edit*: I found this other video which demonstrates the capabilities in a little more detail:

I managed all of the typical compass requirements for a ‘school task’ easily:

The only struggle I had for these is that I was using a pad with a ring-binding – which meant that as I swung the tool around, it sometimes got stuck and couldn’t get past. Whoops. So don’t use ring-bound pads! Again, this wouldn’t affect a school student I imagine. I did wonder if it would get stuck on the central binding of an exercise book, but I tried it with a couple and it got past quite easily with a little prodding. I’m not sure this would even be a likely scenario in many schools as lots use individual pieces of paper rather than squared exercise book paper for constructions work.

You may have noticed that the tool has individual holes that are set such that you can adjust the radius of a circle by the millimetre, so there’s almost no loss of ability to resize the circle with a radius of between 10 and 110mm. The nice rubber segment where you place your finger as the compass ‘point’ is comfortable to use and very easy to push down and hold the tool in place. It has never slipped in all the times I have used it. It’s also easy to put a standard pencil in the holes, although it needs to be pretty sharp. If you want to use a pen you really need a a fineliner.

What was really nice was that you can happily stop and start any arcs or circles and you don’t get any wobbles or double lines in your diagram. It’s also really easy to swap fingers around when you get most of the way around a circle, and it doesn’t ‘upset’ the diagram. This is a great advantage over the traditional compass, especially for schools.

In summary, for schools, the advantages are pretty clear – it’s only one tool instead of several, it doesn’t have sharp bits, it doesn’t get ‘loose’ in the way compasses do, it doesn’t slip, and it does all the things required from the UK curriculum very easily. I would prefer to have them in my department rather than a myriad of protractors, rulers compasses. I don’t know if they’d save you money overall, as I don’t know the current cost of a class set of rulers / protractors / compasses, and I doubt many schools stock class sets of set squares. Either way, they are available with discounts for bulk purchases here:

Buy From Tarquin

As a massive geometry geek, I thought I’d try out a few more advanced things with it, to see if it could replace a compass for what I use them for… I suspected the accuracy of a compass was going to win overall, and I was right.

I tried a little quadrature of a rectangle first:

IMG_6628

I was happy with the result. Not perfect, but that’s more me than the tool. So far so good.

Then the ultimate test, this tricky Islamic Art style thing:

IMG_6629

You can see I was far less successful with this one! Where three circles are supposed to overlap on one point, they often miss a bit. This was largely because I struggled to line up intersecting lines with the little cross on the tool, compared to just sticking a compass needle on it.

Here’s a compass version:

IMG_6630

Now in fairness, I’ve used a compass a LOT more than I’ve used this new tool, and I tried again the following day with much better results. This time I was a little more strategic and lined up not only where the cross should go, but also checked that the pen lined up with all the intersections before just blindly drawing it from the cross. The result was much better:

IMG_6631

But I still felt a needle was easier.So while I will definitely put this tool in the back envelope of my notebook (another handy advantage!), I don’t think I will be throwing away my compass just yet.

Either way, I’m pretty sure this tool is NOT designed to replace a compass for more advanced geometry, and it of course has limitations.  It’s designed to replace 4 school tools with one, and it does that, in my opinion, very well. Two thumbs up.

 

 

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:

comp.PNG

Or at least as a/b.

A second student’s work showed this:

work2

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:

w3

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:

24

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):

prop

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:

dnline

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.