Yesterday I presented at the latest LaSalle Maths Teacher conference in Bristol. I essentially ran a problem solving workshop with some of my favourite puzzles and tasks for students and teachers alike. The presentation is maths spa 2
One thing I’ve noticed more and more recently is how out of sync the transition is for mathematics students leaving primary (grade?) school and starting secondary (high?) school. I’ve worked in, and visited, a large number of schools at this point, and although certainly not always the case, I’d comfortably say the majority of schools I visit follow very similar, and in my opinion, flawed, year seven programmes for their eleven year olds.
Typically (I should be cautious using that word), students arrive on day 1 and are placed in mixed groups for the first few weeks. This period can last anywhere between two and six (!) weeks. This period is effectively a holding cell until departments determine the ability of each child according to their own internal tests and assessments, despite the fact that students are coming in with a lot of maths data attached to them from primary. The reasoning behind this is often cited as standardising across groups who have come from different schools, or protecting against students performing well or poorly in their final assessments in primary school, but not performing consistently at that level. Cynically one might say that this is all really about not trusting the data they come in on…
Once the students are finally re-assessed in Year 7 – which could be anything from a single test to three or four, students are then placed into sets – which could be anything as strict as a class-by-class hierarchy, or (my personal preference for what it’s worth) loose setting with three ‘bands’ that are essentially mixed within a couple of grade boundaries. Now I should reiterate that this is just what I’ve experienced, and certainly does not speak for the entire English school system, but as I said, I’ve seen a lot, and they usually fall into some kind of version of this structure. I’m really interested to hear about alternative approaches in the comments.
So what do students *do* in this interim period? Well, again, it differs, but often it’s looking at the basics of numeracy – number operations, proportion work, fractions etc. This in part is to help prepare them for the assessments, but also acts as a testing ground to see if students stand out as being higher or lower achieving than their data suggests. It can be completely detached from any kind of scheme of work or curriculum. I hear a lot about students ‘going backwards’ or ‘standing still’ in this time, simply because they (most of them) aren’t learning anything new, and much of what they’re being taught, or are revising, is stuff they did to death, successfully, at primary school.
At this point I should add a little balance. I’m not bashing teachers, and I understand that there is some truth when I’m told “they *should* know that stuff, but often they don’t, or they’ve forgotten it”, so perhaps this time is not wasted as such. In fact, in my own teaching experience I have frequently found that to be true with a lot of children, (but also false with others!). Certainly, if this entire process takes place over a week or two, then the implications of ‘not moving forwards’ are fairly light – the end justifies the means and all that. However I also feel that with all the massive overhauls of the primary curriculum, and the remarkable work primary teachers are doing (under impossible pressures I might add) with the new maths programs everywhere, that times have simply changed. Is a review needed? I think so.
What I’m seeing and hearing now is that Year 7 students coming into Secondary schools are, in general, much stronger at maths. They have a better conceptual understanding and ability with number. Of course not *all* of them, and of course there will be some areas where this difference is less obvious than others, but I do think the changes are having the desired effect. We could argue about the costs of those changes to everything else, but that’s not what I’m focusing on here.
With those changes in mind, the second thing I’ve noticed is that the entire Year 7 scheme of work is often blind to the new primary curriculum, which has been in place a number of years now. I don’t want to keep using anecdotal evidence here, so let’s take a look at an actual scheme side by side with the primary programme of study.
The picture above is taken from a scheme of work for Year 7 which is freely available on TES. A few disclaimers: anyone can put anything on TES. This was uploaded in 2010, but was updated in 2014. It has reviews from as recent as a month ago, so it is being used. It has been downloaded 20,000 times. There are more detailed breakdowns of the topics, differentiated etc but this is the broad overview.
From September to January there is literally nothing on there that isn’t taught at primary, several times over six years. Furthermore, almost all of it was taught at primary (several times across the six years) BEFORE the curriculum changes.
One could argue that we need to reteach things all the time (I agree), but typically we’d reteach it and add more content, go deeper, expand the concept. The above is taken from the Year 5 programme of study. It involves composite shapes, and is therefore arguably more advanced than what is listed in the Year 7 scheme of work (point 9).
Above is part of the Year 6 programme of study. It seems a lot more advanced than fundamental concepts of arithmetic using fractions (point 10), which is taught in January in Year 7 (for the scheme I’ve posted).
I won’t keep comparing, you get the point. The Key Stage 1 & 2 Maths Programme of Study is available here.
Many schools have fantastic schemes of work for Year 7, I have no doubt about that, and if you’re reading this thinking ‘yeah but we dont do that, this is rubbish’ then great, clearly you’re one of those schools.
All I’m trying to put across is that times have changed, and the Primary curriculum should, I think, feed directly into the Secondary one. The gap between Year 6 and Year 7 is no bigger than Year 5 and Year 6, so students aren’t forgetting anything more than they would normally. If you’re in secondary and aren’t familiar with the new (not that new now) programme of study, it’s well worth a quick read – particularly the Year 5 and 6 parts.
Any interesting contributions about how you structure your scheme of work (and your groups) in Year 7 are most welcome in the comments.
Someone tweeted me this puzzler a couple of days ago (I forget who it was, apologies):
“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:
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:
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:
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:
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:
So now I have some ugly trig to work through:
[/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:
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. “
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:
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:
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:
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:
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:
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:
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.
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:
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.
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:
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.