Monday, 23 March 2015


I used to be the worksheet king. Making them clear, uncluttered, as simple as possible. They're all, hundreds of them, on the system at work. Here: I've found one from February 2005:

That must have been Year 6, ten, eleven year olds, I was teaching that year.

But these days I seem to be using them less and less. And the Abacus textbooks, with lots of sums and things in,
I've hardly touched them at all this year.

What is happening?

Partly it's that I'm more and more wanting the kids to be creative in their maths work, and neither the worksheets, nor the textbook seem to give enough space for this. With creative work, there's usually a stimulus at the beginning of the lesson (after we've done a bit of number circle or estimation or some starter or other) and then there are constraints (often the manipulatives we use are part of this) and a requirement that I give by speaking to the class, maybe an example. Then off you go...

I'm not against either worksheets or textbooks. It's just the way I've been moving. You can see why perhaps in my earlier posts. (You can see some of our work on the Year 4 blog.) And, besides, as well as my own ideas I've been getting so many from the great people in the Math-Twitter-Blog-o-sphere #mtbos that there's really not enough time to do half of the things I want to do.

So I was interested to read a post by Andrew Gael on different kinds of worksheets that approach the same task. I immediately wanted to do a worksheet-free lesson based on the same premise. But I also wanted to see what the worksheet would do for us. I chose the one I liked best, and passed it out without too much explanation to my Year 4s (8 and 9 year olds). I had to translate a little. I don't call it graph paper; I call it squared paper. But most of them could see the idea, though I think some them were a bit fazed by getting a worksheet with written instructions out of the blue like that. A few asked for a bit of supplementary explanation.

Quite a few found it straightforward, like this:
For one it was just so too easy, he went all one-dimensional:
Some got the side lengths wrong:
And one got a bit confused:
Aside from anything else, you can see it's a worthwhile activity just to draw a grid of a certain size (just as it's a worthwhile activity for young kids to create their own number line) and about half the class need a bit more practice at this. I think about a third of the class could do with repeating this activity and getting it right. So it's been informative. All of them could do with annotating their grid with a bit of explanation. Checking wouldn't be a bad thing either! Tomorrow.

What did we do next? More of an open-middle kind of thing. They cut rectangles form squared paper. Wrote how many squares on one side and their name on the other. Then they cut bites out.

We looked at them a bit together, and worked out the area of the original rectangles of some of them. We'll return to them, and the worksheet, tomorrow.

Anyway, what do you think? I've changed Andrew's question to fit my case. What are the advantages of the worksheet here? (Should I be using them more?) And how about the task afterwards? I'm interested in your thoughts.

The next day I gave everyone their work back to check, and change if need be. I also gave one of these orange sheets (with the orange taken out to make space for writing) to everyone.
We'd been talking about 16 X 5 type questions - and will talk more.
Most people seem to have got the idea; though I think I'll throw a few "draw me a rectangle with 45 squares on it"-type questions when we all have our whiteboards for a lesson starter.

Thank you, all you brilliant commenters! I've certainly got one or two things a lot clearer in my head about worksheets good and bad, and more besides.

Sunday, 15 March 2015

A problem from Sweller

This is just a draft - not for general consumption. I've backdated it so it won't show up unless you actually have a feed that follows my blog. This post will self-destruct in seven days. Meanwhile a less combative and more positive version will appear.

I don't usually do negative. I mainly report on my own experiences and steer away from saying what other people do is bad. But I'm going to make an exception today. I don't feel very comfortable doing it; I don't really like controversy. But maybe I'm a bit of a coward and should stand up for what I know a little bit more...

Knowing how influential his ideas are, I recently read John Sweller's Story of a Research Progam.

And there, right at the foundations of his cognitive load theory, was an experiment that seemed so poor, that I wanted to say why it's so poor:
The Beginnings of Cognitive Load Theory 
 I, along with research students Bob Mawer and Wally Howe, was running an experiment on problem solving, testing undergraduate students (Sweller, Mawer, & Howe, 1982). The problems required students to transform a given number into a goal number where the only two moves allowed were multiplying by 3 or subtracting 29. Each problem had only one possible solution and that solution required an alternation of multiplying by 3 and subtracting 29 a specific number of times. For example, a given and goal number might require a 2-step solution requiring a single sequence of: x 3, - 29 to transform the given number into the goal number. Other, more difficult problems would require the same sequence consisting of the same two steps repeated a variable number of times. For example, a 4-step problem always had the solution: x 3, - 29, x 3, -29 while a 6-step problem required 3 iterations of x 3, - 29. Accordingly, all problems required alternation of the two operations a variable number of times.
My undergraduates found these problems relatively easy to solve with very few failures, but there was something strange about their solutions. While all problems had to be solved by this alternation sequence because the numbers were chosen to ensure that no other solution was possible, very few students discovered the rule, that is, the solution sequence of alternating the two possible moves. Whatever the problem solvers were doing to solve the problems, learning the alternating solution sequence rule did not play a part.
Cognitive load theory probably can be traced back to that experiment. My objections to the many variations of discovery and problem-based learning also have a similar source. While the puzzle problem-solving task used had no direct educational relevance because such tasks do not form part of any curriculum, the results seemed to say something about how students learned and solved problems. It was obvious to me that if I had simply informed students to solve each problem by alternating the two moves until they reached solution, they would have immediately learned the rule and would have been able to solve any problem presented to them no matter how many moves were required for solution. Of course, since these were problem-solving experiments, I had not informed participants of the alternation rule and most failed to discover the rule for themselves.
I find this really astonishing. Sweller doesn't seem to understand what learning is. He thinks the students shouldn't be working on the structure of the problem, but on a pattern that's in the question poser's mind, in this case the alternating x3, - 29 pattern. He thinks what needs to be learnt is something in the task setter's mind rather than in the exploration of the problem.

When I produce a pattern in my students minds that's a product of my limited example choice rather than the phenomena themselves I'm concerned.

For instance, my class of 8 and 9 year olds looked at angle using the angles on pattern blocks as an example. These all have angles that are multiples of 30°, which makes things easier for a first sweep. But I was surprised to find later, that, when they were asked to estimate the angle of a slice of eaten pie that was 72°, they estimated 90°. I said it was clearly less than a right angle, and they switched to 60°! I realised at that moment that they had taken the limited example space of our first foray as being the whole story: angles come in 30° multiples!

But here Sweller is concerned that the students aren't picking up on his limited example choice.

I don't think he sees that this - mathematics - is something that can be investigated by the students. In fact what he's attempting to communicate is not mathematical. It's just an arbitrary choice of the experimenter. It may as well have been something in any subject - say a list of Babylonian kings - for all the arbitrariness and lack of mathematics in the learning.

There is mathematics in this type of problem, but it lies in a different direction to the one Sweller is pointing.

I wondered how my class might approach it.

The -29 makes the arithmetic a bit heavy. But what if we switched to -3 instead, and ×2 instead of ×3? That would make the arithmetic more manageable.

I needed to make the whole thing intelligible to the students, so we looked at function machines first.

Emily Allman had shared pages from one of Mitsumasa Anno's books where there are a wonderful series of function machines. We looked at them closely. This one features doubling:

We tried out a few other machines, where I asked for inputs and gave the outputs. We did the same in pairs creating our own machines and functions. We got inside a big cardboard box to be the function machine, receiving numbers through a hole on one side and posting them out modified through another hole.

The next day, we were ready for this:
Children were in pairs, with whiteboards, some doubled and subtracted three, some just doubled. The ones that chose to double were getting to some pretty big numbers!

After a while I collected in some of the numbers that the pairs had found:
We returned to our search for a while longer. I went and got a hundred square. Stopping the class, I asked which numbers they had managed to find now. We turned the ones we had found over to the red side, leaving the others blue. They were starting to tumble now.
Then AP noticed something. There were some blue diagonal lines left, the ones we hadn't managed to get to. LD noticed the red-red-blue pattern. MT noticed that the blues were the numbers in the three times table.

Time had run out, so we returned to it the next day, to record what we'd been exploring, and extend it too. The students were starting to feel confident with ways of reaching numbers:

Some were still looking at doubles. One pair noticed that the last digit of the doubles was in a pattern after 1: 2,4,8,6,2,4,8,6.
 I encourage the students to write down what they had found:

We were on the verge of finding out not only what was obtainable from the two machines starting at 1, but also why. We could see that a certain pattern of numbers was obtained from doubling. We could also see that there were three sets of numbers, the multiples of three, the numbers one less than a multiple of three and the numbers one more than the multiples of three. But I felt that pushing further would be overstretching things, and we should quit while we were ahead. It had been an interesting and profitable exploration.

We did return to the idea of repeated doubling later in the week with a reading of the wonderful book One Grain of Rice by Demi:
There's a real wow! in this book when the commonplace doubling 2,4,8... soon becomes... millions. Actual gasps from the students and the pull-out spread of elephants delivering hundreds of millions of grains of rice!

With older students, I might be inclined to push for the structure of the network of numbers that are obtainable:
What other questions could we ask? How would varying the initial number affect this structure? How would changing the amounts by which we multiplied and the amount subtracted alter the pattern? It would be good to view using mod 3. What pattern of numbers in mod 3 do you get by doubling? Subtracting 3 doesn't alter these, so it becomes clear why some numbers are unobtainable.

All this, I hope you'll agree, is very different to the impoverished view of learning presented by Sweller.

The problem isn't just with Sweller's particular research though. It's partly an artifact of scientific method. Experimental method rests, famously, on changing one thing and keeping everything else the same. But what happens if the phenomena are more complex than this allows for? Well, you can, quixotically, plough on. Like Nasurdin looking for his keys under the lamp post. His friend helps him search, and after a while asks him exactly where he dropped them. "Over there, in the dark," Nasrudin answers. "Why are we looking for them here then?" asks the friend. Nasrudin: "Because there's more light here."

Great learning can happen within a certain ecology: in a pleasurable search, in a supportive culture, a succession of individual, paired, group and whole-class thinking, a to-and-fro between teacher and students - a gathering and recording of ideas, crediting of those who take the thinking further, a return to search further, a respect for the powers of the learner.

Our educational research experiments are just not up to the job though. They can't search for the key here. And so they end up looking "where there's more light" and making recommendations in their own restricted image: "Teach just one thing at a time, test it after."
Knowing the structure of the numbers obtainable with ×2 and -3 is of course, as Sweller says, not an essential piece of information. But this kind of task brings with it all sorts of other benefits. Students starting from a simple question, deciding which direction to go in, exploring, discovering new territory, noticing features and patterns, becoming familiar with the territory, proposing generalisations, investigating further. This is the gift from mathematics - we must guard it jealously!

Saturday, 7 March 2015

Folding fractions

Fraction Fortress was good for building up familiarity with fractions. But Year 4 will need more soon. Last year we used Cuisenaire rods and made fraction flags. What will we do this year? Estelle mentioned fraction faces. That's a nice idea.

And then I saw somewhere (where?) the idea of folding a square in different ways to make different fractions. I like that. Different ways of getting a half. A quarter. That seems worthwhile. And gives a space for children to explore in.

I saw this great page on folding fractions too by Rachel Thomas at Plus Magazine. It shows a way of folding a third that young children are not going to discover.
Thirds are usually a bit of a surprise to 8 year olds. They don't get to see them very much; so it's worth meeting them in various ways, including this one, even if they don't see why this is a third. It's easy to fold too, just three folds.
 And you get a ⅜ thrown in free too. And that other fraction, whatever it is (that would be a good one for older kids).
I thought I'd try a fifth too,. I started by halving the paper like this, and then folding over a quarter:
- - - That's neat - the orange part is a half!
Anyway, do that four times and you get:
And in the middle is a fifth. There's a neat visual demonstration of that. It would be worth seeing what else the children notice about that last picture. What do they wonder too? (There are some good fractions here for older children to work out too.)
That seems like a good way to spend half an hour.

Another idea: fill up those Cuisenaire rod squares in different colours to show different fractions.
The hundred would do halves, quarters, fifths well. The 36 square would be good for halves, quarters, thirds, sixths. And so on.

Friday, 6 March 2015

One and a half small lessons

I've been trying counting circles in my Year 4 class. Mostly it's gone fairly smoothly, but interestingly when I tried it with jumps of one and a half a lot of the children found it a bit tricky.
We got there, but it was interesting to see there was a fair amount of faltering. I thought, I need to return to this. Maybe just on a plain number line.

Then I saw something interesting on twitter. It led to Kassia Omohundro Wedekind's blog post on  a similar thing. She had used "brownies" and half-brownies to help the same age children make connections.

It didn't provoke quite the same ideas in my mind, but it provoked ideas. Whole biscuits and half-biscuits would create an interesting pattern as they were laid down. Perhaps we could see what was noticed?
Well, it was just too jam-packed this week to fit this in. But I did have 15 minutes with the other Year 4 class, and I thought I'd give it a try. It's introduced here as a decimal activity, but I'd definitely started it off as a fraction one!

Once again, there were some hesitations and mistakes among the correct moves. Which tells us that it's worthwhile doing this. I'll recommend that both classes spend a little time counting in one-and-a-halves in a number of different ways.