Monday, 14 January 2019

High Context, Low Context

Thanks to Michael Pershan for writing a lovely piece about ancient Greek mathematics. He links to and comments on aspects of an article, The two cultures of mathematics in ancient Greece by Markus Asper. 

Asper says:

The words of, for example, Euclid or Archimedes appear to be of timeless brilliance, their assumptions, methods, and proofs, even after Hilbert, of almost eternal elegance…Recently, however, a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.
The part of the iceberg that is underwater, so to speak, is practical mathematics, the kind that builders need for construction, that commercial calculatiors performed on a sand tray with pebbles and that surveyors used when stretching out ropes to measure land.
Some of the books of the theoretical, abstract mathematicians have come down to us; the everyday mathematics of craftsmen, engineers, accountants and surveyors is all but lost to us.
Euclid distilled the discoveries of the two centuries before him into a careful tower of propositions, each careful built on the one below, logically building upwards by detailing one step at a time:

The fragments of builder wisdom that have turned up in rubbish tips in Egypt couldn't be more different: recipes for how much stone you'll need for a particular sized house. There's not even a general formula.

High Context, Low Context

At this point I want to bring in the anthropologist Edward T Hall's description of high context and low context cultures. For Hall, “a high-context communication or message is one in which most of the information is either in the physical context or internalised in the person, while very little is in the coded, explicit, or transmitted part of the message”. In a low-context communication on the other hand, meanings are explicitly stated through language. Hall was interested in cross-cultural mis-understanding and understanding. I want to mention maths education in the light of his distinction.

Euclid is low-context: he tries to make every little step explicit for us.

The builder's recipes on the other hand were high-context. The builder works with his apprentices on site and such recipes as were written down would be adjusted and reworked within a social, oral, gestural, practical context.

Can you see where I'm going with this?

Euclid has a priveleged position in the history of mathematics and of learning mathematics, and it's style, impersonal, logical, abstract, has permeated mathematical writing. Moreover, the textbook and the test have, for additional reasons of their own, adopted the same low-context mode - everything you need to know must be contained in the writing

This kind of low-context writing, that has to contain all the information that would normally be carried by a social situation, the possiblity of clarificaton, questioning, gradually tuning in, is hard to read for the uninitiated. It's not our normal kind of communication.

Why do we funnel children through a low-context communication, the word problem, the odd-one-out verbal reasoning question, the test question, when we're in a high-context situation? We're like the builders, the students are like our apprentices, we have a social, oral, gestural context in the classroom, we have lots of materials. We can present fragments - an image, a numberless problem, an equation. There doesn't need to be a question. Through the unfolding of the lesson in the high-context culture of the classroom we can gradually, through dialogue and practical work, build the knowledge together.

Tuesday, 27 November 2018


Rather than triangles, which was on the Grade 3 (English Year 4) plan, I felt that hexagons would give us more scope for focusing on properties. It's worth looking at Christopher Danielson's lessons on this. I wasn't really expecting to get on to proof, but if a conjecture came up, I would be happy (it didn't this time!). We started with a Which One Doesn't Belong?
I mentioned that three of them were hexagons; they didn't have to be regular to be hexagons. 

We asked the classes to make hexagons from pattern blocks, to make one whose shape was not chosen by anyone else. This had advantages. The hexagon, kind of, 'belonged' to the student. We got a range of different hexagons that we could construct and measure easily, and would also, mostly, tile. But I'm still wondering about the gains and losses from this approach, and I'll come back to that.
Finally, after a number of repeats and non-hexagons that we didn't spot straight away, we had a set:
which I reproduced for the class without the colour, so that we'd be focusing on the component shapes less:
- still time for another pattern block hexagon WODB though:
Notice that 'looks most like a hexagon'! We were going to shake off that regular hexagon thing...

We looked at side length, counting the length of almost every side of the pattern blocks as one. I also talked about perimeter. (BTW: my opinion is that it's really good to introduce this separately to area. Although it goes together with area in our minds like salt-and-pepper, my feeling is that some students get a little which-is-which?) We documented this for our hexagon, later adding on parallel lines and symmetry:
Then, starting from the known fact that a square's corner is 90°, I asked the class to find the angles on the other pattern blocks, which, together, step-by-step, they did. The big breakthough was when a pair realised that three of the acute angles of the thin rhombus fit into the squares corner. Then that was a measure for all the others.

So, the next day, we applied this to our hexagons. I'm pleased about this part of the sequence, because it seemed to make it possible for everyone.

Time for another WODB then next day, with four of our hexagons with angles shown.
I hesitate to add this bit, because I'm unusually lucky to have a colleague in secondary who laser cuts me shapes out of acrylic, and that's not accessible to many. But the kids were really excited when these arrived...
 Immediately they began tessellating them:
 We recorded this later:
 This one AK couldn't get to tile:There's a couple we had difficulty tiling:
Martin helped out with one:

And Hana helped out with the other with a tiling that includes some other pattern block shapes:
Rod did the same:

All this led on nicely to our other tiling work with the pattern blocks.

To review what we'd done, I made little books about the 25 hexagons (taking too long on all the images, which was useful in class but I don't recommend to others!) and asked children to write about different aspects of them.

 The books showed that the students had a lot to say about the various properties we'd looked at.

We also finished off with a WODB that students annotated individually. This told a mixed story. I'd chosen some hexagons that weren't within the pattern block family.  The students did notice lots, and showed their knowledge through their annotations:
But some of them also treated these hexagons as if they had the same uniformity that the pattern block hexagons had. Some saw, for instance, the diagonal lines of the green shape as parallel. Some tried to see the hexagons as composed of pattern block shapes. And some tried to measure the perimeters - oh dear, I hadn't thought of that, and hadn't built in whole-number lengths! I kind of feel this was unfair of me, but it has made me more aware of something.

It seems we have some things to unlearn now. Another thing: for our brief estimation actiivity the following week I used Estimation 180's What degrees of the pie have been eaten? So many students put 90° or 60° - and suddenly I realised, now they think all angles are going to be in mutliples of 30°!

So we may need to unlearn:
  • all lines that look vaguely parallel are parallel;
  • all hexagons are made up of pattern block shapes;
  • all side lengths are whole numbers;
  • all angles are multiples of 30°.
Luckily, it's not hard to find counter-examples that undo most of these overgeneralisatons that I've led them into. The undoing could even be an impactful way of learning these important points. So, I'm not too disheartened.

But I'm still mulling this over. Would it have been better to have given a selection of more varied, non-pattern-block hexagons, like Christopher Danielson's
They would have lost the personal creation, the ease of measuring, the tiling perhaps. But they would have had a bigger, more representative example space.

Or I could have chosen a different playground for hexagons. A geoboard maybe, or as in Lana Pavlova's lovely hexagon work, the tangram. Or moved more quickly through differrent embodiments of hexagon.

All this connects with some of my recent thinking about examples and generalisation. Have a look at this, from Thinkers by Chris Bills, Liz Bills, John Mason and Anne Watson:
I'm now much more aware of how our choice of example, the variation we include, carries its own potential misinformation. Students at the start of this felt that a hexagon was a regular hexagon - that's what they'd been shown. We show a range of rectangles and don't include a square. We call the blue and light brown pattern blocks rhombuses, but (natually) call the square a square, even though it's a rhombus too. We look at fractions as circles. Students are going to see features of the examples we present as essential to the properties we want them to become familiar with. Some of those features will be, but others won't. 

So the 'problem' with my hexagon lesson is perhaps a more general one...

How would you do a set of lessons on hexagons? What do you think about this problem with examples? As usual, I'd value any comments. 

Saturday, 20 October 2018

A hundred pattern

Back in Grade 3 (English Year 4) now. We've been doing quick dot images once a week, looking at the different ways the students see the images without having to count every dot.
We've presented it to our parents and the school, from Grade 1 to 5, getting them to talk to each other about how they saw it.

It was time to try and count something bigger, to give the classes a bigger challenge. Could they use their ability to 'conceptually subitise' to help solve a more challenging problem? For a while I thought this would be a good one. The holes in the drain cover are conveniently - almost - in groups of ten
But then I remembered this pattern with Cuisenaire rods. It was one that had started off a wonderful discussion on Twitter (See Dan Finkel's vlog of it). But it had those triangles, a bit like the triangles of dots we'd just looked at. Would the students be able to use some of the pattern in this to make their counting easier? I was not sure whether this would be a bit too much of a leap...
 So we could write on it more easily, I photographed it again on white.
The question: if the white rod = 1, what is the whole thing?

Students worked in pairs to try and work it out.
Some pairs broke the image down to help to see it.
I didn't expect that some pairs would rebuild it...
... and some would rearrange it.

It was really interesting to get an insight into thinking. Tracy Zager has just tweeted a flow chart about mathematical thinking that includes the need for the teacher to see the students' thinking in some way.

This student had worried me a few days before when she'd asked me what half of 20 cm was. But now she was working out what three times 15 and three times 18 were. She didn't quite get to the correct answer, 100, because she didn't include the one in the middle. But she's done all but a tiny bit of the thinking! In fact, most of the students who got it wrong, were just getting one small thing not quite right. Her partner thought it was 102 because he counted the middle one three times!
Another approach was to add up all rods of each colour separately and then total them:

Interesting how they'd written the threes down as twos first of all, and then appear to have compensated for this mistake at the end by adding another nine on.

I pulled the class together and shared these two main strategies. I think in hindsight I would like to have dwelt on this more. I would have liked the class to have looked at what people were doing with the rods too, but it was all really busy! Perhaps we can return for a recap after the two week break.

Someone muttered something about failure. I'm not sure how convincingly I answered this. I said that I thought that even when people didn't get 100, they'd done a lot that was right on the way to 100. He seemed to accept this, but I'm sure there's more work to be done establishing a culture where we focus on showing our thinking rather than thinking about right/wrong.

What we did do was really interesting. I asked the class to make their own hundred pattern, so that it wouldn't need to be counted. (Not just ten orange rods, I said!)

There were all sorts of great approaches:
And some more:
We had a look at that double staircase afterwards; it bothered someone that there was only one orange rod!
The student with the yellow circle was making more hundreds, this time making his fives differently. I wish we'd looked at this too. I would have liked to give him credit for not just doing what I'd asked, but for taking it further and trying a variation. We talked about the person, and some students asked me if we could have a go at doing some like this. There wasn't time this time, but I'm so pleased how many different kinds of creations there were, and also that there was talk of coming back to it in a different form! (Time for some #hudnredface images too perhaps?!) I was pleased that there'd been a lot of choice in the two tasks - first in how they solved the question of what the picture represented, then of how to represent 100 differently. There had been so many different approaches to both!

One thing. There were mistakes. But a lot of them were what you might call silly mistakes, ones that the students realised were wrong as soon as it was mentioned. Counting the central white rod three times. Putting just eight tens in their design rather than ten. We all make mistakes like this, and maybe we can as a class get more forgiving about them. Also, having more practice with seeing numbers and counting flexibly should make the mistakes happen less often.

Saturday, 26 May 2018

Multiplication with Cuisenaire

In our number of the day sessions (which we haven't done recently but various students keep asking me that we do again) we evolved to writing the number of tens in the day number as one way of saying something about the number. 94=9x10+4. Most of the students seemed comfortable with this.

When we used letter names for Cuisenaire rods (which we stopped doing about half way through the year), it was natural to write 2r=p for 'two red rods are the same length as a pink rod'. Notice how that hardly seems like multiplication at all; more like counting.

I'm intrigued by how this seems to be easy for young children. Multiplication is meant to come later. What is going on?

The answer seems to be that, after the children have got to know the rods, they work as "units". I've just been reading Christopher Danielson's excellent Teacher's Guide to his wonderful How Many? and it's been helping me to think about this idea of a unit.

If we used cubes instead:
there are other units involved, units we could count. There are maybe four lots of counting going on at the same time. Counting how many cubes in the red two. Counting how many cubes in the brown two. Counting how many twos there are. And counting how many cubes there are in the four. It's too crowded; too much counting at the same time. We all know how patting yourself on the head and drawing circles on your stomach at the same time is a lot more than two times harder than doing them separately.

But, when through familiarity we've got used to say how the yellow rod can be seen as "five" (being five of the smallest white rod in length), it begins to behave as a unit. So, if we have four of them:
it's like having four apples. That the five is composed of five ones is in the background, not interfering with our count.

Which is my guess about why everyone in K3 (5 ad 6 year olds) was able to think about multiplication in this lesson (we call it 'times'; every now and then I throw in a "lots of"):

 There's an album of these photos. (The tracks are available from Numicon. They were developed by Tony Wing, starting from the work of Catherine Stern.)
I'd be interested in your thoughts.