## Tuesday, 27 November 2018

### Hexagons

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
 source
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.

1. So many great explorations in these lessons!
And I agree with your observations on how our choice of representations affects the learning. It always does. No matter what set of hexagons (or any other concepts) students end up investigating, there is always a potential for overgeneralizations. It's like, there is always WODB going on, and the features that are common for all WODB elements are at risk of being overgeneralized.
And it could have been a negative thing if 1) I am not becoming aware of it as a teacher 2) I am done teaching this group of students and don't have an opportunity to continue this work.
I really loved how pattern blocks allowed to extend the exploration of hexagons to introduce perimeter and angles in different contexts (tangrams didn't quite lead into these). You brought it non-blocks WODB at the end that provided a different hexagons models. Now you can do the same with perimeter and angles. Students already know what angles and perimeters might be like, so now you can build their understanding of what else they might be like.
After reading your post, I thought that we always have some incomplete models and definitions in our heads that we adjust as we get exposed to the new examples and counterexamples. The awareness of my students' models can help me create experiences for them that will alter their ideas. And I think these moments of cognitive dissonance, wheb students realize that something they thought true no longer holds are very memorable and lead to the learning that persists.
Yesterday, I discovered in an unrelated activity that students were treating an equal sign as "next comes the answer". We discussed what an equal sign means. In their reflection at the end of the lesson, many mentioned that it was their "wow" moment and the biggest learning of the lesson. I suspect this will stick well in their memory.

1. Thanks Lana!
I like your thought about cognitive dissonance, and I agree: my feeling is that these moments are memorable; we just have to make sure that they're pleasurable as well. I won't 'correct' any 'misconceptions' (and I think this is not a great word; often they're just preliminary conceptions), I'll try and find ways they can see the critical counterexamples.

John Mason and Anne Watson have written lots about 'variation theory' - which is to do with the range of examples we present - and it's good to get to grips with it within my own experience as a teacher. I guess we're always sort of conscious of our example choice, but it's good to bring the consequences of our choices into closer focus and become more conscious of the process.

2. Hi Simon

I really enjoyed reading this post. A great example of how there are lots of things that can be attended to, some of them unexpected, embodying for you the bit from Thinkers that you quote.

By curating such a rich experience, the possibility of learners attending to things other than you expected/hoped increases. By reducing the range of things that could be attended to you might possibly reduce the chances of learners attending to other things, but the experience may become less rich. (That's assuming that it is possible to direct a room full of children's attention in such a way...)

When you ask "Would it have been better if...", it would certainly have been different. I imagine if you do one of these alternatives next year, there will be 'gains' and 'losses'.

In any event, there seems to be sustained interest for this work, for you and the children. From whence did this arise?

Thanks for writing, I almost feel as I have had the experience too!

Danny

1. Thanks Danny.

For me, I want to help the children find their way into geometry, but in a way that they are creating the objects they work with. The challenge is to find a 'playground' where the range of hexagons they make is wide, but they also have some that have properties that are interesting. Although there were 'drawbacks', pattern blocks were actually quite a good space to explore this in. I've tried before with geogebra, giving different templates for different types of hexagons, but there's not the same ease of play with that. I think geobards would work well and would give a range of hexagons.

The sustained interest for the children? I think the way we personalised the hexagons helped. I think they also sensed that when we learnt, or revisited the vocabulary (parallel, obtuse, acute, perimeter, symmetry), it was to do a job, it wasn't just learning the name of one shape, but was terminology that could be used extensively. Keeping it hands-on probably helped too.

Thanks for sharing in the experience!

2. Yes, and I would like to draw your attention to *your* part in sustaining this interest... the interest that comes from responding to the learners, the unexpected, and then becoming inspired, adapting ideas from others and making them your own, experimenting with approaches, feeling you have learned something about how (these) learners learn, through to the validation gained from sharing your experience with others during and afterwards...