Saturday, 25 July 2015

Equality, equivalence, sameness

I've already blogged about equivalence. But there's a philosophical discussion children could have here.

What does it mean when we say things are the same?

Heraclitus famously said, "You can't step into the same river twice." The water, of course, has changed.

This statement, I think, could be a great starting point, stimulus, for discussion. Peter Worley suggests the line could be taken away from Heraclitus and given to Tina, talking to her brother Timmy, while they're visiting a river.

There does seem to be a lot to think about here, and it relates to the idea of equality in maths.
Peter James Jackson in one of his wonderful videos wants us to say = as "is equivalent to".

Why? Maybe to emphasise the idea of the balanced equations, rather than, as with the  = button on the calculator, "gives the answer..." This is good, but I'd like to get equals to be associated with balanced equations too.

Maybe too, to stress that the two are different but the same. But then I'd say that equality has the idea of difference in it.

Parmenides famously "answered" Heraclitus by saying, "You can't step into the same river once."

For a long time I thought he was just being ridiculous. Well, he was in a way, but within that craziness, there is, for me now anyway, a serious kernel.

Whenever we call two things the same, they are also different in some way. They're maybe in a different place, or a different time. They may be in a different form.

For instance, if we say that two celebrities arrived at an event in the same dress, we never mean that they've both squeezed into the one dress. They are wearing two separate dresses in all-too-slightly different places, with all kinds of subtle differences, definitely taking different shapes, possibly different sizes.

So same never means "completely the same". That's what Parmenides is saying: you have to have two (at least) different things to have sameness. Those things will be the same in some important way, but will be different in others.

Now I wouldn't say all this with my class, of course. Or even lead towards any of it particularly. But I know it's there, a linguistic and thought territory to be explored, every bit as "out there" as the little forest behind the houses across the road from school where children come back with all sorts of insects I can't immediately identify.

I think it would be good to have the philosophical discussion and the maths discussion about equality at roughly the same time, preferably near the beginning of year. I'm looking forward to hearing what my new class have to say!

Wednesday, 15 July 2015

Which One Doesn't Belong?

I really like the Which One Doesn't Belong? idea, that Christopher Danielson has made such a good book of! It's becoming a phenomenon, with a twitter account and a great website created by Mary Bourassa devoted to assembling the growing body of WODBs.

Here's one of mine:
Something's niggling me about them, and I must just get it clear in my head. I wondered if I'd got it wrong, especially when John Golden asked
Perhaps I should be making my differences more different?

 What do I mean? Well take this one, by Barb Seaton:
It works like a good WODB - you can find a reason for any of the four. Take the top right one: you could say, "It's the only brown one. The rest are white." How would that be, if they were all different colours? Like this:
Could you still say, "The top right one - because it's dark brown"? I've been - in my slow way- pondering the difference between these two cases.

 In the first one there's a binary difference: there's the brown dog and considering the category of colour the others are all the same - white. In the second case, as far as colour is concerned, you could pick on any of them and say it's different because it's such and such a colour and the others aren't. It's not binary in quite the same way: there are four values for the category of colour.

Christopher Danielson goes for both kinds of difference in his original example:
He says (my notes in brackets):
  • The bottom left shape doesn’t belong because it’s not shaded in. (Binary)
  • The top left shape doesn’t belong because it only has three sides, while the others have four. (Binary)
  • The top right doesn’t belong because it is the only square. (Not binary - there are three shapes. He could have said, "It has right angles." Somehow that feels a little more binary as right angle- not right angle is such a major distinction with angles.)
  • The bottom right doesn’t belong because it’s the only one resting on a side. (Binary)
I'm kind of pleased that even this one has a not binary example in it. It seems to open up the possibilities a bit, to relax the whole thing. Of course, binary is satisfying.

I only did one lesson on Christopher Danielson's book. It was worthwhile, but I didn't pay attention to the distinction I'm making, and how the students were relating to it.

Just recently, Dani  Ruiz Aguilera has posted a good pattern-block example:

I wonder if he had in mind the not-binary category of order of rotational symmetry? Certainly there are lots of other features you could pick as well, some of them binary.

 I'm intrigued by these shapes and see lots of interesting things in them. Like, that you can transform the top right one into the others with a bit of internal rotation:
I wonder, what is the proper formal language for this binary-not binary distinction in differences?
Does it matter? Do you have a preference?

Sunday, 12 July 2015

#tiling again

Well, John Golden has posted about his pattern block explorations, so I will too. Along with Dani Ruiz Aguilera he should accept a little of the "blame" for me spending so much time shuffling small shapes around. Dani has been doing some industrial-scale pattern block workshops himself:
I don't know whether I should put this post here or over on I put my own things over there usually, including some of the things I've done with pattern blocks. But, although I've kind of "got the bug" with this, hopefully there are spin-offs for the classroom, like Teflon in space travel and the kitchen.

I've posted about some of the work I did in the class with a particular set of pattern blocks, the square, the hexagon and the thin white rhombus, already.

We've used Math Toybox's great Pattern Block tool - and a great thing about this is that you can save your creations onto the gallery, and even edit them again later on. John used it to respond to what we'd done in class:
I liked it that there were some interesting rules for these line patterns:
We looked at that and one or two other things, and then I asked them to work in bigger groups to extend one of them:
Meanwhile, something was intriguing me: a non-periodic, almost free-form, tiling using these shapes:
It has a really interesting balance between being forced to put the tiles in certain places, and having some freedom. To start with, if you begin with a hexagon, there are a lot of different ways of surrounding it. Here are some of them:
Similarly, there are a lot of ways of surrounding a square (finding them all might be an interesting task). You can see I've restricted myself. I never put two rhombuses against each other. Or two squares. I've got a lot of questions about this, and here are some of them:
  1. It feels like it would carry on forever in all sorts of ways. Can we prove it does?
  2. Once there's a "line" of squares and rhombuses, the line won't go away. What are the rules for it's behaviour?
  3. I've got an intuitive feel for an algorithm for making it carry on. Can we write down an algorithm for making sure it continues to grow without getting into any "impossible" situations?
  4. It feels like this could make a good game. What would the rules be?
  5. You can get areas of different regular tessellation - how many kinds of these are possible? 
  6. Can we - it feels like we can - create writing or pictures with this?
  7. Can I have a large public space, St Mark's Square in Venice for instance, to tile in this way? Or failing that, a playground?
Meanwhile, I was exploring other patterns:
Another thing that caught my attention was dodecagons.
Those holes can hold a dodecagon:
These can "point" in one of twelve directions. The idea of embedding these pointers and maybe other dodecagons, in the tessellatin grabbed me, and for a while I thought about how numbers can be patterned in this kind of space.
(At the same time, I've been reading Roger Penrose on, among other things, his work on non-periodic tiling. I guess I'm thinking about those 17 wallpaper groups, and trying to push other symmetries embedded within a simple translation.)

On a more practical note, I saw that you can get pattern block stickers!
These would really help the students recording a creation directly into their books. And something they could do this for - why hadn't I thought of it before? - popped into my head: pattern block equations:
 As Mary Pardoe tweeted:

Tuesday, 7 July 2015


Everyone teaches coordinates, and these are the three lessons I did with my class just before the end of term.
First of all, battleships. This is usually played in the spaces between lines rather than on the intersections of lines, so I adapted it to be more like Descartes would have wanted:
Here we are playing. (Somehow the logistics of playing two games at once, and marking what you're attacking as well as what you're defending was a tiny bit confusing for some children.)
I took the same grid and suparimposed it on an aerial photo of the school, so that we could do a coordinates treasure hunt game.
One player writes down the coordinates of where they've "hidden" the treasure. The other player then guesses where it is, giving coordinates. If they're right next to it ("hot") they're given a red cube, if they're a bit further ("warm") a yellow cube, and further away still ("cold") a blue cube.
But the activity I like best is to use Desmos to draw something by entering the coordinates. My stipulation was that their designs should "go round in some kind of circle sort of thing". It didn't matter what, as long as it returned to the start again. This, I think, took the pressure off thinking too hard about exactly what to draw. I especially like this for the freedom it gives, and the instant feedback on how you're doing!