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.
2r=p

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.

Wednesday, 25 April 2018

Playful Math Education Carnival #116

Welcome to the 116th Playful Math Education Carnival! I'm Simon, your host this month. I'm giving the Carnival a bit of a Twitter twist, because Twitter's such a great way to find out about lots of the  playful mathematics that's happening.

Math Camp

Sometimes people need spaces outside the usual classroom to get playful. Meredith asked:
and got a lot of good answers. Mike came up with 15 (+1 bonus) ideas. Lybrya reminded her about Jo Boaler's Week of Inspirational Maths. Paula recommended flexagons. Kent recommended  Exploding Dots. Melissa recommended mathematical art - Celtic knots or compass work - and the sort of math puzzle table activities that Sarah specialises in. Kathy added Mathlicious and Desmos. And Brian replied with Yummymath. Michaela went for activities from MathforLove.

And, getting really physical, Malke of Math in Your Feet fame has a Moving Patterns Summer Pilot too that explores mathematical ideas in a playful and energetic way.


Geometric patterns

Somehow, geometry lends itself to play especially well. Like Annie Perkins making this islamic pattern, based on a 'jali' or screen with holes in it:
Ferozkoh Jali by Annie
Turquoise Mountain made it in wood for their exhibition. Samira Mian taught it on one of her online courses. Annie painted it, Paula embellished it, Martin made it dance, John Geogebra-ed it. Then Paula used the pattern to create all sorts of tetrahedron-y creations, including an amazing folding version. And Vincent saw a link with the 'Pavage de Diane' He animated the tiling:

And, noticed that at one point it looks like a triaxial weaving pattern. Alison, who explores the mathematics of triaxial weaving in depth found the one:

Talking of Vincent and geometry, have you seen the delicious book Geometry Snacks which he co-authored with Ed Southall?


Other arts

Jazz musician and mathematician, Marcus Miller writes about the commonalities in creativity in jazz and mathematics.



Malke continues to crochet exquisite hyperbolic forms:

The 13th Gathering for Gardner took place this month. Read an interview with Fields Medal winner Manjul Bhargava on how games inspired him as a mathematician. And see this older video for how his cultural heritage of music and poetry played a part too:



More on Games

Kent Haines blogs about Peg Solitaire.

Jon Orr sees a lesson about tricks and understanding, playing Nim with his daughters.

Alison Kiddle blogs on playful mathematics at the BCME Conference. Becky Warren blogs about a session on a non-routine activity with counters, and also on using pattern blocks for the first time.
Becky's pattern block fish
LC Dawson used Q-bitz to create fraction talks (and which one doesn't belong? images).
And, returning to patterns, she's also been busy putting Christopher Danielson's '21st century pattern blocks' to brilliant use:

And talking of Christopher, his new book and teacher guide How Many? is out (preview here).


Denise

Denise Gaskins orgainises this monthly carnival.  But she needs monthly hosts: why not volunteer to host an edition?

Let's finish with her consideration of the place of curiosity in learning mathematics,
and for good measure her appreciation of Keith Devlin on mathematical thinking.

Sunday, 15 April 2018

some thoughts on estimation

I'm intrigued by a recent article by Philip Ball, How natural is numeracy?

There are a lot of things in there that are really interesting, but it got me thinking about something I've pondered before: how we put numbers on a number line. I haven't really got any implications from this for classroom practice; it's more just something I've pondered because I do estimating a certain way with my classes: draw a blank number line, mark a number on the right that you're sure is too big (be brave!) and one on the left that you're sure is too small. Then mark on your estimate.
It's a good routine. A question arises: would we expect the estimate to be half way between the two other points, the bounds within which we think the actual number must be?

As Ball writes:
Some researchers have argued that the default way that humans quantify things is not arithmetically – one more, then another one – but logarithmically. The logarithmic scale is stretched out for small numbers and compressed for larger ones, so that the difference between two things and three can appear as significant as the difference between 200 and 300 of them.
In 2008, the cognitive neuroscientist Stanislas Dehaene of the Collège de France in Paris and his coworkers reported evidence that the Munduruku system of accounting for quantities corresponds to a logarithmic division of the number line. In computerised tests, they presented a tribal group of 33 Munduruku adults and children with a diagram analogous to the number line commonly used to teach primary-school children, albeit without any actual number markings along it. The line had just one circle at one end and 10 circles at the other. The subjects were asked to indicate where on the line groupings of up to 10 circles should be placed. 
Whereas Western adults and children will generally indicate evenly spaced (arithmetically distributed) numbers, the Munduruku people tended to choose a gradually decreasing spacing as the numbers of circles got larger, roughly consistent with that found for abstract numbers on a logarithmic scale. Dehaene and colleagues think that for children to learn to space numbers arithmetically, they have to overcome their innately logarithmic intuitions about quantity.
Attributing more weight to the difference between small than between large numbers makes good sense in the real world, and fits with what Fias says about judging by difference ratios. A difference between families of two and three people is of comparable significance in a household as a difference between 200 and 300 people is in a tribe, while the distinction between tribes of 152 and 153 is negligible.
Yes, it does make good sense, to me at least. So does that affect how we estimate? Is our uncertainty around our estimate logarithmic?

So I tweeted:

and got some replies. 

Professor Smudge pointed out that we would never do this though we might estimate how large a portion this would give. It's a good point, and it's worth thinking about whether an estimate we ask students to do is 'natural'. It also explains why we're quite likely not to get this right.

Here are the estimates and bounds. The estimate is where the blue and red meet, the lower figure is the left end of the blue and the upper figure is the right end of the red.
The top six are what I was expecting: the red is bigger than the blue. On the other hand, two (Lana and Kit) have red and blue the same length, and two (ThinkMath! and Erick) even have the red smaller than the blue. What other factors are at play here, I wonder?

As some estimates were still coming in, I sent out a neutral kind of tweet, with the third sentence adapted according to the answer I'd got:
The number counted is 344. What I was looking for is how your estimate is positioned between your lower and upper number. You've put yours closer to the lower one. Can you say anything about that? (No problem if nothing comes to mind.)
ThinkMath! said:
Using the phrase, you’re sure of, I chose 100 and 500, because I would be pretty confident (betting $ on it) that the number would fall into that range. By just looking at the jar, I divided into fourths, and figured 80-90 strands per section. 320-360.
Mary said:
My tendency is to be surer of lower limit than higher - not sure why. Interesting!
Miss Primary said:
I think I was more confident of counting a minimum number from looking so kept lower number close to my estimate but not so confident with the upper estimate so went bigger to be on the safe side!
Lana said:
Guess it looks like a lot of noodles, so I reeeeally wasn't sure abt the upper bound and pushed it up significantly.
Vincent said:
That's a good question, and don't think I have a scientific answer. I feel that I can grasp what roughly a 100 is, and I see there's between 100 and 400. But maybe up to 500. I think I had the 500g in mind, and a spaghetto weighs more than a gram.
The Chalkface said:
Tried not to think too hard about it at the time. In think maybe it has to do with thinking multiplicatively. As in, it's probably no bigger than twice my guess. So I guess on a log scale, I'm closer to my upper bound. Should have gone with 450 and been symmetric and less wrong!
I asked him more about the logarithm thing.
Probably depends on what's being estimated, but, say, guessing a person's salary we already talk in log 10 (5-figure salary, etc). I'd also argue that 600k is a better estimate for Nottingham population (300k) than 30, which it is multiplicatively if not additively.
That's interesting, isn't it. Our number system makes us think logarithmically: 1-digit numbers, 2-digit numbers, 3-digit numbers. And I like the example of population.

It would be interesting to try this on a larger scale, again with a sample of people that know how this works. (I tried looking at some of the Estimation 180 data, but it was clouded by some unthinking  and some completely out of the ballpark responses.)
It's not hard to estimate how many sleepers there are from the photographer to the 39 sign.
But it it gets harder to estimate, the bigger the number.

Sunday, 8 April 2018

Number of the Day

How to encourage agency in mathematics learning?

I would say:
  • Connect to play, inquiry, choice
  • Have materials available that give lots of options and connect with mathematical structure
  • Activities that encourage thinking and talk
  • Students writing to express their mathematical ideas
I'm impressed by how Madeleine Goutard got her students writing maths. It was important to her that her students had agency, had their own thoughts and mathematical experiences, which they wanted to express.

On Twitter I saw classes celebrating their hundredth day. I'd never done that, but it comes about the right time of the year for my 5 and 6 year olds; just as lots of them are really ready to think about numbers around 100. We celebrated last year, and it was good. 

I'd also heard of people doing a kind of number of the day thing, where every day the students got to say something about that number. I had mixed feelings about it. I didn't want the students who were still not sure of their teens numbers to feel out of their depth or not contribute when we got to the bigger numbers. But, on balance, I thought it would give the students lots of experience with gradually increasing numbers, giving them a predictable format, and a supportive context for writing their own equations, scribed by me. I would try not to 'steer' them too much, just watching what developed. We'd give about ten minutes to it each day, more if we did some practical work linked to it.

So we began with 1. I wrote what they said on the whiteboard and usually tweeted it and kept it.

I added the subsequent days to that thread. It's hard to follow on Twitter because it's branched. The days are also here in this album. A vital tool in all this was our magnetic hundred square, which we gradually filled with the numbers, swapping the blue-red sides as we discussed different aspects.
At the beginning there were a lot of 'it looks like'-type observations. After a few weeks of them, it was getting a bit repetitive, and I wanted to see more of those equations and I gently discouraged them. We also had a lot of inequalities, bigger than, smaller than. I wrote this out at in words at first, and then thought, actually, > and < is easier for the students to read than the words. It's funny, inequalities hardly came up at all with my class last year. This year they like them. I didn't manage to scribe everything. For instance, for 9, SB said he knew 4+5=9 because 8 is 4+4. (Why didn't I make more of that?!)

I made an effort to write equations with the sum at the beginning sometimes. Students often get the idea that = means "and here comes the answer", like when you press equals on a calculator. Sometimes I'd say "is the same as" instead of "equals" too.

I added in manipulatives at various points, to keep it different, to make connections with other knowledge, and to provoke different ways of looking at the number. At eighteen, we arranged eighteen pegs on pegboards
and I asked how they had arranged them:
At 19, a little number-writing practice. I didn't call it that. We looked at different ways we could number these hexagons.
Some people did it in various spirals, some in lines.

For ten and twenty, we got the ten frames out first:
For twenty-four, we first counted out twenty-four pattern blocks.
You can see a thing starting to develop here. Perhaps it was the materials and the counting in tens, but the students began to express the numbers as a number of tens plus a single-digit number. I wasn't asking for this particularly, but, despite the reduction in variety, I was pleased because that way of seeing numbers reflects our place value system and helps to make sense of what we say and write.

For twenty-five we gave them homework over the weekend (we don't give this often): to number the square tiles any way they liked.
It seemed to help to tune in to equations. (Now, looking at AF's response, I wonder why I didn't make more of that!)

For twenty-nine and thirty, I threw in a tray of eggs. The students were starting to see numbers in different kinds of groups now.

I think I added in that 30-1=29. We hadn't seen any subtraction yet, and I wanted to open up that possibility.

On day 30 there was a flood of ways of seeing:
(Here X arrives. SB said it and explained it both ways, so I introduced the sign. Early I know, but it turned out to be useful. I was conscious, as we progressed, that there would be children who wouldn't grasp this, so I often paraphrased it as "lots of" and gave the choice of writing it either as repeated addition or as a multiplication.)


This was hotting up! I quickly printed out an empty tray of eggs and got everyone to write their way of grouping the eggs as an equation.

We sometimes counted our numbers. (I'd do that thing we do where they had to follow my finger which would trick them sometimes by going  backwards.)
At thirty-six, a good opportunity to count in twos came up:
At day 44, AA and EV both said an equation that had a subtraction in it. I highlighted them:
Most days there was something to remark on. Look at this lovely series:
HA enjoyed the tautology 45=45. MM was enjoying big numbers, something that was to continue, and spread. Infinity!
We often got the little whiteboards out first, so that everyone was writing now.
(Look at that 47 <>47 at the top. Inventive.)

By fifty-six, the equations were starting to get a little unwieldy.
and the next day, I asked if I could just write 5x10 to say 5 lots of ten. 

I wanted to make sure this tens and ones thing was making sense to everyone. We used the rekenrek to represent numbers a few times:
 And also the Dienes tens and ones inside a 100 square:
At sixty-three we shook it up a bit, using Cuisenaire rods:
We looked at how the ones, threes and sevens fitted exactly:
 The next day, sixty-four, I asked if the blacks would fit exactly again:
 And what about green?
Seventy-five came, and MC was on fire, first of all talking about patterns in the hundred square:
 then counting the empty green squares at the top:
Students were very particular about it being an equation, not just an expression. But they were cool about whether the = came near the end or near the beginning.
We had to take numbers off our hundred square to keep track of TT's meandering. Putting them back on is always interesting:
The excitement at approaching one hundred was building.

We arranged a hundred things. For homework over the holidays, we asked students to arrange a hundred things at home. We were getting books ready for the big day too, and beginning to read them.

We weren't going to fit this all in in one day - there would have to be at least a week of celebrations and investigatins!

One is a Snail, Ten is a Crab is a great book - all about seeing numbers in different ways. We made some more ourselves, finding ways to make ten with just the animals in the book:
We counted to twenty using Cuisenaire rods to represent the animals:
and then made one hundred, using Cuisenaire rods:
We did similar things with the wonderful All the Little Ones and a Half.
'It's day one hundred!"
It's been a joyful ride. And I've been able to see my students develop in their thinking and inventiveness. Of course we went a little beyond 100. But that's another story. There's lots I've left out and it's already a long post. (See the album if you want to see all the days.)

You must have a certain endurance if you've made it this far in the blog post. As always, I'm interested in your thoughts. Just writing this, I've seen things I might have done differently. Maybe there's something you don't agree with - I'd be interested in that too!