Mathematical reasoning

I've been thinking more about articulating claims, and proof, this term so I was pleased when Tracy tweeted:

Looking for pushback. How's this draft descrip of math reasoning? Synthesizing several sources. (Imagine as flow.) pic.twitter.com/KhjW5k9UkJ

— Tracy Johnston Zager (@TracyZager) June 27, 2015

Sometimes tweets are too short. Kristin Gray needed to respond with a blog post.

Mike Flynn had an addition:

@MathMinds @TracyZager No pushback but a suggested add-on. "Applying the rule (or claim)" Putting these ideas to work in novel situations

— Mike Flynn (@MikeFlynn55) June 27, 2015

Let me go through this and add a few of my thoughts.

• Of course, all this happens within a situation where the initial pattern sniffing is encouraged. The slow-time question, "what do you notice?" is great for catalysing this kind of thinking Another set up is to give out resources and constraints and ask for exploring and noticing. And there's the kind of number talk when you're looking at something like this, and someone notices another way of constructing a number sequence.

• If we hadn't already had a packed lesson, I might have got the Cuisenaire rods to try and construct the sequence in the way the red numbers show. Then everyone would have had the chance to experience that understanding.

• Which brings me on to my next thought, which is that Kristin is right about the “Extending Pattern Using the Pattern”stage. Students need to play a little here. Does it work if I carry it on? It's great with rods or other manipulatives, but whiteboards and pens or pencil and paper are fine for experimenting too.

I was privileged to talk with Tracy, Mike, Kristen, Elham and Virginia Bastable (co-author of the great Connecting Arithmetic to Algebra) tonight!  I also encountered Wendy for the first time. Here we are, looking very pensive:

• And talking with Tracy tonight (talking!) she says words to the effect: we don't have to expect children to work through these somehow programatically or sequentially. I know what she means. The teacher would be taking back the torch from the children. Virginia Bastable agreed, you can just focus on part of this progression for a while, then later take on another part. This is liberating for anyone, which includes me, who feels getting the "full house" might be wonderful, but also wonderfully rare. All this can be diachronic -happening over time. Record it on the claims board, come back to it later, when you and the students have all had time to reflect, or to come at it with fresh eyes.

Jumping, Sliding, Swapping

We've just  completed four lessons that involve jumping, sliding and swapping. This was Julie's idea; not something we'd tried before, but full of great maths. They all have a playing phase; and then with three of them there was the phase of looking at number patterns. (Click on the heading links to go to the Year 4 blog for more detail.)

➊ Solitaire

This was one we just played, on different-shaped boards. The aim is to end up with just one ball. You remove balls by jumping over them into an empty space (not diagonally though). In hindsight we could have looked at how many moves are necessary on different-sized boards. I'd have to investigate this a bit myself first, but maybe there's mileage in this.

I mentioned to the class that my solitaire board can't be played down to just one ball if the empty space is in the centre. Perhaps I might have shared the proof of this as it's a really simple satisfying one. I don't know. I'll share it with you anyway.
If assign three colours to the holes like this, each colour is present twelve times (and so with an even number of each colour):

Whenever there's a jump the number of each colour either increases or decreases by one. For instance:

And so there are now odd numbers of each colour. As this continues, there will always be all even, or all odd numbers of all the colours. So, there can never be just one ball left - one odd, and two even colours.

Isn't that neat? And very similar to the kinds of proofs I was thinking abaout with Cuisenaire rod filling.

➋ Three Cats Three Dogs

There were three things about this activity (source), which involved swapping pairs of cats and dogs to sort them into all-cat  and all-dog groups.

1. It was explicitly about groups of threes (randomly chosen "thinking threes" as we call them) working well as a group, and making sure each member contributed. This was successful, even though my boys aren't always great at working with the girls.
2. I asked the groups to invent a notation to record what they'd done. This I was really pleased with: it was simple to do, they needed to do it completely on their own, and there was more than one way to do it.
3. We started to look closely at the minimum number of swaps necessary with varying numbers of animals. This turns out to be a pattern of triangular numbers, which the class could understand, even though we didn't make the link with triangular numbers (which we'd looked at before explicitly).

❸ Towers of Hanoi

This time in pairs, again inventing notation. We used Cuisenaire rods to represent the disks.

 There was a different pattern of numbers that emerged here, this time powers of two, minus one:

This powers of two really grabbed the class, and we took them up, and down to fractions together.

A few people wanted to carry on doubling while the rest of the class were doing some coordinates work, and I let them go for it:

❹ Jumping Frogs

Here frogs may either slide one place, or hop over one frog into an empty space; the aim being to have the pinks on the right and the blues on the left. There's a great nrich app that helps with this, but I also printed out some lilly pads for work with counters.

This puzzle was perhaps the hardest for the class to solve, but they kept at it!

Again we looked at the number pattern for different numbers of frogs, this time based on square numbers. Wanting to link in to our coordinates work I used desmos to tabulate and graph our results for the minimum number of moves for one frog at each end, two frogs at each end, and three frogs at each end. Then we looked at the three points on the graph, saw that they weren't in a straight line, and estimated where the curved line would cross the line going up from four on the x axis. I then added in the line. Samyak saw that you could get the progression by adding successive odd numbers:

I've now, following a really thought-provoking discussion with Paula Beardell Krieg, decided that I want to drop in this kind of graphing more often, to really get the feel of how we do this and the relationships look. Estimating was a good idea, and that it went well makes me feel like we could do this with more curves.

All-in-all, I was surprised how much maths we got from these four puzzles. I knew they would be playful and need lots of hard thinking, but the different number patterns that we uncovered added enough to make it really worthwhile. You could do this with older year groups and take it further.

I'll be on the lookout for puzzles like this which embody different number sequences, and probably use these activities next year.

37 + 25 = ■ + ●

I've posted about Connecting Arithmetic to Algebra before. This slide, based on the book, is one of a number of useful ones from Kristin Gray's blog:

Today, before the main part of the maths lesson, we looked at an equation on the board. I asked them not to calculate it (although some couldn't resist of course);

37 + 25 = ■ + ●

I asked, "What do you notice?"

Aditi started us off by saying it was balanced. Brilliant! We'd done some work on balanced equations earlier in the year, so I was really pleased to hear this as the first thing.

There were some other good points, and then Justus said the two numbers could be instead:

38 + 25

We looked slowly at what he'd done. Justus said that you could always do this, take it from one addend and give it to the other. Most of the class agreed. 

That's going up on the Claims Board.

"What other pairs of numbers could we have?" I asked.

And then there were a flood of answers.

Alonso gave us:

-1 + 63

and then there were lots of sums with minus numbers in too!

James gave us:

-38 + 100

And Annie:

It's another reason to start a lot more writing in their maths journals next year. There is a lot of great thinking, and it's getting wiped out, rather than pondered over for a bit longer and looked back on. The class were evidently enjoying the freedom of the exploration and talk, but not everyone joined in the discussion. Writing would give more time for everyone to get their thoughts together and put them into words. In September...

On other occasions (like this), I've used Cuisenaire rods to help in representing the pattern. Today the ideas seemed to flow so well without this, but another time I might use them again for something like this.

#tiling

It's the #mathphoto15 summer challenge on Twitter and it's tessellations week. There have been lots of great tiling patterns shared already, and it's been really interesting because we inevitably stretch the definition of what a tessellation is. Does there have to be traslational regularity? Can a tiling have a centre? Does it have to be regular even? Do there have to be a finite number of different tiles? Do they have to be on a flat surface?

It's just the week for me, and my favourite tool for exploring tiles is pattern blocks, so I returned to some earlier thoughts I'd had, and a particular pattern.

I took this one a bit further on Math Toybox:

But of course I wanted to share the week with my class a little. So I showed them this and talked about some of the different kinds of tilings you might get. Then I asked them for some of their own. With a constraint. It's usually good to give a constraint. ("Can you make a pattern without a centre?" is a good one; our first impulse is to build out from a centre.) This time the constraint was to use the same three tiles. So here's some of their ideas:

Interesting how they've taken it round a corner. Will this work?

Samyak thought there wouldn't be any more hexagons in this one.

This one was the most interesting to me, but its creators, Marie and Rose
thought it wasn't regular enough!

The Prisoners' Dilemma

One of my Year 4 students asked me to explain a Nash Equilibrium a few weeks ago! I've been thinking about it. I didn't really want to do much explaining. I could only really see myself doing the lesson if I was going to get lots of ideas from the class. So how to open it up?

In the end I decided to go for it. I explained, with a big stone, what an equilibrium is (made easier by lots of the children being French and Spanish), and, briefly, what a dilemma is. Although I'm not overly fond of teaching vocabulary (see comments on Paula Krieg's great post on functions) I'm fine with teaching these words, because they're very easy to explain, and useful words all over the place.

I gave as an example, the famous prisoner's dilemma. I told a story about Albert and Bert, how they stole lots of gold and were arrested. How the police didn't have enough evidence to really put them in prison for a long time. How they took the two off to separate cells and offered to do a deal with each ("It's called 'interrogation," as Rose pointed out).

Another misgiving I'd had was about the morality of all this, the assumption about what a self-interested person does. In the end, I think the whole situation shows how a narrow view of what self-interest is gets us into a pickle, but I didn't want to have to be pressing towards this end.

After the story-telling and word-explaining, we looked at the graphic. Most of the class seemed to get it. In fact there was a lot of un-asked for spontaneous debate about what they would do. I knew I was onto the good stuff! And there we left it for the weekend.

Coming back today, I thought it best to dramatise it, to get the situation really clear in our minds. We split into fours and were cops and robbers. Then I asked them to draw some pictures to explain an equilibrium and a dilemma, and then write a little about what they thought about the situation. While they were doing this I went round and asked them what they thought. Here's a video of one of the drama pieces and some of the thoughts.


I didn't press the point about the Nash equilibrium, though I think it's a fantastic thing to think about. I love how there's a difference between the game theory and what happens in real life (like Annie talking about how she wouldn't rat on her brother). There's lots to explore here for any age. And then there are behavioural economics experiments that are great to explore, like the ultimatum game. I'd like to see more of this intersection between maths, psychology and economics in schools.

Proof

I've been thinking lots about children articulating claims in maths lessons. Luckily, I'm not thinking alone.A claim came up,

S claim: U can add the same number to both #s in a sub prob and get the same difference.His representa. @MikeFlynn55 pic.twitter.com/GRmDUqG4f5

— Kristin Gray (@MathMinds) May 13, 2015

and naturally I turned to Cuisenaire rods:

@MathMinds @kassiaowedekind @ekazemi or Cuisenaire rods pic.twitter.com/KeiQlICv5L

— Simon Gregg (@Simon_Gregg) May 13, 2015

A lot more Twitter conversations happened. Tracy Zager was tweeting out lots of pointers. For instance:

@kassiaowedekind @Simon_Gregg @MathMinds @Zakchamp @MikeFlynn55 From Mason & Burton: pic.twitter.com/iIACvLMFok

— Tracy Johnston Zager (@TracyZager) May 21, 2015

Blogs were helping me think about it too. Kristin Gray's post The Meaning of Subtraction Kassia Omohundro Wedekind's blog post Counting, Conjectures, and Claims

On Twitter I got a great tip off that I've been following up:

@Simon_Gregg I like the practical, intuitive approach this enables. Caroline Ainsworth has done some great work on this - on NCETM website

— WellingtonPrepHead (@WellingtonPHead) May 17, 2015

So I went to Caroline Ainsworth's work on using Cuisenaire rods.

Of course I tried this approach with the Year 4s (I had the whole year group working in the two adjoining classrooms because my colleague was away) and you can see the results here on the Year 4 blog.

And Marie and Samyak made a claim, I'm pleased to say.

The next day when I reminded them about this claim (which was by then up on the new Claims Board) almost everyone agreed with it. So I asked them to show why it's true on paper, using pictures of the rods and words that would explain it to someone that didn't know about it. (I mentioned proof, but didn't stress this word.)

There were a few that didn't get very far. But there were a lot of good explanations, all different in their presentation:

Some of these explanations seem to amount to a visual proof for me. OK, they are not showing a generalised case for every difference, but taking Angiolina's example just above, she's adding the same thing to the two same left hand ends of the rods. Euclid had it as an axiom, and we know without articulating it that "if equals be added to equals the sums will be equal."

To me, the quality and individuality of the children's proofs justify this activity. They're returning to something very basic, but they're looking at it through more algebraic and logical eyes. And they're taking possession of their knowledge by experiencing it physically and articulating it in words and pictures.

Tracy pointed me towards Avery Pickford's great posts about proof. (I've been playing a 2-digit version of Mastermind with the class since reading them.)

 I've got lots of questions. Like:

Are these 'proofs'? Where does explanation end and proof begin? Which is the most valuable?

The understanding is the key thing; what does proof add?

How does the social dimension enhance this learning?

What is the place of un-articulated experience and learning in all this?

What other activities will lend themselves to this kind of thinking?