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?

Scaling

A scale, from the Latin word scala, a ladder

We've just finished a series of seven lessons on scaling with the Year 4 classes (8 and 9 year olds). It's not been a very prominent part of any curriculum I've worked with, but for a long time it's seemed to me such a ubiquitous bit of maths, that I was keen that our children have a handle on it. The immediate prompt that made me finally plan the lessons was a comment by Paula  Beardell Krieg on an interesting post by Tracy Johnston Zager:

"What I am wondering about these days is when are children developmentally ready to understand the concept of ratios and relationships? Personally, I love having been able to memorize all my “math facts,” but I thought, really really believed that doing calculations was doing math. If I had understood that math was more about the discovery and description of relationships it would have made a difference. I think that this is where the Wall might be: when students need to shift from “getting an answer” to understanding relationships. The calculations mind-set is so deeply hard-wired into the brain that this shift never happens.
So I am wondering, from your work with young children, when do you think you can start talking about math as if it’s tool for discovery?"

Paula puts relationships and ratios together; you could include proportion and scaling. You could include fractions or percentages. It's all part of one concept. I chose scaling as the word we would use, as it seems like the most everyday way of talking about it. We talk about scale models, the scale of a map, scaling up a recipe or picture. I dropped in  the words "ratio" and "proportion" occasionally, but not in any way I wanted the kids to remember. It was about the experiences primarily.

Reflecting on the lessons made me think a little more about what scaling is. It's not that easy to encapsulate what it's essentially about, and I'd appreciate other people's views on this. Thinking through what's common to all the activities, it seems it's a kind of "I'm taking you with me" thing. Change one thing, other things have to change too.

It's much easier for me to think about in particular examples. The longer the car journey is, the bigger other things get. You need a bigger bag of travel sweets for instance.

So, how well did these lessons do? (Click on the numbers in circles for more detail on each activity.)

➊ Scaling up shapes on grid paper. Here, what goes together is the horizontal and the vertical scaling. Some of the children at first scaled one way but not both. But in the end they all got it. It's a really satisfying lesson for the children this, because they're creating their own shape, and the challenge level is just right.

Of course, there's something really interesting going on with the area within these shapes - there are square numbers involved - but though we touched on this, the thrust of the lesson was just to be able to create the shapes.

Julie, did something good with it in the other Year 4 class: she got the kids to add numbers to label all the lengths. This makes the numerical patterns much more obvious, and I'd do that next time.

There was triangle grid paper and some of the children wanted to use that. I went with it because I wanted to see how they'd scale up the three dimensions. But it's such a wonderful thing being able to draw 3D things on flat paper that some of them got a bit carried away with the first shape, and couldn't really scale it up easily!

If I had the same children again in later years, I would start to look at the areas and the volumes involved, how if you graph them, you don't get a straight line. This kind of non-linear scaling is really important in science, engineering and industry, not to mention maths itself!

➋ Using the pattern blocks is a nice easy activity, and kids a few years younger could do this, making sure there were twice the number of triangles as squares. Because it was easy I could use it to link this scaling to its graphical representation as a diagonal line.


This incidentally led to a nice exchange of ideas on Twitter. John Golden, who'd originally suggested pattern blocks, came up with a brilliant ratio chart. This is something older children could investigate, as well as his suggestion of more complicated mixtures.

❸  Folding and halving A-size paper is something younger children could do. It could also go off in the directions of fractions and even the idea of infinite series. (I didn't mention in the post how we looked at other rectangles that don't scale up this way - and got into an interesting discussion about squares in particular.) We touched on the fact that the A-sizes have a 1:√2 ratio, but this is something older year groups could explore more.

❹ The 1:100 and 1:500 scale models of the A380 came along at just the right time. Children have so much experience with scale when it comes to models. And as we were looking at maps of runways for our work on headings, it was natural to ask, if this is the plane scaled down, how much would the runway scaled down be? We could have done more on this if time allowed, and scales in maps is such an obvious place to deal with scaling.

➎ Cuisenaire rods gave us the chance to pass by lesson 1 again. I think younger children could do this too. What's special about this approach is that it makes the square numbers really apparent, so it would help older kids too. You can actually stand blocks on their end in the trays, so you could scale up in three dimensions too, and think about cube numbers.

➏ Kids should cook regularly anyway, but this time we especially focused on the scaling up the recipe. (I had to contrive it a bit, because the recipe was right to start with!) The online activity was good too. It showed me that scaling from a recipe for 2 to a recipe for 3 was very challenging for them. That's where to go next year and beyond. This activity also has the advantage that you can eat it!

➐ The Zoolander question from Robert Kaplinksy was, in hindsight, quite challenging for this age group, maybe too challenging. The class got the humour of Zoolander not understanding a scale model though!

❦

Strangely perhaps, I'm not satisfied with all this. And I don't even know why. Is it that scaling is such a slippery concept? Is it that the work we did didn't involve much calculation? Is it just that we need to be doing things like this every year? Maybe it's this... I don't know.

I'd be interested in your ideas about this. What, for you, is the essence of scaling? Do you think it's worth devoting time to? Are there other ways of approaching it you would choose? Let me know!

I like how lots of people have different ideas

I first came across this book in Kristin Gray's blog. The subtitle Strategies for Building Algebraic Thinking in the Elementary Grades gives more of a picture of what the book is about.

Chapter 1 is available as a PDF, so I read it - and liked it,. And bought it. I'm reading it now.

The idea is, we teach arithmetic, sure, but what about noticing the patterns in what we do, making generalisations, or claims, about the ways numbers behave?

It's not about using algebraic notation - far from it - it's about noticing patterns.

An example from our Y4 classes this year: we asked children to look at patterns if you add consecutive numbers. Quite a few children noticed and made the claim that you always get an odd number if you add two consecutive numbers. Some of them could justify the claim. We'd already had a lesson about the addition of odd and even numbers, so they could refer back to that. Eventually we had a whole load of claims, which I tweeted:

the patterns in adding consecutive numbers: add two & you get an odd sum... pic.twitter.com/1oU21aKYSr

— Simon Gregg (@Simon_Gregg) March 30, 2015

There's more going on here too. It's not just about meta-thinking as subject matter, but the emphasis of the book - and often my approach - is the way this thinking is done. Here's an excerpt of one of the many conversations in the book:

Ms. Diaz: ... Have you ever spent time thinking about how you participate in discussions? Like what do you do when something is hard for you to think about? Or when you don't get what someone is saying? 

Kathryn: Well, no one has ever asked me to think about this before. Usually, it is like we just have to have silence. 

Ms Diaz: When I was in school, we didn't spend time talking. Only the teacher did the thinking. But I want us to be a team so that we can all contribute. 

Will: I like how we show our thinking, you know like we come up to the overhead and show our thinking. I like how lots of people have different ideas. 

Brent: It is like there are lots of teachers.

What do you see in this snatch of conversation?

I'm reminded of Freemont's four freedoms mentioned in my last post: freedom to make mistakes, to think for yourself, to ask questions and to choose a method of solution. Conversations like this are places to find these freedoms. Where the teacher asks questions, and genuinely wants to hear what the students think about the question, rather than something that they've told them to learn. It's worth reading Ilana Horn's post on Asking the Right Question.

How do we do that in math class, the place with the most deep-seated rituals of recitation and mindless calculation? How do we move from what English mathematician and philosopher Alfred North Whitehead called ‘inert ideas’ — those which have been received and not utilized or tested or ‘thrown into fresh combinations’?

(I should add, I also love Ralph Pantozzi's coin-tossing activity in the video in the last post, even though, it's not about reflective conversation. It's a teacher-orchestrated activity, and I do plenty of those. One like this can clearly be really enjoyable and memorable and can lead on to opportunities for reflection and conversation. Nicole Louie however mentions "good task worship" in her comment on Ilana's post, and I've been thinking about that lots. You see, I'm always looking for good tasks. Am I a worshiper? I'm still pondering that...)

Four Freedoms... and a Lesson

I like this:

How to Teach Mathematics in Secondary Schools, Herbert Fremont Published June, 1969 "Four Freedoms" for the student of mathematics #mathchat

— Ralph Pantozzi (@mathillustrated) April 30, 2015

Here - thank you, Ralph Pantozzi, thank you Herbert Freemont - they are:

1. The freedom to make mistakes. 
2. The freedom to think for yourself.
3. The freedom to ask questions.
4. The freedom to choose a method of solution.

And have you seen Ralph Pantozzi's great (and prize-winning!) probability lesson?

Careless

When I put this model of learning down this morning it was fine.

I go away for five minutes, and when I come back someone had knocked it over:

Whoever it was - I don't want to know - just be more careful with my models.

_________________________________________________________________

Perhaps I should explain myself. This model (link to source) seems to start at the bottom and build up. But the first layer, Remember, looks to me to be a poor start to a lesson. Some present experience - an image, a video, manipulatives - that embody things learnt before would be a much better starting point. The very question, "Do you remember when we learnt about..." seems to me to suggest forgetting.

Luckily the makers of this diagram have actually built the instability of the model into the picture!

Perhaps I've got it wrong, and misunderstood the whole thing. If you think so, let me know.

Why?

In maths teaching it seems we often feel we're being asked the question Why? What is maths good for?

Derek Haylock gives five good reasons for teaching maths in Mathematics Explained for Primary Teachers:

See here.

But I'd like to look in a different place for answers, back to Antiquity, to the Greeks, who in some ways started it all.

The story, of "the Delian Problem", retold in Have You Been to Delphi?  by Roger Lipsey gives a surprising answer to the why question:

Devastated by a plague, the people of Delos sent an embassy to the Delphic oracle. "What must we do to halt this terrible plague?" they asked. 

Double the size of your altar. 

The instruction seemed straightforward. The embassy returned to Delos, gathered the city's surviving craftsmen, and explained the task. 

The craftsmen set to work. Their first impulse was to add a second altar of the same size to the existing one. They may well have done this — the sources are uncertain. What is certain is that the plague raged on. "The god," they concluded," must mean for us to replace the altar with one twice as large."

They set to work again. Measuring each side of the existing altar, they doubled all of the measurements and began construction — only to realize after the first courses of marble had been laid that they were actually making an altar eight times as large as the previous one. There was something wrong with the calculations. 

The plague raged on. Hearing that Plato was returning from Egypt, they cheered up. Plato was one of the foremost geometers of their generation. They sent a delegation to overtake him at Karia. They explained their difficulty and the utter urgency of solving the problem. Smiling, the philosopher assured them that the answer was near at hand. Gathering the men around, he took a stick and sketched in the sand how to derive two mean proportionals from the existing dimensions of the altar. These lengths, and these alone, would permit them to double the altar. 

"You know," added Plato, I cannot agree that the oracle was saying only 'do this thing, meet this challenge, and all will be well.' The message is larger than that. The god is telling you and all Greeks, that if we cultivate geometry and all branches of learning we will have greater welfare. We will not be just warriors but students of the Muses, men of peace."

I love what Plato says. Not just warriors but students of the Muses, men of peace. Where that fits in Derek Haylock's five I'm not quite sure. Intellectual development I guess, though it seems broader than that. (I also find the linkage between this new kind of thinking and the decline of violence really interesting.)

I've been asking, "What do you notice? What do you wonder?" lots as starting points for discussion this year. I've even decided these questions were so important that we needed mascots for them - and we spent a little time creating them: Nautilus and Wombat.

Wondering isn't just a handy tool in the classroom. Proper wondering, and real noticing allow us to be our own people rather than someone or something else's tool. Aristotle said it really well:

"That it is not a science of production is clear even from the history of the earliest philosophers. For it is owing to their wonder that men both now begin and at first began to philosophise; they wondered originally at the obvious difficulties, then advanced little by little and stated difficulties about the greater matters, e.g. about the phenomena of the moon and those of the sun and of the stars, and about the genesis of the universe. And a man who is puzzled and wonders thinks himself ignorant (whence even the lover of myth is in a sense a lover of wisdom, for the myth is composed of wonders); therefore since they philosophised in order to escape from ignorance, evidently they were pursuing science in order to know, and not for any utilitarian end. And this is confirmed by the facts; for it was when almost all the necessities of life and the things that make for comfort and recreation had been secured, that such knowledge began to be sought. Evidently then we do not seek it for the sake of any other advantage; but as the man is free, we say, who exists for his own sake and not for another's, so we pursue this as the only free science, for it alone exists for its own sake."

 Metaphysics 1.1-2, (982b)

Pythagoras is supposed to have compared humankind to the three kinds of people that came to the Olympic Games. First, there are those who came to buy and sell, next those who came to compete. But best of all, there are those who came to spectate.

It's quite an odd perspective from our point of view (maybe it was in Antiquity too). We're usually more impressed by the sportspeople than the spectators: they've put a lot of training in and they're showing what the human body is really capable of. I certainly prefer doing sport to watching it. We tend to think that doing is better than watching, in all sorts of ways. But the idea of spectating, or watching, was really key to these philosophers, and we've inherited some of its importance in our education as well as in our entertainment. The word for spectate here is θεωρεῖν, theorein, from which we get our word theory:

from Greek theoria "contemplation, speculation; a looking at, viewing; a sight, show, spectacle, things looked at," from theorein "to consider, speculate, look at," from theoros "spectator," from thea "a view" (see theater) + horan "to see," possibly from PIE root *wer- (4) "to perceive" (see ward (n.)). 

So, standing back and watching, considering... perhaps coming up with a theory...

This all contrasts very markedly with the view of maths, and all subjects, as what Aristotle calls "sciences of production".  A view in which the subject is like a road that starts off with learning numbers, goes through addition facts, times tables, algorithms and definitions, onto formulas and nomenclature to memorise, tests and exams, a certificate, a job, a useful contribution to the economy. A road that must be embarked on early and hurtled along swiftly.

Let's see where this ancient view can take us. Spectating... noticing and wondering. Take your time, enjoy the scenery...

(Images form seekecho.blogspot.fr)