Magnitude

It came as a shock to me that there are people without numbers and counting. I first came across this in Daniel Everett's wonderful book about his time with an Amazonian people called the Pirahãs, Don't Sleep, There are Snakes:

But bigger surprises were in store. 

One of the first was the apparent lack of counting and numbers. At first I thought that the Pirahãs had the numbers one, two, and “many,” a common enough system around the world. But I realized that what I and previous workers thought were numbers were only relative quantities. I began to notice this when the Pirahãs asked me when the plane was coming again, a question they enjoy asking, I eventually realised, because they find it nearly magical that I seem to know the day that the plane is arriving. 

I would hold up two fingers and say, “Hoi days,” using what I thought was their term for two. They would look puzzled. As I observed more carefully, I saw that they never used their fingers or any other body parts or external objects to count or tally with. And I also noticed that they could use what I thought meant “two” for two small fish or one relatively larger fish, contradicting my understanding that it meant “two” and supporting my new idea of the “numbers” as references to relative volume – two small fish and one medium-size fish are roughly equal in volume, but both would be less than, and thus trigger a different “number” than a large fish. Eventually numerous published experiments were conducted by me and a series of psychologists that demonstrated conclusively that the Pirahãs have no numbers at all and no counting in any form.

Because they're sometimes cheated by traders, the Pirahãs asked Everett for regular lessons in counting to ten. But despite a year's worth of lessons they don't really learn to do it! These are people that are in many ways a lot smarter than us, people who can walk with nothing into the jungle and come back with all sorts of food, some of it carried in baskets that they've woven on the spot from wet palm leaves. These are happy self-sufficient people. But they don't do numbers.

The ATM has as its first guiding principle that, "The ability to operate mathematically is an aspect of human functioning that is as universal as language itself. Attention needs constantly to be drawn to this fact."

How does this square with the Pirahãs?

At first it bothered me. Could it be that in some way mathematics is not a universal? But then I realised that I have a kind of blindness with numbers too. Because numbers aren't there, I wasn't seeing the mathematics. The two go together for us, numbers and mathematics.

Talking about two, what about the Pirahãs' two, hoi? Everett says it's really about relative volume. And this is key: our lives, all human lives (and the lives of animals too) are full of thinking about magnitudes - volumes, areas, distances, times, weights - usually continuous magnitudes, ones where an exact whole number doesn't come into the thinking.

That basket that the Pirahãs made on the spot to carry what they'd gathered, the Pirahãs had to select the frond of the right width and break off the appropriate length. They had to lay it alongside other fronds and weave in and out until the beginnings of roughly the right size of basket appeared. All sorts of mathematical thinking here, even if numbers don't come into it. And of course, every step, every reach, every move in fact has all sorts of magnitudes involved. Our experience of our bodies in the world is full of them - so full that they're kind of invisible!

I've been thinking about this more after some tweets with Tali Leibovich-Raveh. She shared some articles that she has co-written: Magnitude processing in non-symbolic stimuli and From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition (pdf).

Both of them discuss how early number sense is studied. Often dot images are used:

OK - there's four dots and there's three dots, and four dots is more. But there's more going on here. The one on the left also has a bigger pink area, and covers more space (normally this 'convex hull' isn't coloured, but that doesn't mean it's not perceived). And though the amount of black is the same in the two images, the total length of the circumferences in the left hand image is greater. So when a young child indicates that there's more in the left image, we don't know whether they're solely responding to the number of dots. There's all sorts of magnitude 'mores' that they could be influenced by.

Tali Leibovich-Raveh goes on to argue that there's evidence that in fact it's the sense of continuous magnitude that is primitive, and that number sense is built on this.

When I read this, I started to think again about all sorts of things in this light.

Seeing a boy piling blocks up in the sandpit in these few seconds here:


He's not counting them. He's interested in height, specifically I think in how high up he can make it go. He knows (and here is one of those so-common it's invisible bits of mathematics) that if he adds to his height, he'll be able to add to the tower's height.

You start thinking about it, and magnitudes are everywhere. Taking a common list of play schemas:

• Transporting
• Enveloping
• Enclosing
• Trajectory
• Rotation
• Connecting
• Positioning
• Transforming

every one of them involves magnitudes of some kind or other (as well as arrangements and geometry and topology and patterns - but counting not so much). If we have a number-skewed idea of mathematics, we don't credit these play types for all the mathematics they contain.

As Tony Cotton writes:

I have described previously the enjoyment in observing my grandchildren creating patterns, experimenting with filling and emptying containers or loading toy trucks with rods. Interestingly, what they don’t do is count. They only count when asked by teachers or other adults. Counting is their lowest mathematical priority.

Watching funny cat videos afterwards, all sorts of jumps, cats squeezing through gaps and into boxes. They're funny when they go wrong, but in the wild a sense of magnitudes, a sense of timing, distance, volume is essential for survival. Will this branch be strong enough to hold me? Can I get through this hole? Can I jump this gap?

source

And what about maths teaching beyond the early years? This takes whatever innate mathematical abilities we do have and goes beyond. It's a cultural activity that co-opts mental processes that haven't evolved for school mathematics and uses them to build mathematical understanding.

I've posted before about how Cuisenaire rods in some ways bypass counting. In the light of magnitude thinking, I see their use as a kind of extension of the play we do so much of: judging lengths, filling containers, loading and unloading, putting things end-to-end and side-by-side, making arrangements based on size and shape and pattern.

Getting young children to play with the rods is always fascinating. I recently showed my Grade 3 class some pictures of when I visited them with Cuisenaire rods when they were in K3 and got them playing.

Building on the understandings that young children have is so important, and here there's a kind of natural transition between the world of playing with objects with continuous magnitudes to playing with wooden rods with discrete magnitudes. We're still in the realm of length and area and volume, still using our knowledge of placing things, of lining up, or building, of balancing.

Children are in familiar territory when they lay rods side-by-side. They see that the length of the red rod + the length of the red rod again is equal to the length of the pink rod.

Or they might say that two of the red rods are equal to the pink rod. Or that the red rod is half the length of the pink. Or they might get to know the numerical equivalents and see that four is double two. All this with very little counting.

I think those of us who emphasise physical and spatial resources in our mathematics teaching for other things than simply counting should take courage from these ideas about magnitude. Even experientially, apart from any research results, once we decide to see it, we can see how full our behaviour is of magnitudes. How ready we are to think in this way.

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.

Generalisations

Mike Flynn's Beyond Answers has a nice long list of general claims primary students might make in math class (p158-9):

Begininning with:

Numbers

• Numbers have a set order.

• Our number system is organised by powers of ten (base ten).

• If 10 is added to a two-digit number, then tens digit increases by 1. 

• Numbers can be added and subtracted by their place value. 

As Mike says, these structures in numbers might come up incidentally, or the teacher might take a dive into one in particular, according to the interests and needs of the class.

I feel like his list should be more widely available, and expanded upon, so that together we're really looking out for when students come to these structures.

In class I'm on the look out for, if not articulated general claims, at least general explorations. When I see one, I take a picture and bring it to the class's attention. They're getting used to this. On Friday I was getting the class to hold up successively (if w=1) the Cuisenaire rod that's 10, 8, 6, 4... "The even numbers!" someone says. BT comments that you add 2 to each of the even numbers to get to the next. We look at the odds; it's the same.

So how do these general explorations come about? The students know there's a certain latitude in tasks. You know by now that I use Cuisneaire rods a lot. Because they preserve numbers intact and visually show their relationships, they're particularly suited to uncovering and showing structures in our numbers.

For example, back in January (when we were still calling the rods by colour letters; now we're revisiting some of the same situations with numbers) EW wrote how different trains were equivalent to trains of just white rods.

O + b = 17w
O + B = 19w
O + O = 20w

So we look at this, and I ask the class to try their own similar equal trains.

It's natural to try the reds after that.

What generalities are there here, now that we're onto using nuimbers? 

Some numbers are and some are not equivalent to a certain number of twos.

They alternate, one number that is, then a number that isn't.

Another time, AN made this series of equal trains:

There's an implicit generalisation here:

When you add 1 to an amount, the result is the next counting number.

What's the genearalisation here?

So, on Friday we laid the staircase of even numbers on the carpet and checked the common difference is two with red rods. We did the same with the odd numbers. We moved on, looking at pairs of rods that equal 11. But a few children were still playing with the odds and evens...

"Look, I've made a Which One Doesn't Belong!"

"Me too!"

Mike is emphasising MP7 - Look for and make use of structure. Here's his summary (p154):

• Recognizing when students are working on an idea connected to structure and bringing the idea to the surface through questions and comments
• Giving students opportunities to see and explore a structure in their own work or in tasks designed to highlight particular structure
• Providing time for students to build an understanding of it
• Supporting students as they apply their knowledge of the structure in novel tasks

How do you look out for generalisations, implicit or spoken? How do you nurture them, promote them?

Cardinality, ordinality and developments with the Cuisenaire rods in K3

A few posts ago, I talked about asking children to use a particular manipulative, thinking of the Cuisenaire rods in particular. I got some great replies, most of which emphasised asking children to select the appropriate tools is an important part of the problem-solving process. I've been pondering this lots, which is why I haven't replied to the replies. I'm also reading Mike Flynn's brilliant new book Beyond Answers, which outlines how the CCSS Standards for Mathematical Practice can be brought to life in K-2 classrooms (I hope to blog about the book soon).

I'm trying to make problem-solving more and more part of my class, and I want to have more lessons where the children select the right tool, whether it be cubes, number line, hundred square, ten frames or whatever to help the solve problems.

But I've also got the provisional conviction (if such a thing can exist!) that the Cuisenaire rods, used right, can give young children an environment to explore in a more open-ended way, and give them a really robust and flexible number sense. It's a conviction I want to test through my reading and thinking, and also in practice.

There's a few things the rods do really well. One is being solid colourful things that children enjoy making things with. That's a great starting point.

Another is that, in some ways, they bypass counting.

Now counting is important and I've been putting more emphasis on counting collections this year as well as choral counting.

But counting is complicated. As Young Children's Mathematics by Carpenter et al summarise it,

• There's an ordered sequence of counting numbers, and numbers are always assigned to items in a collection in the same order starting with one.
• The one-to-one principle. Exactly one number from the counting sequence is assigned to each item in the collection.
• The cardinal principle. The last number in the counting sequence assigned to the collection represents the number of objects in the collection.

And when it comes to counting for addition or subtraction there are added complications.

Alf Coles contrasts this way of knowing numbers, cardinality, with one that isn't based on a set of objects counted, ordinality, which is represented as a teaching approach in Gattegno's use of Cuisenaire rods:

"One clear hypothesis to emerge is that students’ awareness of ordinality may be distinct from awareness of cardinality and, in terms of developing skills needed for success in mathematics, that ordinality is the more significant."

I see it as, once the different lengths become familiar, children can think about addition and subtraction without having to count. Like here, you can see that the pink plus the white are the same length as the yellow. You get to see pretty quickly that if you switched the white and the pink they'd still equal the yellow in length. And lots more besides. You can hold the whole relationship in your head, without any smaller units distracting.

What I've been able to do is help the class to gradually build up a familiarity with this, and say to the students, 'you go and make something now, and write it down.' And what's really exciting, they're starting to explore patterns in this, starting to systematise what they're exploring.

There were some great developments this week. Here's T looking at how if you repeatedly add a white rod you move up through all the different lengths in turn:

C had a variation on this:

 F was exploring the fact that something equals itself, enjoying the tautology of it:

D knew that he could generate lots of trains equal in length to the orange rod, just by creating the now-familiar staircase that children are making again and again:

 M was looking at what you have to add end-to-end to a pink rod to move up through all the lengths of rod:

I asked if anyone wanted to be videoed reading reading what they'd written - and some did.
The ability to set out all the information systematically is a great skill in maths, aside from any breakthrough that it helps you make. I hadn't asked for this and of course not everyone was doing it. And a few children were getting muddled.

Or needing to go back to the rods and fix what they'd written:

But luckily my TA and I were able to get round to everyone, and I think all of us are getting the basic idea, and that all is one of the my main considerations for when I'm happy to point to new developments and try new things.

One of the things we did move on to this week was based on E's work:

E was measuring these trains with a long line of white rods and counting them up. That's hard to read; it says:

o + b = 17w

o + B = 19w

o + o = 20w

It might look too simple to be a development, but this implicit measuring of the rods by another rod both connects with cardinality and leads onto lots of other things. It can lead to measuring with other rods, and crucially to talking about the rods as a number rather than simply as a colour, things that we'll be doing soon.

So the next day we looked at E's work and I asked the class to do similar things. And off they went:

Interesting with the bigger ones - a few children getting tangled with the troublesome teens - twelve, thirteen, fourteen, fifteen - it's a tongue-twister, and so easy to miscount at this age.

One group wanted to try it with ten oranges!

(We're going to have to come up with a convention to distinguish o from 0.)

We looked at one creation as a class:

I asked the class how we'd write this, and everyone wrote their way on little whiteboards. I chose a few to come to the whiteboard and share their ways:

We also looked at another lovely bit of systematisation, this time from A.

She was measuring all the rods with the whites. We're about ready with this to start talking about the rods "as numbers". A little pinch of cardinality, and our ordinality has new wings...

------------------added a little later---------------------

Gattegno:

Mandating the materials

Kim was kind enough to comment on my last post. Here's part of what she said

I've been working with tape diagrams for several years now, so I do feel like I know them (though I can always learn more), but it's true that they were foisted upon me by EngageNY and so it wasn't a voluntary process. I guess one question I have about that as a teacher, though, is: is there ever a time when we should make a particular tool or model mandatory because we are trying to help kids become familiar with it, so that later they can have a choice about whether to use it or not? My leaning has always been toward not mandating any tool or model ... toward always leaving it open to the child to explore and choose the representation that makes sense to him/her. But when I started working with younger kids, it seemed like it might be necessary to "mandate" certain models (like the tape diagram, or the number bond, or the "quick tens" drawings) for at least a couple of days as a way to lay a foundation. I wonder deeply about this, because it goes against my instincts to mandate, but then I think that maybe as a 4th grade teacher I was just benefitting from the groundwork that my colleagues had already laid through some of their "today we're all going to try this model" work. Would love your thoughts on this.

It's a really interesting comment, and I've been pondering it through the weekend.

It's been a delight to work with Graham Fletcher's 3-Act tasks over the last few years. Is there anyone who doesn't know these yet? Just in casre, let me tell you, it's an excellent approach to problem-based learning, where, in "Act 1" children watch a scenario, and then are asked to notice things and ask questions about it. Hopefully, and usually, there's a "how many?" question that comes up naturally, just the right one to pursue. Children can then estimate an answer to the question, and think about what additional information they need. Then, in Act 2, they're given some supplementary information that should help them to get started on calculating. While they're doing this, they can select the resources, the materials that will help them best, whether it be pencil and paper, number line, hundred square, snap cubes or whatever.When they've had plenty of time to struggle with the question and come up with answers, there's Act 3, which shows them the answer being revealed.

So, to repeat Kim's question, is there another type of lesson, where students simply get to know and explore what a material or a representation can do?

I  would say a definite yes to this. Especially with younger learners, I want them to spend time getting to know number lines, ten frames, counting, snap cubes, counters and of course Cuisenaire rods, just to see what they can do. I still want there to be a degree of openness in the task; I never want students to just follow instructions, but the challenge can be to achieve certain things, or explore possibilities with the manipulative.

Take a recent lesson with number lines. We'd had a really interesting discussion about where numbers should go on a line, and I then had a great lead in to an idea I'd seen on Kristin's blog, children themselves placing numbers onto an empty strip of tape (a literal one this time).

And we're regularly doing this with the Cuisenaire rods. Just on Thursday, the task was to find different ways of making a "train" of two rods that is the same length as the orange rod.

I've gone into more detail about what we've been doing in our Cuisenaire lessons here; one things's for sure - I've definitely mandated their use. I want my students to be really familiar with them and use them for exploring how numbers work, for asking questions and investigating. And hopefully to have a real "feel" for numbers because they've navigated the model in lots of ways.

A couple of examples. During the Thursday lesson, M, who knows I like his questions and observations, called me over. He had something to say.  He said making the same-length train with the blue and the white rod was "fussy". I'm puzzled, and, after a bit of questioning, he asks a neighbour about it, in Spanish. No, it's not "fussy", it's "easy"! I ask if I can video him talking about it:

I love it when students bring up something we can explore further, in this case how easy making a set of equivalent trains is when the white rod is part of one of them.

Earlier on, in free play with the rods, T had made a German flag.

Something must have really intrigued T about this, because he then started trying to make same-colour trains in other colours, adding a white rod at the end if necessary.

I'm looking forward to sharing this with the class again and exploring this idea together.

Now, this isn't problem-solving in the way we most often talk about it in maths lessons, but this sort of inquiry is, to me, worth sharing with the class and pursuing together. 

As we build up an understanding of the way numbers work like this, I'm expecting that the children will be flexible thinkers with numbers, because they've seen how they work. I'm hoping that, eventually they won't need to pull the Cuisenaire rods out every time, because of the number sense they've developed through them. There will be new things we explore where we'll need them again, so they will always be a place for them. But there will also be the "mastery of structure" as Goutard calls it, that means students carry all sorts of acquired understandings with them mentally.

But I'm interested in Kim's instinct about mandating. How would it respond to the kind of work I'm talking about?