Saturday, 2 March 2019


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.

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?
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.


  1. Really like this direction of thought. Maybe the difference comes later from play and application? Our natural play is constructive, organizational, building... But the standards are computational and abstracted.

    I see with college students that some don't play. But that seems more rare with kids - is that your experience? Maybe magnitude and pattern lead naturally to number and computation, as we learn to play with the ideas?

    Anyhow, thanks - a lot to think about?

    1. Thanks John. Yes, young children do just play. It's a nice thing to stand in the middle of a playground of four and five and six year olds, maybe with a few older visitors, and see everyone busily engaged in something - making volcanoes out of sand, getting inside tyres and piling them up around themselves in a big cylinder, laying out planks and tree slices in a kind of path and following it, building the giant Polydron into houses... even in this short sample there is so much mathematics, and lots that I'm probably not sensitive too. For me too this post is just headlines, things to think about more and watch out for. And then the challenge of how to use what we do 'naturally', to keep and develop both the freedom and pleasure of play and the mathematical understandings that are bound up in it.

    2. There's one of those viral animal videos where there's a dog with a stick, trying to get the stick along a passageway that's too narrow for the stick. All sorts of mathematical diagrams appear round the dog's head, and then it tilts its head so that the stick is diagonal and can now go through the gap. Part of the humour is the ridiculousness of all this heavy maths for something that seems so obvious. The obviousness is the wonderful product of all the playing with things, including sticks, that we all do, and the solidity and no-need-to-even-think-about-it-ness is such a gift that play gives us. But the obviousness is also a problem for us as observers: that twist of the head easily passes without notice.

    3. Tali Leibovich-Raveh17 May 2019 at 06:15

      Yes! My theory is that to understand the concept of number (i.e., quantity that is correlated with but independent of continuous magnitudes), one must first (1) understand the natural relationship between continuous magnitudes and discrete quantities. This, I think, happens by means of play and experimenting with the environment and (2) understand that this relationship is not perfect and there are exceptions. In other words, sometimes you need to ignore the continuous magnitudes and pay attention to the number. it's only after that one can control how to use continuous magnitudes, and not be controlled by them (like in number conservation tasks) we can say that the concept of number is acquired. This is still a theory. I'm working on testing it.

    4. It's a really interesting theory Tali. Thank you for inspiring thought about teaching - and about play and experimenting too!