If you are a teacher, what do you think about the following (real life) situation?

**Teacher:**"It’s 1 o’clock. Today, school ends at 3 o’clock. How long do you have to wait?"

**Student:**"Two hours."

**Teacher:**"What’s 3 minus 1?"

**Student:**(After considerable struggle, cannot answer.)

—

**Teacher: "**What’s 4 divided by 2?"

**Student:**Cannot do this, but announces the fact "Four times three is twelve."

**Teacher: "**Here’s four marbles for you to share among two students."

**Student:**Without hesitation, separates the marbles so that he holds two in each hand.

The interaction described above is an abridged version of an account related by

**Stanislas Dehaene**in his book "

**The Number Sense**." (Links provided below.)

**Stanislas Dehaene is a cognitive neuroscientist with a strong background in math and computer science. The student is M, is one of his patients. M is not a child, but is a retired and talented artist who has a brain lesion in the inferior parietal lobe of his brain.**

Mr. M has some knowledge of numbers. He knows the history of certain numbers and can give lengthy lectures about the significance of some historical dates. In his mind, he can move back and forth among the hours of the day. He is sporadically successful when dealing with numbers in other very concrete settings. He has some rote verbal memory and can recite such things as "Three times four is twelve," but he has no comprehension of what that means. He has absolutely no understanding of numbers when they are presented as abstract quantities. Despite considerable effort, Dehaene has been unable to teach M how to do even the most simple arithmetic computations in an abstract setting.

For a platonist like me, numbers exist independent of my own mind. They have meaning without any concrete context whatsoever. Five plus four is always nine, whether it is five apples plus four apples, or five cars plus four cars, or five miles plus four miles. The concrete setting — apples, cars, or miles — has nothing to do with what 5 + 4 is. Five plus four is nine, and that’s it. It has meaning all by itself. It is one of the ultimate permanent properties of number.

But Mr. M shows us that, whether or not numbers are real things, the platonist’s concept of numbers as abstract entities is not independent of one’s brain. For Mr. M, such numbers do not exist.

Dehaene’s book is an account up to 2011 of what cognitive psychology and neuroscience has revealed about number sense and how we create it. Much of this is made possible by advances in brain imaging techniques that provide highly detailed pictures of the functioning of the brain (see the notes at the end of this post).

Thousands of studies have shown that the adult brain has an intricate network of neural circuits that react to numbers. The studies show that numerical undertakings activate localized regions throughout the cortex, and that these regions are found in the basically the same physical location in every brain.

One fairly specific region that reacts to numbers is located at the back of the brain in the groove between the two hemispheres. It is called the

**HIPS**region (short for the horizontal part of the intraparietal sulcus). It seems to be almost exclusively devoted to number size and magnitude. In other words, it is one of the regions that reacts to

**cardinality**— to numbers as quantity as opposed to numbers representing order or sequence.

The HIPS region doesn't appear to react to anything other than numbers. And it reacts no matter what mode is used to input them — spoken words, printed Arabic numerals, collections of dots. You cannot even sneak a number into the brain without activating that area. It reacts to numbers even if you are not consciously aware of them — the briefest subliminal exposure of an Arabic numeral hidden amongst a stream of other visual stimuli activates the HIPS region while the same stream without the numeral has no effect.

It's satisfying to discover that a certain part of the brain is tuned to a notion so fundamental as cardinality. But there are some surprises: the relationship between the activity in the HIPS region and cardinality is not so neatly packaged as one might hope.

To illustrate the complexity, take a few seconds to solve the following simple puzzles. In each puzzle, you have to place the correct symbol, either = , or < , or > , in the square to make the math correct. As usual, the symbol < means "less than," while > means "greater than."

Puzzle A: You solved it instantaneously.

Puzzle B: You solved it quickly, but you performed a brief double check.

Puzzle C: You counted the dots in some manner (or you just skipped this puzzle altogether).

Puzzle D: You solved it instantaneously.

Puzzle E: You counted the dots.

The speed with which you solved these puzzles echoes the activity in the HIPS region, and the correlation is quite strong. The greater the activity in the HIPS region the quicker you will have solved the puzzle.

What is happening is this: when you are presented with two numbers that are small and different (puzzle A) or numbers are large and fairly far apart (puzzle D), there is strong activity in the HIPS region. For numbers that are close in magnitude, the activity in the HIPS region diminishes as the numbers get larger (puzzles C, and E, and even B to some extent).

What you have just experienced is three separate ways that we compare magnitudes. As adults, we typically do it by counting to obtain the exact magnitude, but that is not the only way.

We solve puzzle D

*without actually knowing the exact magnitudes*. Cognitive neuroscientists call this

**the approximate number sense**and describe the quantities as

**analogue magnitudes**. They use a pan balance (an analog device) to model what is happening. A pan balance can readily distinguish between, say, a collection of 60 one-gram tokens and a collection of 80 one-gram tokens without knowing the actual number of tokens in each collection. However, when comparing 60 tokens with 59 tokens, the pan balance cannot so easily distinguish the difference, just as the HIPS activation is weak when two large numbers are close in magnitude.

Puzzle A illustrates yet a third way that we compare magnitudes. Small numbers seem to have a special treatment by our brains. Although you may think that you just rapidly counted the dots, it is unlikely that you really did. Psychologists refer to the ability to accurately estimate small quantities without counting as

**subitizing**.

Young children can subitize the difference between 1, 2, or 3 objects. Many animals also have this ability, and both children and some animals have an approximate number sense. What young children do not have, and what animals do not have, is the adult human’s precise sense of cardinality. Older children acquire it but animals never do. Humans can eventually learn to compare magnitudes of virtually any size — we learn how to count, and we learn what counting means. This is a number sense that we are

*not*born with — it is a cultural artifact — we acquire it through education provided by our parents and teachers.

If we are given a bag of marbles and we count them 1, 2, and so on, until we reach the last marble with a count of 31, then we learn that 31 represents precisely the quantity of marbles in the bag. We learn that every bag of marbles has a precise quantity associated with it. And if for a second bag the last number reached turns out to be 34, we know that second bag has more marbles than the first one.

Tribes or groups of humans that do not count, or that only have a very rudimentary numbering system, cannot discern the difference between even moderate quantities when the quantities are close in magnitude. For such people, there is absolutely no difference between a bag of 31 marbles and a bag of 34 marbles.

Don’t you find this intriguing? One of the foundational aspects of mathematics, the notion of a precise cardinality, is a human creation.

### Some additional notes

**Brain imaging**

In the past two decades new brain imaging techniques such as

**fMRI**(functional magnetic resonance imaging) and

**MEG**(Magnetoencephalography) have been developed that allow the brain to be studied in great detail. These techniques examine the functioning of the brain, not just the anatomy. With fMRI it is possible to scan activity across the entire cortex with a spatial granularity of 1 to 6 millimetres. With MEG we can monitor duration and response times with a resolution in the order of milliseconds. These techniques are noninvasive and hundreds of images can be gathered in a few minutes without danger to the people and patients who have graciously allowed researchers to test them.

**The distance effect**

**When numbers are large but close together, it is difficult to discern a difference between them. This is called**

**. For puzzles A through E above, using dots enhances this effect, but brain scans and psychological studies show that the distance effect remains even when numbers are presented using Arabic numerals. It takes a bit longer to decide which of 18 or 19 is larger than it does to decide which of 4 or 5 is larger.**

*the distance effect***Ordinal numbers**

**As well as using numbers to describe quantity, we also use numbers to describe order. When we want to distinguish between the two different uses we typically refer to them as**

**and**

*cardinal numbers***.**

*ordinal numbers*As children learn to count, they are also acquiring an ordinal number sense. When a child counts "1, 2, 3, 4," she must understand that order is important, that 1 is always before 2 which is always before 3 which is always before 4.

Brain imaging studies have been used to explore how the brain reacts to ordinal numbers. Instead of asking "Which of these two numbers is the larger?" the question becomes "Are these three numbers in order?" In one study, the ordering question was presented both symbolically using Arabic numerals, and nonsymbolically using dots. With Arabic numerals, the brain regions that are are activated for the ordinality task are separate from the regions that are activated for the cardinality task. With nonsymbolic representations, there is considerable overlap.

**Implications about education**

Education strategies based on cognitive psychology are often viewed as some sort of

**. Indeed, I react somewhat negatively when I see phrases like "brain based teaching methods."**

*snake oil*But current research, which combines "hard" neuroscience with "soft" psychology, seems to be very solid, There are already important conclusions about how we acquire number comprehension, and I expect that as the science progresses, we will have to rejig current theories about learning.

For example, we now know that children are born with neural circuits that can be adapted for number comprehension, and that they are ready for such accommodations quite early. The research show that infants have an approximate number sense and can subitize small numbers. And contrary to the widely accepted belief about "object impermanence," young children do not think that objects cease to exist when they are out of sight. In other words, children are ready to begin acquiring arithmetical concepts much earlier than prescribed by Piaget’s constructivism.

Another result, perhaps more disturbing, is that recent fMRI studies indicate that not all brains are tuned with the same acuity (thereby raising the spectre that there is such a thing as "math brain"). It is not known whether the differences are due to nurture (practice and training) or nature (biological). The current hypothesis is that it is due to both.

The new cognitive neuroscience does not have all the answers, and there is danger of being too optimistic about what it offers. But it tells us that we should paste

**labels on our lesson plans.**

*handle with care*###
**Links etc.**

Stanislas Dehaene,

*The Number Sense [How the mind creates mathematics], revised and expanded edition*, Oxford University Press, 2011.Here are two related online videos by Stanislas Dehaene. The first is an 8 minute interview (French with English subtitles). The second one is long and in French and covers the material in his book.

Where do mathematical intuitions come from?

La bosse des maths

The following 30 minute video (in English) shows the detail possible with modern brain imaging techniques. Although it is about reading rather than math, it is what led me to buy Dehaene’s book about the number sense.