New Yorker article on number sense

From March. Actually, the topic of the article is Dehaene, but it talks about some studies too. Excerpts after the break, interspersed with hyperlinks to citations that I looked up.

…Mr. N, had sustained a brain hemorrhage that left him with an enormous lesion in the rear half of his left hemisphere. He suffered from severe handicaps: his right arm was in a sling; he couldn’t read; and his speech was painfully slow. He had once been married, with two daughters, but was now incapable of leading an independent life and lived with his elderly parents. Dehaene had been invited to see him because his impairments included severe acalculia, a general term for any one of several deficits in number processing. When asked to add 2 and 2, he answered “three.” He could still count and recite a sequence like 2, 4, 6, 8, but he was incapable of counting downward from 9, differentiating odd and even numbers, or recognizing the numeral 5 when it was flashed in front of him.

When he was shown the numeral 5 for a few seconds, he knew it was a numeral rather than a letter and, by counting up from 1 until he got to the right integer, he eventually identified it as a 5. He did the same thing when asked the age of his seven-year-old daughter. In the 1997 book “The Number Sense,” Dehaene wrote, “He appears to know right from the start what quantities he wishes to express, but reciting the number series seems to be his only means of retrieving the corresponding word.

Dehaene also noticed that although Mr. N could no longer read, he sometimes had an approximate sense of words that were flashed in front of him; when he was shown the word “ham,” he said, “It’s some kind of meat.” Dehaene decided to see if Mr. N still had a similar sense of number. He showed him the numerals 7 and 8. Mr. N was able to answer quickly that 8 was the larger number—far more quickly than if he had had to identify them by counting up to the right quantities. He could also judge whether various numbers were bigger or smaller than 55, slipping up only when they were very close to 55. Dehaene dubbed Mr. N “the Approximate Man.” The Approximate Man lived in a world where a year comprised “about 350 days” and an hour “about fifty minutes,” where there were five seasons, and where a dozen eggs amounted to “six or ten.” Dehaene asked him to add 2 and 2 several times and received answers ranging from three to five. But, he noted, “he never offers a result as absurd as 9.”

Actually the patient is referred to as N.A.U. in the paper:

Stanislas Dehaene and Laurent Cohen. Two mental calculation systems: A case study of severe acalculia with preserved approximation. Neuropsychologia, 29:1045–1074, 1991.

How do we know whether numbers are bigger or smaller than one another? If you are asked to choose which of a pair of Arabic numerals—4 and 7, say—stands for the bigger number, you respond “seven” in a split second, and one might think that any two digits could be compared in the same very brief period of time. Yet in Dehaene’s experiments, while subjects answered quickly and accurately when the digits were far apart, like 2 and 9, they slowed down when the digits were closer together, like 5 and 6. Performance also got worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and 8. When Dehaene tested some of the best mathematics students at the École Normale, the students were amazed to find themselves slowing down and making errors when asked whether 8 or 9 was the larger number.

Dehaene conjectured that, when we see numerals or hear number words, our brains automatically map them onto a number line that grows increasingly fuzzy above 3 or 4. He found that no amount of training can change this. “It is a basic structural property of how our brains represent number, not just a lack of facility,” he told me.

Quoting Dehaene, Dehaene-Lambertz, Cohen ’98, “The numerical distance effect … refers to the finding that the ability to discriminate
between two numbers improves as the numerical distance between them increases
… the number size effect … refers to the finding that
for equal numerical distance, discrimination of two numbers worsens as their numerical size increases.”

Note that the “number size effect” is an instance of Weber’s law (also known as the Weber-Fechner law).

The “distance effect” has been known before Dehaene, but he published (at least one) paper on it that presents a summary of past research as well as a couple of experiments to firm up support for the conclusion that numbers are indeed mapped to an internal “number line” representation before making the comparison (at least, for moderately sized numbers; numbers with four or more digits are compared symbolically). The paper is:

Stanislas Dehaene, E Dupoux, and Jacques Mehler. Is numerical comparison digital ? Analogical and symbolic effects in two-digit number comparison. {Journal of Experimental Psychology: Human Perception and Performance}, 16:626–641, 1990.

Interestingly, both effects are found in non-human animals as well:

Stanislas Dehaene, Ghislaine Dehaene-Lambertz, and Laurent Cohen. Abstract representations of numbers in the animal and human brain. {Trends in Neuroscience}, 21:355–361, 1998.

The number area lies deep within a fold in the parietal lobe called the intraparietal sulcus (just behind the crown of the head).

This is one of the areas. Quoting Wikipedia, Dehaene and Cohen “identified patients with lesions in different regions of the parietal lobe with imparied multiplication, but preserved subtraction (associated with lesions of the inferior parietal lobule) and others with impaired subtraction, but preserved multiplication (associated with lesions to the intraparietal sulcus). This double dissociation suggested that different neural subtrates for overlearned, linguistically mediated calculations, like multiplication, are mediated by inferior parietal regions, while on-line computations, like subtraction are mediated by the intraparietal sulcus”:

Stanislas Dehaene, Laurent Cohen. Cerebral Pathways for Calculation: Double Dissociation between Rote Verbal and Quantitative Knowledge of Arithmetic. Cortex. Volume 33, Issue 2, , 1997, Pages 219-250.

Wikipedia also states that, “In addition to these parietal regions, regions of the frontal lobe are also active in calculation tasks. These activations overlap with regions involved in language processing such as Broca’s area and regions involved in working memory and attention.”

…working with Jean-Pierre Changeux, he set out to create a computer model to simulate the way humans and some animals estimate at a glance the number of objects in their environment. In the case of very small numbers, this estimate can be made with almost perfect accuracy, an ability known as “subitizing” (from the Latin word subitus, meaning “sudden”). Some psychologists think that subitizing is merely rapid, unconscious counting, but others, Dehaene included, believe that our minds perceive up to three or four objects all at once, without having to mentally “spotlight” them one by one. Getting the computer model to subitize the way humans and animals did was possible, he found, only if he built in “number neurons” tuned to fire with maximum intensity in response to a specific number of objects. His model had, for example, a special four neuron that got particularly excited when the computer was presented with four objects. The model’s number neurons were pure theory, but almost a decade later two teams of researchers discovered what seemed to be the real item, in the brains of macaque monkeys that had been trained to do number tasks.

The model referred to appears to be this one:

Dehaene, S. & Changeux, J. P. Development of elementary numerical abilities: a neuronal model. J. Cogn. Neurosci. 5, 390–407 (1993).

The model predicted log-Gaussian tuning curves for single-neurons responsive to numerosity. The connection between the model and the later experiments is detailed in this review paper:

Stanislas Dehaene. The neural basis of Weber-Fechner’s law: Neuronal recordings reveal a logarithmic scale for number. {Trends in Cognitive Science}, 7:145–147, 2003.

The experimental paper is:

Nieder A., Miller E.K. (2003) Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex.

Here’s a review by Nieder:

Nieder, A. (2005) Counting on neurons: The neurobiology of numerical competence. Nature Reviews Neuroscience 6:177-190.

Here’s a later paper by Nieder reviewing how ordinal (ordering, ranking) ability fits in:

Jacob, SN, Nieder, A. (2008) The ABC’s of cardinal and ordinal number representations. Trends in Cognitive Sciences 12:41-43.

But the brain is the product of evolution—a messy, random process—and though the number sense may be lodged in a particular bit of the cerebral cortex, its circuitry seems to be intermingled with the wiring for other mental functions. A few years ago, while analyzing an experiment on number comparisons, Dehaene noticed that subjects performed better with large numbers if they held the response key in their right hand but did better with small numbers if they held the response key in their left hand. Strangely, if the subjects were made to cross their hands, the effect was reversed. The actual hand used to make the response was, it seemed, irrelevant; it was space itself that the subjects unconsciously associated with larger or smaller numbers. Dehaene hypothesizes that the neural circuitry for number and the circuitry for location overlap.

This effect is called the SNARC effect and was apparently first found in this paper:

Stanislas Dehaene, S. Bossini, and P. Giraux. The mental representation of parity and numerical magnitude. {Journal of Experimental Psychology: General}, 122:371–396, 1993.

However, the crossed-hand aspects of the effect have failed to replicate in 2006:

Guilherme Wood, Hans-Christoph Nuerk, Klaus Willmes. Crossed Hands and the Snarc Effect: Afailure to Replicate Dehaene, Bossini and Giraux (1993). Cortex, Volume 42, Issue 8, , 2006, Pages 1069-1079.

However, that article is careful to note that “The spatial numerical association of response codes (SNARC) effect denotes the association of number magnitude with left-right responses, namely that the left hand responds faster to smaller numbers while the right hand responds faster to larger numbers… The SNARC effect has been found consistently over a wide range of experimental manipulations and participant groups…”. They postulate that “If the saliency of hand-based coordinates is high, it seems likely, that both hand-based and hand-independent frames of reference influence the SNARC effect and the underlying spatial representation of numbers”.

A review article from 2005 is:

Edward M Hubbard, Manuela Piazza, Philippe Pinel, and Stanislas Dehaene. Interactions between number and space in parietal cortex.. Nat Rev Neurosci, 6(6):435-48, June 2005.

The Mundurukú, an Amazon tribe that Dehaene and colleagues, notably the linguist Pierre Pica, have studied recently, have words for numbers only up to five. (Their word for five literally means “one hand.”) Even these words seem to be merely approximate labels for them: a Mundurukú who is shown three objects will sometimes say there are three, sometimes four. Nevertheless, the Mundurukú have a good numerical intuition. “They know, for example, that fifty plus thirty is going to be larger than sixty,” Dehaene said. “Of course, they do not know this verbally and have no way of talking about it. But when we showed them the relevant sets and transformations they immediately got it.”

Pierre Pica, Cathy Lemer, Véronique Izard, and Stanislas Dehaene. Exact and approximate arithmetic in an Amazonian indigene group. Science, 306(5695):499-503, October 2004.

Incidentally, there’s an interesting, more recent paper about the Munduruku — but that deserves its own post.

English is cumbersome. There are special words for the numbers from 11 to 19, and for the decades from 20 to 90. This makes counting a challenge for English-speaking children, who are prone to such errors as “twenty-eight, twenty-nine, twenty-ten, twenty-eleven.” French is just as bad, with vestigial base-twenty monstrosities, like quatre-vingt-dix-neuf (“four twenty ten nine”) for 99. Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum of terms. Consequently, the average Chinese four-year-old can count up to forty, whereas American children of the same age struggle to get to fifteen. And the advantages extend to adults. Because Chinese number words are so brief—they take less than a quarter of a second to say, on average, compared with a third of a second for English—the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers. (Speakers of the marvellously efficient Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)

(I didn’t bother to look up the study for that one)

Finally, here are some general links about the topic:

CR Gallistel, R Gelman. Mathematical cognition. The Cambridge handbook of thinking and reasoning, 2005.

Stanislas Dehaene. The number sense. Oxford University Press, Penguin press, New York, Cambridge (UK), 1997.

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