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What Counts: How Every Brain is Hardwired for Math
Though he admits to not being particularly good at math, Butterworth (cognitive neuropsychology, U. College, London), the founder of the Mathematical Cognition journal, contends that we all possess an inherent "numerosity" sense developed to different degrees of course. The author bases his case on empirical research and historical speculation. Annotation c. Book News, Inc., Portland, OR (booknews.com)
- GenresMathematicsNeuroscience
432 pages, Hardcover
First published August 27, 1999
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About the author
Brian Butterworth
21 books9 followersEmeritus Professor of Cognitive Neuropsychology,
Institute of Cognitive Neuroscience,
University College, London
Fellow of the British Academy
Institute of Cognitive Neuroscience,
University College, London
Fellow of the British Academy
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Displaying 1 - 5 of 5 reviews
November 3, 2023
I found the part on brain damage and bugs in arithmetic particularly interesting. Some of the histories of numbers and fundamental theorems would have been a bit scant had I not already known them, but I did enjoy the breadth of the chapters.
June 13, 2022
I liked it but did not love it. A little dry. Working my way through his other books, though.
August 11, 2009
Cognitive neuroscientist Brian Butterworth asserts that the left parietal lobe of the human brain contains a genetically wired-in “Number Module” that serves as the basis for learning all mathematics. This module allows babies only a week or two old to distinguish between numbers of objects (up to 4) just by looking at them (this is called “subitizing,” in contrast to “counting”). All later mathematical operations are built on this simple core that allows us to relate mathematical abstractions to objects in the real world. Butterworth does not acknowledge inherited mathematical talent or gift: he believes that after you inherit the basic capacity for “numerosity’” it’s all motivation and hard labor—even prodigies, he claims, are simply folks who become obsessed with numbers and spend all their time thinking about them (he found the movie Good Will Hunting highly implausible—the hero was a “natural” mathematical genius who didn’t care about numbers and didn’t work at them).
To make his case, Butterworth cites evidence from many fields: notably archaeology, anthropology, ethology, history, and neuroscience. He goes into great detail, sometimes tediously so, but he makes many interesting points—for example, he claims that archaeological evidence indicates that not only did Cro-Magnon Man show evidence of “numerosity” but so did the Neanderthals and perhaps even our precursor Homo Erectus. The (to me) most interesting discussions involve pathologies: brain damage that causes malfunctions in some mathematical operations but not others. For example, stroke victims: Signor Strozzi could still count, but couldn’t carry out the simplest calculations; Signor Bellini could still calculate but couldn’t remember his multiplication tables; and Signora Gaddi remained sound in verbal and reasoning ability but was completely unable to recognize any number above 4.
Here are some of Butterworth’s key points:
*The Number Module. “The Number Module, which I claim human infants are born with, and which we share with non-human species, is known to function only for numerosities up to about 4, 5 at the very most. That is, our brain is genetically programmed with the capacity to represent numerosities up to 4.” The Number Module is very basic. It allows us to recognize numerosities (one object? two objects? three? four?), to detect changes in a “collection” of objects based on adding or removing objects, and to order numbers by size. Butterworth demonstrates that numerosity is independent of language, verbal memory, and reasoning.
*Specialized Memory System. Butterworth cites many studies to show that the brain must have a specialized system for remembering numbers. “We think that, rather than using the general-purpose, verbally coded short-term memory, arithmetic uses a special-purpose memory.”
*Specialized Circuits. The author also claims that there are separate, specialized circuits for handling arithmetic “facts” (like the mulitiplication tables), procedures (how to divide), conceptual knowledge (how multiplying and dividing are related), and “transcoding” from words to numbers and numbers to words (changing “five” into “5”). “Numerical abilities separate themselves from other, apparently similar, skills. Writing words and writing numerals, reading words and reading numerals, all involve distinct brain circuits, despite having common input pathways from the eyes and common output pathways to the hands.”
*Counting. “The question, then, is how do we get beyond 4? Historically, a key development has been the universal use of fingers, even where there are no specialized words for numbers.” Butterworth hypothesizes that the Number Module developed in the rear of the left parietal lobe because of that area’s proximity to the motor cortex—which, among other things, governs the movement of fingers. Finger-counting, he believes, was perhaps the first step (beyond the Number Module’s subitizing) into calculation. As evidence he cites the practices of still-primitive tribes, such as some of those in Papua New Guinea, who even now use not only fingers but other body parts for counting. (One tribe counts to 33, with 31 represented by the left testicle, 32 by the right testicle, and 33 by the penis.) Almost all children learn to count on their fingers, and many well-educated moderns still secretly rely on their digits while counting. Butterworth thinks that finger-counting helped pack neurons into the “Mathematical Brain.” Another bit of evidence supporting his hypothesis: people with Gerstmann’s Syndrome suffer from finger agnosia (they can’t feel their fingers or tell where they are), and also tend to suffer from alcalculia (inability to calculate). “Where the finger representation fails to develop normally, it can have knock-on [English expression:] effects on the development of number skills.” “My hypothesis is that, without the ability to attach number representations to the neural representations of fingers and hands in their normal locations, the numbers themselves will never have a normal representation in the brain.” “My hypothesis, therefore, is that as the child grows and develops, the subitizing circuits in the inferior parietal link up with the finger circuits in the interparietal sulcus. The fingers therefore gain and extend representation by this link: they come to represent the numerosities.”
*Learning by Doing. “I will argue that differences in mathematical ability, provided the basic Number Module has developed normally in our Mathematical Brains, are due solely to acquiring the conceptual tools provided by our culture…. To become good at numbers, you must become steeped in them.” “Good math skills go together with having memorized the most facts. This certainly reinforces the role of practice and learning.” Even Sir Francis Galton, author of Hereditary Genius (1869), acknowledged the importance of motivation and hard work: “By natural ability, I mean those qualities of intellect and disposition, which urge and qualify a man to perform acts that lead to reputation. I do not mean capacity without zeal, nor zeal without capacity, nor even a combination of both of them, without an adequate power of doing a great deal of very laborious work.” [Butterworth’s italics:]
*Dysfunction. “Without properly functioning brain circuits for numerosity, the upper bound to numerical abilities is very low: roughly what they can count on their fingers.” “The Number Module in our Mathematical Brain is very specialized and very hard-wired. If it is damaged early on or if it fails to get built, no other part of the brain can satisfactorily take over the job.” “Without the meanings—the numerosities—the ability to repeat the words is little help in arithmetic.”
*Teaching Math. “The criterion I have used to identify good educational practice is this: is it supported by psychological research?” “One of the absolutely fundamental points about arithmetic is that there are many equivalent ways of reaching a goal, and understanding their equivalence is a key to understanding the concepts and principles of arithmetic.” “My suggestion is that teachers, and parents, can help children understand what they are doing by showing them how to transform arithmetical problems into collections and numerosities.” “For people to enjoy mathematical activites, understanding is the key." "The key to being good at something is reflective practice—hours, days, years of it.”
*Tidbits. Some interesting micro-goodies:
—Braille: In one-finger proofreaders of braille text, there are vastly more nerve cells in the brain for that finger than for the others. However, if the reader stops proofing, the cells are gradually reassigned to other functions—evidence for “use it or lose it.”
—Dyscalculia: Dyscalculia is the math equivalent of dyslexia, and afflicts about 3% of the population.
—Auditories vs. Visuals: Math prodigies who are “auditories” (hear numbers) tend to develop their gifts very young (as early as age two or three), whereas “visual” prodigies usually develop their gifts much later—the teens or even adulthood.
—10-Year Rule: Based on a German-American study, “The highest levels of performance and achievement [in any discipline—sports, music, mathematics…:] appear to require at least around 10 years of intense prior preparation.”
—Joke: “A lawyer, an artist and a mathematician are arguing over whether it is better to have a wife or a mistress. The lawyer argues for a wife, stressing the advantages of legality and security. The artist argues for a mistress, emphasizing the joys of freedom. The mathematician says, ‘You should have both, then when each of them thinks you are with the other, you can get on with some mathematics.’”
Butterwoth concludes his book with a discussion of “easy numbers and hard numbers”—maintaining that the latter (such as negative numbers and infinite numbers) are “hard” because they either do not connect with or are difficult to connect with the underlying numerosities or our normal calculation procedures. An appendix contains the author’s explanation of “Gödel’s Theorem”—the mathematics equivalent of Heisenberg’s Uncertainty Principle—which shows that, in mathematics, nothing can really be “proved.”
To make his case, Butterworth cites evidence from many fields: notably archaeology, anthropology, ethology, history, and neuroscience. He goes into great detail, sometimes tediously so, but he makes many interesting points—for example, he claims that archaeological evidence indicates that not only did Cro-Magnon Man show evidence of “numerosity” but so did the Neanderthals and perhaps even our precursor Homo Erectus. The (to me) most interesting discussions involve pathologies: brain damage that causes malfunctions in some mathematical operations but not others. For example, stroke victims: Signor Strozzi could still count, but couldn’t carry out the simplest calculations; Signor Bellini could still calculate but couldn’t remember his multiplication tables; and Signora Gaddi remained sound in verbal and reasoning ability but was completely unable to recognize any number above 4.
Here are some of Butterworth’s key points:
*The Number Module. “The Number Module, which I claim human infants are born with, and which we share with non-human species, is known to function only for numerosities up to about 4, 5 at the very most. That is, our brain is genetically programmed with the capacity to represent numerosities up to 4.” The Number Module is very basic. It allows us to recognize numerosities (one object? two objects? three? four?), to detect changes in a “collection” of objects based on adding or removing objects, and to order numbers by size. Butterworth demonstrates that numerosity is independent of language, verbal memory, and reasoning.
*Specialized Memory System. Butterworth cites many studies to show that the brain must have a specialized system for remembering numbers. “We think that, rather than using the general-purpose, verbally coded short-term memory, arithmetic uses a special-purpose memory.”
*Specialized Circuits. The author also claims that there are separate, specialized circuits for handling arithmetic “facts” (like the mulitiplication tables), procedures (how to divide), conceptual knowledge (how multiplying and dividing are related), and “transcoding” from words to numbers and numbers to words (changing “five” into “5”). “Numerical abilities separate themselves from other, apparently similar, skills. Writing words and writing numerals, reading words and reading numerals, all involve distinct brain circuits, despite having common input pathways from the eyes and common output pathways to the hands.”
*Counting. “The question, then, is how do we get beyond 4? Historically, a key development has been the universal use of fingers, even where there are no specialized words for numbers.” Butterworth hypothesizes that the Number Module developed in the rear of the left parietal lobe because of that area’s proximity to the motor cortex—which, among other things, governs the movement of fingers. Finger-counting, he believes, was perhaps the first step (beyond the Number Module’s subitizing) into calculation. As evidence he cites the practices of still-primitive tribes, such as some of those in Papua New Guinea, who even now use not only fingers but other body parts for counting. (One tribe counts to 33, with 31 represented by the left testicle, 32 by the right testicle, and 33 by the penis.) Almost all children learn to count on their fingers, and many well-educated moderns still secretly rely on their digits while counting. Butterworth thinks that finger-counting helped pack neurons into the “Mathematical Brain.” Another bit of evidence supporting his hypothesis: people with Gerstmann’s Syndrome suffer from finger agnosia (they can’t feel their fingers or tell where they are), and also tend to suffer from alcalculia (inability to calculate). “Where the finger representation fails to develop normally, it can have knock-on [English expression:] effects on the development of number skills.” “My hypothesis is that, without the ability to attach number representations to the neural representations of fingers and hands in their normal locations, the numbers themselves will never have a normal representation in the brain.” “My hypothesis, therefore, is that as the child grows and develops, the subitizing circuits in the inferior parietal link up with the finger circuits in the interparietal sulcus. The fingers therefore gain and extend representation by this link: they come to represent the numerosities.”
*Learning by Doing. “I will argue that differences in mathematical ability, provided the basic Number Module has developed normally in our Mathematical Brains, are due solely to acquiring the conceptual tools provided by our culture…. To become good at numbers, you must become steeped in them.” “Good math skills go together with having memorized the most facts. This certainly reinforces the role of practice and learning.” Even Sir Francis Galton, author of Hereditary Genius (1869), acknowledged the importance of motivation and hard work: “By natural ability, I mean those qualities of intellect and disposition, which urge and qualify a man to perform acts that lead to reputation. I do not mean capacity without zeal, nor zeal without capacity, nor even a combination of both of them, without an adequate power of doing a great deal of very laborious work.” [Butterworth’s italics:]
*Dysfunction. “Without properly functioning brain circuits for numerosity, the upper bound to numerical abilities is very low: roughly what they can count on their fingers.” “The Number Module in our Mathematical Brain is very specialized and very hard-wired. If it is damaged early on or if it fails to get built, no other part of the brain can satisfactorily take over the job.” “Without the meanings—the numerosities—the ability to repeat the words is little help in arithmetic.”
*Teaching Math. “The criterion I have used to identify good educational practice is this: is it supported by psychological research?” “One of the absolutely fundamental points about arithmetic is that there are many equivalent ways of reaching a goal, and understanding their equivalence is a key to understanding the concepts and principles of arithmetic.” “My suggestion is that teachers, and parents, can help children understand what they are doing by showing them how to transform arithmetical problems into collections and numerosities.” “For people to enjoy mathematical activites, understanding is the key." "The key to being good at something is reflective practice—hours, days, years of it.”
*Tidbits. Some interesting micro-goodies:
—Braille: In one-finger proofreaders of braille text, there are vastly more nerve cells in the brain for that finger than for the others. However, if the reader stops proofing, the cells are gradually reassigned to other functions—evidence for “use it or lose it.”
—Dyscalculia: Dyscalculia is the math equivalent of dyslexia, and afflicts about 3% of the population.
—Auditories vs. Visuals: Math prodigies who are “auditories” (hear numbers) tend to develop their gifts very young (as early as age two or three), whereas “visual” prodigies usually develop their gifts much later—the teens or even adulthood.
—10-Year Rule: Based on a German-American study, “The highest levels of performance and achievement [in any discipline—sports, music, mathematics…:] appear to require at least around 10 years of intense prior preparation.”
—Joke: “A lawyer, an artist and a mathematician are arguing over whether it is better to have a wife or a mistress. The lawyer argues for a wife, stressing the advantages of legality and security. The artist argues for a mistress, emphasizing the joys of freedom. The mathematician says, ‘You should have both, then when each of them thinks you are with the other, you can get on with some mathematics.’”
Butterwoth concludes his book with a discussion of “easy numbers and hard numbers”—maintaining that the latter (such as negative numbers and infinite numbers) are “hard” because they either do not connect with or are difficult to connect with the underlying numerosities or our normal calculation procedures. An appendix contains the author’s explanation of “Gödel’s Theorem”—the mathematics equivalent of Heisenberg’s Uncertainty Principle—which shows that, in mathematics, nothing can really be “proved.”
June 5, 2010
A little repetitive for me, after having read Keith Devlin's THE MATH GENE, but still interesting, and I can never get enough on the brain and math.
April 26, 2012
Very good book, I loved it all the way!!
Displaying 1 - 5 of 5 reviews