Customer Reviews

The Number Sense: How the Mind Creates Mathematics

5 star:		(11)
4 star:		(3)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

I immediately gave this book a glance-through upon seeing the title. The resemblance to Steven Pinker's 'The Language Instinct' and his quote at the bottom hooked me, and the inside didn't disappoint. A lot of people have written books questioning why we don't understand math; here's someone who wonders why we do.

Regarding the part about memorization - I assume that the numbers shown to the test subjects were our European numerals in all cases. I wonder what would happen if Chinese digits were used -- they all look distinct, in comparison to, say, our ambiguous 6 and 9, which can be confusing (ever see "1 2 3 4 5 SIX 7 8 NINE" on a gambling table to avoid this?). Can people recognize Chinese digits faster?

(And Ronald, I too immediately formed a Japanese mnemonic upon seeing the string of digits in that chapter. Unconsciously, in fact. The five/nine ambiguity disappeared!)

One quibble is that Dehaene seems to fall into the trap that many people - mathematicians included - blindly accept as fact; the idea that the European numerals that we use every day are superior to anything else. 'It's hard to see how they could be improved upon', he says, (or something to that effect - I'm doing this from memory). Arabic numerals (by which I mean those used by Arabic-speaking people, not the European variations that 'we' use) have the advantage of all being written without lifting your pen, and Chinese digits, for which trying to distinguish between, say, "160" and "180" in very small print is no problem. When you think about it, any place-value system with a zero is equally effective regardless of the forms of the numerals.

All in all a fascinating and informative look at a subject that's been largely neglected; at least in the popular press. Well worth reading. Does Dehaene have another book in the works?

(Recommendations from me? 'The Great Mental Calculators' by Steven Smith, which is tough to find, 'Innumeracy' by John Allen Paulos, and most of all 'From One to Zero/The Universal History of Numbers' by Georges Ifrah. All fantastic.)

I have not yet finished Stanislas Dahaene's excellent book "The Number Sense". But I would like to add an observation on chapter 4. The author discusses many studies which show that persons whose native language uses number names stemming from Chinese, such as Chinese, Japanese, or Korean, can remember much longer strings of numbers, on average, than speakers of such western languages as English or French. He attributes this to the shorter length of the spoken names of the numbers in the eastern languages. However, another important factor in Japanese, at least, must be the ease with which meaningful mnemonics can be made. Japanese effectively has three different ways to name each digit. One stems from Chinese (ichi, ni, san, shi, go, roku, shichi, hachi, kyuu, juu), another is the native Japanese counting system predating Chinese influence (hitotsu, futatsu, mitsu,yotsu,itsutsu,mutsu,nanatsu,yatsu,kokonotus,too) and the third is from English (wan, tsuu, suree, foah, etc.). The digit zero can be named as "oh" from the English letter "O", or "ma" from "maru" meaning circle, etc. It is almost always possible to make an easily remembered mnemonic. This way commercial telephone numbers are made easier to remember in advertisements. (Japanese telephones have only digits on the buttons, no added letters.) Telephone numbers for pet shops and veterinarians often have pairs of ones, "11". Because "wan wan" is the Japanese equivelant of "arf arf". The dentist downstairs in my building uses the number "1818" because "ii ha, ii ha" means "good tooth good tooth". Mr. Dehaene does not make it clear whether studies have been done attempting to measure number memory span, isolated from the effect of mnemonics. If this could be done I would be very interested in learning of the results. As of the end of seven of nine chapters, I say this is a very well written, and extremely interesting book. I highly recommend it.

I am very grateful to the friend who directed me to an article in last July's issue of "Discover" that describes Stanislas Dehaene's new book "The Number Sense: How the Mind Creates Mathematics." The article highlights the examples Dehaene gives of people who have brain injuries that destroy their ability to do parts of arithmetic, while leaving other skills intact. Dehaene combines these examples with evidence from reaction time experiments and from new brain imaging techniques to make a compelling case that we share with other animals an analog method for dealing with quantitative information. He uses the metaphors of a mental number line and an analog accumulator, and notes that these may be more than just metaphors. Anyone interested in the teaching and learning of arithmetic must read this book. And it is so well written -- in English by a Frenchman! -- and contains so much new informnation that it can be recommended to everyone.

Dehaene goes beyond the biological heritage we share with other animals to consider how the language processing parts of our brain contribute to our ability to do arithmetic. He also gives a clear and complete description of why hindu-arabic numerals are now universal, noting that place value systems arose independently in four different civilizations. In all, he makes a compelling case that those of us interested in the teaching of arithmetic have to pay attention both to evolution and to the intelligent design of numeral systems.

Dehaene gives examples of how our non-linguisitic, linguistic, and cultural heritages interact in our doing arithmetic, and of what can go wrong when they are out of sync. He notes that speakers of English fall considerably behind speakers of languages that use the Chinese way of saying numbers, first in learning to count beyond twelve and later in skills such as "borrowing" and carrying." In Japanese, "thirteen" is "ten three" and "twenty-one" is "two ten(s) one," etc.

My current interest is in introducing young children to the numbers between the whole numbers that are needed for measuring things. Dehaene's book encourages me to continue searching for ways to delay fraction talk and fraction ways of saying decimals. But that is another story. I am sure that others interested in education will find ideas in this book that will help them in their work. And that everyone can enjoy the exciting story that Dehaene tells.

If you are interested in mathematics, psychology, neuroscience / cognitive science, biology, linguistics, and/or all of the above, you will love this book. Don't get me wrong, not only is this book for those with technical interests but is well written enough and accessible enough for even laymen to get into. Read this great book!

Fascinating. That's the word I've been using to describe this book. From the information on numerical experiments with animals to the solid evidence presented for the hardwired concept of numbers from birth, the author ties together an amazing work.

The clear organization of the book, described from the onset, was my first clue that this was a bit more polished than other works. From a gradual examination of an innate "number sense" in even lower animals, through babies identifiying that 1+1 is not 1 or 3 but 2, up to the examination of innate calculatory facilities, this book covers it all. By the end, you will have been presented with practically all the evidence you need to make your own conclusion--and the case for the existance of the "number sense" is incredible. Psychology and neurology are both linked to the examination.

As a math teacher, I've even gleaned a few points to help in the presentation of the topic to my students. While I don't agree with all of the authors views on education, I do think he made some very profound points (the comment that standardized education has probably stunted natural development ofthe culture--specifically language--is a good example.)

Probably my only real disappointment with the book is the author's overemphasis on evolution throughout the text. Yes, most technical types unfortunately seem to be passionate about evolution's role in everything nowadays; it wouldn't have been that big a deal had the last few pages of the book not delved wholeheartedly into a philosophical expository of how mathematicians of the past were silly to have attributed the apparent design of the mind to match the universe numerically to a God, when evolution is an available alternative. However, this small distasteful thread didn't really detract from the research. (In fact, in practically all other maters the author is refreshingly objective and unbiased, examining counterpoints to his theories and conclusions.)

In all, the book still earns its five stars for rigor, fascinating material, objective examination, and clear presentation. Well worth the read, well worth the purchase.

This is a masterful account of how people, and animals, use their number sense to bring mathematics into their world. It uses recent advances in neuroscience and evolutionary theory to create a coherent account of how we evolved our mathematical abilities. For instance, he shows how and why we have a marvelous analog capability, a subitization capability for immediately recognizing the numerosity of small collections, and a natural knack for counting.

But he also explains how and why we are very bad at digital computations and 'unconnected' memory tasks, like remembering multiplication tables. Not much use for knowing what 7 * 8 is amongst hunter gatherers! His solution -- let children use calculators! This pragmatism, and caring attitude, pervades the book relieving math anxiety like a wine French wine reduces stress.

He attacks the "Bourbaki" school for demolishing mathematical education in France (and elsewhere) by forcing axioms down children's throats instead of gently stimulating their intuition and allowing them to use their inborn number sense. Piaget also comes under attack for not allowing that children are born with number sense, and can do 'hard sums' much earlier than Piagetian dogma allows for. This attack is backed up by references to dozens of research papers that make his case conclusive.

The final chapter is a tour-de-force overview of mathematical philosophy , with a devastating attack on Platonism and a harsh critique of formalism. He comes down (mostly) on the side of the intuitionist mathematics of Poincaré, but is quite harsh on the constructivist Brouwer.

All in all, a jaw-droppingly brilliant book on the philosophy and psychology of mathematics. I've read a few such books, and this is by far the best I've read. Start here if you want to begin to find out what mathematics might really be about. He also has a superb bibliography to lead you to more mathematical wonders.

Let me start by saying that I have a large interest in both the fields of mathematics and neuroscience. And so when I came across this book with its outstandingly positive reviews, I decided to give it a try.

After having read it I can say that this work definitely deserves the positive attention it has garnered thus far. This book is a very comprehensive start to what may be an ultimate understanding of the human (and indeed even animal!) affinity to numbers. Like others who reviewed before me have said, it is very cleanly organized, presents a wealth of compelling evidence from a variety of fascinating and ingenious experiments, and is a pleasure to read by both experts and laymen alike. In short, I recommend Dehaene's The Number Sense to anyone who is curious about how our brains deal with math.

Dehaene covers a wide range of topics throughout the course of this book, discussing what he calls the "number sense" in infants, adolescents, adults, and animals in the context of both classic and more modern experimentation. Fascinatingly, he talks about how even animals have a basic ability to approximate numbers and how some (such as chimpanzees and macaques) can even be trained to perform rudimentary arithmetic with Arabic numerals! Furthermore, he asserts that even human infants are born with an innate knowledge (albeit extremely limited) of numbers that can be detected using very clever experiments. In this way Dehaene has perhaps dispelled forever the long-held notion that mathematics is a purely human science learned after birth by means of human language. In all cases he provides solid, detailed evidence supporting his arguments and clearly explains every conclusion he reaches such that the information is readily accessible by even those modestly educated in the subject.

One of the more interesting and readily appreciable points Dehaene makes is the animal (humans included) inability to comprehend large numbers. He posits that we innately understand and grasp numbers only up to the number four; thus we are to able estimate and differentiate these discrete quantities quickly and accurately. Naturally Dehaene provides satisfactory experimental evidence to this conclusion. Beyond this however, our ability to discern exact quantities fades dramatically. What's more is that our ability to discern the difference between two nearly equal quantities drops rapidly as a function of quantity magnitude and the distance between them. More concisely, given two sets of chocolate chips, many higher life-forms can distinguish between the set that contains one versus the set that contains two, but none can distinguish 99 from 100 if they were laid out randomly. However, one could probably distinguish between sets of 50 and 100, but would be unable to accurately approximate the number of chips in either. This is perhaps something that we've all thought about (I know I have), but Dehaene takes it several steps further by giving it a name, establishing its ubiquity in all intelligent organisms, and by providing an evolutionarily sound explanation to this phenomenon.

Among other topics, Dehaene talks of the notion that a small percentage of people associate numbers with color and position in space, the ability of humans and some monkeys to understand fractions, differences in mathematical abilities based on the language one learns to count in, whether or not the human brain is a logic machine that calculates based on set algorithms like a computer, and why the number crunching capacity of a trained prodigy is still vastly inferior to that of a modern calculator. In all, there were a great number of highly fascinating topics about the science, psychology, neuroscience, and philosophy of mathematics to be read about in this work.

Perhaps the part I least enjoyed about this book was reading through all of the data that Dahaene provides. Make no mistake, Dehaene writes in great form and provides ample support for every claim he asserts. Equally importantly, he clearly labels his own conjecture as such. He writes how a scientist should write in my opinion. However, he provides so much detail in the way of experimental evidence that I found the reading to be rather slow at points. Long after a point Dehaene is trying to make is firmly established in my mind, I still find myself reading evidence in support of his argument. But among the list of things that could be considered wrong with a book, my grievance ranks pretty low, and by no means did the reading get slow all the time.

In summary I highly recommend this book to both experts and laymen alike. It is full of fascinating information and interesting experiments that elucidate some of the neuoscientific basis of mathematics. Furthermore, it uses clear, concise, and at times humorous logic to explain the number sense that is so obviously present in humans and animals alike. My only complaint is a very minor one about a slight excess of detail in some areas. But I would nonetheless say that this was a highly enjoyable read and a great learning experience for me. If I've piqued your interest in this review, then make sure to read this book!

I received this book as a gift. I skimmed it and left it unread for months. I thought it was a cash-in on Steven Pinker's excellent The Language Instinct. But no!

When I finally read it was as impressed with Dehaene's science as with his literary flair - not to mention the excellence of his self-translation. His thesis is fascinating, his references authoratitive and his story telling masterful.

This book should fascinate everyone who can count up to two or beyond. Dry it is not. Beyond the interest he develops in the subject are the insights he gives into the ingenuity with which psychological experiments can be designed to eliminate the flaws that a layperson would think unavoidable.

There is a darker side, however. Like that other fellow with his lethal experiments with chicks to test for chemical changes relating to memory, there are allusions to intrusive experiments on living subjects . True, these were some time ago and Dehaene exhibits a proper degree of unease about them. None the less I am much more comfortable with the ingenious ape and baby tests. In my view, no matter how intriguing the question, and how beguiling arguments of research potential leading to new therapies, there is a line that should not be crossed. Dehaene, I'm glad to say, seems aware enough not to cross it.

Fascinating. Highly recommended.

This book is a tour de force of neurology as it applies to mathematics. From anectodes and case studies of mathematical greats and autistic calculating prodigies to brain scans and theory of the evolution of our calculating ability.

The only reason I give it four stars is its obvious political correctness is parts where it fails to even consider taking group differences into account (read genes) before ascribing differences to culture. For instance, the bulk of Chinese superiority in math at grade school level is ascribed to the fact that chinese sounds for numbers is smaller than english, allowing chinese students to keep longer strings of calculations in their short-term memory. Well, this could very well be a contributive factor, but what about American second or third generation chinese who speak only english who also show heightened mathematical ability? Not a peep about that!

Of course the above is just a small complaint in the end. The book on the whole is fascinating and well worth a read.

I got this book as a gift and wasn't expecting too much. It's a little dry -- you'll be exercising your number sense while you count the pages until the end -- but it's very comprehensive and, even if it's not engaging, it's interesting in many parts. Overall it's a nice summary of the research going on in psychology and neuroscience about how the brain represents numbers and performs calculations.