The Math Instinct by Keith Devlin is a must-read for anyone who struggles with math, and for parents whose children struggle with school math. According to Devlin, not only do we as humans have inborn math instincts but so do many other species. The beginning half of the book is devoted to interesting stories illustrating the performance of various math principles including a dog and a frisbee chase as well as an ant's daily travels.
The second part of the book is a discussion of how people use math and his defining of different types of math. According to Devlin, the subject taught to children in schools is ABSTRACT math. Innate math such as real world math is NATURAL math. Both are just mathematics. "The distinction lies in how the math comes to get done. Abstract math is symbolic and rule based. To do abstract math you have to learn what the symbols stand for & you have to learn how to follow the rules. Natural mathematics arises, well, naturally."
(This part of the book is fuzzy in my memory so the details may be somewhat different when you read the book.) He shares a scientific experiment where researchers interviewed people in the grocery store and asked them to carry out various math calculations when comparison shopping. The researchers found that the participants were much better doing the math when they used their own calculation methods to arrive at an estimation and when the situation felt relaxed and informal. When the situation was more structured in terms of providing exact answers and the participants felt their math was being tested or evaluated (as experienced while in school) the higher their anxiety and the worse they performed. It was discovered that “when asked for approximate answers to math questions the greatest brain activity occurs in the brain regions related to number sense & support spatial reasoning skills. For questions requiring an exact answer the brain activity took place in the frontal lobe where speech is controlled. This is also the area of the brain that specifically seeks meanings.” As it was pointed out, most “school math” is focused on exact answers whereas in “real life math” the focus is usually more on estimation. "The problem is that humans operate on meanings. In fact, the human brain evolved as a meaning-seeking device. A computer can be programmed to obediently follow rules for manipulating symbols with no understanding of what those symbols mean. People do not function this way. Mastery of school arithmetic involves the acquisition of some kind of meaning for the object involved & the procedures performed in them."
Devlin also shares a math research project involving young Brazilian street vendors and their "numerical dexterity". According to Devlin, when the vendors were totaling up orders they showed “considerable familiarity with numbers. To simplify computations, they used properties of the specific numbers they were faced with. Their standard approach was to find a way to transform the problem into one involving numbers & arithmetic operations they could recognize & handle. Sometimes involved rounding, sometimes they would break the initial problem into 2 or more subproblems.” What was interesting was that after completing a transaction, the researcher would ask the vendors to solve the exact same problem in abstract form, meaning just asking the vendors to add certain numbers (which were the exact same numbers and computation the vendors had just performed in relation to the produce). The vendors were unable to do math this way. “An obvious key factor for the street traders is that within the computations, both the numbers & the operations they performed on them had meaning & the operations made sense. Indeed the children were surrounded by physical meanings of the arithmetical procedures they performed...In short, street mathematics is all about carrying out meaningful operations or meaningful objects, whereas school mathematics is about carrying out purely formal manipulations of symbols whose meaning, if any, is not represented in those symbols. For most people, $27.99 means something, but 27.99 doesn’t – it is just a number.”
The underlying conclusion shared among all the math research projects described in the book is that, “It’s not that someone who gives a ridiculous answer to a math problem can’t see how silly it is when it is pointed out. If asked to repeat the same calculation in a context or fashion with immediate practical significance, they generally get the right answer, or at least an answer that is plausible. And they do even better if presented with a “real life” problem that they can handle using “street math” – arithmetic methods they have developed themselves “on the job”. In some cases there is almost no procedural difference between a street method and a schoolroom counterpart. And yet the difference caused by the absence of meaning in school math is significant. Complexity is not an issue since most can cope with considerable complexity in their mental arithmetic.”
“In contrast to street math, the essence of school math is that it is entirely symbolic. In performing a standard school procedure for addition, etc. you carry out the very same actions, in exactly the same order, regardless of what the actual numbers are and what they measure. This is the whole point. The methods taught in school are supposed to be universal. Learn them once and you can apply them in any circumstance, whatever specific numbers are involved.”
For students to succeed in school math, the student has to understand what the symbols represent. If they understand what they are doing (what is going on) they don’t forget the procedure. For example 3,844 means 1.62 squared. Because of this meaning to someone (the relationship between the numbers), the better s/he is at math. People good with numbers have relationships with them – the numbers have meaning. "Just how successful a person is at mastering school mathematics is largely a matter of how much meaning they can construct for the symbols manipulated & the operations performed on them. The problem many people have with school arithmetic is that they never got to the meaning stage, it remains forever an abstract game of formal symbols."
Devlin has great respect for school math and believes, “In the hands of a person who can master the abstract, symbolic procedures taught in school, those procedures are extremely powerful. Indeed they underlie all our science technology & modern medicine & practically every other aspect of modern life.
But that doesn’t make them easy to learn or to apply….Since the arithmetic procedures taught in school are designed to be universal – to apply in all cases, whatever the actual numbers – the first think a student has to do in order to master those procedures is to learn to ignore temporarily any possible meanings in terms of actual numbers or real objects in the world. The second thing the student must do to acquire mastery is to construct a different, more abstract kind of meaning.”
Devlin delves into the differences of math test scores among the various countries and believes that within countries such as America, England and much of Western Europe there is a math language barrier which further complicates the math process. According to Devlin, the various scores remain pretty equal throughout the younger grades but when higher numbers are introduced the disparity in test scores begins. “In Chinese & Japanese their number words are much shorter & simpler making them easier to recite, either aloud or internally & that in turn makes them easier to learn. The Chinese rule for making number words for numbers past ten is simple: 11 is ten one, 12 is ten two, 13 is ten three, two ten for 20, two ten two for 22. American kids encounter the various special rules for forming number words. Chinese children don’t, they simply keep applying the same ones that worked for 1 to 12. Also, Chinese language rules closely follow the base-10 structure of the Arabic system. A Chinese pupil will see from the linguistic structure that the number two ten five (i.e. 25) consists of two 10’s & one 5. American kids have to remember that “twenty” represents two 10’s.”
The Math Instinct concludes with a 4 step method of how to improve your math skills. These are:
“1. Be aware that math activity is a natural thing that occurs all the time in nature. Knowing that math is natural should help overcome any math fear. Note: this is what Devlin attempts to do in the beginning of the book which makes it invaluable for math phobics.
2. Approach abstract (i.e. school) math as merely a formalized version of your innate mathematical abilities – that is, as formalized common sense. In math as in most things in life, your approach can make all the difference in how well you do.
3. Recognize why the school methods were developed, what their advantages are & what it is about them that makes them hard to learn. Knowing why something is done a certain way generally helps us to handle it.
4. Practice. Given sufficient repetition, our minds can become skilled at performing practically any new task i.e. swimming, speaking a foreign language, etc. For the human brain familiarity breeds a sense of concreteness. And there we have the key to learning how to handle abstract entities: become sufficiently familiar with them for them to become (i.e. to seem) more concrete. No particular skill is required to do this. All it takes is sufficient repetition. It is only by repetition that the brain can be taught a new skill or to regard the abstract as concrete. Practice basic arithmetic, at least up to decimals & fractions, until you become proficient. Although boring, keep in mind it is a remarkable feature of the human brain that it can acquire such a wide range of new skills.”
Reviewed by Sally Leathers 11/4/05 on the LivingMathForum, shared with permission.