Everyone has the genetic ability to do maths, claims Keith Devlin. So why don't most of us use it?
How's your maths? Most people assume that it takes a special kind of brain to be able to do mathematics. It does not. Mathematical thinking is an amalgam of nine basic mental abilities that each of us has - abilities our ancestors acquired hundreds of thousands of years ago in order to survive in the world.
The puzzle is this: mathematics is at most 5,000 years old. That is far too short a time for any major evolutionary changes to the human brain. Hence, when we do mathematics, we must be using mental faculties that were acquired long before maths came along. What these faculties, when did we acquire them, and what survival advantages did they confer? Here are some possible answers:
* Number sense. This is not the same as being able to count. It is restricted to "knowing" the size of very small collections of objects. It includes the ability to recognise that a collection of three objects has more members than a collection of two. This sense is not something we learn. We are born with a number sense. In terms of evolution by natural selection, there are several advantages conferred by possession of a number sense that would have led to its incorporation in the human gene pool. For example, knowing whether your group outnumbers a potentially hostile group, or whether this tree has more fruit than that one and is thus better worth the effort of climbing.
* Numerical ability. Number sense does not require a concept of numbers as abstract entities or an ability to count. Numbers and counting are learned.
Early methods of counting, by making notches in sticks or bones, go back at least 30,000 years. With numbers, humans can count to determine the number of objects in very large collections. Counting involves our capacity for language, and indeed numbers only "exist" because we have words or symbols to denote them. Thus, our numerical ability is in large part a consequence of our possession of language, which our ancestors acquired between 75,000 and 200,000 years ago.
* Spatial reasoning ability. This forms the basis for geometry. Good spatial reasoning had survival advantage for our ape-like ancestors in the jungle and for their descendants who took to the open savannas, where they needed to judge how far away that sabre-toothed tiger was.
* A sense of cause and effect. The survival advantages of being able to recognise how one thing causes another are obvious, and many of our fellow creatures exhibit this capacity.
* The ability to construct and follow a causal chain of facts or events. (A mathematical proof of a theorem is a highly abstract version of a causal chain of facts.) Our ancestors required this ability to plan sophisticated hunts and to design tools.
* Algorithmic ability. An algorithm is a step-by-step procedure for performing a certain mathematical task - the mathematician's analogue of a recipe for baking a cake. In elementary school we are taught algorithms for adding, subtracting, multiplying and dividing numbers.
* The ability to handle abstraction. We acquired this ability with our acquisition of language.
* Logical reasoning ability. The ability to construct and follow a step-by-step logical argument is fundamental to mathematics. It is another abstract version of the ability to construct and follow a causal chain and hence was also acquired together with language.
* Relational reasoning ability. Much of mathematics deals with the relationship between (abstract) objects. It was the necessity to keep track of increasingly interpersonal relationships in ever larger and more complex societies that prepared the brains of our ancestors for the mathematical thinking (some of) their descendants would one day engage in.
Everyone has these nine abilities, regardless of how poor their score was the last time they took a maths test. So what does it take to put them together and do maths?
The key is the ability to handle increased abstraction. We can all reason about physical objects in the world. You get mathematical thinking when you take those same nine reasoning processes and apply them to the totally abstract objects of mathematics.
In other words, mathematical thinking involves taking thought processes that evolved for everyday activities and applying them to abstract objects created by the mind. The trick is to make those abstract objects seem real - to fool the brain into thinking it is dealing with real objects. Once you do that, the rest is comparatively easy. After all, the mind is then doing something for which it evolved over millions of years and which it finds natural and instinctive.
So am I saying that to do mathematics you have to treat it like reading a novel or watching a movie? In fact, I am going one step further. When you start a new novel, you have to familiarise yourself with the characters and the situation in which they find themselves. In the case of mathematics, the characters never change, only the situations in which they find themselves. You only have to familiarise yourself with them once, and from then on everything amounts to finding out new things about them.
What does that remind you of? It reminds me of soap opera - long-running television series such as Coronation Street. And that, I contend, is the secret to being able to do maths. A mathematician is someone for whom maths is a soap opera.
The "characters" in the mathematical soap opera are not fictitious people but (fictitious) mathematical objects - numbers, geometric figures, topological spaces, and so on. The facts and relationships of interest are not births, deaths, marriages, love affairs and business relationships, but mathematical facts and relationships about mathematical objects, such as: Can you find an object X that has property P? (eg solving an equation); are objects A and B equal? (eg are those two triangles congruent?); what is the relationship between objects X and Y? (eg is this line tangent to that circle?). Mathematicians think about mathematical objects and the mathematical relationships between them using the same mental faculties that most people use to think about other people.
Mathematicians do not have different brains. They have found a way to use a standard-issue brain in a slightly different way. What distinguishes a mathematician from a high-school student struggling in a geometry class is the degree to which each can cope with abstraction. The mathematician learns to create and hold an abstract world in his/her mind, and then experience and reason about that world as if it were real. In short, the mathematician approaches mathematics as a soap opera.
Keith Devlin is dean of science, St Mary's College of California and author of The Maths Gene: Why Everybody Has It But Most People Don't Use It, Weidenfeld and Nicolson, Pounds 9.99.
* COCONUTS ARE A BREEZE, BUT NUMBERS...
In Brazil a few years ago, a group of researchers went to a busy street market in the city of Recife and tape-recorded child stallholders as they carried out the arithmetic involved in selling coconuts and other goods.
Even though they were surrounded by noise and bustle, with the constant distraction of having to keep watch over the entire stall, the children - all of school age - maintained a 98 per cent accuracy rate, despite being faced with some formidable mental calculations.
When the researchers visited the children in their homes and presented them with a paper-and-pencil test involving the same arithmetic problems they had breezed though at their market stalls, they failed miserably - a mere 37 per cent accuracy. They simply could not do with abstract numbers what they had been able to do with ease with coconuts and money. The step from the physical reality of a market stall transaction to the abstraction of the standard "maths test" was too much for them.