Devlin's Angle

June 2011

Wanted: a Mathematical iPod

For some years now, I have been using the term "everyday mathematics" to refer to the following collection of topics: basic number concepts, elementary arithmetic, arithmetical relationships, multiplicative and proportional reasoning, numerical estimation, elementary plane geometry, basic coordinate geometry, elementary school algebra, quantitative reasoning, basic probability and statistical thinking, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use.

These are the basic quantitative and logical abilities that are essential for anyone to function well in, and contribute to, modern society. The world we live in requires a degree of mastery of each of these mathematical topics.

"Mastery" in this context means not just being able to perform calculations with fluency. It is also important to have a good conceptual understanding of numbers, arithmetic, and reasoning, particularly in the context of real-world applications. A person can have computational skills without much conceptual understanding, though that invariably leads to problems if any further progress in mathematics is required - and many people find that it is.

For example, a child can learn the multiplication tables by rote to the point of rapid, automatic recall, without having any understanding of what multiplication is or how it relates to things in the world. Indeed, many successful adults never fully understand multiplication, though they can use it correctly in certain circumstances. This is demonstrated dramatically by the widespread belief in the adult population of the US that multiplication of positive whole numbers is repeated addition. (See my column for January 2010 and the earlier columns referred to there.)

When something as basic as multiplication of positive whole numbers turns out to be conceptually fairly complex, you know that everyday math involves a whole lot more than rote learning of a few facts. You can learn to calculate with numbers without any real understanding of the underlying concepts. But applying arithmetic to things in the world, to quantities, and understanding the relationships between those quantities, requires considerable understanding of those underlying concepts.

One obvious (and characteristic) feature of everyday math is that it can all be done in the head. You don't need a paper and pencil. In large part this is because the objects it deals with are all in the physical environment we live in. Indeed, everyday math is a collection of mental skills for understanding and reasoning about our environment.

A less obvious feature of everyday math, well known to mathematics educators, is that practically everyone is able to achieve mastery of those skills if they find they need it in their everyday lives or jobs. (See my 2000 book The Math Gene for a summary of the main evidence.)

Though it is possible to express everyday math in formal symbolic terms, you don't have to. When you do, however, a curious thing happens. Ordinary people who become proficient in everyday math in their daily lives to the point where their accuracy rate is around 98% when they do it mentally in a real setting, find that their performance level drops to around 37% when they try to do it symbolically. (I describe this in The Math Gene as well, where I also provide an explanation.)

Given the huge importance of everyday math in today's society, it clearly makes sense that we should teach it in the most effective way. Quite clearly, teaching it symbolically does not work for a majority of people. So why do we do so?

The answer is technology. For two-and-a-half thousand years, symbolic representation was the only way we had to store and widely distribute mathematics - even basic, everyday mathematics. But what if there were an alternative method of storage and distribution for everyday mathematics, from which people could learn how to do it. One that would produce that near perfect 98% accuracy performance exhibited by ordinary citizens when they need it in their everyday lives.

Such a storage/distribution device would be equivalent to the iPod for storing and distributing music so anyone could listen to it and appreciate it, a storage device that circumvented the formal, symbolic musical notation that for many centuries was the only medium we had for storing music. Wouldn't it be cool if there were such a device?

Well, it turns out the technology already exists to build such "mathematical iPods" - devices that store everyday math in a native fashion, without the need for symbolic representations written on paper. How do I know? One of the advantages of living in Silicon Valley, as I do, is having an opportunity to see, and try out, new technologies. Unfortunately, the limits on the length of my column mean there is not enough space for me to give any details here. Unlike Monsieur Fermat, however, I will be able to pick up the story next month.

To be continued.

Devlin's Angle is updated at the beginning of each month. Find more columns here. Follow Keith Devlin on Twitter at @nprmathguy.

Mathematician Keith Devlin (email: is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition. His most recent book is Mathematics Education for a New Era: Video Games as a Medium for Learning, published by AK Peters/CRC Press.