What is mathematics education good for?

Mathematics in life. 

First, mathematics is a basic tool of everyday life. When coffee comes in 13 oz. packages and 16 oz. cans, can you take that information in stride (walking slowly by the shelf), along with the prices and the prominent signs claiming ‘contains 23% more’ on the cans, to help decide which you’d prefer to buy? In buying a new car with various gimmicks in financing, rebates, and features, do you understand what is going on? If most people did, the gimmicks would be pointless.

Second, mathematical reasoning is an important part of informed citizenship. Can you understand the reasoning behind studies of health risks from various substances, and can you judge how important they are? When listening to politicians, can you and do you use your reckoning powers to help decide how important some statistic is, and what it means? Can you measure and calculate adequately for simple sewing and carpentry? Can you plan a budget? When you see graphs in newspapers and magazines, do you understand what they mean, and are you aware of the several devices frequently used either to dramatize or to play down a certain trend?

Third, mathematics is a tool needed for many jobs in the infrastructure of our increasingly complex and technological society. These uses are pervasive and varied. The dental technician, the fax repairperson, the fast food manager, the real estate agent, the computer consultant, the bookkeeper and the banker, the nurse and the lawyer, all need a certain proficiency with mathematics in their jobs.

Fourth, mathematics is intensively used (and sometimes abused) in most branches of science. Much of theoretical science really is mathematics. Statistics is one of the most common uses of mathematics. Many scientists use the widespread computerized statistical packages, which alleviate the need for computation. However, people who use statistical packages are often shaky in their understanding of the basic principles involved and often apply statistical tests or graphical displays inappropriately.

Mathematics is alive. 

To me, these utilitarian goals are important, but secondary. Mathematics has a remarkable beauty, power, and coherence, more than we could have ever expected. It is always changing, as we turn new corners and discover new delights and unexpected connections with old familiar grounds. The changes are rapid, because of the solidity of the kind of reasoning involved in mathematics.

Mathematics is like a flight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process of sharpening our perception so that we discover patterns that are everywhere around. In his famous apology for mathematics, G.H. Hardy praised number theory for its purity, its abstraction, and the self-evident impossibility of ever putting it to practical use. Now this very subject is applied very widely, particularly for encoding and decoding communications.

My experience as a mathematician has convinced me that the aesthetic goals and the utilitarian goals for mathematics turn out, in the end, to be quite close. Our aesthetic instincts draw us to mathematics of a certain depth and connectivity. The very depth and beauty of the patterns makes them likely to be manifested, in unexpected ways, in other parts of mathematics, science, and the world. To share in the delight and the intellectual experience of mathematics—to fly where before we walked—that is the goal of a mathematical education.

Testing and “accountability”. 

Unfortunately the goals of school mathematics have become incredibly narrow, much narrower even than the first set of utilitarian goals listed above, let alone the others. It is popular lately for politicians and the public to demand “accountability” from the school systems. This would be great, except that educational accounting is usually based on narrowly-focused multiple-choice tests.

It is as if students were considered to have mastered Shakespeare if they could pass a timed vocabulary test in Elizabethan English, or that they had learned to write when they could correctly choose the grammatical form of a sentence from four possibilities. The state and regional boards of education these days hand out a laundry list of skills which students are supposed to know at a certain age, rather than a curriculum:

horizontal addition versus vertical addition, addition of 2 digit numbers to 2 digit numbers with a 2 digit answer versus addition of 2 digit numbers to 2 digit numbers with a 3 digit answer, etc.

A front-page article in the newspaper, contrasted the elementary schools in two similar difficult neighborhoods . The first was a ‘successful’ school, with two reading lessons a day, the second lesson in ‘test-taking skills’ and practice for the standardized reading test. In this school, 80.5% scored at or above grade level. The other school was an ‘unsuccessful’ school where they prepare for the test for ‘only’ 3 months. In that school, 36.4% score at or above grade level.

The reporter cited an example of how the principal sets the tone in the ‘successful’ school:

She is not satisfied with just the right answers; she wants the right steps along the way. In one fourth-grade class, she noticed that pigtailed Keanu had made a wild, though accurate, stab at a problem requiring her to average which of two stores had lower prices.

“It looks like you were going to do it without doing the work,” she told Keanu.

As she watched Keanu go through the calculations, she stressed fundamentals beyond arithmetic, like putting numbers in clearly ordered columns.

I can’t tell which school is actually more successful without seeing them for myself, but one thing I know: neither the test scores nor the cited incident are demonstrations of greater success.