Mathematics: essential learning?

Are there things everyone should be required to learn? If so, what are they?

 page of logarithms from the Handbook of Chemistry and Physics, 44th edition, 1962-1963

There are lots of things that are useful to know or be able to do. Reading and writing are fundamental. Knowing how to count, add and subtract. Grammar can be useful, and spelling too. So is recognising street signs. The list could go on.

These are things that are useful to know, but they are not identical to things students have to study. In high school in the US, I had to take two years of a foreign language in order to get into a good university. French was my worst subject. Then, at Rice University, I had to take two years of a language to graduate, even though my major was physics. I chose German this time around, and despite studying hard, was lucky to pass. For me, studying foreign languages was challenging, and I retained little of what I learned.

I vaguely remember some of the things learned in school mathematics classes, like interpolating in a table of logarithms. To multiply or divide numbers, we would look up the logarithm of each number, add or subtract the logarithms and then find the number corresponding to the result. For greater accuracy, we would interpolate in the tables, namely estimate the number between two entries in the table.

I learned how to use a slide rule, which is basically two rulers with logarithmic scales that can be used to multiply and divide. I remember in year 8 daring to use my slide rule in an exam, and then checking the result by calculating it longhand.

These skills became outdated decades ago, after the introduction of pocket calculators. No one says today that anyone should have to learn how to interpolate in tables of logarithms or to use a slide rule. Most young people have never heard of a slide rule.

Some knowledge becomes obsolete and other knowledge is never used. So is there anything that everyone must study and learn?

The math myth

These reflections are stimulated by Andrew Hacker’s new book The Math Myth. He is greatly disturbed by the requirement that all US students must study math (or maths as we say in Australia) to a level far beyond what is required in most people’s lives and jobs.

Hacker, a political scientist at Queens College in New York City, actually loves maths, and shows his knowledge of the field by dropping references to polynomials and Kolmogorov equations. He is ardent in his support of learning maths, primarily arithmetic (requiring addition, subtraction, multiplication and division) and practical understanding of real world problems. His target for criticism is requirements for learning algebra, trigonometry and calculus that damage the morale and careers of many otherwise capable students.

Andrew Hacker

In the US, according to Hacker, the most common reason students fail to complete high school or university is a maths requirement. Everyone has to pass maths courses, and learn how to solve quadratic equations, whether they are going to become a hairdresser, truck driver or ballet dancer. His argument is that many people have talents they are prevented from fully developing because of an absurd requirement to pass courses in mathematics. Even when students pass, many of them quickly forget what they learned because they never use it.

Hacker makes a bolder claim. He says that in many professions in which maths might seem essential, actually most practitioners use only arithmetic. This includes engineering. Hacker interviewed many engineers who told him that they never needed to solve algebraic equations or use trigonometric functions.

On the flip side, Hacker cites studies of some occupations, like carpet laying, in which workers in essence solve difficult equations, but they do it in a way passed down from experienced workers. The irony is that many of these workers never passed the maths classes mandated for finishing high school.

The resulting picture is damning. Millions of students struggle through maths classes, some of them falling to the wayside, others developing maths anxiety, yet few of them ever use the knowledge presented in these classes.

Why maths requirements?

How has this situation arisen? Hacker puts the blame on leaders of the mathematics profession, mostly elite pure mathematicians, who sit on panels that advise on high school and university syllabuses. Few of these research stars have any expertise in teaching, and indeed few of them spend much time with beginning students. Not only do they seldom visit a high school classroom, but most avoid teaching large first-year university maths classes. Educational administrators defer to these gurus rather than consulting with teachers who actually know what is happening with students.

It might be argued that being able to do well in maths is a good indicator of doing well in other subjects. Perhaps so, but this is not a good argument for imposing maths on all students. Research on expert performance shows that years of dedicated practice are required to become extremely good at just about any skill, including music, sports, chess and maths. The sort of practice required, called deliberate practice, involves focused attention on challenges at the limits of one’s ability. This sort of practice can compensate for and indeed supersede many shortcomings in so-called general intelligence. In other words, you don’t need to be good at maths to become highly talented in other fields.

Hacker argues that the test most commonly used for entry to US universities, the SAT, is unfairly biased towards maths, to the detriment of students with other capabilities. Not only do maths classes screen out many students with talents in other areas, but selection mechanisms for the most prestigious universities, whose degrees are tickets to lucrative careers, unfairly discriminate against those whose interests and aptitudes are in other areas.

Education as screening

Hacker’s analysis of maths is compatible with a wider critique of education as a screening mechanism. Randall Collins in his classic book The Credential Society argued that US higher education served more to justify social stratification than to stimulate learning. In other words, students go through the ritual of courses, and those with privileged backgrounds have the advantage in obtaining degrees that give them access to restricted professions.

In another classic critique, Samuel Bowles and Herbert Gintis in Schooling in Capitalist America argued that schooling reproduces the class structure. Their Marxist analysis gives the same general conclusion as Collins’ approach. Then there is The Diploma Disease by Ronald Dore, who described education systems worldwide, but especially in developing countries, as irrelevant in terms of producing skills that can be applied in jobs.

Schooling, up to teenage years, remains one of the few compulsory activities in contemporary societies, along with taxation. (In some countries, military service, jury duty and voting are compulsory.) There is no doubt that education can be a liberating process in the right circumstances, but for many it is drudgery with little compensating benefit, aside from obtaining a certificate needed for obtaining a job, while what is learned has little practical relevance.

A different system would be to set up entry processes to occupations, ones closely related to actual skills used in practice. Exams and apprenticeships are examples. Attendance at schools and universities then would be optional, chosen for their value in learning. There is one big problem: attendance would plummet.

Some teachers set themselves the task of stimulating a love of learning. Rather than trying to convey particular facts and frameworks, they see that learning facts and frameworks is a way of learning how to learn. The ideal in this picture is lifelong learning.

The trouble with schooling systems is that they undermine a love of learning by imposing syllabi and assessments. Students, rather than studying a topic because they are fascinated by it, instead learn that studying is tedious and to be avoided, and only undertaken under the whip of assessment.

How many students do you know who keep studying after the final exam? On the other hand, people who are passionate about a topic will put in hours of concentrated effort day after day in a quest for improvement and in the engaged mental state called flow.

The paradox of educational systems is that they are designed to foster learning yet, by subjecting students to arbitrary requirements, can actually hinder learning and create feelings of inadequacy. The more that everyone is put through exactly the same hoops — the same learning tasks at the same time — the more acute the paradox.

A different sort of education

Taking this argument a step further leads to a double implication. Education should be designed around the needs of individual students, as attempted in free schools and in some forms of home schooling. The second implication is that work should be designed around the jointly articulated needs of workers and consumers. Rather than students having to compete for fixed job slots, instead work would be reorganised around the freely expressed needs and capacities of workers and local communities.

Whether this ideal could ever be reached is unknown, but it nonetheless provides a useful goal for restructuring education — including maths education. This brings us back to Hacker’s The Math Myth. There are two sides to his argument. The first, as I’ve described it, is that US maths requirements are damaging because few people ever need maths beyond arithmetic and the requirements screen talented people out of careers where they could make valuable contributions.

The second element in Hacker’s argument is that for the bulk of the population, there are useful things to learn about maths and that these can be made accessible using a practical problem-solving approach. To show what’s involved, Hacker describes a course he taught in which students tackled everyday challenges.

Hacker’s course shows his capacity for innovative thinking. The Math Myth is not an attack on mathematics. Quite the contrary. Hacker wants everyone to engage with maths by designing tasks that relate to their lives.

Whether Hacker’s powerful critique will lead to changes in US educational requirements remains to be seen. Although Hacker talks only about pointless maths requirements, his arguments challenge the usual basis for screening that helps maintain social inequality. If maths cannot be used to legitimise inequality in educational outcomes, what will be the substitute?

Whether you respond to maths with affection or anxiety, it’s worth reading The Math Myth and thinking about its implications.

Brian Martin