Mathematics – are we on the right track?

David Pavett writes

There was an interesting piece in the Guardian by Simon Jenkins a few days ago (For Britain’s pupils, maths is even more pointless than Latin). I don’t agree with all his conclusions but the issues he raised deserve discussion. What should be taught in mathematics classes? It is rare for this subject to be raised. In fact in general too little of the time we spend debating education is devoted to the question of its content.

It is not that I want to avoid discussion of the organisation of our school system with that most evasive of slogans “standards not structures” (as if the two could be separated) even in its distilled form according to which it is teacher quality that makes or breaks the quality of education independently of the framework within which teachers ply their trade. All that is specious nonsense which we must continue to challenge.

But that still leaves the vast question of content to be debated.

I taught maths or maths related subjects (e.g. physics) for more than 30 years. During all that time I had the feeling that there was something wrong with what I was doing but I could never quite work out what it was. Over the last few years of retirement I have helped several adults and teenagers trying to come to terms with basic maths. In each case there was clearly an enormous psychological barrier to the subject. When faced with a simple mathematical task tears would sometimes well up. For millions the result of ten years of school mathematics is maths phobia. There has to be something wrong with a system with such an outcome.

Everyone agrees that we need basic arithmetic (although we might disagree about what that entails). That many leave school without even attaining this is clearly a problem. This failure provides a clue to broader issues of maths education. Why do so many people forget nearly everything they encountered in school maths? Learning without understanding, cramming for tests, leads to short-term memory. Things are forgotten as quickly as they are learned. For many (most?) this the story of maths education. The fact is though that taking an interest in the social and political events around is strongly aided by knowledge of maths. For example, we are bombarded with data by politicians which we need to know how to evaluate.

We encounter numbers and mathematical relationships everyday of our lives but most people have learned to switch off when that happens. After a while they don’t even realise that they are doing it. We deal with risk assessments, rates of change, compound interest, ratios, fractions, percentages and much else besides. Simon Jenkins says that all we need is basic arithmetic. Those who agree might reflect how many things they decide not to read/follow just because it involves a little algebra or more than a one-line calculation. How many pathways of knowledge are thereby closed down?

Maths in general education should:

1. be about developing a sufficient feel for the subject to be able to tackle the mathematical issues that can arise in the course of life and in the course of taking an intelligent general interest in the social affairs and the technology that one is likely to interact with (why is a kilobyte = 1024 bytes?). That’s a big ask but in my view a reasonable one. The emphasis here is on “being able to tackle” i.e. being able to work things out on some basic principles rather than simply applying given formulas received as a piece of magic. (Why does the breaking distance of a car double when its speed is increased from 25 mph to 35 mph?)

2. produce an understanding of mathematics as a creative field of human endeavour and not just a vast block of knowledge of which one learns the easy bits without having any awareness of the challenges faced by the subject in the past and the challenges faced today. For example, should pupils not be aware that calculators and computers do not merely do calculations faster than we can but that they are the only sensible way to tackle some problems. Even some very simple equations cannot be solved analytically but are easy to solve with an iterative solution using a calculator (e.g. x = cos x with x in radians). The behaviour of some equations is truly weird. This links with problems of weather prediction. Maths has mystery. This is excluded from syllabuses which seek only to inculcate simple calculating ability.

A perennial problem in education is that the desire to make things as simple as possible can result in educational material that is, frankly, unchallenging (i.e. boring). This natural problem is exacerbated a 100-fold by an education system in thrall to testing mania and competition by league tables. These encourage teaching to the test and syllabuses that can be reduced to mechanical check lists.

Simon Jenkins suggests that politicians are obsessed with maths and science and our national standing in these areas as reflected by PISA tests. He says that part of the reason is that maths is easily testable (things are right or wrong) unlike the rather more “slippery” humanities. I think that he is right about the obsessions but wrong about the nature of mathematics. If maths were taught as a creative, open-ended subject it would become less dogmatic and would, as it should, be an exercise in understanding and imagination. The flip side is that assessing it would become more “slippery” but that is a price to be paid for making its content reflect the true nature of the subject. We all need maths but it should be real maths and not a desiccated selection from it taught as dogma. The time for a fundamental rethink about mathematics education is long overdue.

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5 Comments on “Mathematics – are we on the right track?”

  1. David Pavett says:

    P.S. There is an article in today’s Observer on the need to change what we teach in maths and the way that we teach it (The UK needs a revolution in the way maths is taught. Here’s why…). That may sound close to my case above. However, I think that the solution proposed in the article is pretty much the opposite of what is needed to make a good maths education.

  2. Paul Martin says:

    Thanks for that insightful piece, David.
    I am sure you are right to point out how there is often an emotional component which is frequently left unaddressed but becomes a barrier to learning.

  3. ignacio says:

    Hello, I am 25 years of age, and I live in Argentina. I quite agree with you in some points, but humbly I´ll give my modest opinion. I´d like to mention the importance of teaching children and teenagers, in a different way that the approach needed to teach maths or any other subject to young adults. As any artist would find it impossible to express a surealistic thought in a paper without being taught to excel at making use of a brush, ditto, you can´t make a teenager understand or at least apply the imaginative side of the maths without firstly having been taught the principles, which are vital to have the necessary means to be creative. Therefore, fitstly teenagers should be taught how to handle these bricks, so that they can build whatever wall they might imagine. What I want to emphasize, ( leaving aside certain exceptions i.e. genius) is how important discipline is. And to a certain extent, discipline is likely to be dull and uninspiring, perhaps teachers should work to make this stage more tolerable and manageable. Maybe teachers could be trained to address “math-hater” students, with the aim of making it easier for them to overcome the difficulties in this period, and you never know, a math-hater could become a math-lover. Just I wanted to highlight the fundamental role that discipline and hardworking have, in spite of the fact that this learning procces unavoidably will lead to difficulties , to develop freedom of thinking and critical thinking among our students. So be sensible, don´t go to the extreme, neither left-wingers nor right wingers, strike a balance

  4. ignacio says:

    Sorry I realized that I made some mistakes such us freedom of thought, I am currently studying English ( qué verguenza!!)

  5. David Pavett says:

    @ignacio

    Thanks for the comment. I understand your point. It is one often made with respect to different areas of learning: ‘first you must learn the rules then later you can see if you need to break, modify or add to them’. (I remember my art teacher at school saying that – although to his credit he didn’t actually attempt to teach as ‘rules’. )It sounds very plausible and in the absence of critical examination it is very widely held. All the same I think that it is wrong. Also I do not think that the requirements of learning mathematics are any different to other subjects in this respect.

    Clearly children need to understand key number ideas and arithmetic operations. But, I suggest, it is a mistake to think that this can be achieved by simply presenting children with a package of definitions and techniques (algorithms) and training them to produce the correct answers to problems given (usually for no other reason than to test what they have been taught). This means in effect that children are being taught answers to things that they did not even know were problems. That leads to dogmatic teaching which in turn leads to boredom, failure to understand and ultimately to mathsphobia. I believe that the latter starts to develop from the earliest stages so these too need to be examined to avoid such a terrible outcome.

    I am not a researcher, just a retired maths teacher, so I am not in a position to make detailed recommendations as a result of evaluating different approaches. That said, my conviction is that the lowly place given to mathematical abilities in our general (UK) culture is such that the problem is reproducing itself. To become a primary teacher you need no more than a GCSE grade C. That is not a sufficient level to teach maths to primary children. The result is that many primary teacher are not in a position to teach maths in an imaginative way nor to respond adequately to the questions and problems they encounter. This is the beginning for many of mathsphobia and it continues through the secondary schools.

    So, no, I don’t think there is something called “the basics” which can be taught dogmatically as a basis for a later creative approach. Two Russian thinkers, one a philosopher very concerned with educational psychology (E.V. Ilyenkov) and the other an educational psychologist very well versed in philosophical issues (V.V. Davydov). Have made me go in for a lot of re-thinking about these problems. I recommend their writings.