by Carl Pierer
Thank you for opening my eyes concerning the question whether students should be beaten to study Maths up to the age of 18. Your well-argued and logically impeccable column in the Times establishes beyond reasonable doubt that no one needs to know any Maths further and above the mere basics. It is absolutely clear what those basics are, and they don't need further definition (obviously, knowing times tables is essential and needed, whereas being able to solve quadratic equations is far beyond basic).
Moreover, you successfully avoid the many times rehashed bad arguments in debates about education. Instead, you focus on the points that do indeed form the basis of any good and progressive line of argument. These are: (i) to think about reforms in terms of the education currently successful people have had, (ii) to do away with skepticism about inductive inferences, (iii) to consider a general education system in terms of highly talented and successful people, (iv) to not let yourself be confused by the subtleties of the subject matter as there really is just one thing at stake, (v) to insist that there is something wrong with the subject itself if the curriculum doesn't teach what is “useful”. Unfortunately, the brevity of your column prevented you from exploring the full force of your arguments. Allow me to do so on your behalf.
With one of your examples you solve two age-old problems in philosophy. You write: “The top western country [in the Pisa international league tables] is Liechtenstein. Know anyone who has changed the world who was educated in Liechtenstein? I don't either, but that is the European country we are hoping to emulate.” First off, this solves the problem of induction. The problem is that the inference from “All Swans I've observed so far are white” to “All swans are white” is not necessarily true, i.e. it's logically possible that “All swans I've observed so far are white” is true and “Not all swans are white” is true as well. But why do people wrack their brains over this? Your argument establishes that we merely need to assert the conclusion, isn't it just trivially true that since you don't know anyone who was educated in Liechtenstein and changed the world, there is nobody? At least 200 years of philosophy over and done with.
This conclusion is also a very important one, because obviously if Liechtenstein is doing well in the Pisa league tables and still there is no one who was educated there and changed the world, then the education in Liechtenstein cannot be that good. At least not as good as in Britain, where plenty of world-changing people were educated. Pisa league tables, your argument shows, are not a suitable means of measuring which educational system produces world-changing people. A general education is precisely about the upbringing of exceptional individuals and not the provision of basic numeracy and literacy. Since the Pisa examinations only manage to test the level of the latter, educational policy makers who are concerned with the questions that really matter should stop emulating countries that do well in the Pisa league tables.
This reasoning is related to the cases of two successful scientists you consider. One is a Nobel Prize laureate and the other a knight of the realm, who both underperformed in Maths at high school. This is a wonderful and intricate argument – especially since you preserve the full generality of the two cases by not putting them in the context of any argument at all! A brilliant move. Even so, you show that it is enough to pick a few exceptional stories to cast doubt on the general usefulness of a mathematical education. If two people did extraordinarily well, although they did poorly in mathematics, then in the greatest generality a solid mathematical background is superfluous.
Further, you aptly avoid getting confused by the idea that there could be more than one way of teaching Maths. This idea is so obviously wrong; it is a sad fact that there is one and only one way of how mathematics is to be taught – “America” is leading the way: “But in America, where ‘math' is taught until 18, pupils are even further down the Pisa scale, and the demands of algebra are cited as the main reason children drop out of high school.” This is another excellent argument, which promises to solve many problems – particularly in politics and ethics. The form is such: “America” does A in a particular way, that particular way of doing A has negative results in “America”, hence nobody should do A in any other way either. This way of reasoning comes in very handy: “America” has state-funded public high schools. “American” public high schools have a notoriously bad reputation. Hence, no country should have state-funded public high schools.
Lastly, most convincingly, you suggest a conflict between studying mathematics and creativity. Even though some stubborn mathematicians might deny this, it is painstakingly obvious that what mathematics is about is the acquisition of a set toolbox for a limited set of problems. Particularly, the most promising approach is “to go over the cosine rule” or study times tables by heart. The current curriculum focuses on precisely this. As you rightly point out, many tools (solving quadratic equations for example) aren't needed. So there is no need for further years of mathematical instruction. This clearly proves that the teaching of mathematics cannot possibly be improved to nurture creativity. You are perfectly right in thinking that there is something about the subject that does not encourage any skills other than the application of set techniques.
To sum up, thank you very much for this exceptional piece of argumentative excellence. It is humbling to see how you manage, in a comment on British educational policy, to solve a deep philosophical problem, demonstrate an intuitive grasp of mathematical reasoning and argumentation, as well as establishing that the study of mathematics is not needed for a successful career. It is particularly intriguing how well you made a case in point: you demonstrated that no profound understanding of logical or mathematical reasoning is required to write for the Times.
An enlightened fellow.