Classical education and the math/science problem

Those who design systems of classical education (I’m one of them) tend to stand to math and science like a teacher who can’t get the video projector to work: all are aware that they have to do something but none are sure how to do it, and in the meantime they are painfully aware that they have a class to teach that isn’t being taught.  What to do?

1.) Towards a more classical science:

Classical education seeks to give the principles of things, but the principles of the sciences are primarily their experiments, and there are serious practical problems involved in doing these experiments.  First of all, there simply is no curriculum that teaches the sciences through their foundational experiments. Second, there are impediments to even making such a program: in contrast to the clean, universal, algebraic conclusions that we draw from experiments, the experiments themselves are incomparably more messy, particularized, and inexact; and it is rarely the case that the sort of person who is good at teaching “science” (i.e. the systematically developed set of conclusions) is good at doing science (i.e. showing people how to set up apparatus). Even if one found a good teaching experimentalist, the experiments themselves can be cost-prohibitive, and (on a theme to be repeated frequently) there simply isn’t enough time to do them. Consider what it took for the Mythbusters to experimentally prove something as seemingly evident as the cancellation of momentum.

That said, there is also a critique in all of this about how science is taught, and what a classical education could do to remedy it. We simply aren’t teaching science when all or most of what we present is the sanitized, algebraic, universal law. Natural science is the daughter of abstract theory and chaotic, statistical, and frequently abductive inference, and we too often treat science as though she sprouted fully formed from the head of the theoretical parent alone. One simply can’t deal with a bunch of experiments in a purely theoretical way – there is an irreducibly artistic and intuitive element to them. It is not per se irrational, for example, to run a hundred experiments, find one that agrees with the hypothesis, and then think that the hypothesis is proven (this is, in fact, exactly what the Mythbusters do in the above link).

Conclusion: classical teaching of science should strive to be more experiment based. Cost: less time to do theoretical things and to cover more advanced topics.

2.) Towards a more classical math: 

Classical education sees mathematics as the paradigm of systematic thought, but modern math curricula have only a very general systematic order. The typical math textbook, after a brief nod to number theory in Chapter 1 (natural, whole, integers, rationals, reals…) and some basic manipulation rules, (communitive properly, transitive property, etc.) follow up with something that is anyone’s guess, and that need not have any relation to what came before. Here again, part of the problem is that the math text is presenting material with an eye to preparing one for diverse fields: engineering, physics, chemistry, etc. and so it has less the look of a perfectly assembled engine and more the look of a well-stocked general use tool drawer. The goal in stocking a tool drawer is not the order in which you get the tools or the order you place them in, but simply to have what you need to do the job. The same would be true even for those of us who see algebra as part of a more general art of “problem solving” – it was originally thought up to solve geometric problems and gradually extended its use to the other sciences. But problem solving and systematic exposition are to some extent antithetical: the system treats the end as given while the problem solving art doesn’t. There is a certain logic in the more or less random exposition of topic in the typical algebra textbook – a problem is a situation where one does not know what should come next, and the typical textbook is laid out in exactly this way.

Conclusion: Rename algebra “problem solving” but keep it as its own class. Be clear about how, in its highest use, it is a tool for solving science problems. Teach only techniques relevant to solving scientific problems, in such a way as to make it obvious exactly what the class is good for. Teach another class called “Mathematics” where one gives a systematic account of mathematical things. Cost: You’ve effectively doubled the math load, though you could massage this by putting less in the problem solving class than in is presently taught in algebra. Still, this takes a bite out of the rest of the curriculum and leaves one with far less time to do other important things. But it would be worth it.

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7 Comments

  1. May 13, 2013 at 7:21 pm

    See “Natural Philosophy & the Physical Sciences” by Fr. Wallace, O.P.

  2. Dixi said,

    May 14, 2013 at 4:21 am

    You may have these, but if not, also helpful are J. Klein, Greek Mathematical Thought and the Origins of Algebra; Maurer, Division and Methods of the Sciences; also articles by Maurer Thomists and Thomas A. on the foundations of math., and Neglected T. text on the Foundations of Math. ; Hippocrates Apostle, Aristotle’s Theory of Mathematics (partly a re-construction but a good one of what Apostle thinks Aristotle would have said, and answers some standard objections by moderns, including Heath, to Euclid); Apostle, Mathematics as a Science of Quantity; Apostle, Methodological Superiority of Aristotle over Euclid, could be used as a foundation for better instruction and text writing; from another perspective, articles by E. Nelson Princeton Dept. Math on foundations

  3. Courtney Ostaff said,

    May 14, 2013 at 4:50 pm

    I’d be curious to see your opinion on classically determined curricula. Most classical homeschoolers I know use Saxon Math, for example.

    • May 14, 2013 at 6:11 pm

      I’ve taught many semesters of Saxon. It’s about as diametrically opposed to the classical view of math as one can get – chapter topics are sheerly random and there is no relation of concepts to their basic principles. The author himself tells you that he wrote the text as he did because systematic presentation of mathematics proved too hard for the students, i.e. he is defining his program in opposition to the classical mode (seriously, it’s all right there in the introduction to the book). There are virtues to this approach, and in fact I don’t think he went far enough – in my view we should just commit ourselves to teaching algebra as a sort of problem-solving logic of discovery and teach classical mathematics as another class. This is (perhaps) unworkable, but it is an ideal we might approximate.

  4. Dixi said,

    May 16, 2013 at 1:40 am

    On the practical side of teaching, the Russians seem to have many texbooks that sound better than anything found recently at least in the US. (French textbooks series seem very nice also.) The book Algegra by Gelfand and Shen is worth looking at. Students do at some level want to understand, so a treatment that can convey understanding is very valuable. Saxon doesn’t explain everything as well as he could, even for the audience he addresses. In teaching algebra, I typically observed a lot of students did not seem to be able to “get” algebra. However, they could do the problems in their own way, which often amounted to a series of approximating steps prompted by common sense grasp of how quantities behave. E.g., 5 is too much, but 3 is too small, so try 4 etc. So it was not lack of intelligence or will to work, but some opacity in the presentation of algebra that was to blame. If one could as a teacher understand this phenomenon better, it might lead to better teaching of algebra, ie better bringing the student into act as a knower of mathematics. Btnd. Russell’s anecdote – “Teacher: let x be the number of sheep. Student – But sir, suppose it isn’t …” – contains some of the key to the issue.

  5. Matthew DuBroy said,

    April 10, 2016 at 8:59 pm

    What would teaching classical mathematics look like? What would the order be, etc.?

    • Matthew DuBroy said,

      April 10, 2016 at 9:05 pm

      A book/article recommendation would be very helpful.


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