Tony Gardiner on Exactness and Approximation

It’s been a while. We’ve been trying hard to get out a long, long post on ACARA and the draft curriculum. (Working title: Moby Albatross). We also had a great WitCH planned, but that was torpedoed by Simon the Likeable. For now, we’ll keep readers occupied with some excerpts from the writing of Tony Gardiner.

Gardiner is an English icon, sort of a one-man AMT, but without the foot-shooting. Previously, we wrote about Gardiner’s and Alenxandre Borovik’s beautiful (and free) book, The Essence of Mathematics Through Elementary Problems. Then a few weeks ago, we posted a “puzzle” from Gardiner’s book, Teaching Mathematics at Secondary Level.

Written in 2016, TMSL is Gardiner’s commentary and guide to the English Mathematics Curriculum, with a particular focus early secondary school (Key Stage 3). Although framed around a specific curriculum, much of  TMSL is written from a more general perspective. In particular, Chapter 2.3 of  TMSL is on problem solving and the manner in which problem solving can fit, or misfit, in a mathematics curriculum. We shall excerpt this chapter in three posts, beginning with two short passages and concluding with the main content of the chapter. Our first excerpt is on the importance of exactness in a mathematics curriculum (pp 73-75).


Two further issues warrant comment before we move on …. The first is the matter of exactness and approximation …

Mathematics used to be known as “the exact science”. Mathematical objects sometimes have their roots in the world of human experience; but they become mathematical only when the underlying ideas are abstracted from these roots. Unlike disciplines that work with real data or objects, mathematics studies a world of idealised, mental objects. For example,

  • numbers have their roots in experience;
  • but they soon become “mental objects” with exact properties, and are manipulated in the mind.

In much the same way, a sheet of A4 paper, or a wooden door, may serve as a suggestive model for a rectangle, but

  • a mathematical rectangle is a perfect mental object, whose diagonals are exactly equal—their length being given exactly (in terms of the sides) by Pythagoras’ Theorem.

The mathematical universe consists of imagined objects, which are precisely defined, and hence uniquely knowable. In particular, mathematics, or “the art of exact calculation”, belongs to a completely different conceptual universe from the practical world in which one might

“draw a scale diagram of a rectangle and measure the approximate length of the diagonal”.

Helping pupils to appreciate the difference between these two universes, and to see the advantages—even for the most practical of purposes—of engaging with the exact world of mathematics, should constitute a key (though often unstated) goal of any curriculum.

The process of developing internal methods of calculating with these exact mental objects (whether numbers, symbols, shapes, or functions) is much the same today as it ever was—and is rooted in mental work and written hand calculation. Once these ideas are suitably embedded in the mind, calculators and other tools have much to offer: but initially, the learning process proceeds more naturally without such distractions.

The fact that the world of mathematics operates on ideal objects allows its ideas, its notation, its methods of calculation, and its processes of logical deduction to be exact. This guarantees that the answers and conclusions produced in mathematics are as reliable as the information that was fed into the relevant calculation or deduction. The importance of this aspect of elementary mathematics has been considerably blurred in recent years—for example, by inappropriate and premature dependence on calculators, by reduced emphasis on the need to attain mastery of the art of exact calculation, and by the way “valuing children’s own reasons” has been misconstrued.

In contrast to the exact mental universe of mathematics, the world of experience, of measurements, and of ideas is inescapably “fuzzy”. It should be a goal of any curriculum to convey implicitly this key distinction between the exact world of mathematics, and the approximate world where mathematics is used and applied.

Mathematical exactness is quite different from precision. The very idea of “precision” recognises that, outside mathematics, all measurements incorporate a degree of error, and so are approximate. In contrast, exactness in mathematics allows no scope whatsoever for error; indeed, in an exact calculation an error of any kind undermines the validity of the whole process. Mathematical methods can be applied to values which are only known approximately; but the “exact answer” which mathematics then provides indicates the exact range of values within which the actual answer must lie. To achieve this, we first need to know

  • the maximum extent of potential error in the given data, and
  • how these potential errors accumulate when one carries out exact calculations with numbers that are only known up to this level of accuracy.

For pupils to master the art of approximating arithmetical calculations in integers, they first need to master the art of exact calculation. Only then can they use their knowledge of exactness as a fulcrum for thinking precisely about more elusive approximation, or estimation … And when they come to analyse the errors introduced by such approximations, they will find that this is done via the exact calculations of elementary algebra. Thus, even when seeking to transcend the inherent exactness of arithmetic by cultivating the art of making estimates, there is no escape from the maxim:

Mathematics is the science of exact calculation.

4 Replies to “Tony Gardiner on Exactness and Approximation”

  1. I like it. I would add that even processes that may *appear* approximate, such as asymptotic analysis, probability theory, estimates for solutions to equations, and so on — even these are still *exact*. The fields of numerical analysis and numerical approximation, including things like computational limits and floating point errors, are *exact*. We are calculating precisely and controlling mathematical objects. That is, things that are not existing in the real world.

    Once a computer program is written, hopefully our work in the mathematical world will help us to make the program do what we want. Same story with predictions about the real world using PDE. But these are not the same things, the mathematical objects manipulated in numerical analysis are not actual pieces of computer code, and the functions denoting concentration or temperature are not concentration or temperature itself.

    This is really important, and something I wish more teachers knew at a fundamental level. When I said to one of the teachers at my son’s school at the end of a word problem “Aren’t numbers one of the most useful made up things ever?” they asked me later what I was talking about. Five minutes later and I think they still didn’t understand that the number one isn’t a real thing that exists.

    1. I gave a talk once to the Department of Approximation Theory which was in the Faculty for Exact Sciences of the university.

      1. Approximation Theory was an entire department? Cool (using my very nerdy definition of “cool”)

        And Glen – as a closet philosopher, I could have a lot of fun with your assertion that the number one doesn’t exist (taking a Descartes-type approach is the easy way out) but I kind-of agree with you that numbers are a tool for understanding the world rather than being “things” in and of themselves.

        There is a classic IB joke about a Thermos flask which comes immediately to mind; it involves the “invention” of the alphabet (search it up if you want an intellectual dad-joke)

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