Yesterday, a good friend and colleague, let’s call him Mr. Big, threw a book at us. By Alexandre Borovik and Tony Gardiner, the book is called The Essence of Mathematics Through Elementary Problems. The book is free to download, and it is beautiful.
There is much to say about this book. It is, unsurprisingly, a collection of problems and solutions. By “elementary”, the authors mean, in the main, in the domain of secondary school mathematics. Note that “elementary” does not equate to “easy”, although there are easy problems as well.
The problems have been chosen with great care. As the authors write, the problems are included for two reasons:
- they constitute good mathematics
- they embody in a distilled form the quintessential spirit of elementary mathematics
As indicated by the the Table of Contents, the problems in The Essence of Mathematics are also arranged very carefully, by topic and in a roughly increasing level of conceptual depth, and the book includes interesting and insightful commentary. Their twenty problems and solutions embodying 3 – 1 = 2 is a beautiful illustration.
The Essence of Mathematics also contains an incredibly important message. Here is the very first problem in the book:
1(a) Compute for yourself, and learn by heart, the times tables up to 9 × 9.
Regular readers will know exactly where we’re going with this. Chapter 1 of Essential Mathematics is titled Mental Skills, it includes simple written skills as well, and the message is obvious. As the authors write,
The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight.
In particular, on their first problem,
Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system – in which representing any number, and implementing any operation, are reduced to a combined mastery of
(i) the arithmetical behaviour of the ten digits 0–9, and
(ii) the index laws for powers of 10.
Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.
What does all this have to do with ACARA’s draft curriculum? Alas, nothing whatsoever.
The draft curriculum is the antithesis of Essence. The “problems” and “investigations” and “models” in the draft curriculum are anything but well-chosen, being typically sloppy and ill-defined, with no clear direction or purpose. The draft curriculum also displays nothing but contempt for the prior mastery of basic facts and skills required for problem-solving, or anything.
Essence is not a textbook, but the authors clearly see a large role for problem-solving in mathematics education, and, with genuine modesty, they can imagine their book as a natural supplement to a good curriculum. Such a role can mean slow and open-ended learning, or at least open-ended teaching:
Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time.
The value of such digressions and explorations, however, does not negate the primary goal of mathematics education:
Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students’ performance on the next test, but to establish a firm platform on which subsequent stages can build. …
The effect [of political pressures] has been to downgrade the more important challenges which every student should face: namely
- of developing a robust mastery of new, forward-looking techniques (such as fractions, proportion, and algebra), and
- of integrating the single steps students have at their disposal into larger, systematic schemes, so that they can begin to tackle and solve simple multi-step problems.
Building systematic schemes out of the mastery of techniques. Or, there’s the alternative:
A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy).
There is problem-solving, and there is “problem-solving”. ACARA is shovelling the latter.
Mrs. Big, AKA Mrs. Uncle Jezza, has given the draft curriculum a very good whack in the comments, below. As part of that, she has noted an excellent quotation that begins the Preface of Essential. The quotation is by Richard Courant and Herbert Robbins, and is from the Preface of their classic What is Mathematics?
“Understanding mathematics cannot be transmitted by painless entertainment … actual contact with the content of living mathematics is necessary. The present book … is not a concession to the dangerous tendency toward dodging all exertion.”
“Young man, in mathematics you don’t understand things. You just get used to them.”
Understanding is a fine goal, but it can also be a dangerously distracting goal. ACARA’s “deep understanding” is an absurdity.