You Got a Problem With That?

We’ll take a day off from bashing the draft curriculum, in order to bash the draft curriculum. This one’s not a Crash post, but it gets to the disfigured heart of the draft.

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.


UPDATE (28/05/21)

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.”

While we’re here, we’ll include another great quote, from the About section of Essential, by John von Neumann:

“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.

17 Replies to “You Got a Problem With That?”

  1. It is a beautiful book. Thank you, I’ll be recommending our budding math teachers to study it. Nice change of pace, an *essentially* positive post.

  2. The first quote in the introduction already sums it up:
    “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.
    Richard Courant (1888–1972) and Herbert Robbins (1915–2001), Preface to the first edition of What is mathematics?”

    “Exploring” a vaguely circumscribed topic without stating the minimum insights to be picked up, confirmed and reinforced, as in the draft curriculum, misses the point. So do the step-by-step recipes with lots of fill-in-the-blanks sheets for pretend complex problems in the current curriculum, which go through the motions of problem-solving but don’t involve any actual thinking.
    The one upside I can imagine to these approaches: They don’t require a teacher who knows the material in question, and instead can be implemented as a glorified one-size-fits-all textbook (with videos and whatnot), while being able to be passed off as students learning independently in an individualised way.

    I’d rather they just spent their time on the Euclid Game…

    1. Dear Mrs. Uncle Jezza, thank you. Yes, the quote is a great one, and your comments on ACARA’s “exploring” are spot on. Uncle Jezza had flagged the quote, and I had planned to include it, and at least one other, but the post didn’t run that way. I’ll add as an update now.

    2. “There is no royal road to geometry.” (Attributed to Euclid. Proclus. (1970). A commentary on the first book of Euclid’s Elements (Trans. G. Morrow). Princeton, NJ: Princeton University Press (p. 57).

      The history of collaboration between Courant and Robbins is interesting:

      Click to access cmjRobbins841.pdf

    1. Thanks, Terry. More foundational, but (and) it seems like a good collection. How did you use them, and was your use successful? Also, I’m not sure why attached PDF files are not showing up. I’ll try to look at the settings tomorrow.

  3. I had this class for several weeks filling in for a VCAL teacher. By and large, these students did not have a solid background in mathematics and I had little guidance on what was expected of me. These students had a wide range of interests; some were interested in building, others in hair and beauty, others in hospitality, others were in this class because no other mathematics class was suitable for them. What did they have in common? Nothing.

    So I developed this set of problems. The students could work at their own pace – without a calculator. I collected their workbooks after every lesson, marked them, and returned them at the next lesson.

    In the beginning, the students did not like the idea of no calculator. They quickly got used to the idea, some even liked the idea. One student resorted to asking Siri – unexpected but hilarious.

    I have to boast about one lesson when a teacher came into the room to see a particular student. At that time, two students who were so involved in discussing one of the questions they had resorted to taking over the whiteboard to discuss it between themselves. It was great to see them scribbling and arguing. The teacher was so impressed he wanted the problems to try on his students.

    After about two weeks, I had to relent on no calculators because the teacher whom I was replacing had also set them other work where they could use calculators. It seemed silly to have two sets of work, one in which they could use calculators and one in which they could not.

    Some problems grew out of my own experience. Problem 53 caused considerable debate. Problem 68 was also controversial as one student insisted that Tasmania is not a state of Australia; “It’s an island” I was told.

    The games section grew out of teaching similar students at a different school. I believe that there is room for games in secondary mathematics. The challenge is to get students, and others, to see beyond play.

    The collection has continued to grow so some things have not been used in a classroom.

    Was it a success? I don’t know.

  4. In the current debate, perhaps people are using the phrase “problem solving in the real world” in different ways. Applying mathematics to understand a phenomenon can be simple or difficult.

    A numerate shopper who sees a sign “50% off” will realise immediately that the new price is half of the normal price. This is simple applied mathematics, but not everyone has this ability – even those who have completed high school.

    Understanding that when a ball is thrown in the air it follows a parabolic trajectory involves more difficult applied mathematics.

    I doubt that anyone would disagree with the proposition that you need the basics to be able to solve these problems.

    However, I don’t see anything wrong with giving students *some* experience with open-ended questions in mathematics even at an elementary level e.g. “How much does it cost to keep a dog for a year?” This question is not my question, but I have used it in class with interesting results. Students often begin with “It depends on what you are going to count” – which is precisely the point. The question makes students think about assumptions in a model.

  5. As for solving real world problems let me narrate this story. I was teaching my Year 8 students about the Greek alphabet the other day; all mathematics students should know about it; “Have you ever heard of delta?” I asked. “Yes, it’s a squad.” “A squad?” “Yes, in Call of Duty”. They were amazed that I did not understand.

    The point is that my real world is not the same as the real world of Year 8 students.

    When we talk of problems in the real world, we should clarify what “real world” means.

    1. Hi Terry,

      I’ve played many a game of COD and my reply would have been to stare blankly and say “Ha ha. You know I mean in a mathematical context, right?” Let them have a giggle and move on. Certainly not to agree with them or start bonding over being a virtual soldier :).


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