Ken Clements is an interesting character, of whom and by whom we’ve been reading a lot of late. We plan to write about Clements in the near future, but for now we’ll just post an article by Clements on a really interesting character. Continue reading “NotCH 5: Ken Clements on a Very Young Terry Tao”
We have written about Tony Gardiner and excerpted from his writings a number of times: here, here, here, here and here. We will post an entire article by Tony in a day or so, but we are first posting the problems contained within the article. Unlike the “problem” here, the problems below are genuinely presented by Tony to puzzle over, and are only loosely tied to the text of the article. Have fun. Continue reading “NotCH 4: Five Problems From Tony Gardiner”
This is our final excerpt from Teaching Mathematics at Secondary Level Tony Gardiner’s 2016 commentary and guide to the English Mathematics Curriculum. (The first two excerpts are here and here.) It is a long and beautifully clear discussion of the nature of problem-solving, and its proper place in a mathematics curriculum (pp 63-73). (For Australia’s demonstration of improper placement, see here, here and here.)
Definitely not Sydeney.
OK, in a futile attempt to unPonzi our blogging scheme, we’re closing off our four competitions.* The winners are indicated below, and any winner who has not died of old age should email us to receive their prize, a signed copy of the best-selling** A Dingo Ate My Math Book.***
The Australian Mathematical Sciences Institute has finalised and released its submission for ACARA’s consultation on their draft mathematics curriculum. For eccentric reasons, we haven’t properly read AMSI’s submission. (Seriously.) We understand that it is a good and strong statement. The second and key paragraph is
In early April, AMSI, together with some of its key partners, released a joint statement on the proposed new curriculum “Why maths must change”. AMSI initially endorsed the revised draft curriculum in our joint statement. However, there is now an opportunity to comment on the draft curriculum, and we have revised our position, following extensive consultation with representatives of our member organisations. Many members expressed concern, and indeed alarm, at numerous proposed changes. AMSI and its members believe that the new curriculum should be delayed, and we ask ACARA to halt the current review process.
What is the strongest argument FOR the draft mathematics curriculum?To be clear, what we’re asking for are very specific examples of good things within the draft curriculum, examples of content and/or elaborations that are genuine plusses. So, for example, claiming “the focus on mathematising” as a good won’t win a prize. First of all because the suggestion is really stupid, and secondly because such a generalist statement provides no specific evidence of how the mathematising is good. If you really want to argue that the mathematising is a plus then the argument must be based around very specific examples. Similar to our problem-solving competition, the intention here is not to imply or to prove that there is nothing of value in the draft curriculum. Rather, the competition is intended to imply and to prove that there is very little of value in the draft curriculum. Your job is to try to prove us wrong. Answer in the comments below. The provider of the most convincing evidence will win a signed copy of the number one best-selling** A Dingo Ate My Math Book. *) If anyone is aware of any article/post/tweet/anything in support of the draft curriculum, which also contains at least a hint of evidence, please let us know and we will seek to address it. **) In Polster and Ross households.
Update (29/07/21)We’ve finally ended this. The winner is really nobody, but we’ve awarded it to John Friend. See here for details.
We’ve written about this before, and the point is obvious. But, it’s apparently not sufficiently obvious for some wilfully blind mathematicians. So, let’s go again. Plus, there’s a prize for the best comment.*
ACARA is playing people with a cute syllogism.
- Problem-solving is good.
- The draft curriculum contains lots of problem-solving.
- Therefore the draft curriculum is good.
Yep, the syllogism is flawed from the get go. But in this post we want to focus on the second line, and we ask:
Does the draft mathematics curriculum contain any problem-solving?
Certainly the draft curriculum contains a hell of a lot of something. As we’ve noted, the draft refers to “investigating” or some variation of the word 298 times. And, students get to “explore” and the like 236 times, and they “model” or whatever 264 times. That’s a baker’s ton of inquiring and real-worlding, which some people, including some really clueless mathematicians, regard as a good thing. Ignoring such cluelessness, what about genuine mathematical problem-solving?
The draft curriculum refers to “problem(s)” to “solve” 154 times. But what do they mean? When, if ever, is the draft referring to a clearly defined mathematical problem that has a clearly defined answer, and which is to be solved with a choice of clearly defined mathematical techniques? To the extent that there are any such “problems”, do they rise above the level of a trivial exercise or computation? In the case of such trivial “problems”, is the label “problem-solving” more than a veil-thin disguise for the mandating of inquiry-learning?
In brief, is there more than a token amount of the draft’s “problem-solving” that is not either real-world “exploring/modelling/investigating” or routine exercises/skills to be taught in a ridiculously inappropriate inquiry manner?
Perhaps genuine mathematical problem-solving is there, and we are honestly curious to see what people have found or can find. But, we’ve found essentially nothing.
And so, to the competition. Find the best example of genuine, mathematical problem-solving in the draft curriculum. Answer in the comments below. The most convincing example will win a signed copy of the number one best-selling** A Dingo Ate My Math Book.
*) Yes, yes. we have those other competitions we still haven’t finalised. We will soon, we promise. As soon as we’re out of this ACARA swamp, we’ll be taking significant time out to catch up on our massive tidying backlog.
**) In Polster and Ross households.
We’ve finally ended this. The winner is, hilariously, Glen. See here for details.
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.