I was planning a rant about the dangers of formal assessment of mathematics in primary schools, the insanity of streaming maths classes based on knowledge of times tables at age nine, and suchlike. But there are more than enough depressing stories about the UK education system at the moment, so I've tried to make this a more positive piece, about some of the things I think children should learn about mathematics in primary school, along with a random collection of ideas for actual teaching.
First of all, by age twelve children should have been exposed to something of the breadth of mathematics. This means elements of arithmetic and number theory, geometry and topology and graph theory, logic (including verbal reasoning), algorithms (aka programming), probability and statistics, and so forth. And there should not be a narrow focus on a few topics within those areas: there's a lot more to arithmetic, for example, than multiplication and division.
Secondly, children should come away with some understanding of the depth of mathematics. In particular they should understand that there is a lot of mathematics that is hard. But wait, I hear you say, weren't we all up in arms about Barbie dolls that say "math class is tough"? Do we want to discourage people? Having talked to any number of people who are more or less actively averse to mathematics, from my parents' generation through to that of my nephews, it is clear that it is not so much that they think mathematics is hard but that they think they are no good at it. And even the "better" children — the ones streamed into the top classes — are not told that mathematics is hard, but rather (even if only implicitly) that they are clever. That has a kind of placebo effect on their performance, at least in the short-term and on the narrowly limited syllabus they are exposed to, but gives many of them entirely the wrong mindset to cope later, when they are faced with further levels of abstraction or entirely different kinds of concepts.
So children should learn that there are concepts that are difficult to understand and problems that are hard to solve. There are things no one understands, things they are probably never going to understand, things they will only ever partially understand, things they may only understand after multiple attempts — and that's fine! They should learn that trying to do something and failing is perfectly ok, and that they can always come back to something later, maybe from a different direction or when they need it and are better motivated. And that mostly it doesn't matter: if they never comprehend some particular topic, or take a few years longer than their peers to master some skill, that's neither here nor there. [This is, of course, entirely incompatible with rigid syllabuses and assessment metrics that are going to get children relegated to bottom classes, and potentially their school and teachers marked down as Failing and sanctioned, if they don't acquire the right skills in the right order at the right ages.]
Finally, and perhaps most importantly, children should continue to understand and experience mathematics as a form of play. They should view it as a game, or rather a vast array of different games with interestingly different rules and constraints and possibilities. It is also a vastly powerful toolkit for exploring and understanding the world, natural and social, but it is at the same time a world in its own right, full of surprises and unexpected things waiting to be discovered (along with plenty that is comfortingly familiar).
There follow some fairly random ideas on teaching mathematics, perhaps just for ways to play with Helen, to use if I run some kind of pre-school maths circle, or for an after-school mathematics club if I end up running something like that. This is very much a work in progress, so any suggestions would be very welcome.
A Maths Circle for Pre-Schoolers?
I've been inspired by Zvonkin's Math from Three to Seven, where he describes his experience running a maths circle for his son and some of his friends in Moscow in the 1980s, starting when his son wasn't quite 4. The basic idea here is a kind of guided play — give the kids some suitable props, demonstrate something or point out a problem, and see where they go with that, offering guidance as seems appropriate. The goal is not to produce the next Gauss, but to entertain the children and the teacher. (Perhaps there are some kinds of mathematical intuitions (for spatial orientation?) that (like phoneme awareness or familiarity with musical scales) develop better with early exposure, but I haven't seen any evidence for that and, in the absence of alien overlords with an obsession with Yang-Mills field theories, I'm not planning to try to give Helen a geometric intuition for non-commutative Lie groups.)
I've collected some tools/toys: Diogenes (logic) blocks, pegboards and pegs, a lace-threading board, and so forth. Arvind Gupta has an extensive set of pages that can be cut up to make polyhedra (PDF). But most ordinary toys can be repurposed for mathematical games of one kind or another. A set of six wall-sucker ducks, for example, is Helen's current favourite bath toy: she likes us to take turns "making a new game", which means finding a different way to lay out the ducks, 3x2 horizontally or vertically, as a triangle, partitioned 4-2, and so forth.
I'd be interested in recommendations for puzzles or toys that involve three-dimensional spatial intuition, as that has never been one of my strong points. (Gromov, quoted in Berger, suggests that the non-commutativity of O(3) - the group of rotations and translations in three dimensions - makes it difficult to grasp, even though it "pervades all the essential properties of the physical world".)
It wouldn't be feasible for a school maths club, since the after-school clubs there are essentially child-minding services, but if I run a maths circle for pre-schoolers I'd be tempted to require some kind of parent-participation (like the dance classes I've been taking Helen to). Apart from the parents maybe having some fun too, getting them interested in aspects of mathematics is pedagogically important: they are still by far the most important models for their children.
Basic programming, with no infrastructure at all
Put your pre-schooler or toddler on your shoulders and explain to them that you will only obey three instructions: "forward" (on which you move forward one metre), "turn right" (on which you turn right 90 degrees) and "turn left" (on which you turn left). After carrying out an instruction, and with a short delay, you say some fixed response - "command complete" - to indicate readiness for the next instruction.
Then see if they can get the idea of controlling you like a robot, to navigate around obstacles perhaps, or just to get you where they want to be. That's basically the key ideas of programming: fixed instruction set, limited execution rate, ...
Ancient Egyptian mathematics
Here I'd follow Reimer's Count Like An Egyptian. This basically teaches ancient Egyptian methods for doing addition and multiplication and for representing and manipulating fractions, which are completely different to our standard methods (either of the present or of the 1950s).
One key thing that would be learnt from this is that different methods can exist for solving the same problem, and that there might not be any one best one. Also, if you only know one method for doing something like multiplication, it can be hard to separate the underlying abstraction from the particular method.
I suspect this is most suited to late primary school or early high school, perhaps as additional material for students who are bored. But with the right approach I wonder if it might not be useful for some students who've hit a mental block with the standard methods for doing arithmetic.
The 17 regular plane tiling patterns
I'm thinking of something following Part I of Conway et al.'s The Symmetries of Things. This book nicely illustrates how fast mathematics can climb from elementary foundations: Part II assumes solid group theory and Part III "will be completely understood only by a few professional mathematicians" (and is mostly beyond me), but most of Part I seems like it would be perfectly accessible to primary school children, if not pre-schoolers.
In particular, it doesn't require any arithmetic and it is incremental. So we could just look at the kaleidoscope patterns and have fun with those, then add the gyrations, and leave the "wonders" and "miracles" for later. Also Conway et. al. work backwards, simply assuming a Magic Theorem in their classification — and this is stated in terms of dollars, almost as if they were contemplating school use! — then proving that using Euler's formula, and only right at the end proving that.
There are lots of lovely illustrations of the patterns involved (and Chaim Goodman-Strauss has a web site with versions of these). And much of this could be interactive: finding mirror lines using a mirror, drawing them, playing with a kaleidoscope, pinning copies of patterns and rotating them to find gyration points, and so forth.
This is probably suitable for 6+ olds, but I was so impressed by the presentation that I had to restrain myself from trying to explain it to Helen.
Teach the times tables backwards
All of the effort that goes into memorising times tables does nothing to bring students closer to such wonderful numbers as 37 and 71! Instead of (or as well as) learning the times tables normally, teach them the first 150 or so natural numbers, and whether they are prime or not and what their prime decompositions are. It is at least as important that 16 is 24 as that it is 4x4 or 2x8.
Goldbach's Conjecture, Cantor Diagonalisation, and so forth
There are still unsolved conjectures in mathematics whose statement is quite elementary. Not showing children some of these would be criminal.
Similarly for problems that have been solved but whose solution is difficult or opaque: the four colour theorem, for example.
Conversely, there are some quite striking results that can be demonstrated with relatively elementary mathematics. I went through a phase in my teens and twenties when I used to try to explain Cantor diagonalisation and the difference between countable and uncountable infinities to all my friends, no matter how little mathematics they had done. It just seemed like so much fun, and the only setup (assuming familiarity with fractions and decimals) is explaining comparison of cardinalities by bijection/injection (and perhaps proof by reductio).
During my high school years I sat problem-solving competitions, went to after school classes, went to a National Mathematics Summer School, and so forth. But I've forgotten pretty much everything I did then, so if comes to teaching that kind of thing to high school students I'm going to need a refresher. (I should read George Polya's How to Solve It.) When it comes to that point, though, there's the UK Mathematics Trust to fall back on, and other more formal resources.
Next: see my followup, two and a half years later "mathematics approaching 6".
My experience --- while by no means systematic --- is that "problem solving" has become a hypertrophied, inward-looking, overly-competitive thing in selective American high schools and colleges. (Look up the Putnam competition some time.) My brief encounter with it as a student was pretty repulsive. As a teacher I've acquired the impression that those who get into it need considerable de-programming to actually do research. _How to Solve It_ is genuinely great, but I think the institutional lessons taken from it have been unproductive.