Mathematics education


In response to Jonathan Lange poking me with Lockheart's Lament. It seems that I haven't addressed that in any public space yet.

We looked at this essay in passing in the mathematics portion of my teaching degree. I agree with some of it. Parts of it match my understanding of the Australian mathematics curricula and teaching practice, parts of it don't; these parts may match American curricula and praxis, but I can't speak to that. I'm pretty sure some parts of it are objectively wrong.

Firstly, his essay is transparently written from a pure mathematician's point of view. Now, that's where my sympathies lie, but I'm not sure you'd get the same rant out of an applied mathematician. Furthermore, I disagree with some of his assumptions - particularly the assertion that teaching is some kind of mystical unteachable skill. I suspect that his teaching method is exclusive; when he says

It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them!
I feel that there's an implicit "naturally curious like anyone else (sufficiently like me)" in there. It sounds laudable, but may be exclusionary.

For an essay that's about how mathematics should be taught as an art it seems peculiarly dissonant to deny teaching the same status.

Also in the broad strokes, there's the little problem of the philosophical purpose of education¹. I think I'm broadly in agreement with Lockheart on the purpose of education, but ours is by no means the uniquely correct view. All writing about education is embedded in such a philosophical context, and the context that applies to a given system is mostly a political matter.

Finally, he makes an observation - children go into school all curious and excited, but the adults who come out of school are significantly less curious. This is a common observation, and it's also common to draw the causal link: school eliminates the curiosity of children.

I don't believe this; it seems to be a textbook post hoc ergo propter hoc argumentative fallacy. I'm not aware of any evidence that adults in the pre-compulsory education era were any more curious on average. Sure, there are plenty of historical artefacts showing there were people in ancient Babylon, Greece, the Islamic Caliphate and Renaissance Europe who were interested in mathematical play, but that's not answering the same question. We've got plenty of academic mathematicians - more than we have jobs for, in fact - who are equally interested in mathematical play.

Furthermore, most mammalian species are much less curious as adults than infants. No one is surprised that adult cats tend to be less curious than kittens; I don't see why it's surprising that adult humans are less curious than immature humans.

It's entirely possible that there exists a teaching and schooling method that preserves infant curiosity into adulthood, but I don't think we know what that one is, and not being able to do that is not the same as squashing curiosity.

In the actual mathematical meat, he's quite right that there's a lot more algorithm and definition in the Australian curricula than there is real mathematics. However, that doesn't mean that classroom experiences don't contain many mathematical experiences. The Australian syllabus documents are high-level descriptions of what students are tested on (and hence, are expected to learn) - for example, the NSW syllabus says things like

Data Representation DS4.1 (p 114): Constructs, reads and interprets graphs, tables, charts and statistical information
This does not seem particularly objectionable. Being able to read graphs and tables is an important life skill, and is a natural fit for a mathematics class.

Conspicuously absent is any mention of how this is to be taught; just what (by the end of year 7, in this case) it's expected that students will know or be able to do. There's plenty of scope for teachers to provide authentic² mathematical experiences. There aren't even that many top-level objectives that need to be reached - the whole set of objectives from year 7 to year 10 are there from page 16 to page 27. The crushing bureaucracy mandating rote learning of contextless data might be a peculiarly American phenomenon. Or maybe a product of hyperbole.

Most likely it's the result of his bizarre assertion that "true" teaching cannot be planned.

Lockheart's on firmer ground when describing the cultural understanding of mathematics. It is seen as a tool to other ends, as a collection of disparate arbitrary formulae, as primarily arithmetic. Few people know it for the creative art that it is. But then again, mathematics inarguably is a tool of immense power. As Lockheart puts it, mathematics is the music of reason; it is one of the few ways of knowing that you're thinking correctly³.

He's also more correct in practice than theory. As he says, teaching real mathematics is difficult. It's even more difficult when, as in the classrooms of the world-as-it-is, the teacher needs to take the whole class with them, under a certain amount of time pressure, and get to a pre-determined end state. While there's no particular impediment to exploration-driven mathematics it requires a level of mathematical confidence on the part of the teacher that's not common, particularly in primary school where teachers often only have high-school level mathematics.

So, it seems that my response to Lockheart's Lament is that I agree with it, except in all the details!

¹: This problem occurs over and over and over. It's the halting-problem of education.

²: I really hated the use of the word "authentic" as a buzzword in the teaching literature. Here's where I get to continue that fine tradition!

³: Of course, it doesn't, and cannot, ensure that your axioms are correct. That's what science is for!