I feel like my examples from before (times tables, square and cube numbers, prime numbers) are all zoomed in very close compared with the other examples I gave for other subjects. Well here’s one broader suggestion:
I created these before I started teaching. We use much of this language all the time, and as teachers have a mental image in our minds much like this, but I seriously doubt that anyone I teach has this same overview… and how could they? They won’t just pick it up organically; if I don’t make an effort to teach them, they won’t learn it. How might I do that?
Two ways:
The first would be to have them recreate this structure, leveraging the testing effect until they could reproduce it fluently.
The second would be to ensure that pupils knew which part of the framework each new sub-topic or question type fitted into.
Multiple-Choice Questions
MCQs would be excellent for testing this, since it is an exercise in categorisation/conceptual understanding.
For example:
Which of these questions provides an example use of correlation?
or the other way around
Which parts of mathematics might be used in solving the following question?
In the second case, questions might require knowledge of more than one part of mathematics, and an MCQ would provide the opportunity to select more than one related topic. It might even be interesting for problem-solving – one of the problems with problems is that novices cannot automatically see what kind of mathematics can be used for solving them. For example in the question:
There are some cats and birds in a room. Altogether there are 38 heads and 118 feet. How many birds are there?
Most people do not automatically recognise this as being a problem that can be solved using simultaneous equations. It’s not the only way of solving it, but an MCQ would provide both the opportunity for pupils to see it suggested amongst a list of other possibilities, which might spark a link previously unseen, without giving too much direction. It might also help a teacher see different methods that children might opt for in their class.
Carefully designed, it might help to identify blind-spots in the group’s thinking – for example a bank of fifteen questions could be developed that all look very different, but could all be solved using simultaneous equations. Simultaneous equations could be suggested along with fourteen other options for every question. If pupils in the class were consistently never selecting simultaneous equations as an option, then it would become obvious that they weren’t linking the technique to how it can be used to solve these myriad problem types.
So there’s another suggestion for maths, anyway, very different I think from my earlier suggestions.
Maths teachers – would you be prepared to spend lesson time building up broad and shallow knowledge of a framework such as this one?
...to the real. 
I’ve been thinking about how you could divide up maths recently. Mainly because I think the opportunity was missed to create 2 new GCSEs rather than bloat up the old one. It seems to me that numeracy and algebra are, if done without a calculator, maths. Everything else (until you get to A-level) is the application of maths. I’d want to make that split. I realise this is probably heresy to many in maths education where there is a strong belief that calculation is not proper maths.
Reblogged this on The Echo Chamber.
I just read your article in TES posted yesterday, 6/5/15, about Direct Instruction. Your article hit on the key distinction – the analogy between the theoretical basis of designing effective instruction (engineers of race cars) and the technical aspects of delivering effective instruction (race car drivers). The implications of the analogy are quite radical. As your article suggests, the teaching profession (not to mention education policy and administrations as well as teacher training programs in higher ed) would have to be conceived in a totally different light if teachers weren’t saddled with the abortive expectation that they’re meant to possess the competence or time to design instructional materials in a more or less ad hoc fashion. I found especially interesting your reference to Elizabeth Green’s book, which I haven’t read but will read, because I’ve often thought that taking a medical perspective on problems of education might bring public attention to what the real problems are. I don’t know how it is in England, but here in the U.S., political diagnoses of the sickness of the educational system always fall back on the same rote cliches and false causes: lack of adequate funding, lack of coherent standards, lack of parental and community involvement, etc. Not that these things aren’t vitally important, because of course they are – and who would deny it? But they’re inert premises that don’t address the fundamental problem that so many students receive bad instruction, and this fact is not reducible to the question of whether or not the teachers who deliver it are “good” or “bad” teachers. It’s a Nietzschean dilemma – certainly there are “good” and “bad” teachers out there, but if no one recognizes the logical and functional difference between an engineer and a driver, then the meaning of such normative distinctions is either arbitrary or ideologically-confused.