[keithpeter]

Quotes from OA that struck me as on the button...

"A prejudice that was strongly confirmed was the value of mathematical fluency. Barton says, and I agree with him (and suggested something like it in my book Mathematics, A Very Short Introduction) that it is often a good idea to teach fluency first and understanding later."

Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged 16 and over in a further education college. They were constantly tripping over really quite basic little skill issues and that prevented them from seeing how to tackle the longer and more complex problem solving questions.

"I would go for something roughly equivalent [in the solving of equations such as 4x - 8 = 2x + 2], but not quite the same, which is to stress the rule you can do the same thing to both sides of an equation (worrying about things like squaring both sides or multiplying by zero later). Then the problem of solving linear equations would be reduced to a kind of puzzle: what can we do to both sides of this equation to make the whole thing look simpler?"

The idea of just playing with the notation is one I fully intend to try but getting people to think in that abstract way is hard work.

[zazen]

> Agree fully with Barton and OA here. Until recently I taught GCSE Maths re-take students aged 16 and over in a further education college. They were constantly tripping over really quite basic little skill issues and that prevented them from seeing how to tackle the longer and more complex problem solving questions.

I also agree fully. A little while back I did some support tutoring for A-level maths students. The number of students who turned up who mysteriously "had problems with longer questions"... I wish I'd known the example of calculating the perimeter of the rectangle with fractions. That would have really helped explain why the problem wasn't really the length of the question, it was the fact that the student had never properly learned the component skills separately.

Unfortunately, the problem of building impressive-looking edifices on shaky foundations is absolutely endemic in British high-school maths teaching. Thousands of students who never quite understood fractions are "learning" calculus through being taught recipes, and the easier exam questions are formulaic enough that they get through with Cs at least, without any mathematical understanding.

The A-level statistics modules, in particular, have very impressive _sounding_ syllabuses. Students learn T-tests, Chi-squared tests, all this sophisticated statistical machinery. If all these students really understood this stuff, Britain would have a vast army of highly trained statisticians. But nothing of the sort is true, of course: students are just learning a recipe for processing numbers. I can't imagine the carnage if a statistics exam asked the students to write an essay explaining the principle by which a T-test works.

Pardon my rant, this has been on my mind for a while.

[keithpeter]

Going back around the millennium or before when I last taught A level maths at college, we had them in over the summer before term started for a two week intensive algebra and basics course.

Seemed to help.

The original author (Tim Gowers, a Fields medallist and professor of mathematics at Cambridge) has a totally hilarious blog post about being asked to coach a teenager doing A level maths...

https://gowers.wordpress.com/2012/11/20/what-maths-a-level-d...

[zazen]

Thanks for linking that, it's a great read. I really should read more of Gowers' posts.

The phrase "memory works far better when you learn networks of facts" was a happy find - I've never been able to express that idea so concisely.

I remember discovering they'd moved "differentiation from first principles" away to a further-maths module, as if it's a peripheral, difficult little oddity for the keen kids to hear about. It was the surest, saddest sign that the powers that be had given up on genuinely educating the average A-level maths student.

[mncharity]

> memory works far better when you learn networks of facts

One challenge with teaching a more rough-quantitative Fermi-question-ish introduction to sciences, is it's more sensitive to integration and correctness of understanding. With a Trivial-Pursuit memorize and regurgitate style of "understanding", damage from misconceptions and fragmentation of knowledge is local. Whereas rough-quantitative reasoning benefits from being able to... slide around the knowledge space. Jagged misconceptions and fragmented knowledge seriously impedes the sliding. I imagine memory is similar. Nice phrase.

[zozbot123]

> which is to stress the rule you can do the same thing to both sides of an equation (worrying about things like squaring both sides or multiplying by zero later).

It's a pity that they're being so ambiguous here, because explaining why and when "you can do the same thing" to both sides of an equation is not actually hard! You can apply an injective function that's always defined over the appropriate domain to both sides of an arbitrary equation, and this will preserve the equation entirely because (a = b) is equivalent to (f(a) = f(b)) when f has this property. You can apply a non-injective function with no restriction on its domain, and this may introduce extraneous solutions but will not "miss" any, because (a = b) implies (f(a) = f(b)) if f is always defined. You can apply an injective function, perhaps defined over a more limited domain than the original equality, and this will not introduce extraneous solutions but may "miss" some, because (f(a) = f(b)) implies (a = b) if f is injective, but the converse is not true given any restriction on f's domain. Of course, if these functions are defined in terms of x, then you get to worry about whether the function is injective or well-defined given some value of x. For instance, multiplication by x is not injective if (x = 0) but it is otherwise.

[avmich]

Yes, the first one looks like an important quote :) . When I studied math, I usually had trouble understanding or memorizing a rule unless I had at least a rough idea why it holds. In this case I'd suspect just remembering rules and then using them without understanding would - often - cause inconveniences or later errors, when a rule is remembered incorrectly. So maybe it's subjective - what should be taught first?

[winchling]

As philosopher Daniel Dennett puts it: competence comes before comprehension. It's totally possible to do something well, as animals do, without understanding what one is doing. But for comprehension one needs to have something in place to reason about and make connections.

And this isn't the full picture. Motivation comes before competence. One needs reasons to acquire skills: they have to address problems in one's mind if the mind is to fully engage. Which is why coercive education with its curricula, exams, etc, largely fails.

[ganzuul]

Personally I find understanding, or trying to understand, a very good mnemonic for remembering things.