When people start thinking about math education - as I have at considerable length - they always come up with the idea that since "real math" is about problem solving (which it is) and not about rote algorithms and manipulations (which it isn't) then what we should do is come up with problems to solve, not formulas to manipulate. We should encourage creative thinking, novel solutions, and ways of approaching problems.
This is, actually, all true. Students should indulge in problem solving, analysis, investigation, early proof, ideas of proof, argument, explanation, and all the things that math is really about, and which make math useful.
After all, who except for science and engineering students ever differentiate a polynomial? Knowning qualitatively and intuitively what rate-of-change really means is useful. Differentiating x.sin(x) isn't.
But here's the problem.
To within 20% or so life expectency in the UK is about 70, and there are about 70 million people, so roughly at every age point (up to about 50) there are about a million people.
How do you assess and grade the mathematical standard (whatever that means) of about a million children?
How do you persuade a government full of lawyers and classics graduates that what you're doing is better?
How do you assess the teaching, and the school, if not by the grades they and their students achieve.
It's all about assessment.
So that's the first stumbling block.
The second is the question of getting teachers who are up to the task of teaching in an open and interactive manner, rather than trying to teach rote algorithms and manipulations.
But you don't need to worry about that - schools will never teach proper mathematics. They'll continue to teach arithmetic and manipluation.
If you don't get the fundamentals down, you can't handle the applications. Wax on, wax off - boring but effective. But you also need to spend time putting those fundamentals to work.
The real question is, why do you need assessment? To test that most kids are getting the fundamentals? Easy. Have a few core tests. To check that class time is also devoted to creative stuff, and not just drills? That's a management problem. You can't test whether or not kids have had (say) 50% of their time doing fun non-core work, but you can regulate that through other means.
Are you serious? There are a small number jobs that
1. Are very desirable (lucrative, secure, etc.).
2. Require very high mathematical proficiency.
3. Have drastic consequences if performed poorly.
It's self-evident why assessment is an important function of the education system.
The question becomes: is it more important to correctly identify potential engineers and scientists and to maximize their proficiency at their careers, or more important that Joe the Plumber can balance his checkbook when he goes straight into the workforce after his high school diploma?
I doubt that we can optimize both of these cases simultaneously.
> If you don't get the fundamentals down,
> you can't handle the applications.
Absolutely, which is why I advocatedoing the algebra, doing the manipulation, and doing "the sums" until the act of doing them is almost reflex.
> The real question is, why do you need assessment?
I'm assuming you're serious. Society demands assessment. Government demands assessment. Employers demand assessment (and then ignore it and apply their own tests). Students themselves are conditioned to demand assessment - there are many instances of teachers or lecturers playing with the idea of not giving grades, and mostly the students hated it. (whether it worked better is another question)
FWIW - I advocate a radical shift in what's done, and am working subversively on a ten year plan to make changes. Just becuase I can identify some problems, don't assume I believe they are real, or insurmountable.
> Society demands assessment. Government demands assessment. Employers demand assessment.
My college didn't hand out GPAs and all evaluations were narrative. Only on rare occasions has the lack of numbers attached to my performance prevented me from obtaining what I wanted, and even for those cases I remain dubious about the benefits.
How did they apportion scholarships or determine who received a spot in programs with a limited number of openings? Institutions assess for many reasons, but primarily because educational opportunities are rationed and they need some method for separating the worthy from the unworthy without relying entirely on personal testimonials or politics.
False dochotomy. Pure often leads to applied, applied often leads to pure. What is often regarded today as pure maths has its roots - perhaps long ago - in questions from real life.
Puzzles arise from the real world, and they arise from abstract structures. Regardless, they are puzzles to solve, and the techniques of pure math and applied math can be used.
Thing is - not all students like puzzles, not all students like curiosities, some students actively want stabilty and assurance.
From what I recall of working in academia while the "pure maths" vs "applied maths" is certainly a false dichotomy, there were certainly mathematicians who made it very clear as to which side of that dividing line they were located!
This is, actually, all true. Students should indulge in problem solving, analysis, investigation, early proof, ideas of proof, argument, explanation, and all the things that math is really about, and which make math useful.
After all, who except for science and engineering students ever differentiate a polynomial? Knowning qualitatively and intuitively what rate-of-change really means is useful. Differentiating x.sin(x) isn't.
But here's the problem.
To within 20% or so life expectency in the UK is about 70, and there are about 70 million people, so roughly at every age point (up to about 50) there are about a million people.
How do you assess and grade the mathematical standard (whatever that means) of about a million children?
How do you persuade a government full of lawyers and classics graduates that what you're doing is better?
How do you assess the teaching, and the school, if not by the grades they and their students achieve.
It's all about assessment.
So that's the first stumbling block.
The second is the question of getting teachers who are up to the task of teaching in an open and interactive manner, rather than trying to teach rote algorithms and manipulations.
But you don't need to worry about that - schools will never teach proper mathematics. They'll continue to teach arithmetic and manipluation.