Redesigning Mathematics Curriculum, thoughts?

[lpetrich]

Speaking of "tracking", do you think an organization like Khan Academy could provide us with useful data on what people learn and how?

Yes, I think so.

 

[RaijuRainBird]

Tell you what Mr. Rainbird, I'll have a go at what you're proposing and I'll check back with you to let you know how I'm getting on. I'm a teacher, background in the Arts, speaker of 3 languages other than English and currently trying my hand at AP Chemistry and Physics, primarily with Khan Academy. I'm fascinated by the linguistic elements of learning, semantics etc and recently thought I'd buy a tee-shirt which said "I speak Physics". Currently, I'd say I'm functioning at 8th/9th grade level Math. So, where do you propose I jump in? I'm impressed by your passion and erudition, let's see where we can take this.

Well, take a crack at it man. Most of the materials can be found on Scribd.com, which means you can download pdf's for a small monthly subscription (but you can search them all in one day, download and limit costs I suppose).

I'm finding formal logic, set theory and especially category theory to be essential in abstract algebra and measure theory. Category theory really shines a light on the "black box" that is known as a function or maps, and then allows one to get intimate with the notion of maps -- which are key in, as far as I can tell, nearly every branch of mathematics.

I might suggest Royden and Fitzpatrick (real analysis) and Hungerford/Lang (graduate algebra) as congruent (or instead of) reading to Rudin and Dummit and Foote. The great thing about this list is that the more abstract books take this logically airtight system of sets and categories, topologies, mappings etc, and then go on to build and build until applications are found. Then, when working with applications you have a framework of an airtight logical system which you understand how to use, and also know is the foundation of whatever it is you're working on. It's comforting to know that the rules of sets and mappings are always the justification for whatever it is you are doing, and that you already know all the logical operations and possibilities, and reasons/explanations for anything that is possible in any situation since you already know the set theory, the logic and the category theory. The only statements that are not logically airtight are the axioms, which will need to be accepted as is, without justification. The thing is that many of them basically seem reasonable as is. But some people actually don't accept some or others and right now a good example is the continuum hypothesis and Forcing Axioms vs. V = Ultimate L. The thing about these disagreements is that they are simply a matter of opinion, and since they're axioms of a system, it can be a matter of nothing else other than opinion. This is where metaphysics and epistemology, and philosophy in general, comes into play. These topics help you come to grips with accepting the axioms as they are without reasons other than the plausibility of their statements, or perhaps, by careful argumentation and reflection, an outside philosophical idea may help persuade you to accept the plausibility of an axiom, or what have you. Perhaps the opposite also could happen.

____

It's neat how to really understand the majority of lower division, and even high school, math, it requires nearly all of a University's graduate course catalogue. Which, again, I know the majority of people never even get the chance to take because graduate school is something people typically avoid, and admissions is very competitive. And this trend continues because as new research is published, new applications will be discovered, and will likely be taught to high schoolers one day in the future. The ironic thing is that this cyclic syphoning effect of PhD holders' discoveries being stripped of all the theoretical curriculum which led to these very discoveries are then recycled and re-taught to young kids with as little explanation as possible. I think I just fundamentally take issue with this cyclic progression in math education. But, ultimately these discussions are usually smothered by people who say that teaching PhD coursework to kids is worse than teaching the results of the world's best discoveries to kids without any census as to how these discoveries were discovered and why they are reasonable or correct. I'm not so sure that the latter is any better sounding than the former. Anyway, my opinions are out there.

 

[Beanyboy]

Well, take a crack at it man. Most of the materials can be found on Scribd.com, which means you can download pdf's for a small monthly subscription (but you can search them all in one day, download and limit costs I suppose).

I'm finding formal logic, set theory and especially category theory to be essential in abstract algebra and measure theory. Category theory really shines a light on the "black box" that is known as a function or maps, and then allows one to get intimate with the notion of maps -- which are key in, as far as I can tell, nearly every branch of mathematics.

I might suggest Royden and Fitzpatrick (real analysis) and Hungerford/Lang (graduate algebra) as congruent (or instead of) reading to Rudin and Dummit and Foote. The great thing about this list is that the more abstract books take this logically airtight system of sets and categories, topologies, mappings etc, and then go on to build and build until applications are found. Then, when working with applications you have a framework of an airtight logical system which you understand how to use, and also know is the foundation of whatever it is you're working on. It's comforting to know that the rules of sets and mappings are always the justification for whatever it is you are doing, and that you already know all the logical operations and possibilities, and reasons/explanations for anything that is possible in any situation since you already know the set theory, the logic and the category theory. The only statements that are not logically airtight are the axioms, which will need to be accepted as is, without justification. The thing is that many of them basically seem reasonable as is. But some people actually don't accept some or others and right now a good example is the continuum hypothesis and Forcing Axioms vs. V = Ultimate L. The thing about these disagreements is that they are simply a matter of opinion, and since they're axioms of a system, it can be a matter of nothing else other than opinion. This is where metaphysics and epistemology, and philosophy in general, comes into play. These topics help you come to grips with accepting the axioms as they are without reasons other than the plausibility of their statements, or perhaps, by careful argumentation and reflection, an outside philosophical idea may help persuade you to accept the plausibility of an axiom, or what have you. Perhaps the opposite also could happen.

____

It's neat how to really understand the majority of lower division, and even high school, math, it requires nearly all of a University's graduate course catalogue. Which, again, I know the majority of people never even get the chance to take because graduate school is something people typically avoid, and admissions is very competitive. And this trend continues because as new research is published, new applications will be discovered, and will likely be taught to high schoolers one day in the future. The ironic thing is that this cyclic syphoning effect of PhD holders' discoveries being stripped of all the theoretical curriculum which led to these very discoveries are then recycled and re-taught to young kids with as little explanation as possible. I think I just fundamentally take issue with this cyclic progression in math education. But, ultimately these discussions are usually smothered by people who say that teaching PhD coursework to kids is worse than teaching the results of the world's best discoveries to kids without any census as to how these discoveries were discovered and why they are reasonable or correct. I'm not so sure that the latter is any better sounding than the former. Anyway, my opinions are out there.

Thanks for the well-argued reply. From your first piece I went to formal logic, and read Chapter 1 of Peter Smith's book. Loved it, really loved it. I've ordered the book. So, even if never read anything else on your list, I feel that's going to be a real gem. I'll have a look for Royden and Fitpatrick. Let's see. So many books, so little time.
If I've understood you correctly, are you saying the majority of people never really get to fully understand the math they took in high school? Try explaining the "cyclic progression" phenomenon again to me please. I'm from Ireland and I've never heard the term used before. Well, if you have time. If not, no worries. Thanks again for all your help.

 

[ShayanJ]

The golden rule of education:

 

[Beanyboy]

Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.

I'm not sure Beckett had mathematicians or physicists in mind when he wrote that. He probably did. He probably had everyone in mind. Clever bugger!

Incidentally, the W. B Yeats quote you posted, funny thing, why just a few days ago I used it too myself. Thanks for sharing.

 

[spoirier]

I also have a plan to redesign the math and physics curriculum. For this the way I found best was most often to just rewrite everything from scratch rather than collect external references, as in most cases I don't know good enough references compared to what I can make myself. My work remains quite incomplete but for the start I did care to make texts close to perfection (at least for some kinds of readers). See it here : set theory and foundations of mathematics.