I am curious as to how advanced math is actually taught. Do they give you a real problem that you can actually relate to on some level and have you solve it? I look at long series of numbers and equations and can’t imagine how you could maintain any kind of concentration unless you felt you were going somewhere with it. I know in basic math we are often just doing meaningless excersizes but I can’t imagine doing this on a long drawn out series of any kind.
In the real life world I often find myself faced with problems I can't solve and have to settle for estimations, usually once I can establish some line of logic that leads to a solution I will go back and try and straighten out my estimates by doing some web searching but always within my limits of math.
I started taking the khan academy classes and find I can fly right through the courses using the explanations they give but a few days later I remember nothing, if a month later I need that math I have to go back and relearn it.
I am curious as to the process that best facilitates remembering or excersizes whatever. I suspect it is just a matter of spending the time and practicing, at my age time is running short.
My experience: in advanced statistics courses, almost everything revolved around application problems. Same thing with differential equations. But in a numerical analysis sequence I took, it was almost completely those meaningless exercises, like in an algebra class. Real analysis and abstract algebra were all about “here’s a theorem, now prove it.”
Same way you can look at a paragraph, which is just a long series of words and clauses and keep track of all the relevant bits at the beginning by the time you get to the end.
After a point, math becomes about proving theorems rather than solving equations, and it stops being algorithmic. (In particular, classes aren’t about presenting exercises and trying to copy the procedure in them, like in algebra or calculus). It’s not that the numbers or equations get longer or have more terms; it’s that they’re talking about completely different things.
To answer your question, you might find the MIT OpenCourseware notes helpful. Hereare the ones from MIT’s intro real analysis course, usually the first advanced (i.e., post-calculus) math class students take there. (Those notes happen to be from a great lecturer; a lot of math professors are not great in the classroom.) Notes like that are pretty much what you’d expect to cover in a single math class; the professor writes something like that on the board, answers questions, talks about a few other things he feels like talking about that, and that’s pretty much it.
Most of the real life problems I deal with today are just slightly above basic math. For example, A few years ago I was asked to build this giant bow to power a catapult designed by Leonardo Da Vinci. I had about 10 days to complete the project. I was intitialy caught off guard when I realized how much math I would have to do to make this thing work. The bow powered a rotating drum with a long throwing arm attached to it. I had to figure out how much power the bow would have, how much power it would take to rotate the mass of the drum and the throwing arm, how many g forces would be excerted on the throwing arm, how thick to make the throwing arm to withstand the forces and not waste too much energy being overbuilt, etc. Lots of small basic problems just hard enough to challenge me with my limited education but fairly easy to research and workout. Everything came out right on and I was pleased with that.
It did kind of motivate me to try and learn more advanced math where I quickly discoverd my strategy for learning was not working.
“Advanced math” means different things to different people, and from the replies so far it looks like the range of interpretations is rather large.
FWIW - I read the OP as referring not to specialized topics that a practicing mathematician would study but rather to what you might call “college” math (calculus, differential equations, maybe statistics, maybe linear algebra). My read of the OP is influenced by the Khan Academy paragraph. (You won’t find lessons on Diophantine equations or topology or whatnot there.)
On preview: I see that HoneyBadgerDC has posted more info. The above still seems relevant, so <click>…
It depends on the particulars of the math class topic, and I’ve found also depends on the hosting department of the subject.
For example: When I took Linear Algebra (MA2xx math dept class), it was very procedural and “cookbook” solving, rooted mostly in the proofs of the concepts and not in the application. Leaving that class, I could compute an eigenvalue, but it was without context or usefulness, merely trivia to use to solve other problems.
Taking Linear Algebra from the engineering dept (EE2xx electrical engineering), the math and theory was just as dense, but more time was spent on the why, not the how. I actually cared about the techniques for their future usefulness, not just as trivia to meet a grading requirement.
It’s also significant if the class is still rooted in the old “demonstrate proof and then compute manually by hand” way of learning math, or if it instead doesn’t waste the student’s time with pure pencil and paper but instead opens up usage of computing, application, programming to let the class spend more time exploring the usage of the method instead of writing page after page of number crunching to reach an arbitrary result.
I had actually given up a couple of years ago but decided I want to give it another go possibly using a different approach. Thats what I am looking for here. I can’t even tell you what math I would need to study but I could probably describe the type of problems I would like to deal with.
For the sort of understanding you are after, a mix of lessons on the fundamentals, lessons on how to apply the tools, and lots of practice problems is the name of the game. Working through practice problems is key. The lessons themselves can be a lot of fun and can even be crystal clear the whole way through, but the information quickly fades away, becoming a mere memory of mathematical “story time”. It’s only when you sit down to work out problems without aid that your brain incorporates the information the way it needs to. In a proper course, the problems would be doled out at an appropriate pace so that simple pieces are solidly understood first and then built upon, toward more complex pieces. When done right, each new piece isn’t too much to handle since the older pieces will have been learned. Also note that when done right, this pacing usually spans a couple of years, give or take (n.b.: the duration of college training for engineers). For a more targeted set of goals, this timescale could maybe be shrunk a bit.