I have been ‘planning to learn more math’ for longer than any person should. I’m going to try to outline the roadblocks I’ve run into, and we’ll see if some solutions fall into place. My bet is that the actual reason I haven’t learned much more math is that I haven’t really prioritized it, but who knows? Maybe it’s the world’s fault.
Tricky algebra leaps
I’m unable to find the original source, but I’ve read that a single unfamiliar word in a paragraph can seriously throw off a reader’s comprehension. (the closest thing I could find is this (pdf)). So, when a professor sets out to explain a concept from start to finish, care should be taken to ensure that each step in an explained process is clear and easy to follow — even at the expense of time or space.
Professors often seem to skip why and even jump past most of how to arrive at what a little sooner. I ran into an example of this today, watching the first video lecture offered by MIT for single variable calculus.
The professor poses a problem to the class about finding the area of a triangle made up of the x and y axes and a tangent line to a function. He explains the “calculus” part of the problem very well, but then seems to straddle showing the rest of the problem for completion’s sake and actually teaching us how to solve it.
At one point he needs to find the x-intercept. Rather than explain why it needs to be found and how to find it in concrete terms, he just starts pulling pieces of information from the last five minutes of work into an equation, while changing them slightly and performing transformations on the equation, often skipping multiple steps at a once to save time and space on the chalkboard. After doing all of it, he doesn’t stop and explain why this was done or how it worked — he just jumps into the next piece of the problem.
This is where it gets even worse: there is an opportunity to repeat the work to reinforce it, where the professor could even ask his students to tell him what to fill in and promote active learning, but he decides to be clever instead. Noticing that the function in the question is “y = 1/x”, he calls upon an “argument by symmetry”, and shows very thoroughly that “y = 1/x” means “yx = 1”, which means “x = 1/y”, so everything we just did to solve for the x-intercept can be done for the y-intercept by swapping x and y wherever they occur.
Don’t get me wrong here: that’s a cool logical tool, a cool pattern to show off, a good thing to teach, and this part is even well taught! He shows the trivial step in switching y = 1/x to x = 1/y — but the professor left considerably more complicated transformations unmentioned just seconds ago, and has essentially just said to the class “and what we just did (whether you get it or not), we do again.”
One could argue that the impetus is on the attendants of the lecture to ask for clarification. One could also argue that the bits of math which are unrelated to calculus aren’t worth focusing on in a calculus class, and anyone who is lost should do the remedial work required to get up to speed. It’s also hard for the professor to guess how low level to get. During my teaching experiences, I often felt like I was dragging things out and boring everyone to tears whenever I showed a full piece of code.
Video lectures make this less of an issue, thankfully. I watched the lecture at 2x speed, and rewatched the part of the lecture where the prof worked through the problem multiple times, pausing whenever he got ahead of my head. No single step was too complex, but all together it took some real thought to understand some of the less-trivial changes. It could just be my lack of experience and rustiness in this case, but it seems to be a common issue for teachers.
A dialogue with the class
Continuing that line of thought, I recently took lectures on set theory and proofs. I feel like I got very little out of the second half of the semester, which devolved into the professor writing for 4-5 minutes, turning to the class to say “as you can see, the axiom of choice applies” and turning back to write for 4-5 more minutes. This could happen multiple times in a row each class, sometimes just to solve a single problem.
This may sound like the flip of the instance above — being too low level, going too slowly, showing too much, but I should clarify: the steps being taken here were not all trivial. Often, new concepts and methods of argumentation were being introduced alongside manipulations and substitutions that went by without remark.
This may be an effective method for some people. Maybe it’s similar to what I ran into in my own classroom: the topic seems so obvious to the professor teaching it that it’s kind of like writing prose on the chalkboard. They expect us to just read and understand. Perhaps there’s a sense that “this is the way math is taught”, or a machismo idea of “if you can’t follow this, then you can’t don’t do math”. I’m not sure what the motivation is; I hope it is not something like those.
What I think would work best is a genuine dialogue with the class. Even if there’s 80 people in class, it seems to me that knowledge might be more effectively conveyed by an active exchange, perhaps begun by posing the problem in as non-technical a set of terms as possible, and inviting people’s ideas about how to solve it. When the super-geniuses in the front row respond with a series of symbols, the professor can nod and say “that’s great, but let’s try to put this into simple, every day language.”
From that point on, the regular people who actually need to be in the class have a chance of learning something. The professor can ask leading questions, and try to use real world concepts to provoke thought and inquiry from his students.
When they suggest an idea, it might lead down an interesting — maybe incorrect — path. She could work it out on the board, reiterating the motivations for what’s about to happen, and explaining how what she’s about to do next corresponds to what we just discussed. Again examples from the real world could be used to illustrate why a concept makes sense or doesn’t work as we’d expected. When people know why a thing is being done, it seems likelier that they’ll grok how it is being done. At the end, they’re also far likelier to understand what has been done.
While this sounds great to me, it might not engage anyone other than myself, or those I share a learning style with. It might be a complete failure of an idea that was tried before. Maybe this is what tutorials typically are. I didn’t get this experience in any of my classes, and I haven’t ever heard of math class being like this. It just feels intuitively like it would be a breath of fresh air.
Yet again, I can’t find the research I’m looking for — I think this one came up in a TED talk about khan academy; at the end Bill Gates comes onto stage and has some very awkwardly scripted dialogue with Sal Khan about the topic. Anyway, earlier in the video, Sal shows some of the metrics they had been testing out in grade school programs using khan academy for teaching. They found that some kids rocket through the lessons, while some others struggle, and he spoke a little about the implications of one of their findings on the idea of giftedness in schools.
The students who were struggling often failed to understand just a single aspect of a single lesson, at times because they had missed an even earlier piece of information that the lesson relied upon. Once armed with that single piece of information they needed, they would typically shoot forward and rejoin the students that at first glance had appeared to be gifted. The suggestion was that students that do the best may simply be the ones lucky enough not to have missed the most important parts of class.
This is one of the sticking points in my attempts to go back to learning math: I have no idea where to jump into it. Whenever I do, I either feel like I’m missing vital knowledge, or like I already know everything I’m being taught. It’s kind of like picking up a book that you were reading a year ago: pick a random spot and things seem familiar, almost boring, but some interspersed pieces seem entirely foreign. It’s tough to tell what’s been forgotten versus what hasn’t been read!
For example, I don’t know many trig identities, or projections and the like from discrete geometry in grade 12, but I tend to use radians and remember pieces of plane-intersection math. I also learned a fair bit of combinatorics, but can’t recall the definitions of Pick versus Choose. Say I go back to where I feel fully certain I know everything that’s going on: soh cah toa and simple polynomials from grade 10 — it seems like a mountain range of repeated knowledge to be braved before I could get back to real math.
Having put my thoughts down, reviewed and edited and shuffled them around so they’re neat and tidy, I feel like I’ve detected some things that would be neat to investigate about math education. Real world examples for higher level topics in math would be something excellent to write about (I know that this person has a few great examples for e and an introduction to calculus), but pages about spaces, and metrics, and lie groups, and multivariate calculus — it would be cool to read down to earth explanations of that stuff.
If you’re setting out to learn math (like I, presumably, am), I think it makes sense to have a target you want to hit. Something you want to be able to understand or accomplish. I don’t, though — that’s probably a problem. Too often, I just wander around aimlessly and feeling emptyhanded and emptyminded after a collection of skin-deep first encounters with topics I can’t begin to understand yet. If I chose a specific destination, it’d be a lot easier to find, draw, recognize, and even just to read a map. But good luck even finding a destination!
As a final note, the math I’m talking about in this post is, by and large, not the real math that mathematicians wish everyone knew about. I have read some of the Mathematician’s Lament (pdf), and agree with it as enthusiastically as a layperson justifiably may. Sadly, I know of no place to find better education, and I am not in school any longer. For now, I plan generally to bring myself up to the task of real mathematics, instead of trying to pull real mathematics down to a place where I can try it out.
If you can suggest to me a destination, a resource, or a way to pull real mathematics down from the stars and into reach, by all means, do so. I’d love to hear about it.