Personal Milestones in Learning & Teaching Math (via Programming and Reading); Struggle With Students; Superfluousness of Video Lectures
I think that there were two key, major developmental leaps in my own learning of math:
(1) Writing Computer Programs -- When I learned to program in BASIC on a TRS-80 color computer circa 6th-7th grade, working mostly on my own at home from a book that came with the computer, this increased my precision enormously. When writing code, the fact that the computer complained with a "syntax error" if any single character was malformed caused me to pay attention to the details, and likewise attend to the fact that every single symbol has important, condensed meaning in math/computers. You can have the top-level concepts basically down, but if your details aren't also correct, then your work will collapse into nonsense. As a free and added bonus, it simultaneously got me dealing with logical issues like if/then, not, or, and (I think the book just said "the meaning of these should be obvious"), as well as variables. In junior high school I was writing programs to spit out the results of numerous math homework assignments automatically, so the application was obvious and actually time-saving.
(2) Reading Math Textbooks -- When I was a senior in high school, I started taking a calculus course via experimental technology; namely, a course at the state college that was being delivered by closed-circuit video feed to the high school. In theory, we had microphones on our desks with which we could ask the college instructors questions and participate. In practice, it didn't work well at all: (a) we felt disconnected and intruding when we asked a question (which would hit the instructor by surprise, he'd appear startled, and have to ask where it was coming from), (b) the one TV in our classroom made it hard to see the instructor and what he was writing, (c) the lights in our room were turned off for visibility, as I recall, and (d) I don't think the instructor was terribly good in the first place. It was the first math class I took where I was routinely falling asleep at my desk. In desperation, the only way for me to learn calculus was on the weekend, go down to the basement alone, and read it from the book. And the transformative realization was that it was fundamentally readable, all the information was right there, if I just read it slowly and carefully enough. So while this wasn't how I always approached math classes from then on, it was always my backup plan -- you could learn all of math just from a book if you really wanted to.
Questions I ruminate on: Are these special skills? Can everyone pick up these abilities, or is there evidence that is not the case? Why is it so overwhelmingly difficult to get students to read a math book? Does anyone teach or assess reading a math book? Does anyone teach or assess reading anything in careful detail? Would math instructors everywhere be out of a job if everyone realized that the information is just sitting in a book that you could read on your own time? (What's that Good Will Hunting quote -- "... an education you could have got for a $1.50 in late charges at the public library"?) Part of me very much wants good, open source textbooks to conserve student money and resources -- but is text already dead?
Over the years, I've tried to share these developmental leaps with my students. When I was a graduate student, I tried incorporating some BASIC programming into the algebra course I was teaching (actually, it was included in the book at that time, I think). About 10 years ago I was using MathXL online homework software. A few years ago I was assigning carefully-written algebra homeworks in the standard book format, and grading every symbol/character carefully (which got enormous resistance and hostility).
Again, it seems like the primary struggle we have, with any technique, is the attempt to get students to actually put in the study/ reading/ exercise time required for math (whether in-class, out-of-class, online, etc.), and I've seen all of these initiatives at some point fail when students simply gave up on them. Research seems to show that the more learning students do, the less they like it, because the more work they're doing for it. Perhaps we just have to admit that the primary factor is just how well students are situated in life to actually spend time studying. (Perhaps.)
The point I'm trying to get to is this -- There's a lot of scuttlebutt these days about Khan Academy, other online teaching initiatives, the "inverted classroom" where a video lecture watched before class (we hope) and exercises worked and coached in-class. But I honestly don't see any intuitive advantage to video lectures over having a textbook (like we've had for hundreds of years) and expecting that it be read prior to class. All the same information is there -- Just in a far more efficient format (textbooks). In fact, math notation is fundamentally symbolic manipulation created for the written page -- to me, video seems largely beside the point. If we just taught students to read properly (and truly, my math classes always seem to transmogrify into language-arts classes), wouldn't it be hundreds of times more efficient to just give them a (possibly digital) book? All the reputed advantages of self-pacing, being able to pause and rewind -- is that not inherently possessed by books as well, and more elegantly? Is the idea just that "video" is all the rage and kids are trained to respond better to it? I just flat-out don't see any advantage to video over books -- video lectures go too slowly and make me impatient and irritable -- am I crazy?
Amusingly, one of the recent CUNY research studies on video-lectures had the same problem with their videos as I did in my old calculus class; what the video-instructor was writing on the board behind them was apparently difficult to see, and students complained that they then had to turn to the textbook to understand what was going on (according to the speaker at the recent conference). This being approximately 25 years after my identical experience with closed-circuit video. Written language is still the uber-tool of humankind -- and mathematical writing the most intensely condensed and powerful -- and I'm not seeing any way to avoid embracing it and teaching it as such.