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.
Having watched a handful of Kahn videos (although on finance, not math), I would actually say that their great strength over books is is watching the written language take shape, as with watching an instructor write on a blackboard. Not that I think this is a decisive matter, but it is something I noticed as being sometimes easier to follow than trying to "animate" line-by-line text as I read it.
ReplyDeleteI think the advantages of self-pacing and review in the accompanying exercises are much more significant. Textbooks, dead tree or digital, have exercises as a rule, and sometimes even very well designed exercises, but the possibilities of generating, performing, evaluating, and recording them at the student's pace are sometimes significantly more convenient in a Kahn-style format.
Amen. I am a big believer in books too. I certainly hope they are not dead, because that would mean the death of a whole kind of thinking. The slow analytical thinking, which IMHO is //necessary// for math and science.
ReplyDeleteThe only problems with textbooks is that sometimes they suck! Usually they are too verbose (because you can sell them for a higher price to the student). It all comes from the fallacy that bigger is better.
I am working on a very short book now which teaches math and physics at the same time. I hope kids like it. It is certainly the book I would have liked to have when I was an undergrad.
Well put! I'm totally in favor of shorter, more-to-the-point textbooks. (There was one custom book in particular at my school that went in the opposite direction to my great distress.) One problem is when a book crams together 3 semesters of material and becomes back-breakingly immense. Good luck with your project.
DeleteI love the idea of teaching math and physics together, in fact had been trying to think of whether a whole curriculum could be built around that notion. The idea of not teaching math as an academic subject in its own right but rather in the context of where it is useful. Learning not just how to pass a math exam but how to apply math to real problems. Newton didn't invent calculus to get chicks, he had some physics questions needed answering (quiet, you Leibniz fans).
DeleteCoincidentally, I recently ran into an interesting quote from Stein and Barcellos, Calculus and Analytic Geometry, 5th Edition (1992):
Delete"At the Tulane conference on 'Lean and Lively Calculus' in 1986 we heard the engineers say, 'Teach the concepts. We'll take care of the applications.' Steve Whitaker, in the engineering department at Davis, advised us, 'Emphasize proofs, because the ideas that go into the proofs are often the ideas that go into the applications.' Oddly, mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts..." (To The Instructor, p. xxii)
A 3rd thing item I could add in the same category is: Reading D&D books on my own. Conventional wisdom in most D&D circles is something like "no one learns D&D from a book; you have to be inducted by another live player", to which again I am an outlier. No one in my rural town in Maine has ever heard of the game before (I learned of it through reviews in Games magazine). I got the Holmes Basic D&D set, read it and shared it with friends, and mostly was the catalyst for anyone knowing about in my home time. To whatever degree there were cultural norms assumed and not written in the books, they would be invisible to me; the written text is all I had.
ReplyDelete