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Teaching yourself mathematics

Some thoughts from Mark Palko:

Of all the subjects a student is likely to encounter after elementary school, mathematics is by far the easiest to teach yourself. . . .

What is it that makes math teachers so expendable? . . .

At some point all disciplines require the transition from passive to active and that transition can be challenging. In courses like high school history and science, the emphasis on passively acquiring knowledge (yes, I realize that students write essays in history classes and apply formulas in science classes but that represents a relatively small portion of their time and, more importantly, the work those students do is fundamentally different from the day-to-day work done by historians and scientists). By comparison, junior high students playing in an orchestra, writing short stories or solving math problems are almost entirely focused on processes and those processes are essentially the same as those engaged in by professional musicians, writers and mathematicians. Unlike music and writing, however, mathematics starts out as a convergent process. . . .

This unique position of mathematics allows for any number of easy and effective self-study techniques. . . . All you need is a textbook and a few sheets of scratch paper. You cover everything below the paragraph you’re reading with the sheet of paper. When you get to an example, leave the solution covered and try the problem. After you’ve finished check your work. If you got it right you continue working your way through the section. If you got it wrong, you have a few choices. . . .

I have nothing to add to this except to agree that, yes, doing mathematical research (or, at least, doing mathematics as part of statistical research) really is like doing math homework problems! An oft-stated distinction is that homeworks almost always have a clear correct answer, whereas research is open-ended. But, actually, when I do math for my research, it surprisingly often does work out. Doing (applied) mathematical research is a little bit like waking through the woods: sometimes I get stuck and have to work around an obstacle, and I usually don’t end up exactly where I intend to go, but I usually make some progress. And in many cases the math is smarter than I am, in the sense that, through mathematical analysis, I’m able to find a correct answer that is surprising, until I realize how truly right it is.

Also relevant is Dick De Veaux’s remark that math is like music, statistics is like literature.

10 Comments

  1. Paul says:

    The biggest difference I found going from high-school to college level math is that you started having to figure out which technique to use. Long division is long division. But in proofs, combinatorics, calculus, etc. there's some creativity to seeing the appropriate technique. Beyond that you often have to go so far as to come up with problems.

    It's unfortunate how little any of our classes resemble the work in that field. Literature studies don't revolve around 'what was the name of George's dog?', scientists don't memorize which materials are buoyant, historians don't just recite dates. An occasional teacher would give you a glimpse in the last few years, but I think by that point most students had given up on large fields of knowledge.

  2. K? O'Rourke says:

    The opposite was true for me, but I started in qualitative subjects and do think of math as being more than just doing the calculations/mechanics correctly.

    Of more interest to me "through mathematical analysis, I'm able to find a correct answer that is surprising, until I realize how truly right it is" as I was re-desicovering this by simulating the Monty Hall problem.

    The simulation code is easy and fully explicite about all assumptions

    fsim=function(mp){
    pcar=sample(1:3,1)
    dchosen=sample(1:3,1)
    dopen=sort(setdiff(1:3, c(pcar,dchosen)))
    if(length(dopen) > 1) dopen=sample(dopen,1,prob=c(mp,1 – mp))
    c(pcar,dchosen,dopen)
    }

    fsim(.3)

    MH=t(replicate(100000,fsim(1)))
    tt=table(apply(MH,1,paste,collapse=""))
    round(tt/sum(tt),3)
    MHc=MH[MH[,2]==1 & MH[,3]==3,]
    tt=table(apply(MHc,1,paste,collapse=""))
    round(tt/sum(tt),3)

    so easy to find the right answer (no matter how surprizing) but the realizing how truly right still requires some _fallible_ work.

    As a for instance, the wiki entry on it is long, apparently wrong or misleading on some points – it would be work to do a decent critical appraisal of whats there (and I would be at rsik of making an error).

    K?

  3. Ed says:

    The Monty Hall example shows the divide between math-oriented types and humanities types. I eventually "got" the Monty Hall paradox, but I had to talk my way through it. Its a much messier process.

  4. Bob Carpenter says:

    I don't buy the distinction between math and other fields. If you have a lousy math teacher, math class doesn't look anything like what real mathematicians do — it looks just like a bad history class.

    I have a friend who's a junior high principal, but he was formerly a history teacher. He hired a teacher a couple years ago who taught history like a seminar, not like a recitation of facts. They spent nearly a whole semester on Lewis and Clark, doing all kinds of primary source research and lots of open-ended discussion. My friend previously wound up teaching high school computer science, and his feeling was that his students were better at computers than him but he could get them over their learning hurdles by helping them help each other.

  5. K? O'Rourke says:

    Ed not trying to challenge you or even actually get a response but

    Rhetorically can you prove it?
    With absolutely no fear of being found wrong?

    Running the simulation you can get a statistically well calibrated answer for the assumptions coded in the simulation.

    In some ways not as good, but at least with a well controlled error rate.

    K?

  6. Mark Palko says:

    You've hit upon one of the major flaws in the way we teach math. We spend a lot of time on how to do a particular algorithm and little time on when or why.

    The Saxon method partially addresses this by mixing in review problems so that no assignment is spent doing just one kind of problem. It forces students stop and think about what approach a problem requires.

    I'm not entirely sold on the complete Saxon package but I always tried to incorporate that aspect in my tests and handouts.

  7. Mark Palko says:

    I'll probably have more to say about this later but until then here's one quick thought.

    The curricula for most courses are so overstuffed that teachers who slow down and do the kind of in-depth lessons Bob describes will invariably have to skim over some other sections. Educationally this is a good trade-off but the teacher almost always pays for it with lower standardized test scores and red marks for not having covered all the material.

    If we start firing the bottom 80% of new hires, teachers like this just won't make the cut.

  8. Paul says:

    Definitely agree. Feels like a problem with the tests to me, that they're measuring the wrong thing. Which is understandable, in that it's a lot easier to ascertain if you know when the constitution was signed then if you can read a federalist paper and discuss which sections of the constitution address the concerns. I took the A.P. US History test as a highschool senior, and it was definitely a lot more actual historical analysis/composition, so it is testable.

    Another thought is that for teachers in younger grades the criterion shouldn't necessarily be how they influenced scores that year, but how the students ended up scoring in later highschool/college. If the teacher costs them a few points on an exam but instills a lifelong love of the subject, I'd consider that a resounding success. Statistically a little more complex to measure, but not insurmountably so by any means.

  9. JSE says:

    "By comparison, junior high students playing in an orchestra, writing short stories or solving math problems are almost entirely focused on processes and those processes are essentially the same as those engaged in by professional musicians, writers and mathematicians."

    I'm a professional mathematician and this doesn't seem true to me. As for writing I think it's true to about the same extent that playing teeball is "essentially the same" as playing baseball. I can't speak for music.

  10. ohwilleke says:

    I personally taught myself a lot of mathematics in high school: high school geometry, trig and pre-calculus, three semesters of calculus, linear algebra and discrete mathematics.

    But, I would strongly disagree that it is the easiest subject to teach yourself, which is why so few people do it.

    The basic problem with teaching yourself mathematics from a textbook (as I did) is that it is much easier to get stuck, and once you are stuck, you are dead in the water and can't study anything else until you figure out what you are doing wrong. This happens much more easily with someone else to guide you than it does on your own.

    It is self-contained, but it has lots of pratfalls along the way that have to be slogged through. For example, questions of notation that could be resolved in an instant in a live classroom can take days to figure out yourself. Textbooks don't have enough examples to resolve every possible ambiguity.

    Also, the quality of the study time you need to learn mathematics is much greater than the quality of the study time you need to learn lots of other subjects.