Feeling Mathematics

Posted in all posts, education, mathematics, teaching by coleman yee on February 25, 2006

I came across this from an interesting blog on teaching Calculus:

There are people who have taken Calculus one or more times and still don’t have a feel for what it is they know. They can solve specific problems in context. They have learned which word problems are like which of the completely solved examples in their text. They know how to curve sketch because the rule says that if in this region the curve has a positive first derivative and a negative second derivative then the curve must look like this.

(The Other Camp)

This reminds me of the time when I was doing Calculus in secondary school. For some reason, I still remember my first Calculus lesson where the teacher explained how differentiation works, and I distinctly remeber not understanding what he meant when he talked about “the limit to zero” this and “the limit to zero” that. Everyone else was equally lost, but no one was about to ask for a clearer explanation — the teacher’s body language made it clear that he was just going through the motions of covering the introduction, and it that wasn’t anything particularly important.

And we were right. Although the class never really understood how Calculus works, it made no apparent difference. We knew how to apply differentiation and integration to standard problems, we knew how to mechanically manipulate the numbers to arrive at the final answer, and that was good enough. In fact, most of the class managed to get top grades for that subject the following year in our GCE ‘O’ level exams. All this while without a feel for Calculus.

A year or so after that, I unearthed from a bookshelf dad’s ancient dust-coated Feynman Lectures on Physics. I liked Physics, so I flipped open one of the books in the series (if the title was Lectures on Calculus, things would have been quite different). The page I looked at was an explanation on how Differentiation works, using a distance-time graph of a car to find out its average and eventually instantaneous velocity at certain times, and how the method of Differentiation could be derived from there (are you still with me?).

I was amazed. It was less than two pages of explanation, but things finally clicked. Now, “limit to zero” made perfect sense, and I could see the elegance of alculus. I feel Calculus.

Unfortunately, countless students have gone through Calculus (and life) never having a feel for it. It’s not just Calculus, but most of Mathematics.

Math, for too many, is not much more than a mechanical manipulation of figures — learning and doing what the computer can do much faster and infinitely more accurately. Is there a problem with math education if our students go through their school careers with such an idea, and having no feel for all but the simplest mathematical operations? Most certainly.
I wonder what could or should be done to improve the situation.

5 Responses

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  1. Michi said, on February 27, 2006 at 5:36 pm

    It is, I believe, a rather fundamental property of education in general and good mathematics education in particular that it has to convey the feeling, the mastery, and the beauty of the subject to be really successful. Unless that is done, mathematics learning is the rote-learning of meaningless arithmetical operations with the feel that a calculator or a computer would be much better at it.

    On the other hand, conveying this requires both enthusiasm and mastery on part of the teacher (or the text, as it were); properties that are rare and more or less untrainable. And those being most engulfed in flames of passion for mathematics tend to end up as researchers and not as teachers.

  2. gecko said, on February 27, 2006 at 7:42 pm

    Following from what Michi said, the question to me is whether there are enough passionate educators who end up as teachers.

    My experience in ‘feeling’ Mathematics occurred late – in junior college.

    I remember a JC Maths teacher who taught me and my classmates in our first year. He was from Malaysia and had moved to S’pore to teach ‘F’ Maths. Back in ’93, he was already greying above his ears.

    He would walk into the LT with nothing but a piece of transparency and a Stabilo OHP marker.

    His usual fashion would be to ask us a question e.g. “What is a Riemann integral?”, then tell us in story form the background of the mathematical terms and the people behind the terms, before ‘deriving’ the theorem in front of our eyes to the question “Why is there a Riemann Integral?”.

    At the end of a 1 hour or 2 hour lesson, that single transparency would’ve been filled up, erased and repeated thrice over, our own notes from his writing spilling beyond the same number of pages by the meaningful pace he set.

    Along with other eccentric lesson activities, he quietly gained respect from us all, even those who treated ‘F’ Maths as a means to an end.

    In the 4 years of secondary school Mathematics education prior, no teacher left as deep and positive an impression as he had.

    What could be done to improve the situation?

    On the systemic end, reduce the emphasis on summative assessment and relook formative assessment.

    Passionate Math teachers can only deliver quality ‘teaching’ experiences if they are freed from the workbook-thick rigour demanded of their charges.

  3. zac said, on March 8, 2006 at 10:58 am

    My sentiments exactly. Those people who think they can “do mathematics” but have no idea where it comes from, nor how it is applied, are missing the point. Computers do the algebra better, anyway.

    For the majority of long-suffering students, mathematics is a series of algebraic manipulations – with no idea of whether anyone in the universe ever uses it (mostly because their teacher most likely does not know, either. Good chance it is the music teacher roped in to teach math because they have a free period at that time.)

    When introducing calculus to students for the first time, I take it from the intriguing property of all curves – that the closer you get to a curve, the more it appears to be straight. So we can get a very good estimate for the rate of change of that curve. The closer we go, the better the accuracy. Hence the concept of “limit to zero”.

  4. myownliccleworld said, on November 5, 2006 at 8:53 am

    yea our teacher tried to explane that stuff, nobody really understood it so he gave up and just taught us how to differentiate, i do vaguely remember him drawing a graph on the board though with a line getting closer to something bu never quite reaching it or something. i think that the prob is that these concepts can be harder to understand for some pupils and just learing how do to it is easier. ALso teacher cant really spent too long explaining it as they need to get throught the whole course

  5. janine said, on November 5, 2009 at 7:49 pm

    I can feel that it became more challenging…
    When my teacher calls my name I was very nervous because my classmates is staring at me. Im always shy because my classmates is shouting ” ooooohhhhh ” too. But Ii think that math is very important subject because the day you were born and until the last day of your life there was math.

    Im happy that I can answer what my teacher ask me…Ii was wondering and I always ask myself ” why is my other classmates always saying math is boring?” , ” I cant really understand “, can I copy your answer?” and if theyre number called they always say ” it was difficult ” …
    They say it even they not try it.

    But I have weakness in math , it was division. I cant really understand the pattern in this because you need to divide,multiply and subtract the only missing was addition… but i can divide small numbers…

    Im happy too because Im more active in math than our first grading.

    Thats all…

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