Hi Harold,
>I appreciate the seriousness with which you take my questions, and
>the
time you have devoted to dealing with them. Reading some of your
>responses
to others in the pages you link to, tells me that you
>accord everyone the
same patience and respect.
>Also, I appreciate that you may have limited
interest in engaging in >debate fo!K but for what it!|s worth, what follows is
my reaction to the >things you say.
Actually,
I'm very interested in this subject, and I regard debate as the principal mechanism
by which I learn new things. So fire away.
As I said, if synthetic division
is presented as a mindless activity (as I think it often is), then I completely
agree with you. But does it have to be presented that way? And what level of mastery
is appropriate to require from a student?
Personally,
I would be satisfied if a student could explain what's going on in synthetic division
- why it works, what it means when you do and don't get a remainder - and carry
out one or two simple divisions. At that point, I would feel that I'd discharged
my obligation as a teacher, which is to point out doors that lead to potentially
interesting and useful subjects and help students learn enough to decide for themselves
if they'd like to go through those doors.
In
college, I took a course in numerical analysis, and caught a terrific break when
the professor who normally taught it decided to take a sabbatical. The professor
who substituted for him took the view that it would be unreasonable for us to
have to know more about the subject than he did; and his knowledge was at a level
where he could have an intelligent conversation with an expert in the field, because
he was familiar with the concepts, and with the strengths and weaknesses of the
various techniques, without necessarily being able to crank out the right answers
for any given problem. That's the way he taught the course, and that's the way
he tested us on the material.
I
think that a lot of math should be approached in the same way, and that a lot
of students would get more out of it if it were.
>Also,
according to Seymour Papert - as well as
>intuitively
- the feeling of accomplishment derived from !Ymastering >the environment!| of
the mathematical microworld is empowering; with >all that that implies.
This
is true. And if you know that you're always going to have that micro-world with
you, then perhaps you can make a strong case for never having to be able to get
along without it. But even then, isn't part of mastering a tool having some understanding
of how the tool works? If the micro-world can produce answers, but you don't know
_how_ they're produced, doesn't that put you at a disadvantage - for example,
if it turns out that the code has some errors in it? For some reason, I'm reminded
of the first time I drove a car with a manual transmission. A friend and I had
gone to his parents' house for the weekend, and he was too tired to drive back,
so he told me I'd have to. I'd never driven a stick before, and he was sleeping
about 10 seconds after we got in the car, so he wasn't able to help me. But I
had read a book once about how a clutch works, so I understood the basic idea,
so I was able to get the car to go. But I believe that the _only_ reason it worked
out was that I could model in my head what was happening in the transmission,
which allowed me to make senses of the feedback it was giving me. I think micro-worlds
are most effective when they are used in this way, i.e., when the student understands
what the micro-world is doing. Then the micro-world becomes an amplifier (doing
what you could do without it, except a zillion times faster) instead of a crutch
(doing what you couldn't do without it). (I believe Logo falls into the former
category, does it not? Isn't one of the strengths of Logo that the commands are
simple and straightforward enough that a student can simulate in his head what
the turtle will do, and understand why the turtle will do it?) By the way, what
does the micro-world have to say about the complex roots of the equation?
Given
how it's usually taught, it's easy to see math as a collection >of isolated techniques.
>Here
is the problem I would like to pursue.
>We
must state our objectives up front and convincingly justify them
>in
a language our clients understand.
Actually,
you don't have to do that if you have a monopoly.
The
model you're describing works at a college level, where students decide for themselves
what they want to learn, and colleges lose money offering courses that no one
wants to take. But through high school, students are more or less compelled to
learn whatever their teachers teach them; and the teachers are more or less compelled
to follow whatever curricula are foisted on them by their administrations and/or
school boards. And as far as I've been able to tell, administrations and school
boards are interested mostly in test scores, and hardly at all in what it might
mean to actually educate someone (let alone how to go about doing that). So to
convince them that a topic should be covered, all you have to do is say that students
will be 'expected' to know it in order to do well on standardized tests. I believe
that in a market-driven education system, much of what is covered in math classes
would simply be dropped, or at least taught in a different way, simply because
in a market-driven system, you _do_ have to convince people that what you're selling
them is worthwhile. But that's pretty far in the future, I think.
>Well,
I'd say everyone should be exposed to it.
I
probably used to believe that. In recent years, however, I've become convinced
that the set of things to which 'everyone should be exposed' is vanishingly small,
and perhaps empty. (It might include only the Declaration of Independence and
the Bill of Rights.) Actually, it would be more precise to say that I no longer
believe that I (or anyone else) should have the power to tell people what they
should and shouldn't be learning.
I
wouldn't willingly cede that kind of authority to anyone else, so I find it unreasonable
to ask anyone else to cede it to me. This isn't to say that people (including
me) who believe that the mathematical view is a good thing shouldn't be trying
to convince other people to give it a try! But there is a line between persuasion
and coercion that I'm no longer willing to cross.
>There
is still a question, however!K. Referring to finding
>zeroes
of x^6 etc
>They're certainly valuable for people who want to go on to study
>higher mathematics,
>Could you elaborate on this a bit? Or is this
just "batting practice" >too?
First,
let me say that by 'higher math' I mostly mean math that is over my head! So I
don't have any great examples at my fingertips for why finding the roots of a
6th degree polynomial would be useful. Here is an example where the Rational Root
Theorem is useful in solving a problem in number theory: http://mathforum.org/library/drmath/view/51565.html
Other examples
would include showing that certain constructions are impossible with a straightedge
and compass, because they would be equivalent to finding rational roots of higher-order
polynomials. The latter, by the way, points up one of the interesting distinctions
between micro-worlds and 'by-hand' methods: the latter can be generalized by using
variables instead of specific values, to provide proofs in addition to demonstrations;
and solutions reached by hand are often exact. For example, 1.7324 (one of the
roots from your graph) is awfully close to the square root of 3. Is it sqrt(3)
one of the roots of the equation? The micro-world isn't going to be able to distinguish
between the exact root and something close to it. For certain applications in
number theory, the difference between exact and approximate is like the difference
between pregnant and not pregnant. (This isn't to say that you couldn't build
a micro-world that would use something like Mathematica to solve problems symbolically
while displaying the solutions numerically. It's just to say that it's not often
done that way.)
However,
even though I'm not exactly brimming with examples here, I will say that as far
as I can tell, if a technique is taught in a math class, it's because mathematicians
have found that it turns out to be useful for doing something else later on. Of
course, in many cases, 'later on' turns out to mean 'in graduate school', and
in cases like that, it seems perverse to force high school students to learn it,
and dubious to think that they'll still remember it when it comes time to use
it. I think this is probably one of those cases. My own view is that the proper
time to learn about synthetic division is just before you'll need to know it in
order to do something you actually want to learn how to do. But that view is part
of a larger view of education in which each individual student would take a different
path up the mountain of learning. It's largely incompatible with the prevailing
view, i.e., that it's a reasonable thing to think that 20 (let alone 20 million)
students should be learning about the same topics, in the same order, at the same
time.
