Re: question for math teachers

From: k wallace (wallace.k_at_engr.orst.edNOSPAMu)
Date: 11/23/04 

Date: Tue, 23 Nov 2004 09:10:43 -0800

Herman Rubin wrote:
> In article <20041122033559.06111.00000577@mb-m23.news.cs.com>,
> Chergarj <chergarj@cs.comhaho> wrote:
>
>>rphenry@home.com comments on the sequence of jr.hi and hischool math courses:
>
>
>>>The way geometry is taught, you need to know some algebra.
>
>
> This is a bad idea. Euclid's students knew no algebra; it
> had not yet been invented.

I have to disagree- I worked with my daughter (and several of her
friends) through last year's geometry class. Other than a few "solve for
x" problems involving the Pythagorean Theorem (light pole this tall,
shadow this long, how tall is the man casting the shadow sort of stuff)
there was no "algebra". There were, however, a lot of theorems, but
really understanding *why* was not (to my mind) adequately taught.
That's what I spent a lot of time with these girls on. As a result, I
believe, they all scored directly at the top of the class. (as a result
of course of their understanding, not just my tutoring and help).
>
>
>>>The way algebra 2 is taught, you need to know some geometry.
>
>
>>>Etc.

while currently this is true, there is no reason I can think of that it
needs to be.
As for the forgetting of things over the intervening year- I disagree
that it doesn't mean that they never understood it in the first place. I
am talking about kids who, unlike me and several other nerds of
then-high-school-age, do not spend their summers and/or free time on
science and math; they spend it on basketball and socializing and camp.
The fact that my daughter forgot how to exactly use exponent rules, how
to simplify alebraic fractions, etc- as soon as she did a few examples,
she recalled the way things work, yes. But I can bet that next year,
when she starts trig, she'll have forgotten the sine-cosine relations,
the rules for angles, etc- because she's not using them this year at all.

For example- I do a lot of math in my school and work life. However,
yesterday I had to derive an expression for a general form of
deformation of columns and beams, and in integrating ended up with a ODE
  , second order, nonhomogeneous. I had to think for *quite a while* to
recall the part about setting my particular equation equal to Ax + B,
before solving it became simple again. Not because I didn't understand
Diff Eq's the years ago when I took that class-but because I hadn't
*used* that skill in a while. I think that happens to most people.

                -k wallace
>
>
> This is an indication that those involved in designing the
> elementary and high school programs have no understanding
> of mathematics.
>
>
>>Actually, Geometry in high school requires at least introductory algebra
>>knowledge.
>
>
> It should not.
>
> Also, the proof-based Euclidean Geometry in high school using has
>
>>returned, and very strongly.
>
>
> Not really. Students mainly memorize theorems and proofs,
> instead of getting an understanding of what is involved.
>
> Of course, any decent mathematical program would have
> taught the most important part of algebra in first grade,
> or at the latest second, and proofs in general in
> elementary school. But we cannot teach it to most of
> the teachers, and it is quite possible that the students,
> taught to memorize definitions and facts, and taught how
> to solve specific types of problems, can no longer learn
> it unless they have somehow or other kept the mental
> abilities alive after the educationists have done their
> damage.
>
>
>>The usual sequence of course, sometimes switching the order of
>>Algebra-Intermediate and Geometry, is necessary because the subject matter
>>build as one studies each successive course. Strong algebra skill is used in
>>Trigonometry;
>
>
> Algebra SKILL, not any understanding of the fundamentals
> of algebra. With any understanding, the knowledge of
> what methods are needed is there, and the rest is just
> practice.
>
> strong algebra skill and some details of Trigonometry are used in
>
>>Calculus.
>
>
> Not if someone is going to understand calculus.
>
> The basic concept of limit should be introduced no
> later than infinite decimals.
>
>
>>The Geometry is a foundation course that may help in understanding of
>>trigonometry,
>
>
> You mean five or six facts?
>
> but most students will forget most of so many different theorems,
>
>>and such forgetting seems not to be a serious problem in learning trigonometry,
>>since the numeric sense of algebra secures the students progress.
>
>
> Numeric sense? Numeric understanding can be important,
> particularly induction and the structure of integers,
> but nothing computational is more than a minor asset.
>
> Most
>
>>students will find trigonometry to be easier and more fun to study than
>>Euclidean geometry.
>
>
> Because it is not needed. But unless one is going to
> spend a lot of time on computational tricks, there is
> little in the course.
>
> We are never going to progress as long as those who think
> that one can get understanding from computation and
> examples, or even that those help in understanding, are
> teaching anything of mathematics. The educationists have
> no understanding of concepts or how they can be learned.

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