Math Forum Discussions

Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05

Re: New Post at RME: Who Was George Polya's Intended Audience? (Or More Mathema
Posted: Jun 4, 2010 9:05 PM

Plain Text

Dear All,

Here is a link that Michael had sent me that shows their claim about
Polya's book intended for math majors or graduate students and not
K-12 math students:

http://mathforum.org/kb/message.jspa?messageID=1479090&tstart=0.

However, this claim is false since the introduction to Polya's book
"How to Solve It" clearly says that Polya wrote the book for all
teachers and students of mathematics.

Section 1. Problem Solving in Mathematics.

Suck problem solving out of math and reduce it to rote and cookbook
recipes for solving problems, and what is left is definitely not math
but only a ghost of math. Failing to teach genuine problem solving
and failing to teach the beauty behind math misses the point of what
math truly is: Math is a creative human endeavor done for pleasure.
Yes, math is also useful in a wide variety of fields, and we should
not neglect to show that either. But showing students what math is
means that we should show them all that gives math its character
and not just a piece of that.

Mathematically Correct complains that students under reform programs
don't learn computation and basic skills, but they fail to mention that
students who are taught math in ways they support don't learn these basic
skills either. And this fact was noted even before the beginning of the
Math Wars and the work of the NCTM. In college I had gotten a book that
Tom Hamel, a math professor at Austin Peay used to own and was giving away,
called "Guidelines for Teaching Mathematics" (2nd edition) by Donavan A. Johnson
and Gerald R. Rising (published in 1972). Here is their view on computational
skills (as quoted from the beginning of Chapter 13):

"To learn mathematics and to use it requires a mastery of computation. To master a
skill such as computation requires practice, repetition, and drill. However, this
practice does not need to be given by countless time-consuming and boring exercises.
When practice becomes a meaningless activity, it causes unfavorable attitudes and
habits to develop. Even mathematicians dislike rote computations. Consequently, our
first responsibility in teaching computational skills is to make this practice as
palatable as possible." Beautifully said! I would not doubt that mathematicians
dislike rote, meaningless computations. So why is meaningless rote practice okay
for kids and other students of mathematics?

Here is their take on genuine problem solving (as quoted from the beginning of
Chapter 14):

"Learning to solve problems (that is, finding an appropriate response to a situation
which is unique and novel to the problem solver) is perhaps the most significant
learning in every mathematics class." And this book includes ideas on how to use
problem solving to practice computational skills! Why must emphasizing problem
solving necessarily exclude practice with developing computational skills?
In fact, such problems that interest the students can help motivate them to
work on developing their computational skills.

Here is their take on so-called problem solving posed as "word problems" in
arithmetic and algebra:

"To many teachers, problem solving means only solving the verbal problems that
form an important part of arithmetic and algebra. Many attacks have been made on
this particular genre of problem. [Examples of techniques are then listed such as
geometric approaches and tables.]

"It is much better to attempt to give the student a general method of attack on
such problems; otherwise, he can solve only those problems that fit neatly into
the specific problems provided." And we know that, in the real world, problems
that arise do not resemble these textbook problems. So what good does it do
for the student to know how to solve these kinds of problems but cannot
transfer his knowledge to solve other problems?

Though Johnson and Rising do not explictly mention what audience George
Polya intended for his book "How to Solve It," it is clear that they realize
his book was intended for all students and teachers of mathematics.
In short, Johnson and Rising believe that problem solving is an integral part of
EVERY math class, not just those in college with math majors or just classes for
"mathy" kids. And they realize that George Polya would agree wholeheartedly.

Section 2. The New Math Era.

Johnson and Rising wrote this book at the end of the New Math Era and before
the Math Wars in America. They discuss the programs of the New Math Era
in Chapter 2. I think it is worth mentioning some things they say about the
New Math Programs of the 1950s and 1960s. Here is an interesting quote at the
beginning this chapter:

"New Math. To a great many laymen this phrase has a negative connotation.
The public thinks of new math as something exotic and impractical: Students know
how to do problems but they never get the right answer. They know the difference
between number and numeral, but they cannot calculate with accuracy--let alone speed.

"This is a false and pernicious view of new mathematics curricula: false because it
implies that the older programs did a better job, pernicious because in its focus
on side issues it misrepresents the many important changes that have virtually
universal support from mathematicians and teachers." These changes were
supposed to update the K-12 math curricula to include new discoveries in
mathematics and new uses for mathematics and to meet society's increasing
need for mathematically competent citizens. Furthermore, the New Math programs
were developed to try to improve the math education of students because the
previous traditional ways of teaching were failing to help scores of students
learn math. The New Math programs were supposed to emphasize problem solving
and conceptual learning and de-emphasize rote, meaningless drill and
specific recipes for solving math problems.

Thus, Johnson and Rising have noted that older programs of math education (the
ones that existed before the New Math Era and the ones Mathematically Correct
support) do not teach students how to compute. Johnson and Rising later note
in that chapter that most of the New Math programs failed for these reasons:

1. Content focused on curricula problems rather than on teaching problems.
So-called teaching manuals were full of discussions about the math itself
but very little on teaching suggestions.

2. The texts were written only for top students; the average and below
average students found these texts impossible to understand.

3. Rigor, precision, and symbolism were overdone and often became the ultimate
goal of learning math rather than as an aid to learning and communicating
mathematically.

4. The volume of content was too great for students and teachers to handle.

5. Applications of math were ignored. Johnson and Rising mention that
Morris Kline had heavily criticized the New Math programs for this reason.

6. Mathematicians did not communicate very well to the teachers what they
were trying to accomplish with the New Math programs. Many schools had
implemented these programs without a clue as to where they were supposed to
lead to. Thus, schools tried to teach math without any real goals in mind
other than to "cover" a certain set of curricula.

Jonathan Groves

On 6/2/2010 at 4:40 pm, Michael Paul Goldenberg wrote:

> Please read the latest post at
> RationalMathEd.blogspot.com: "Who Was George Polya's
> Intended Audience? (Or More Mathematically Correct
> Lies)"
>
> Excerpt: One of the more difficult aspects of wars,
> even ones where the main ammunition is words, is
> separating lies from facts. Every side in a war has a
> proclivity for propaganda. Inconvenient facts are
> brushed aside. Inaccuracies, petty or gross, become
> the coin of the realm. The Big Lie rules.
>
> Of course, sometimes, it is possible to sort through
> the fog of war to arrive at what appears to be
> incontrovertible truth. It may take years, even
> decades, to find the facts, even when they are
> readily available to anyone who bothers to look in
> the right place for them. Sometimes, they've been
> staring everyone in the face for a very long time.
>
> Thus, it is with no small embarrassment that I
> present a long-overdue and clearly definitive retort
> to one of the lies frequently promulgated a decade or
> so ago by Professor Wayne Bishop and some of his
> Mathematically Correct and HOLD anti-progressive
> allies, namely that George Polya's work on heuristic
> methods (from the Greek "???????" for "find" or
> "discover": an adjective for experience-based
> techniques that help in problem solving, learning and
> discovery) was intended only for graduate students or
> perhaps undergraduate mathematics majors, not for the
> general student of mathematics, and certainly not for
> high school students or younger children.
>
> Of course, in the Math Wars, it is of the utmost
> importance to the counter-revolutionaries and
> anti-progressives that nothing that broadens access
> to mathematics be allowed to stand unchallenged or
> unsullied. Any curriculum, pedagogy, tool, etc., that
> is brought forward by reformers as "worth trying"
> must be smashed. That has been the tireless task of
> members of groups like Mathematically Correct and
> HOLD: to undermine any and all efforts to change what
> they view as immutable approaches to the teaching and
> learning of mathematics.
>
> Read the entire post at: http://tinyurl.com/2bl5f7r