Elementary pseudo-algebra

The Jan. 2002 issue of The College Mathematics Journal contains the
following item in Ed Barbeau's column, "Fallacies, Flaws, and
Flimflam," page 39.
==============================

FFF#183. Dimensions of a yard.

Problem. The distance around a rectangular yard is 228 feet. If the
length of the yard is 6 feet more than the width, find the dimensions
of the yard.

Solution (by a student). Divide 228 by 2 to get 114. Knowing that the
length is 6 feet more than the width, I subtracted 6 from 114 which
gives me 108 and add the 6 I subtracted onto the length which makes
120. I then divided the 108 by 2 to get the result which is 54 feet
long. By dividing the 120 by 2, I would get 60 which is the same as the
length.
================================

This student solution is a variation of the "strategy/algorithm" that
is becoming more and more widespread under the guise of "solving
algebra problems" and "algebra for all." In my opinion, far from
teaching any meaningful concepts, these mechanical recipies are doing
little more than enhancing the pseudo-education of students.

In the U.S., this type of pseudo-education is being promoted--at
conferences, workshops, minicourses, and training sessions--by people
who promise to boost scores on assorted "mastery tests" and other
standardized tests. These promotions are being adopted mindlessly by
administrators and teachers, whose bonuses and other financial rewards
are based on the results of these tests.

As long as these types of recipies continue to be promoted, the
pseudo-education of American students will continue unabated.

Dom Rosa

 

I vaguely remember that sort of solutions when I was in elementary
school, *before* we were ever introduced to algebra. It was rather
frustrating since every solution was sort of ad hoc. Once we got into
algebra, that crap disappeared like a nightmare.

I think you should never ever give any problem that can't be solved in
some systematic way but only with some ad hoc approach, before you've
learned that systematic way.

 

That's a great phrase. Maybe the government should have a Department of
Pseudo-Education. Or maybe we already do.

 

Yes, we already do! In October 1999, U.S. Secretary of Education
Richard Riley announced his department's endorsement of 10 K-12
mathematics programs selected
by an "Expert Panel" of educators. Subsequently, 200 mathematicians and
scientists signed a highly critical letter addressed to Riley. This
letter, published as a full-page advertisement in The Washington Post
on Nov. 18, 1999, called the recommendations "premature" and asked
Riley to withdraw them. "These programs are among the worst in
existence," said David Klein, a professor of mathematics at California
State University Northridge, who helped write the letter and organize
the signing
campaign. "It would be a joke except for the damaging effect it has on
children."

What is even more disturbing is the extensive funding that the National
Science Fondation has been awarding to the promoters of these "reform
math" programs.

DR

 

It's very popular to criticize "mechanical recipes", but isn't it
exactly the _lack_ of a "recipe" that has you hot under the collar here?
(I mean, what we expect the student to do is to follow Standard Operating
Procedure: let the width be w so the length is w+6 and the
perimeter is 2w + 2(w+6) ; set equal to 228, solve for w .)

I absolutely agree that IF the student is unable to follow SOP here,
he or she will be unable to solve more complex problems later.
But that's a big "if". What if the student saw the answer immediately
in his/her way? I have to say quite frankly that my reaction to
the problem was almost the same as the student's: it's clearly a
nearly-square yard, but for the 12 extra feet, so the short ends
are (228-12)/4 feet long and the others of course are 6 feet longer.

Kind of a sneaky twist to use feet and yards in the same problem,
I have to say! I'll remember that.

dave

 

The key part of algebra, the use of variables not just for
numbers, can be taught with beginning reading, and should
be so taught. Once this is done, and one uses the one rule
of equality, that the same operation performed on equal
entities gets equal results, the reduction to straight
arithmetic is not only easy but natural.

One does not have to memorize lots of special procedures and
rules to understand something, and generally that memorization
interferes with understanding.

 

The key part of algebra, the use of variables not just for

I agree 100%. The algebraic approach to solving the problems should be
taught as early as possible. I see no reason why one should spend a
couple of years doing things in a hard way only to learn much simpler
way and never going back. It's a very practical thing to learn.

 

The solution seems fine. It simply appeals to the obvious method
to recover two numbers from their sum S and difference D, namely

S,D = L +-W -> L,W = (S +-D)/2


S = L + W L = (S + D)/2
i.e. ->
D = L - W W = (S - D)/2

Here it is clear that the sum S is equal to half of the perimeter.

Why does Barbeau think the solution is Fallacious, Flawed, or Flimflam?
Why does Dom Rosa think the solution is based upon pseudo-mathematics?
Does anyone understand their objections?

--Bill Dubuque

 

Teaching students recipes like the above completely defeats the reason
for doing these "silly" problems--which is to teach elementary algebra
concepts: identifying the given information in terms of standard
variables, determining the applicable formula, using the substitution
principle, performing elementary algebraic operations.

Dom Rosa

 

The logic of equality is governed by more than just one rule.
Namely, it employs the rules of Substitution and Replacement
in addition to the Equivalence Relation rules for equality.
See my prior post [1] for further discussion.

While I'm being pedantic, I should add that it is sad
that budding algebraic minds are being corrupted with
overloaded language such as "variable". The damage
caused by this is so tremendous that many students
find it impossible to later appreciate the powerful
concept of purely *formal* objects such as polynomials,
rational functions, power series (generating functions).
Apparently the active viewpoint of X as a "variable"
becomes entrenched so deeply in their mathematical mind
that they cannot help but envision F(X) as a function.
The consequent loss of generality leads to embarrassing
algebraic impotence (or, even worse, dense remedies
achieved by popping little blue topological pills)
Emmy Noether would be rolling in her grave (or ROTFL).
Evidence that this variable plague is pandemic can be
seen almost every day on any mathematical newsgroup,
e.g. see the current thread [2] on partial fractions.
My posts there contain further discussion and references.

--Bill Dubuque

[1] http://google.com/groups?selm=
[2] http://google.com/groups?selm=

 

I think though the point is that the student was NOT taught this as an
algorithm, but figured out the solution using his own strategy and
method.

Why is it better to teach a recipe like let w be the width, then the
length is w +6 so the perimeter is 2(w) + 2(w+6) and then solve
for w to get the answer. If the student doesn't understand the
procedure they are no better off with this algorithm then with the
strategy the student used.

--
Dorothy

There is no sound, no cry in all the world
that can be heard unless someone listens ..

The Outer Limits

 

It's the problem with NCLB. Now all kids are being taught to do is,
well, pass a test.

 

Based on my experiences, students ARE "taught this as an algorithm."
Too many American students are being cheated out of an education in
elementary algebra. In my opinion, this is a national disgrace, and it
is important that people speak up. DR

 

I regret to say that, looking at Barbeau's recent FFF, I have the same
question about one of the items. It's FFF#250. Minding the technology.
CMJ 37:2 (March 2006) pp. 122-123, sent in by Paul H. Schuette. He used
three computer algebra systems (CAS) to compute the indefinite integral of
1/(x sqrt(1 - x^2)) wrt x. Someone casually reading FFF#250 would think
that Maple, Derive and Mathematica all give results which are incorrect.
But in fact, all three give results which are perfectly correct!

Before examining that integral however, let's consider a simpler and far
more common one which poses exactly the same "problem": the indefinite
integral of 1/x wrt x. All CAS known to me give log(x) as the answer, while
all elementary calculus texts known to me give instead log|x|, plus an
arbitrary constant of integration. Why the discrepancy? CAS are normally
designed to give results which are applicable in the complex, not just the
real, realm. There, log(x) is a correct answer, while log|x| is not. But of
course, for the purpose of elementary calculus, log|x| is perfectly fine
and arguably nicer than just log(x). Nonetheless, log(x) is not wrong in
that context. The two expressions, log(x) and log|x|, differ when x < 0,
but that difference is just a _constant_ (albeit an imaginary one, pi i)
and so we may think of it as being harmlessly "absorbed" in the arbitrary
constant of integration.

Now let's go back to FFF#250.
For the indefinite integral of 1/(x sqrt(1 - x^2)) wrt x, Schuette reports
that Maple (versions 7 and 9.5) gives -arctanh(1/sqrt(1 - x^2)), while
Derive (version 6) gives ln((sqrt(1 - x^2) - 1)/x) and Mathematica (version
unspecified) gives ln(x) - ln(1 + sqrt(1 - x^2)). These are _all_ correct
antiderivatives. In particular, for 0 < |x| < 1, they have the same real
part. Their only differences, for -1 < x < 0 or for 0 < x < 1, are
merely _constants_, albeit imaginary ones.

There are at least two lessons to be learned here.
1. Warn students in elementary calculus, if they use CAS, that those
systems are designed to work with complex numbers. As such, an
antiderivative from a CAS may well differ from an antiderivative in the
text (or as obtained by the student or instructor) by a _complex_ constant.
2. Be careful about what you claim to be FFF. (Otherwise, it may be your
claim itself which is FFF.)

Regards,
David W. Cantrell