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.
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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.

hrubin...

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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.

rusin...

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 .)

hrubin...

It is a mistake to have the student use only one variable.
Use variables as linguistic, and do not worry about how many.
So let k be the length (I am not using l because it may be
confused with 1) and w the width. Then the equations are

k = w + 6,
2k + 2w = 228,

and there are lots of correct methods to use.

If the problem is discussed in class, show how many
of them can be used to solve the problem, all derived
by using the one rule of equality.

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.

Bill Dubuque...

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?

Dom...

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.

toto...

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.

Dom...

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

Dom Rosa

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

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.

fishfry...

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

man_in_black529...

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

Dom Rosa