**A Note on Teaching Secondary School Mathematics**

In
Chapter 9, ** Strategies in Teaching Mathematics**, with heading,

"When
the students understand these few problems, the class can move on to rectangles,"
they then explain, and a rectangle whose area is marked "?" has sides
listed as x and x+2, while beside it is another rectangle with area marked x^{2}+5x
and one side x and the other side "?". So far so good, since knowing any two of area, side and side of a
rectangle will give the other measurement, whether expressed (rather
mysteriously) as polynomials or not.
Now the authors become more ambitious:

"When
the class has mastered these basic rectangle problems, we can move to a
worksheet (or demonstration lesson) of more complex problems of the same
general type..." and they offer a picture of a square of sides x+3 and x+3
and area "?", and another announced square of area x^{2}+8x+16
and sides "?" and "?".
Then comes a rectangles with binomial sides, e.g. x+2 and x+3, pictured
as partitioned into four sub-rectangles of sides x, 2, and 3, to illustrate
the product x^2 + 5x + 6, and inversely, the quadratic area and one side given,
to find the other side. Again, so far
so good.

The triumphant conclusion, with no
further partitioning, consists of the exhibition of a rectangle marked as
having area x²+ 11x + 18 and sides "l =?" and "w=?", and
another, smaller, one of area only x²+7x+12, with the same labels on the
sides. Clearly he intends l = x+9 and
w=x+2 in the first, and l =x+4 and w=x+2 in the second, though curiously enough
the second has a visibly smaller width in the illustration. They continue:

"In
the last two problems, the students are factoring trinomial expressions of the
form x²+ bx + c into (x+d)(x+e). As we
have previously mentioned, many students find it easier to proceed from a
concrete model to an abstract one. Thus,
this approach appears to be a good alternative to that used in most
textbooks."

A
"concrete model"? The
expression x²+ 7x + 12 can just as easily represent an area (assuming x is not
between -3 and -4) of a rectangle whose sides are 3 and (x^{2} + 7x +
12)/3. (For x between -3 and -4 the
area represented by the polynomial is negative.) The "model" is idiotic; it not only teaches nothing
about factoring that cannot be taught without the rectangles but layers the
whole subject with false and irrelevant notions.

Moreover, the authors here were not
the first to offer this fatuous "concrete model" for factorizing
polynomials; they took it straight out of earlier texts for school algebra (and
perhaps previous methods texts as well?).
The idea that one can determine the dimensions of a rectangle from
knowing its area, provided the area is expressed as a polynomial, seems to have
had a long history:

From __General
Mathematics__, by F.E. Grossnickle, Professor of Mathematics at New Jersey
State Teachers college, a textbook published in 1949 and apparently intended
for use at the 9th Grade level (p.231):

"The
area of a rectangle is 3w²‑ 2w.
Since each term contains w, one of the factors of the product 3w^{2}
‑ w is w and the other is 3w ‑ 2.
The dimensions of the rectangle are thus w and 3w ‑ 2." The author has here an illustration showing
a rectangle inside which is printed "A = 3w²‑ 2w". He nowhere states that "w" is
meant to stand for "width," and possibly one can forgive him for
setting an apparently ambiguous problem with this implied assumption, for if w
does stand for width the other side of the rectangle is indeed determined. But no; the "w" is just a letter,
as we see from the following page (p.232), where he has a list of exercises,
among them this:

"Problem 9.
The area of a rectangle is 8a²‑ 6a. iSince the width is one
factor of the expression and the length is the other factor, what are the
dimensions of the rectangle?"

Thus
Grossnickle in 1949, and Krulik and Weise in 1975, separated by a revolution
known as "The New Math". Plus
ça change...

Now a
third example exhibits what might be called evolution of this concept of
rectangles with polynomials for sides. __Integrated
Mathematics, Course 1__, by Isidore Dressler and Edward P. Keenan (Amsco
School Publications, Inc., 1980) is intended for the New York State
"Integrated Curriculum", in which algebra, geometry and other topics
are not segregated into different years, but all appear in all three of the
high school program years. __Course 1__
is partly geometry, partly algebra, partly logic, etc. Probability and statistics, too.

Chapter
14, __Special Products and Factoring__, is much more careful about its
definitions, and is explicit about the integers being in question when integers
are to be factored, and it includes all the definitions it intends when it is a
question of factoring polynomials (with integral coefficients). Thus there is no question, in this chapter,
about what the factors of 2L+2W are:
the definitions given require the answer to be 2(L+W), and no wiseacre
will be able to embarrass the teacher by claiming L(2+ W/L) as another
possibility. On the other hand, to say
only that "The area of a rectangle is x²+2x; what are the sides?" --
even in this chapter -- is just as ambiguous as anything in Grossnickle, or in
Krulik and Weise, since the sides are, in those books, not explicitly required
to be polynomials with integer coefficients.
Dressler and Keenan do not make this mistake.

On page 399, then, Exercises 1-51 ask, "Write the expression in factored form," and this is straightforward. Ex. 52 says, "The perimeter of a rectangle is represented by 2L+2W. Express the perimeter as a product of two factors." Again, 2(L+W) is the answer clearly enough, since "factors" in this chapter can only be polynomials with integer coefficients. Why the expression should be the perimeter of a rectangle, however, is not explained. Still, that is the correct formula, and the problem, such as it is, is not mistaken. The exercise list then concludes with:

** In
53-56, the expression represents the area of a rectangle. Write this expression as the product of two
factors.**

** **

**53.
****5x + 5y **

**54.
****18x + 6 **

**55.
****x² + 2x **

**56. 4x ^{3}+6x²**

Well,
they got away with it, since they only asked for "two factors",
which in the present context is unambiguous, and they didn't ask for
"width and length", which would be silly. But the echo of earlier generations of textbooks is still there,
in the conceiving of the problem in the context of rectangles. Each polynomial has two factors, by the
definitions they gave (I didn't repeat it all here, but this is the case), and
the pedagogical background, of representing a product of two things as the area
of a rectangle, is probably not dead in their imaginations, or that of the
students. But there is no need to
invoke rectangles in stating these factorization problems, and in the hands of
a teacher who learned her Methods from Krulik and Saunders there is no doubt
that these last four exercises will be so interpreted.

In fact,
Dressler and Keenan themselves skate on very thin ice on page 411 where they
write:

** **

** In
49-51, the trinomial represents the area of a rectangle. Express the dimensions of the rectangle as
binomials.**

** **

**49. x² + 8x +
7 **

**50. x² + 9x +
18 **

**51. 3x² + 14x +
15**

Had they asked for "the dimensions" they
would have failed to make sense. An
answer to 49, assuming x¹0, could then have been, e.g., “x and (x + 8 + 7/x)”. As
it is, they made the non-explicit assumption that the dimensions were __expressible__
as binomials with integral coefficients.
This is of course not necessarily the case.

The evil
in all these examples, taken from textbooks of 1948, 1975, and 1980, is
still being repeated in school mathematics textbooks. It all derives from more
than a certain ignorance, or inattention, concerning the logic of mathematical
statements, though there is little excuse for error or fuzzy definition in a
textbook and no excuse at all in a book designed to teach future teachers how to
teach. These errors proceeded from a deeper source, the artificiality of the
contexts in which these "area" problems were set.

Teachers of school algebra have long had, or have thought they had, very
few ways to make their teaching interesting, or "relevant", for the
students they were teaching. The
writers of these books did not consider polynomials interesting as objects in
their own right, probably because they themselves didn't know very much about polynomials. (There is a lot to know,
and much of it is very interesting indeed, though the techniques of factoring
the usual high-school examples are probably tiresome to teach to unwilling students.) So, they seek to dress the lessons up with
"real-world problems," to put life into what
they are indirectly telling their students is a boring subject.

A
product of two polynomials? Well, if a
product of two numbers is an area, and if a polynomial sometimes represents a number,
let's not fill the page with a long list of motiveless products, but make it a
bit more interesting by having some of the products be areas of
rectangles. In more recent books, even
rectangles aren't interesting enough, and have to be swimming pools or
gardens. But real gardens have sides
whose
lengths are positive numbers, and they are not
necessarily integers, let alone polynomials. A strained translation of
a problem concerning polynomials into a problem concerning gardens requires
some change of assumptions, and some new attention to the language used. Gardens are not mathematical objects. Without great care, then, one can find oneself talking nonsense as a result of a hasty
translation of an abstract question into a real one, as well as ** vice
versa**.

A good
course in algebra, or statistics, or geometry, should naturally pay attention to the relationship of the mathematical model with the real thing
modeled whenever this is possible. When this is not possible, or even when
it is merely not illuminating, the invention of a fatuous real-world
interpretation is no help and must not be attempted. Children can
recognize fatuity as easily as adults, often more easily. It is bad for
their general attitude towards education, and their respect for truth and
precision, when the teacher or textbook
fails to recognize this fact.

Ralph A. Raimi

Revised
3 February 2005