A Note on Teaching Secondary School Mathematics by Stephen Krulik, of Temple University, and Ingrid B. Weise, of the Montgomery County (Maryland) Public Schools, published by W.B.Saunders at Philadelphia in 1975:
In Chapter 9, Strategies in Teaching Mathematics, with heading, Factoring, the authors think they have a geometric approach of value. A square is pictured, with area given as x2; what is the side? Clearly, the factors, x and x. Next is pictured a square with area 9x2; the sides are each 3x.
"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 x2+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 x2+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 (x2 + 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 3w2 ‑ 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
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