Mathematically Correct presents
The Pythagorean Theorem
G. D. Chakerian and Kurt Kreith
At a recent school meeting, a group of Davis parents and teachers used the Pythagorean theorem to illustrate the difference between a constructivist vs. traditional approach to teaching. Their goal was to provide other parents with a basis for responding to a recent decision by the Davis Board of Education. For in Fall, 1996 Davis junior high schools will offer a choice between two different courses in Algebra 1, one emphasizing constructivist pedagogy and the other relying on a more traditional deductive approach.
As set forth in the currently used text Themes, Tools and Concepts, one constructivist approach to the Pythagorean theorem is based on the use of geoboards. A traditional approach appropriate to Algebra 1, one found in many algebra texts, is based on the dissection of a square.
The former calls on students to use rubber bands to build a right triangle on a geoboard, use rubber bands to enclose the squares defined by the triangle's legs and hypotenuse, and then look for patterns in the areas of the squares so generated. The latter asks the student to visualize two different dissections of a square of size (a+b) x (a+b). Using the usual notation of a, b, and c for the legs and hypotenuse of a right triangle, one such dissection corresponds to a^2 + b^2 + 2ab and the other corresponds to c^2 + 2ab . Equating these two expressions yields the usual symbolic representation of the Pythagorean theorem: a^2 + b^2 = c^2. [^2 notation indicates squared terms]
Laudable as the use of experimentation as a prelude to mathematics may be, there are serious dangers hidden in this constructivist approach to the Pythagorean theorem. For instance, while it is easy to construct right triangles on a geoboard by orienting the legs of the triangle along the horizontal and vertical axes, the example given in Themes, Tools and Concepts suggests that the student should use more general orientations. The question that then arises is, “how is the student to know whether a triangle with such general orientation is, or is not, a right triangle?” (It is very easy to construct geoboard triangles with one angle imperceptibly close, but not quite equal, to a right angle.)
The only mathematical solution is to use the Pythagorean theorem itself (or more precisely, its converse) to confirm that such a triangle is in fact a right triangle. However, this is precisely the knowledge that the student is being urged to construct! Thus, implicit in this particular constructivist approach to the Pythagorean theorem is the notion that the student should build his or her own knowledge by “eyeballing” right angles.
Training a generation of carpenters to rely on “eyeballing” right angles would be a national disaster (none of us would let such a carpenter touch our house). Yet, in the name of constructivism, we seem to be encouraging a generation of children to erect this pillar of mathematical knowledge on just such a basis.
Another problem arising in this constructivist approach to the Pythagorean theorem is that of calculating the areas of the squares built on the sides and hypotenuse of a geoboard triangle. The usual formula “Area = Side x Side” requires that we first determine the lengths of the sides of these squares. However, unless these squares are aligned with the geoboard’s vertical and horizontal axes, finding the lengths of their sides also requires the Pythagorean theorem!
An alternative way of finding the areas of “tilted squares” is to use an advanced mathematical result called Pick’s theorem. While children can be taught to use Pick’s theorem at an early age, any semblance of a mathematical understanding of this tool is well beyond the traditional high school curriculum. It may also be that students will be taught to calculate areas by counting unit squares, and pieces thereof.
Another possibility is to enclose such a “tilted square” within a larger square whose sides are parallel to the axes of the geoboard. While this provides an appropriate way of calculating areas, note that it corresponds to the dissection “(a+b)^2 = c^2 + 2ab” arising in the traditional proof of the Pythagorean theorem cited above. That is, this particular approach to implementing the “discovery process” takes the student half way to actually understanding the Pythagorean theorem. However, on ideological grounds, it stops short of conveying the gift of understanding.
Some may argue that it doesn’t really matter which method is used to teach the Pythagorean theorem – i.e., that both methods lead to the same result. However, this is not true. Experiments with the geoboard correspond to a cumbersome verification of the Pythagorean theorem in rather special circumstances (the geoboard’s discrete structure is well suited to experimentation, but it fails to represent the more general structure of the Euclidean plane). The traditional dissection approach corresponds to a proof of the theorem, providing an answer to the question “why.” One approach sets the stage for discoveries that lead to conjectures; the other emphasizes properties of area that lead to the understanding of an important truth.
Many parents believe that the most important end of education in any field is to raise the question “why,” to know when an answer might or might not exist, and to demand an answer when it can be given. Having children “discover” a hodge-podge of mathematical properties, without providing answers to which properties are true and why, is to deny them a real mathematical education.
While manipulatives can be powerful tools for leading students through a discovery process that reinforces mathematics, the haphazardly planned use of manipulatives can be destructive. An essential adjunct to “hands-on” mathematics is an effort to organize ideas and develop the capacity for mathematical thought and reason. Experiments performed under the tutelage of unskilled guides can lead students into a chaotic jungle, one in which their minds become entangled in an underbrush of mismatched concepts to which they, their parents, and their future teachers will be hard pressed to bring order.