Authors of recent texts and resources, even beyond the study of geometry, have developed effective implementations of learning theory in the design of their les-sons. Let’s look at an example of a hands-on problem and see how it can be used to help students arrive at their own conclusions.
Take a sheet of paper and fold it in half. Then take the half sheet and fold it in half again, and again, and again. How many times can you continue to fold the
paper in half before it becomes impossible to fold it again? Is this true for all pieces of paper or just the one you were folding? What if your “piece” of paper was a 25-foot stretch of cash register tape? What if it was a sheet of newspa-per? What patterns do you see?
If you have actually tried this, you might have found it simple to make the fi rst and second folds, but by the fourth or fi fth fold, it becomes diffi cult. Regardless of what kind of paper you used, you were probably unable to fold it more than about 8 times. Why does this happen? Let’s organize the data in Table 3.2 :
Table 3.2
Paper-Folding DataNumber of Folds Number of Layers of Paper
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1,024
As is evident in this table, by the time you try to fold the paper for the fi fth time, you are folding 32 pieces of paper, and by the eighth time, you are folding more than 250 sheets. A standard phone book is about 900 pages, so folding the piece of paper in half 12 times would be like attempting to fold a stack of four phone books in half. But let’s look at the problem with some more depth. How would we deter-mine the number of thicknesses of paper if we knew only the number of folds?
Students may recognize the numbers in the right-hand column of Table 3.2 as powers of 2 and determine that the fold number, n, used as the exponent, deter-mines the thickness in terms of the number of sheets of paper (e.g., sheets = 2 n ). So, if we wanted to fold the paper in half 17 times, we would fi nd that it requires 2 17
= 131,072 layers of paper, which is the thickness of about 145 phone books—a stack of paper that is more than 25 feet (7.6 meters) high.
Looking at the table again, what do we get when we take 2 0 ? Because the exponent tells us how many times to fold the paper, and the simplifi ed result equals the number of thicknesses of paper, it is logical to assume that 2 0 = 1, and now we have a reason for why this is true. If we trust the structure of mathematics and the consistency of patterns in problem solving, then raising to the zero power makes sense. Perhaps when you were in high school or middle school, you were told that any number raised to the zero power is equal to 1 but could not explain why this is the case. A simple paper-folding activity such as this can help students reach that conclusion for themselves.
What patterns do you see as the n values decrease in Table 3.3 ? Think about it before reading on.
Students may recognize that the values double as you move up the table, but they are divided in half as you move down the table. We can compare, for example, the fact that 2 2 = 4 but 2-2 = 14, and that 2 3 = 8 but 2-3 = 18. Can we predict the relation-ship between taking 2 4 and 2 -4 ? Using the logical order of mathematics, students will discover — without being told—that negative exponents pro-duce reciprocals and will be able to generalize this to simplify, for example, 5 -3 . Again, the effect of negative exponents does not need to be taught as a rule to be memorized; instead, the relationship can be discovered by students as they explore patterns in a table.
The paper-folding problem is used to teach an inquiry lesson in which students work through the activity and essentially invent their own mathemati-cal rules. The teacher’s role is not to provide direct instruction but to select a rich task and to guide the Now, let’s probe even deeper into the mathematical implications of the problem.
Suppose that you were to extend the table further, in both directions ( Table 3.3 ):
Table 3.3
Extended Table for the Paper-Folding Problemn 2n
-3 0.125 = 18
-2 0.25 = 14
-1 0.5 = 12
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1,024
11 2,048
12 4,096
Students use hands-on materials as an aid to construct their understanding of mathematical principles.
students in their exploration of that problem. An inquiry lesson can produce a deeper and longer-lasting conceptual understanding than can traditional lecture-type methods. It is consistent with a theory of teaching and learning known as the constructivist model . The constructivist model is an outgrowth of the work of Jean Piaget, although he himself did not use the term. There are three central compo-nents to the constructivist viewpoint. First, a constructivist believes that knowledge cannot be passively transmitted from one individual to another. Rather, knowl-edge is built up or constructed from within as we have experiences in our lives.
Second, a constructivist believes that children create knowledge not only by doing but also by refl ecting on and discussing what they have done. A hands-on lesson is not inherently constructivist—it only becomes so when accompanied by signifi cant discourse or processing along the way. Finally, a constructivist generally views learning as a social process in which students compare and contrast their ideas about the patterns they see and what they believe about particular problems or concepts.
In a way, constructivism appeals to common sense. Most people would agree that it is more desirable, for example, for a middle school student to discover the need for common denominators when adding fractions than to be told that “you must fi nd a common denominator any time you add two fractions.” There are many mathematical rules, such as “invert and multiply,” “count up the total number of decimal places and move it over that many in the answer,” “FOIL to multiply the binomials,” and “multiply the exponent and the coeffi cient and reduce the exponent by 1 to fi nd the derivative,” that many people have memo-rized, but few can explain why they work because most have been educated in traditional, lecture-oriented environments. Most people’s mathematical knowl-edge, thus, relies on rote, pencil-and-paper procedures rather than on conceptual understanding.
However, putting constructivist research into practice requires a great deal of skill, both in selecting appropriate activities and in guiding students as they explore new concepts. Paul Cobb, in his research on the constructivist model of teaching and learning, stated that “although constructivist theory is attractive when the issue of learning is considered, deep-rooted problems arise when attempts are made to apply it to instruction” ( Cobb, 1988 , p. 87 ). Because using an inquiry approach is generally more diffi cult than “teaching as telling,” more classroom teachers are still using traditional approaches even though they often acknowledge that the teaching techniques are not working. A publication from the Association for Super-vision and Curriculum Development (ASCD) ( Brooks & Brooks, 1993 ) outlines the difference between what might be observed in a more traditional classroom versus what is seen in the constructivist classroom ( Table 3.4 ).
It should be noted that the constructivist classroom can and should include some individual work, pencil-and-paper tests, and even lectures. In fact, the model is often misconstrued as one in which it is “never appropriate to lecture or pro-vide direct instruction,” but that is simply not the case. The issue is the frequency with which various teaching techniques are used and whether the student’s back-ground (knowledge and beliefs) and thinking processes are the focal point of the classroom. Essentially, whereas the curriculum has been historically content cen-tered, the constructivist approach is student centered. The examples used through-out this textbook are rooted in constructivist theory because it underlies much of the reform effort in curriculum, teaching, and assessment.