Exploring
New Knowledge: by
Liping Ma, from Knowing and Teaching Elementary Mathematics |
||||||||||||
Scenario
Students bring up novel ideas and claims in their mathematics classes. Sometimes teachers know whether a student's claim is valid, but sometimes they do not. The perimeter and area of a figure are two different measures. The perimeter is a measure of the length of the boundary of a figure (in the case of a rectangle, the sum of the lengths of the sides of the figure), while the area is a measure of the size of the figure. Because the calculations of both measures are related to the sides of a figure, the student claimed that they were correlated. The immediate reactions of the U.S. and Chinese teachers to this claim were similar. For most of the teachers in this study, the student's claim was a "new theory" that they were hearing for the first time. Similar proportions of U.S. and Chinese teachers accepted the theory immediately. All the teachers knew what the two measures meant and most teachers knew how to calculate them. From this beginning, however, the teachers' paths diverged. They explored different strategies, reached different results, and responded to the student differently. HOW THE U.S. TEACHERS EXPLORED THE NEW IDEA Teachers' Reactions to the Claim Strategy I: Consulting a Book. While two of the U.S. teachers (9%) simply accepted the student's theory without doubt, the remainder did not. Among the 21 teachers who suspected the theory was true, five said that they had to consult a book. Four of the five explained that they needed a book because they did not remember how to calculate perimeter and area:
With no idea how to calculate the perimeter and the area, these teachers found it difficult to investigate a claim about the relationship between the two measures. So they chose to consult a textbook or another authority. Ms. Francesca, a beginning teacher, did know the formulas for calculating the perimeter and area of a rectangle. Believing that the student's claim would not hold in every case, she thought that the only way she could explain it to the student was "to take other examples that did not hold true." However, she explained that because she did not understand why the formulas worked, it was hard for her to develop a counterexample on her own. What she would do would be to find someone to tell her, or "go home and look it up and check it out":
It was obvious that Ms. Francesca knew more about the topic than the other four teachers. Yet she also noticed that she lacked specific knowledge related to the claim. She would turn to a textbook or to those with more knowledge, hoping that would help her to find a correct answer for the problem. Strategy II: Calling for More Examples. Thirteen U.S. teachers proposed another strategy to explore the claim -- calling for more examples:
These teachers' responses to the claim, that it needed more examples, were based on everyday experience, rather than mathematical insight. Most adults will not be persuaded to accept a proposition with only one example. The teachers' comments on the student's mathematical theory, in fact, paralleled general statements such as "Even though I see two white swans, I would not believe that all swans are white." However, how many white swans do we need to see in order to believe that all swans are white? Themselves concerned about the number of examples, these teachers ignored the fact that a mathematical statement concerning an infinite number of cases cannot be proved by finitely many examples -- no matter how many. It should be proved by a mathematical argument. The role of examples is to illustrate numerical relationships, rather than prove them. Although the teachers were able to point out that one example is not sufficient to prove a theory, they were not able to investigate the claim mathematically. A few of them suggested trying arbitrary numbers, for example, "one through ten," or "strange numbers such as threes and sevens." These suggestions were based on common sense, rather than mathematical insight. Strategy III: Mathematical Approaches. The remaining three teachers investigated the problem mathematically. Ms. Faith was the only one who achieved a correct solution. Her approach was to present an example that disagreed with the student's theory:
The student used a square with sides of 4 inches and a rectangle with the width of 4 inches and the length of 8 inches to prove her statement. The perimeter of the square was 16 inches and that of the rectangle was 24 inches. The area of the former was 16 square inches and that of the latter was 32 square inches. The student concluded that "as the perimeter of a figure increased, the area increases correspondingly." Ms. Faith would ask her to try another example, a rectangle with a width of 2 inches and a length of 16 inches. The perimeter of Ms. Faith's rectangle was 36 inches, 12 inches longer than that of the student's rectangle. According to the student's claim, the area of Ms. Faith's rectangle should be bigger than that of the student's. However, it was not true. Ms. Faith's rectangle had the same area as that of the student's, 32 square inches. With a single counterexample, Ms. Faith disproved the claim. Ms. Francine also tested the claim by trying a long, skinny rectangle. However, she was not as successful as Ms. Faith:
Ms. Francine came close to finding a counterexample. However, she failed because she followed the pattern in the student's example -- changing the perimeter by changing a pair of opposite sides and keeping the other pair of sides fixed. She reduced the perimeter by reducing the length of one pair of opposite sides from 4 inches to 2 inches, but kept the other pair of sides unchanged. Contrary to her expectation, the student's claim still held: the area of her new figure decreased as well. Then she was confused. She decided to give up her own approach and look it up in a book -- the response of a layperson rather than the response of a mathematician. Mr. Felix was the third teacher who approached the problem mathematically. He would explore why the student's claim was true:
Mr. Felix's approach explains why Ms. Francine failed to disprove the student's claim. When the increase (or decrease) of the perimeter is caused only by the increase (or decrease) of only one pair of opposite sides, the area of the figure will increase (or decrease) as well. The area of the increased (or decreased) new figure is the increased (or decreased) length times the length of the unchanged side. Using this pattern one can generate infinitely many examples that support the student's claim. Mr. Felix, however, did not completely examine the student's claim. He stopped after explaining why the claim worked in this case and did not investigate the cases in which it would not work. Of the 23 U.S. teachers, Ms. Faith, a beginning teacher, was the only one to successfully examine the student's proposition and attain a correct solution. Table 4.1 summarizes the U.S. teachers' reactions to the student's claim. TABLE
4.1
Teachers' Responses to the Student Ball (1988b) indicated three possibilities that teachers might use to respond when they are confronted with a new idea proposed by a student:
The teachers in the study chose the second and third alternatives. The teachers who took the second alternative reported that they would "tell" or "explain" the solution to the student. The teachers who took the third alternative reported that they would invite the student to investigate or discuss the claim further. In addition, most teachers explained that they would first give a positive comment to the student. Therefore, the teachers' responses to the student fell into two main categories: praise with explanation, and praise with engagement in further exploration. Sixteen U.S. teachers (72%) described an intention to engage the student in a further proof of the claim. However, without an understanding of the proof themselves, their attempts to engage the student in such a discussion could only be superficial. Three teachers reported that they would "look it up with the student":
These teachers were the ones who did not remember how to calculate the two measures of a rectangle. What they suggested that the student should do was the same as what they themselves wanted to do -- to find the knowledge that is stored in a book. Six teachers said they would ask the student to try, or to show them more examples, to prove her own claim:
These teachers merely asked the student to try more examples, but did not think mathematically about the problem or discuss specific strategies. Five other teachers offered to try more examples with the student, but did not mention specific strategies either:
Five teachers mentioned specific strategies for approaching the problem. However, except for that mentioned by Ms. Faith, the strategies were not based on careful mathematical thinking. When they suggested trying "different numbers" or "strange numbers," they were not considering different cases in a systematic way as we shall see the Chinese teachers did. Rather, the strategy they proposed was based on the idea that a mathematical claim should be proved by a large number of examples. This misconception, which was shared by many U.S. teachers, would be likely to mislead a student. HOW THE CHINESE TEACHERS EXPLORED THE NEW IDEA Teachers' Approaches to the Problem The Chinese teachers' first reactions to the problem were very similar to those of the U.S. teachers. About the same proportion of the Chinese teachers (8%) as of the U.S. teachers (9%) accepted the claim immediately, without any doubt. The other Chinese teachers were not sure if the claim was valid or not. It took them a while to think about it before they began to respond. Of the four interview questions, this took them the longest time to think over. And, once they started to discuss the problem, their responses differed considerably from those of their U.S. counterparts. The Chinese and the U.S. teachers' responses differed in three ways. First, many Chinese teachers showed an enthusiastic interest in the topic, the relationship between the perimeter and the area of a rectangle, while the U.S. teachers tended to be concerned with whether the claim that "as the perimeter increases, the area increases as well" was true or not. Second, most Chinese teachers made mathematically legitimate explorations on their own, while most of their U.S. counterparts did not. No Chinese teacher said that he or she should need to consult a book or someone else,^{2} and none ended up saying "I am not sure." The Chinese teachers' explorations, however, did not necessarily lead them to correct solutions. Consequently, most U.S. teachers who held a "not sure" opinion avoided a wrong answer, but 22% of the Chinese teachers, because of their problematic strategies, gave incorrect solutions. The remaining 70% solved the problem correctly. Third, the Chinese teachers demonstrated a better knowledge of elementary geometry. They were very familiar with perimeter and area formulas. During their interviews, many discussed relationships among the various geometric figures that were not ever mentioned by any of the U.S. teachers. For example, some Chinese teachers said that a square is a special rectangle. Some also pointed out that a rectangle is a basic figure -- that perimeter and area calculations for various other figures rely on using rectangles.^{3} Figure 4.1 summarizes the reactions of the teachers of the two countries to the problem. Justifying an Invalid Claim: Teachers' Knowledge and Pitfalls. Sixteen Chinese teachers who investigated the problem mathematically argued that the student's claim was correct. Twelve teachers justified the claim by discussing why it was the case, the other four teachers addressed how it was the case. These teachers tended to build their arguments on the correspondence formed by identifying the length, width, and area of the rectangle with two numbers and their product:
Their strategy, although incorrect, was grounded in appropriate, although incorrect mathematics. First, the teachers identified the student's claim as a numerical relationship -- the relationship between two factors and their product in multiplication. Then they drew on an established principle of this relationship -- that between the factors and the product -- to prove the claim. The flaw was, however, that they failed to notice that the claim involved two different numerical relationships, not just a multiplicative one. While the relationship of length, width, and area of a rectangle is multiplicative, that of its length, width, and perimeter is additive. The perimeter of a rectangle can increase while two of the sides of the rectangle decrease in length. The teachers who said that the claim was true had explanations similar to Mr. Felix's:
Like Mr. Felix, they failed to consider all the ways in which the perimeter of a rectangle may increase. Therefore, they only explained how the student's case was true, but did not explore the real problem: if it is always true. Although these sixteen teachers did not attain correct solutions, they showed the intention to explore the problem mathematically. Instead of making general comments about the student's claim, they investigated the problem and reached their own conclusions. Moreover, these teachers were aware of an important convention in the discipline: any mathematical proposition has to be proved, and they tended to follow this convention. They did not just opine "the claim is right," rather, they gave proofs of their opinions. The arguments they made, although deficient, were grounded in legitimate mathematics. In addition to a solid knowledge of the calculation of the two measures, these teachers displayed sound attitudes toward mathematical investigation. Of course, their approaches also revealed an obvious weakness -- the lack of thoroughness in their thinking. Disproving the Claim: The First Level of Understanding. Fifty of the 72 Chinese teachers gave correct solutions but their different approaches displayed various levels of understanding. The first level was to disprove the student's claim. The 14 Chinese teachers' approach at this level was similar to Ms. Faith's -- looking for counterexamples:
To disprove the claim, the teachers created two kinds of counterexamples. One consisted of figures with longer perimeter but smaller area or shorter perimeter but bigger area, than one of the student's figures. The other kind consisted of figures with the same area but a different perimeter -- or the same perimeter but a different area -- as the student's figures. Identifying the Possibilities: The Second Level of Understanding. Eight teachers explored the various possible relationships between perimeter and area. They gave different kinds of examples that supported, as well as opposed the claim, to show the various possibilities:
Mr. A. revealed that increasing the perimeter may cause the area to increase, decrease, or stay the same. Ms. E. described two cases in which the two measures changed in different ways -- while the perimeter increases, the area decreases, and while the perimeter stays the same, the areas decreases. At this level of understanding, teachers discussed various facets of the relationship between the perimeter and the area of a figure. In particular, they examined different kinds of changes in the area of a rectangle that can be caused by changes in the perimeter. The teachers did not simply disprove the student's claim, rather, they presented a wider perspective in which the student's claim was included. Clarifying the Conditions: The Third level of Understanding. In addition to displaying the various possibilities, 26 teachers clarified the conditions under which these possibilities held. These teachers tended to explore numerical relationships between perimeter and area and elaborate specific examples:
Tr. R. articulated the strategy that she and several other teachers used to explore the conditions under which the student's claim held. They first examined the cause in the student's claim -- an increase in the perimeter. They investigated the situations that would produce an increase in the perimeter of a rectangle and found three patterns. Then they analyzed the changes these patterns would produce in the area. Through a careful examination, Tr. R. attained a clear picture of how the area might be affected by an increase in the perimeter of the different ways:
The solution that these teachers attained was: when increase in the perimeter is caused by the increase in either or both the length and the width of a rectangle, the area of the figure will increase accordingly; but when the increase in perimeter is caused by increasing length and decreasing width, or vice versa, the area will not necessarily increase as well. About two thirds of the 26 teachers elaborated their discussion in the manner of Tr. R. They addressed both situations -- when the claim holds and when it does not necessarily hold. The remaining third of the teachers focused on one of these situations. The teachers who reached this level of understanding did not regard the claim as absolutely correct or absolutely wrong. Rather, they referred to the concept of "conditional." They argued that the claim was conditionally correct:
In clarifying the different conditions under which the student's claim would hold or not hold, the teachers developed different relationships between the perimeter and area of a rectangle. The student's claim was not simply abandoned; rather, it was revised and incorporated into one of the relationships. Explaining the Conditions: The Fourth Level of Understanding. Six of the teachers who reached the third level of understanding went even further, explaining why some conditions supported the student's claim and why other conditions did not. Their approaches varied. After a detailed and well-organized discussion of the conditions under which the student's claim would hold, Tr. Mao said:
Tr. Mao's argument was based on a geometric representation of the situation. He also applied the distributive property to add another proof to his approach. Tr. Xie's argument about why rectangles with the same perimeter can have different areas was also very insightful. He first indicated that for the same perimeter one can form many rectangles of different lengths and widths, because there are many different pairs of addends that make the same sum. Then he argued that when these pairs of addends become factors, as in calculating the area of the figure, obviously they will produce very different products. Finally, using the fact that the closer the value of the two factors, the larger their product, he claimed that for a given perimeter, the square is the rectangle with the largest area:
Tr. Xie and Tr. Mao did not draw on the same basic principles of mathematics for their arguments. However, both developed solid arguments. In fact, a basic principle of mathematics may be able to support various numerical models. On the other hand, a numerical model may also be supported by various basic principles. A profound understanding of a mathematical topic, at last, will include certain basic principles of the discipline by which the topic is supported. Passing through various levels of understanding of student's claim, the teachers got closer and closer to a complete mathematical argument. ^{1} The term "a closed figure" used in the scenario was intended to invite the teachers to discuss various kinds of figures. However, during the interviews teachers talked exclusively about squares and rectangles. A few Chinese teachers said that closed figure is a concept introduced at the secondary school level in China so they preferred to focus the discussion on the particular figure mentioned by the student. back ^{2} Stigler, Fernandez, and Yoshida (1996) reported a similar tendency on the part of Japanese elementary teachers. back ^{3} In the Chinese curriculum the area formulas for other shapes such as squares, triangles, circles, and trapezoids are derived from that for rectangles. back ^{4} In Chinese elementary math textbooks, a stands for length of a figure, and b stands for width of a figure. back |
||||||||||||
From Knowing and Teaching Elementary Mathematics by Liping Ma. © 1999. Pages 84-98 used by permission of Lawrence Erlbaum Associates, Publishers, Mahwah, New Jersey. |