Exploring New Knowledge:
The Relationship Between Perimeter and Area

by Liping Ma, from Knowing and Teaching Elementary Mathematics

Scenario

Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure1 increases, the area also increases. She shows you this picture to prove what she is doing:

How would you respond to this student?

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:

[Pause of about 5 seconds] I forgot my perimeters and my areas here. [Frank looked intently at the problem for about 10 seconds] Well, let's see now the area...[pause of about 10 seconds]...I have to look it up and I will get back to students. (Mr. Frank)

I think I would be looking up formulas, first. To give me the basic formula, for the perimeter and area. And then see if they might even give some examples of the perimeter expanding in one way, and see how they formulated their problem, and see if hers meshed up with what they had in the book. I could also say maybe we could contact someone who has more background in that area, another teacher. (Ms. Fay)

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":

Let us see, perimeter is [she mumbles the formula to herself]. How would I explain it to her that, that does not hold true? I guess the only other way I would, right now from the top of my head, is to take other examples that did not hold true, and illustrate to her that...that it does not hold true. And I cannot remember exactly why...I would go back and research it and find out why, and then come back to her and show her. And probably if like somebody were to come up to me and tell me this right now. I would tell them, I do not believe this is true, but let me find out for sure and go out on my own and look it up and do problems, and then come back and tell her why.

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:

I am not sure. I would say probably that it may work in some cases, but may not work in other cases. (Ms. Fiona)

What I would need to do is probably have enough examples. (Tr. Blanche)

We should talk about whether it worked in every case, if it proves true in every situation. (Ms. Florence)

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:

I would say, "Now tell me though what happens when you have got 2 inches on the one side and 16 inches on the other side." I would ask her what the perimeter is, then I would ask her to figure out the area. Aha!

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:

I would say that by this picture that is right. How about, though, draw another picture, but skinny, long...then showing her that maybe it would not always work...Like that [she drew some figures on paper]. Four and 8...I am trying...the area is when you multiply, 32. So, yes, that is right...Let us say this one, 4 by 4, and let us say this is 2 by 4...oh, oh, wait a minute. I do not know. I do not know if she is right or not...I guess we would have to find out,...look it up in a book!

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:

I would...confirm that indeed in the case of these rectangles and squares that is true; that it does get bigger. I would talk about why that is the case. What the relationship between the area and the perimeter is, and how to use something like a squaring off grid method, to talk about how adding that extra perimeter adds to the area.

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
U.S. Teachers' Reactions to Students' Claim (N = 23)

Reactions % N
Simply accepted the claim 9 2
No mathematical investigation 78 18
Investigated the claim 13 3

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:

1. Divert the student from pursuing ideas outside the scheduled curriculum.
2. Be responsible for evaluating the truth of the student's claim.
3. Engage the student in exploring the truth of her claim.

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":

OK, what I would do is go, go with her to a math book and look up perimeter, look up area, and how, how perimeter and area are related, and go through it together. (Ms. Frances)

I think I would say, "I am not real sure but let us look it up together, and see, and see if we can find a book that would show us whether you are -- your discovery is correct or not." (Ms. Fay)

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:

She's right. Have her try, encourage her and say, I think you are right and have her maybe show the class or show me -- try it with different examples and make sure that she can support her hypothesis. Put her in a position of that "I really found something out" -- make her feel good. (Ms. Fleur)

Oh, most likely, oh yes. Now I just want to make sure it is right. Well, I would praise her for doing work at home...I would then use these as examples on the board. Maybe ask her to be my teacher's helper, give other examples. (Tr. Belinda)

I'd be excited. I really do not have a comment on it. I'd probably like to have her do a few more to prove it. (Tr. Beatrice)

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:

I am not sure. I would say probably that it may work in some cases, but may not work in other cases. I would say, well you know, this is very interesting. Let us try it with some other numbers and see if this works as well. (Ms. Fiona)

I think the best, you probably have to go through and start with, again, even a different group of numbers and bring her on all the way through. In other words, well maybe it would work with one case but it would not work with the next case. So maybe showing the girl working with not just the 4 by 4 and then the 4 by 8, but say 3 by 3 and try it with other numbers. Well, let us say she continues in this vein... (Tr. Bernadette)

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:

I think the student is right. As the perimeter of a rectangle increases, its area increases as well. We know that the area of a rectangle is the product of its length and width. In other words, the length and the width are the two factors that produce the area. Unquestionably, as the factors increase, the product will increase as well. (Ms. H.)

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:

The student's claim is true. Let's have a look at how it is true. If we overlap the square on the rectangle, we will see another uncovered square. That will be the increased area. One pair of opposite sides of the increased areas is actually the width of the two original figures, the other pair of opposite sides of the increased area is the difference between the length of the original rectangle and the side of the original square. Or, we can say that it is the increased piece of the length...(Ms. B.)

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:

Her claim was not true. I will say nothing but show the student a counterexample. For instance, under her square (with sides of 4 cm), I may want to draw a rectangle with the length of 8 cm and the width of 1 cm. She will soon find that my figure is of longer perimeter but smaller area than hers. So, without saying, her claim is wrong. (Ms. I.)

This claim does not hold true in all cases. It is easy to find cases which can disprove the theory. For example, there is a rectangle, its length is 10 cm and its width is 2 cm. Its perimeter will be the same as that of the student's rectangle, 24 cm. but its area will be only 20 square cm, smaller than that of the student's rectangle. (Tr. R.)

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:

I will present several figures to her and ask her to calculate their perimeter and area:

By comparing these figures, she will learn that as the perimeter increases, the area does not necessarily increase as well, such as in the case of figures a and b. Also, when the perimeter remains the same, the area may not be the same, such as in the case of figures c and d. So she will know that there is not a direct relationship between perimeter and area. What she has found is one of several solutions of the problem. (Ms. E.)

I will first praise her for her independent thinking. But I will also let her know that there may be two other situations as well. For example, when the perimeter increases, the area can increase, but it may also decrease, or even stay the same. Then I will show her an example of each case to compare with her rectangle (with length of 8 cm and a width of 4 cm). I will first give an example of her claim, like a rectangle with a length of 8 cm and width of 5 cm. The perimeter will increase from 24 cm to 26 cm, the area will increase from 32 square cm to 40 square cm. Now, the second example will be a rectangle with length 12 cm and width 2 cm. Its perimeter will increase to 28 cm, but its area will decrease to 24 square cm, only three quarters of the area of her rectangle. Another example might be a figure with length 16 cm and width 2 cm. Its perimeter will also increase, up to 36 cm, but the area will stay the same as that of her rectangle, 32 square cm. So I will tell her that mathematical thinking has to be thorough. This is one feature of our thinking that gets improved in learning mathematics. (Mr. A.)

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:

It is obvious that in some cases the claim holds but in some other cases it does not hold true. Yet when does it hold true and when does it not? In other words, under what conditions does it hold up, and under what conditions does it not? We had better have a clear idea about it. To clarify the specific conditions that cause the various possibilities, we can first investigate the conditions that will cause an increase in the perimeter, and then explore how these conditions affect the change in the area. (Mr. D.)

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:

I would say that the student's claim holds up under certain conditions. We know that changes in the length and width of a figure may cause an increase in perimeter. There are three ways to change the length and the width of a rectangle that would cause an increase in its perimeter. The first is when either the length or the width increases but the other measure remains the same. Under this condition, the area of the figure will increase accordingly. For example, given that the length of the student's rectangle increases to 9 cm and its width remains unchanged, the original area, 32 square centimeters, will increase to 36 square centimeters. Or, given that the width of the original rectangle increases to 5 cm but its length remains unchanged, its area will increase to 40 square centimeters. The second way to increase the perimeter is when both the length and the width increase at the same time. Under this condition, the area will also increase. For example, given that the length of the rectangle increases to 9 cm and the width increases to 5 cm at the same time, the area of the rectangle will increase to 45 square cm. The third condition that causes an increase in perimeter is when either the length or the width of a figure increases but the other measure decreases; however, the increased quantity is larger than the decreased quantity. Under this condition, the perimeter will also increase, but the change in area may go in three directions. It may increase, decrease, or stay the same. For example, given that the width increases to 6 cm and the length decreases to 7 cm, the perimeter will increase to 26 cm and area will increase to 42 square cm. Given that the length increases to 10 cm and the width decreases to 3 cm, the perimeter will also increase to 26 cm, but the area will decrease to 30 square cm. Given that the length increases to 16 cm and the width decreases to 2 cm, the perimeter will increase to 36 cm, yet the area will remain the same, 32 square cm. In brief, under the first two conditions, the student's claim holds true, but under the last condition, it does not necessarily hold. (Tr. R.)

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:

So, now we can say that the student's claim is not absolutely wrong, but it is incomplete or conditional. Under certain conditions it is tenable, but under other conditions it does not necessarily hold. I am glad that you raised the problem. I have figured out something new today which I haven't thought about before. (Tr. J.)

After the discussion I may want to give her a suggestion to revise her claim by confining it to certain conditions. She may want to say that under the conditions that the increase of the perimeter is caused by the increase of either the length or the width but the other side remains unchanged, or by the increase of both the length and the width, the area of the rectangle increases as well. That will be a safe statement. (Ms. G.)

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:

At last, we can have an examination of why these conditions are tenable. Imagine how the area of a figure changes when its perimeter changes. Under the first two conditions, the original area remains but a new area is added to it. For instance, when the length increases but the width remains the same, there will be an extra area expanding horizontally from the original one. On the other hand, when the width increases but the length remains the same, there will be an extra area expanding vertically from the original one. If both the length and the perimeter increase at the same time, the original area will expand in both directions. In any of these cases, the original area is still there but some other extra area is added to it. We can draw figures to display the cases. In fact, it can also be proved by using the distributive property. For example, when the length increases 3 cm, it becomes (a + 3) cm.4 The area will be (a + 3) b = ab + 3b. Now, compared to the original area, ab, we can see why it is larger. 3b is the increased quantity. However, given that one measure increases and the other one decreases, the original area of the first figure will be destroyed. There is no reason that guarantees the new area will be bigger than the previous one.

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:

The area of a rectangle is determined by two things, its perimeter and its shape. The problem of the student was that she only saw the first one. Theoretically, with the same perimeter, let's say 20 cm, we can have infinite numbers of rectangles as long as the sum of their lengths and widths is 10 cm. For example, we can have 5 + 5 = 10, 3 + 7 = 10, 0.5 + 9.5 = 10, even 0.01 + 9.99 = 10, etc., etc. Each pair of addends can be the two sides of a rectangle. As we can imagine, the area of these rectangles will fall into a big range. The square with sides of 5 cm will have the biggest area, 25 square cm, while the one with a length of 9.99 cm and a width of 0.01 cm will have almost no area. Because in all the pairs of numbers with the same sum, the closer the two numbers are, the bigger the product they will produce...(Tr. Xie)

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.