Chapter 6. Teachers and Reform

One goal of this study was to determine the extent to which U.S. teachers have been influenced by current ideas about the teaching of mathematics. Reform documents--most notably the NCTM Professional Standards for Teaching Mathematics (1991)--provide guidance on how to teach mathematics in the classroom, or at least on what features of instruction ought to be evident in the mathematics classroom. Many of our codes were inspired by these reform ideas. In this section we will examine how the teachers in our sample think of themselves in relation to current reform ideas, both in general and in relation to the lesson we videotaped. Analyses presented in this section are based on answers given in the Videotape Classroom Study teacher questionnaire.

GENERAL EVALUATIONS

Although the question clearly has a different meaning across cultures, where ideas about education are communicated to teachers in quite different ways, it nevertheless is interesting to see how teachers responded to the question, "How aware do you feel you are of current ideas about the teaching and learning of mathematics?" The teachers could answer "Very Aware," "Somewhat Aware," "Not Very Aware," or "Not at All Aware." The results are presented in figure 81. The distributions of responses to this question differed significantly across countries. Thirty-nine percent of teachers of U.S. lessons report being "Very aware" of current ideas; 5 percent of those teaching Japanese lessons indicated this level of awareness.

Figure 81

Teachers' ratings of how aware they are of current ideas about the teaching and learning of mathematics

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

We asked teachers how they usually hear about current ideas about the teaching and learning of mathematics. Responses were open-ended; we coded them into five categories: School-Based Programs, Information from Colleagues, External Seminars, Publications, and Other. Percentages of teachers in each country who included responses in each category are shown in figure 82. Significantly more Japanese than U.S. teachers mentioned school-based programs. Significantly more U.S. teachers than German teachers, and more German teachers than Japanese teachers, mentioned attending external seminars or workshops. More German teachers than Japanese teachers mentioned their colleagues as a source of knowledge.

Figure 82

Teachers' responses when asked where they get information regarding current ideas about the teaching and learning of mathematics

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

U.S. teachers were asked what written documents or materials they had read to stay informed about current ideas. Dozens of documents were mentioned. Thirty-three percent mentioned the NCTM standards by name (either the Curriculum and Evaluation Standards for School Mathematics or the Professional Standards for Teaching Mathematics). Forty percent mentioned some other NCTM publication by name. And 59 percent mentioned either the Standards or some other NCTM publication.

EVALUATIONS OF THE VIDEOTAPED LESSONS IN TERMS OF CURRENT IDEAS

Claiming to be aware of current ideas is one thing; implementing those ideas in the classroom is another thing entirely. Still, a large percentage of U.S. teachers reported that the lesson we videotaped was in accord with current ideas about teaching and learning mathematics. Bearing in mind that "current ideas" may differ between Germany, Japan, and the United States, we nevertheless asked teachers to specifically evaluate their own videotaped lesson in terms of current ideas. Teachers could say that it was "not at all" in accord with current ideas, "a little" in accord, "a fair amount," or "a lot." Twenty-seven percent of the U.S. teachers responded "a lot," and 70 percent responded either "a lot" or "a fair amount." None of the German or Japanese teachers responded "a lot," and 37 percent (German) and 14 percent (Japanese) responded "a fair amount" (figure 83).

Figure 83

Teachers' perceptions regarding the extent to which the videotaped lesson was in accord with current ideas about the teaching and learning of mathematics

 

NOTE: Due to a clerical error in which an unweighted average was mistakenly substituted for a weighted average, the numbers in this graph differ slightly from those reported in Peak (1996: page 46). Specifically, the total of "a fair amount" plus "a lot" for the U.S. teachers was reported as "almost 75 percent." The correct total should be 70 percent, as shown in this graph.

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

Teachers who said that the videotaped lesson was either "a lot" or "a fair amount" in accord with current ideas about the teaching and learning of mathematics were further asked to justify their responses by citing specific aspects of the lesson that exemplified these ideas. This gave us the opportunity to see which aspects of the lesson teachers focused on and to see what, in the video, could be found to connect with their descriptions.

We analyzed these responses only for the U.S. teachers. Although the range and variety of responses to this question were great, the vast majority of teachers' responses fell into three categories:

• Real-World/Hands-On. Thirty-eight percent of the U.S. teachers gave answers that (1) focused on the application of math to daily life, or (2) involved the use of physical or manipulative representations of mathematical concepts. For example: "The four problems dealt with temperature in Anchorage, Alaska. This gave me a chance to relate mathematics to everyday life."
• Cooperative Learning. Thirty-one percent of the U.S. teachers mentioned cooperative learning in their answer. One teacher, for example, mentioned her practice of having "study buddies" where students pair up to work together; other teachers pointed to their use of peer tutoring, having students explain answers to each other.
• Focus on Thinking. Finally, 19 percent of the U.S. teachers mentioned a focus on thinking, specifically conceptual understanding, a focus on process over product, or a focus on problem solving. Some of these teachers specifically contrasted this focus with one that emphasizes computational skills.

In general, we can see what teachers are talking about when we review their videotapes: All teachers who pointed to real-world applications did include such applications in their lessons, and the same was true for cooperative learning. Whether or not these features led to lessons that were, in fact, more in line with those envisioned by reformers is a question we shall return to.

U.S. REFORM IN CROSS-CULTURAL PERSPECTIVE

Although it is unclear exactly what is meant by "reform," or even "current ideas" in the context of Germany and Japan, it is quite clear in the United States. Thanks to the influence of the National Council of Teachers of Mathematics, we have some fairly well-specified ideas about what the mathematics classroom should look like, and many teachers claim to be familiar with these ideas. Furthermore, the majority of teachers in our video sample believed that we would find evidence of these current ideas in the lessons we had videotaped. Is there any evidence, in our data, that U.S. teachers are, in fact, implementing these ideas in their classrooms?

Although most of the current ideas stated in such documents as the NCTM Curriculum and Evaluation Standards for School Mathematics (1989) and the NCTM Professional Standards for Teaching Mathematics (1991) are not operationalized to the extent that they could be directly coded, it is possible to use some of the indicators we have developed in conjunction with these current ideas. When we view our data in this way, we come to this conclusion: Japanese classrooms, on average, appear to more closely exemplify current ideas advanced by U.S. reformers than do classrooms in the United States and Germany. Because the reform ideas considered here emerged from the United States, we limit our discussion to a consideration of the contrasts between the United States and Japan.

Let us take a couple of examples. In both of the NCTM documents just mentioned, problem solving is proposed as the central focus of curriculum, teaching, and learning. The Curriculum and Evaluation Standards for School Mathematics states that, "In grades 5-8, the mathematics curriculum should include numerous and varied experiences with problem solving as a method of inquiry and application so that students can use problem solving approaches to [among other things] investigate and understand mathematical content; ...develop and apply a variety of strategies to solve problems, with emphasis on multistep and nonroutine problems..." (NCTM 1989, page 75. Italics added.). Similarly, the NCTM Professional Standards for Teaching Mathematics proposes the posing of "worthwhile mathematical tasks" as the first standard for teaching. Worthwhile mathematical tasks are further defined as tasks based on "sound and significant mathematics" that "engage students' intellect; develop students' mathematical understandings and skills; stimulate students to make connections and develop a coherent framework for mathematical ideas; call for problem formulation, problem solving, and mathematical reasoning" (NCTM 1991, page 25).

Several indicators in our study point to the greater consistency of Japanese lessons in terms of these criteria. Content analyses showed that Japanese lessons included more advanced levels of mathematics, and that the mathematics was presented in a more coherent way than in U.S. lessons (see, for example, the analyses by the Math Content Group). Japanese lessons included more emphasis on concepts than U.S. lessons, and were more likely to develop instead of merely state the concepts. Japanese teachers also were more likely than U.S. teachers to make explicit the connections within a lesson. These facts would appear to give Japanese students an advantage in the quest to "make connections and develop a framework for mathematical ideas." Finally, our analyses of performance expectations of tasks posed during seatwork showed that Japanese students, more than U.S. students, were engaged in genuine problem solving during the lesson, rather than simply the application and practice of routine problem-solving skills.

Let us take another example from the reform documents. Both of the NCTM documents we are discussing place communication and discourse at the center of their proposed reforms. The Curriculum and Evaluation Standards for School Mathematics states that the study of mathematics should include opportunities to communicate so that students can, for example, "reflect on and clarify their own thinking about mathematical ideas and situations" (NCTM 1989, page 78). The Professional Standards for Teaching Mathematics (1991) devotes three of its six teaching standards to discourse. Teachers, according to the document, should "orchestrate discourse by posing questions and tasks that elicit, engage, and challenge each student's thinking" (page 35). Students should "listen to, respond to, and question the teacher and one another," and "make conjectures and present solutions" (page 45).

Although we have only completed a rudimentary analysis of classroom discourse, we already can find some evidence that Japanese teachers, more than U.S. teachers, orchestrate the kind of discourse called for in these reform documents. For example, we find Japanese teachers asking more describe/explain questions, and fewer yes/no questions, than U.S. teachers. Also relevant is the analysis of student-generated solution methods, which occurs more frequently in Japan than in the United States. The reason for this pattern is clear: Japanese teachers often have students struggle with a problem for which they have not yet been taught a solution, then present the solutions they generated to their classmates. Presentation and discussion of alternative solution methods may provide a natural opportunity for engaging in the kind of mathematical discourse reformers are seeking to foster.

Of course Japanese teachers may not teach the way they do because they are following the recommendations of U.S. reformers. And it is also worth pointing out that there are some respects in which they do not appear to teach in accordance to the proposals of U.S. reformers. For example, Japanese teachers engaged in far more direct lecturing/demonstration than U.S. teachers--a practice frowned on by reformers. And, contrary to specific recommendations made in the NCTM Professional Standards for Teaching Mathematics, Japanese teachers never were observed using calculators in the classroom.

REFORM IN THE U.S. CLASSROOM: OBSERVATIONAL INDICATORS

We have seen that on many of the variables coded, Japanese teachers teach more in accord with current U.S. reform ideas than do U.S. teachers. But what about differences among U.S. teachers? Do some teachers show more evidence of reform than others? We have already shown that the majority of the U.S. teachers in our sample felt that their lesson was in accord with current ideas. Of course, there were some U.S. teachers who did not feel this way. Could we find some differences among teachers who did and did not report implementation of current ideas in their classrooms?

For purposes of analysis we defined two groups of U.S. teachers: One group (N=22) responded "a lot" when asked the degree to which the videotaped lesson was in accord with current ideas (i.e., those responding "a fair amount" were excluded from this analysis); the other group (N=19) answered either "a little" or "not at all." We can call the first group Reformers, the second, Non-Reformers. We compared the classrooms of these two groups of teachers on all of the variables discussed earlier in this report.

Overall, the analyses revealed very few significant differences between the two groups of teachers. Although this lack of differences may be due, in part, to the lack of statistical power given the small size of our sample, we do not believe this is the primary reason. In our own viewing of the tapes, we did not see a strong distinction between these groups.

Statistically significant differences did emerge, however, in organization of the lesson and materials.

Organization of the Lesson

As a group, Reformers spent a higher percentage of their time in seatwork than did Non-Reformers. Reformers spent 43.1 percent of lesson time in seatwork, compared with 28.6 percent for Non-Reformers. (The average for all U.S. teachers was 37.3 percent.)¹

Figure 84

Percentage of lessons among Reformers and Non-Reformers in the United States in which seatwork of various kinds occurred

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The two groups also differed in the kind of seatwork in the lesson: individual, group, or a combination of the two. In figure 84 we show the percentage of lessons in the two groups that included individual seatwork only, group seatwork only, both kinds of seatwork, or no seatwork. We can see that the Reformers were far more likely to include both individual and group seatwork in their lessons than were the Non-Reformers, while they were less likely to use individual seatwork only.

Instructional Materials

Finally, we found one significant difference in the tendency of U.S. Reformers and Non-Reformers to use certain types of materials in the classroom. Specifically, Reformers were less likely than Non-Reformers to use textbooks in the lesson: 18 percent of lessons for Reformers, and 63 percent for Non-Reformers.²

REFORM IN THE CLASSROOM: QUALITATIVE ANALYSES

Other than these areas of differences, we have little quantitative evidence that reform teachers in the United States differ much from those who claim not to be reformers. Most of the comparisons were not significant. But it is useful to look more qualitatively at the lessons taught by Reformers and Non-Reformers. Our conclusion is interesting: It is true that teachers who cite features of instruction, such as the use of real-world problems or cooperative learning, do implement such features in the lessons we videotaped. However, these features alone do not necessarily indicate the quality of instruction as intended by the NCTM standards, and in fact may only bear a superficial relationship to the quality of instruction. Similarly, a high quality lesson could be constructed that did not contain these features. Quality of mathematical activity depends on how features are implemented. Let us explore a few examples.

Example 1: Airplane on a String (US-060)

US-060 was taught by a teacher who judged it to be a good example of current ideas about the teaching and learning of mathematics. The lesson included a real-world problem situation and a period of cooperative group work. It also included a writing assignment in which students were asked to reflect on what they learned in the lesson. We agreed with the teacher's assessment of the lesson, and we will try to explain why.

The lesson started with the teacher asking for a volunteer to come to the front of the room and swing a model airplane on the end of a string around her head in a circular motion (figure 85). Everyone appeared attentive, and enthusiasm was high as students wondered what the airplane on the string would have to do with mathematics.

Figure 85

Frames from the video of US-060

 

SOURCE: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.

The teacher then posed a question: How fast is the plane going? Many students gave answers, which the teacher wrote on the overhead projector: "About thirty miles per hour." "Fifteen miles per hour." "Two-hundred thirty." And so on. Why are the answers in miles per hour, the teacher wondered aloud. What would be a rate we could measure? Revolutions per minute? Per second? If we measured such things, could we convert them to miles per hour? Discussion of these questions took up the first 10 minutes of the lesson.

After the teacher could see that most students in the class understood the problem and had thought through some of the issues involved in solving the problem, she had them work in groups for the next 30 minutes. Each group had its own airplane and string, and each worked to measure the speed of the plane. The general strategy adopted by all groups was the same: Measure the radius of the circle from the plane to the point around which the plane was swung; then, calculate the circumference in inches, count the number of times the plane traversed the circumference in a set amount of time, and get an answer in inches per second or per minute; finally, convert the answer into miles per hour. Students appeared to be very involved in the activity.

As students worked, the teacher circulated and posed a new question to each group: If a bird were sitting on the string, halfway between the plane and the center of the circle, would it be traveling the same speed as the plane, faster, or slower? Groups actively began to discuss this possibility, with differing opinions offered and justified. One student noticed that her hand was not completely still as she twirled the plane, and that this must have affected the radius of the circle. The teacher asked the group to consider how they might need to adjust their method to take this problem into account.

Thirty-nine minutes into the lesson students convened again as a class. Over the next 12 minutes all groups presented their answers, then two groups presented their solution methods to the class. Someone brought up the bird: how fast would the bird be going? The teacher told them that they would discuss this tomorrow, then handed out a writing assignment for homework. The writing assignment asked students to describe the problem they had worked on, then summarize the approach their group took to solving the problem. It also asked them to write about the role they played, specifically, in the group's work.

We thought this was in line with NCTM standards for several reasons. Students were engaged in a rich mathematical problem that appeared to be perceived as a real problem by most of the students in the class. The problem was closely tied to mathematical concepts--circumference and radius of a circle, and rate. The task encouraged students to make connections among these concepts and between these concepts and a real-world domain. Students were encouraged to come up with their own ways of solving the problem, and much of the lesson focused on discussing the validity of the methods they devised. Finally, students were left to reflect on their activity and to ponder a new dimension, the addition of the bird to the string. There was a clear sense of what the next step would be as the class pursued the topic further.

Example 2: The Game of Pig (US-071)

US-071 was also judged by the teacher to exemplify current ideas about the teaching and learning of mathematics. On a superficial level, the lesson had much in common with the one we just described: It included a hands-on learning experience, working in groups, and a writing assignment. However, we judged this to be less in line with NCTM standards than the previous lesson. Let us try to explore the reason for our judgment.

The lesson started by asking students to recall the game of Pig, which they had played previously. Take 5 minutes, the teacher told the students, and write down everything you remember about the rules of the game.

The game of Pig is played with dice. Students work in groups. One student rolls a pair of dice, and all students who are playing receive the number of points that is the product of the two numbers rolled. The process is repeated, and the number of points on each turn is added to each player's total. However, if a one is rolled, players receive no points for that turn; and if two ones are rolled, players lose all of their points and have to start over. Players may elect to stop playing at any time, in which case they are left with whatever number of points they have accumulated up to that point.

After students completed the 5-minute writing assignment, the teacher went over the rules of the game. She then told the students that today they would play the game twice, first with 6-sided dice, then with 10-sided dice. "We will try to see if you can get a higher score playing with 10-sided dice than with 6," the teacher remarked. She told students to prepare a score sheet, then handed out the 6-sided dice to each group of students and the play began. Halfway through the period the teacher handed out the 10-sided dice. At the end of the lesson, students were given a 10-minute writing assignment:³

This teacher reported, on her questionnaire, that this lesson dealt with probability and uncertainty. When asked what she wanted students to learn from the lesson she wrote: "How does theoretical probability compare with actual experimental data?" And indeed, it would seem possible to develop a lesson on probability that involved comparing 6-sided with 10-sided dice. However, when we watch the lesson and read the transcript in detail, we find no evidence that this lesson involved probability. Virtually all of the mathematical talk that went on in the lesson concerned multiplication of single-digit numbers and addition (i.e., those operations required to keep score). In our judgment, that was the mathematics that students were getting out of this lesson.

Twice the teacher brought up probability for discussion, but in each case, she failed to pursue the discussion for more than a few seconds. The first such instance occurred near the beginning of the lesson when the teacher was explaining the rules of the game. She was discussing what happens when double-ones are rolled on the dice:

There was no attempt on the teacher's part to explain how knowing the probability of a dice throw might influence the playing of the game. No additional problem was posed for which students could use probability, and there was no discussion of probability throughout the rest of the lesson.

The other point at which the teacher asked a question that could have led to a discussion of probability happened 34 minutes into the lesson. The teacher suggested that students who kept ending up with a score of zero might want to reconsider their strategy. However, the point was dropped after this isolated comment and did not lead into any discussion of probability, game strategy, or the relation between the two. It is interesting that the teacher raised the question of strategy. One could, from there, potentially get into a discussion of probability theory. However, there really is no clear strategy for this game, which makes it harder for the teacher to get a discussion going about strategy.

At the end of the questionnaire we asked the teacher which part of the lesson exemplifies current ideas about the teaching and learning of mathematics and why. This teacher wrote:

• Students were involved in a hands-on, interactive activity. They were allowed to come to their own conclusions about their experience. They were required to communicate their experience to others, both verbally and in writing.

It is clear to us that the features this teacher uses to define high quality instruction can occur in the absence of deep mathematical engagement on the part of the students.

Example 3: A Non-Reformer (US-062)

Let us briefly present one more example, this from a teacher who, although she claimed to be "very aware" of current ideas about the teaching and learning of mathematics, judged her videotaped lesson to be "only a little" in accord with these current ideas. The lesson dealt with factoring of polynomials, and discussed factoring in the context of slope and the solving of simultaneous equations. This teacher stated the goal of the lesson in terms of student understanding: "I wanted the students to arrive at an understanding of what factoring is, and to be able to use the language." In fact, this lesson showed a great emphasis on student thinking and understanding and appeared to us to be in line with NCTM standards. It is an interesting example to consider because the teacher did not see it as in accord with current ideas.

At the beginning of the lesson students were given back a test they had taken the previous day that dealt with factoring of polynomials. She asked students to go over the tests in their groups, for 5 minutes, discuss the questions they got wrong, and then decide on one item, presumably the most problematic one, to present to the class. As the students deliberated, the teacher went from group to group answering questions and facilitating discussion.

Eight minutes into the lesson, the class reconvened. Each of the five groups sent a representative to the front, one by one, to present and discuss a problem from the test with the class. Let us look at an excerpt from the transcript to get a sense of what the class discussion was like. Here, for example, the student, at the front of the room is presenting problem number 16: "Factor completely, by first factoring out the greatest common factor and then factoring the resulting polynomial: 8x² + 8." We present a somewhat lengthy excerpt.

This excerpt is quite typical of the whole lesson. On the tape we see a class struggling to attach words to their understandings and their solution methods, and the teacher constantly acting to facilitate the exchange. The last ten minutes of the lesson were spent on a new problem that involved finding the slope of the line crossing through two points. The slope is negative, which some of the students find confusing. Again, there is a lively discussion as the teacher tries to mediate between two different solution methods, only one of which ends up being correct.

This teacher explained that the lesson was not in accord with current ideas because "current mathematical thinking is that factoring is not an important concept." She then added the comment: "I disagree."

¹ Standard errors for Reformers and Non-Reformers were 3.99 and 5.26, respectively.

² Standard errors for Reformers and Non-Reformers were 8.56 and 8.44, respectively.

³ A double slash mark (//) on two succeeding turns in a transcript indicates overlap in speech. Thus, the // in the beginning of the turn at 40:29 indicates that this utterance (which was inaudible, as indicated by the blank space between the parentheses) started right after the teacher said "Venn diagram" in the preceding turn. An explanation of this and other transcription conventions is included as appendix F.