# «SE 062 649 ED 431 621 Stigler, James W.; Gonzales, Patrick; Kwanaka, Takako; AUTHOR Knoll, Steffen; Serrano, Ana The TIMSS Videotape Classroom Study: ...»

The results of this coding are displayed in figure 80. The highest incidence of both kinds of linkingacross lessons and within lessonswas found in Japan. Indeed, teachers of Japanese lessons linked across lessons significantly more than did teachers of German lessons, and linked within lessons significantly more than teachers of both German and U.S. lessons.

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

Our observations of the videotapes reveal that the Japanese teachers appear to use linking in a systematic, almost routinized, way. They tend to start the lesson by recalling or reviewing what was done in the previous lesson. JP-036, for example, opened with the teacher saying, "Then, urn yesterday the last part on ratio we did two characteristics, but characteristics that can be changed into a multiplied form, right? Please summarize what we practiced?' The same lesson closed with the teacher saying, "Well we could not do concrete practice so I would like to do it using this in the next class period. I also want to review and go over number five in the next class period:' Twenty-six of the 30 Japanese lessons included linking to a past lesson, and 19 included linking to a future lesson.

Twenty-nine of the 30 Japanese lessons included linking to different parts of the same lesson. In one example, the teacher referred back to a statement made by a student several minutes earlier: "Just now you've heard the opinion of your friend and um while using that as a reference you can continue on your computer or urn you can continue on your computers by reading ahead yourselves or you can prove it while getting hints from the computers. Okay? Or you can prove it by um expressing your opinions urn with your friends:' These kinds of statements were far less common in German and U.S. lessons.

** 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 documentsmost 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.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.

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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).

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 "reform7 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. teachersa 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.