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CHAPTER 3 :
TEACHING
KEY POINTS:
- The content of U.S. mathematics classes requires less high-level thought than classes in Germany and Japan.
- U.S. mathematics teachers' typical goal is to teach students how to do something, while Japanese teachers' goal is to help them understand mathematical concepts.
- Japanese teachers widely practice what the U.S. mathematics reform recommends, while U.S. teachers do so less frequently.
*Although most U.S. math teachers report familiarity with reform recommendations, only a few apply the key points in their classrooms.*
Publication and discussion of documents such as these, however, do not change the behavior of all of America's hundreds of thousands of mathematics teachers within a few years. Recommendations for major changes in other areas of American life, such as improving health through regular exercise and proper diet, have required decades of sustained effort by public health organizations at all levels to assist individual citizens in changing ingrained personal habits and attitudes. Indeed, the campaign still continues. Changing our nation's habits of teaching and public attitudes toward mathematics and science may also require a similarly long and concerted effort by many committed people. TIMSS was not designed as an evaluation of the U.S. mathematics reform efforts described in the documents listed above. There are three reasons why TIMSS is unsuitable as such an evaluation. First, because it is an international study, it was designed to measure those aspects of mathematics and science knowledge and practice considered important by the majority of TIMSS nations, rather than those specifically recommended by the U.S. reform community. Second, TIMSS tested U.S. students in the spring of 1995, which was too soon after the publication of the reform documents for states and districts to have designed their own reform programs, retrained teachers in the new practices, and nurtured a generation of students according to the new approach. Third, a proper evaluation requires matching "before and after" measurements between which progress can be judged, and we have no prior measurement which matches TIMSS. For these reasons, TIMSS is not suitable as an evaluation. It should be studied as a baseline measurement against which future progress can be gauged. Until TIMSS, no large nationally-representative study had observed U.S. classrooms to watch how teachers actually teach. To overcome this lack, and to understand how U.S. classroom teaching compares to that of other countries, NCES added an innovative new research methodology to the TIMSS project- videotaping and quantitative coding of a national sample of eighth-grade mathematics classes in Germany, Japan, and the U.S. In the U.S. and Germany, half of the eighth-grade mathematics classrooms in which students were scheduled to take the TIMSS test were randomly chosen to be filmed. In Japan, 50 classrooms from the schools in which the TIMSS test was administered were chosen by the principal and officials at the National Institute for Educational Research. Teachers whose classrooms were chosen and who agreed to participate were videotaped teaching a typical lesson. In this way, videotapes of 230 lessons were collected in the three countries combined. The videotapes were then coded and analyzed to compare the teaching techniques and lesson content typical of the three countries. Teachers also completed a questionnaire concerning the lesson that was videotaped. The findings can be considered representative of the type of instruction received by German, Japanese, and U.S. eighth-grade mathematics students. The results provide a window on actual teaching in U.S. classrooms, and also show how U.S. mathematics classes compare to those in Germany and Japan.
When studying what teachers do in their classrooms, we should first understand what they mean to do. Therefore, the videotape study asked teachers about their goals for the lesson.
When a lesson included a mathematical concept, it was usually simply stated in U.S. classrooms, whereas it was developed in Japanese and German ones. For example, consider a lesson on the Pythagorean theorem. When the concept is merely stated, the teacher or a student might simply say "we find the length of the hypotenuse of a right triangle by using a2+ b2= c2." In contrast, a concept was coded as having been developed if it was proven, derived, or explained in some detail. Figure 10 shows that These findings from the videotape study are corroborated by the TIMSS questionnaire findings. Teachers were asked to choose activities that were characteristic of their teaching from among those listed on the questionnaire. U.S. math teachers were more likely to report asking students to practice computational skills, in most or every class than were their German and Japanese colleagues. Similarly, Japanese teachers were more likely to report they ask students to analyze relationships, write equations, explain their reasoning, and solve problems with no obvious solution in most or every class than teachers in the U.S. and Germany. Linking concepts used in one part of the lesson to ideas or activities in another part of the lesson is believed by experts to improve students' ability to learn and understand a subject in an integrated way. The videotape study found that 96 percent of Japanese lessons included such explicit linkages in comparison to about 40 percent of U.S. and German lessons. Talking about such relationships may help make lessons more coherent for students by showing them the relationships between ideas and activities used in different parts of the lesson. Interruptions present a threat to the coherence of lesson activities. The study found that the flow of mathematics lessons was more frequently interrupted than in Germany and Japan. One U.S. math lesson in four was temporarily halted by an outside interruption, typically a loudspeaker announcement, or a visitor at the door. In contrast, interruptions in German lessons were much less common, and the Japanese lessons observed in the study never experienced outside interruptions. Interruptions coming from within the classroom were also more common in U.S. mathematics lessons, such as substantial discussion of non-mathematical subjects like recent sports events, or extended disciplinary incidents. In the U.S., 23 percent of lessons were broken up in this way, compared to 9 percent in Japan, and 4 percent in Germany.
As noted earlier, the U.S. eighth-grade mathematics curriculum focuses more on arithmetic, while the German and Japanese curricula focus more on geometry and algebra. Furthermore, U.S. eighth graders are studying topics usually learned at the seventh grade in most other TIMSS countries. How does the quality of the mathematical reasoning used in U.S. classrooms compare with that in Germany and Japan? Videotape researchers requested the assistance of 3 mathematics professors and one professor of mathematics education in evaluating the quality of the mathematics contained in the videotaped lessons. This group of four experts was asked to judge the quality of the "story" formed by the sequence of mathematical ideas in a random sample of 90 of the lessons divided evenly among each of the three countries. They studied such factors as the coherence of the sequencing, the type of reasoning required of students, the increase in cognitive complexity between the beginning and end of the lesson, and the way in which the problems and examples contributed to the lesson's central concept. To ensure that the experts were not unconsciously biased toward any country, they were not allowed to actually see the videotapes. Instead, they were provided with a written summary of each lesson's sequence of mathematical statements and equations, as well as how these were embedded in learning activities. The summaries were carefully reviewed to disguise any words such as "yen," or "football," or other hints which might indicate the country in which the lesson was taught. Each expert first independently rated the overall quality of the mathematical content of each lesson as either low, medium, or high. After comparing their ratings, they found high agreement among their judgments. Figure 11 shows their judgments.
These findings that our nation's eighth-grade mathematics classes are based on less challenging material, and lack mathematically rich content suggest that our students have less opportunity to learn challenging mathematics than their counterparts in Germany and Japan.
A great deal of effort has been invested in the reform of mathematics teaching in the U.S. in recent years. There is considerable agreement among experts about what good instruction should look like. The main goal of the reform is to create classrooms in which students are challenged to think deeply about mathematics and science, by discovering, understanding and applying concepts in new situations. For many years, Japanese mathematics educators have closely studied U.S. education reform recommendations, and attempted to implement these and other ideas in their own country. Has the message about mathematics reform penetrated to U.S. classrooms? TIMSS data suggest that it is beginning to, but still only in limited ways. U.S. teachers believe that their lessons are already implementing the reform recommendations, but the findings described so far in this chapter suggest that their lessons are not. When asked to evaluate to what degree the videotaped lesson was in accord with current ideas about teaching and learning mathematics, almost 75 percent of the teachers respond either "a lot" or "a fair amount." This discrepancy between teachers' beliefs and the TIMSS findings leads us to wonder how teachers themselves understand the key goals of the reform movement, and apply them in the classroom. Teachers in the study were asked to describe which aspects of the videotaped lesson exemplified current ideas about teaching and learning mathematics. Most U.S. teachers' answers fall into one of three categories: - Hands-on, real-world math - 38 percent of the teachers mentioned lesson activities that apply math to daily life, such as temperature in Alaska, or that use a physical representation of a mathematical concept, such as geometric blocks.
- Cooperative learning - 31 percent of the teachers mentioned the use of peer tutoring, "study buddies," or math discussion groups.
- Focus on thinking - 19 percent of the teachers mentioned focusing on conceptual thinking about math in preference to computational skills, or mention focusing on problem solving.
Over 80 percent of the teachers in the study referred to something other than a focus on thinking, which is the central message of the mathematics reform movement. The majority of the teachers cited examples of hands-on math or cooperative learning, which are techniques included among the reform recommendations. However, these techniques can be used either with or without engaging students in real mathematical thinking. In fact, the videotape study observed many examples of these techniques being conducted in the absence of high-quality mathematical content.
The videotape study found that, in many ways, Japanese teaching resembled the recommendations of the U.S. reform movement more closely than did American teaching. Japan also scored among the top nations in the world on the TIMSS test. However, until more studies of other high-scoring nations are carried out, we cannot be sure that there is a relationship between Japan's high scores and its style of teaching.
TIMSS provides less data about science teaching than about mathematics teaching, because the videotape study was conducted only in mathematics. However, the TIMSS teacher and student questionnaires included some items about instructional practices which help us understand something about the teaching of science in Germany, Japan, and the U.S.. The questionnaire data has only begun to be analyzed, and more analyses will soon be completed. Preliminary analyses suggest that U.S. science teaching may resemble mathematics teaching in some respects, and differ in others. Therefore, one should not assume that the videotape findings in mathematics apply to science or to other subjects. Taken together, the data suggest that the instruction in typical U.S. mathematics classes is not of as high a quality as that in other countries. Next, we turn to the TIMSS findings concerning the teachers themselves. Do the daily working lives of U.S. teachers provide as much support for their instructional activities as those of other countries? |