• The Complexity of Mathematics Teaching
• Conservatism and the Role of Reform in Mathematics Teaching
• Considerations for Future Research
• A Concluding Thought
• References

The complexity of teaching is particularly evident when one compares the teaching styles in the seven countries that participated in the 1999 video study with the same countries’ achievement levels on the TIMSS 1995 and 1999 mathematics assessments. The United States was outperformed by all six of the other countries in 1995 and by most of the other countries in 1999.² At the other end of the spectrum, neither Japan nor Hong Kong was outperformed by any of the other countries in either year.

Japan and Hong Kong shared some common elements in teaching yet also differed in fundamental ways. In both countries, 24 percent of lesson time was spent on review and 76 percent on new content (including both introducing and practicing new content). No other country’s teachers spent less time on review or more time on new content than teachers in Japan and Hong Kong. In contrast, teachers in the United States spent more time on review and less time on new content than teachers in any other country except the Czech Republic. It is striking that 60 percent of Japanese lesson time was devoted to introducing new content—a higher percentage than in any other country. The second highest percentage was 39 percent (in Hong Kong and Switzerland), compared with 23 percent in the United States. One is reminded of the research by Good, Grouws, and Ebmeier (1983) in which they advocated that teachers spend approximately 45 percent of class time developing new material. Although Hong Kong teachers were relatively high on introducing new content, they also talked more than their Japanese counterparts. Thus, one might get the impression that, although lessons in both countries concentrated on new material, Japanese lessons focused on student thinking, while Hong Kong lessons were very lecture oriented. Yet both Japan and Hong Kong were high-achieving countries.

With regard to the procedural complexity of mathematics problems, Japanese lessons had the highest percentage of high-complexity problems and the lowest percentage of low-complexity problems, although these findings may have been biased because the Japanese teachers were primarily teaching two-dimensional geometry. Even when only two-dimensional geometry problems were considered across all countries, however, no other country had a higher percentage of high-complexity problems or a lower percentage of low-complexity problems than Japan. Considering problems from all topic areas, Japan had not only the lowest percentage of low-complexity problems, but also the lowest percentage of repetition problems for students to solve. One gets the distinct impression that Japanese lessons were forward looking rather than repetitive and review oriented.

The United States had a high percentage of teachers (63 percent) who felt that their assessment of students’ interests or needs played a major role in their decisions on how to teach the content. This result is encouraging although in sharp contrast to Senk, Beckmann, and Thompson’s (1997) finding that there is little support for the notion that teachers base their instruction on students’ cognitions, a finding also consistent with that of Wilson, Cooney, and Stinson (2003). One possible interpretation is that teachers in the United States are sensitive to students’ needs as youngsters but are not particularly attuned to students’ developing mathematical thinking.

Nevertheless, teachers in the United States felt well informed about current issues regarding the teaching of mathematics. Thus, 76 percent of U.S. lessons were taught by teachers who indicated that they were familiar with current ideas in mathematics teaching and learning, compared with only 22 percent of Hong Kong lessons. Consistent with this finding was that 86 percent of U.S. teachers felt that their video lessons were in accord with current ideas about teaching and learning mathematics. (No data were reported for Japan.)

The influence of Hans Freudenthal and his followers (see, e.g., Freudenthal 1991) was evident in the Netherlands, where mathematics problems were more likely to include connections to real-life situations than in most of the other countries. For example, the percentage of Dutch problems per lesson with real-life connections was nearly double the percentage of U.S. problems. The teaching of mathematics in the Netherlands was also characterized by the fact that more lesson time was devoted to students’ private work on problems than to whole-class interaction, the only country for which this was true. This finding represented one of the few real differences in teaching styles across countries.

A certain conservatism pervaded most of the lessons across all countries, with the possible exception of Japan and, to some extent, the Netherlands. For example, the ratio of teacher talk to student talk was high in every country. Some variation was evident across countries, however. For example, the ratio was higher in Hong Kong than in the United States, largely because U.S. students contributed more to classroom discussions. A relevant question is whether the ratio of teacher talk to student talk was related to class size (in larger classes, one would expect teacher talk to dominate).

It was disappointing that technology (computers and calculators) did not play a more prominent role in the teaching of mathematics. Dutch teachers made extensive use of computational calculators, whereas in other countries they were less frequently used. Across countries, graphing calculators were rarely used, and computer use was also sparse or virtually nonexistent.

Another conservative characteristic was that alternative solutions to problems were seldom presented publicly. Alternative solutions to problems were presented for a maximum of 5 percent of problems per lesson in all countries except Japan (17 percent). It seems clear that multiple solution methods per se were not viewed as an instructional goal.

This apparent conservatism raises serious questions about the role of reform in the teaching of mathematics. The experience level of teachers in the TIMSS 1999 Video Study was considerable; they had a median of 7 to 21 years’ experience teaching mathematics. Also, the teachers were reportedly well trained. It seems reasonable to assume that within each country these teachers were a cut above their counterparts. Yet, in their own way, the teachers were remarkably consistent in conducting what appeared to be primarily teacher-centered lessons. The authors noted this commonality on several occasions. So why is it that the teaching looked so similar, allowing for a few exceptions? If these intelligent teachers were defining the teaching of mathematics in mostly similar ways, what factors led to this circumstance? If society is mandating that mathematics be taught in this rather conservative style, why is this the case? I am not sure what the answers to these questions are, but I am fairly confident that the answers are not implied by intrinsic features of mathematics education per se. This gives me great pause when thinking about efforts to reform the teaching of mathematics.

The National Council of Teachers of Mathematics (NCTM) Standards promote a vision about the teaching of mathematics that entails teachers using technology, emphasizing the processes of doing mathematics, and basing their instructional decisions on students’ thinking. Generally, reform in the United States is defined or at least guided by these standards. Not surprisingly, then, most studies of attempted reform have focused on issues within the academic component of mathematics education, such as placing more emphasis on problem solving and students’ conceptual understanding, making greater use of technology, and adopting a broader, more open system of assessment. These studies have generally reported mixed results; some teachers embraced certain elements of reform, while others did not. Usually mathematics educators focus on the more academic aspects of reform at the risk of neglecting the very circumstances that determine the way in which mathematics gets defined and taught. Teachers’ shared culture of mind is shaped by these circumstances, yet this culture is rarely addressed explicitly in most teacher education programs. The literature abounds with findings about teachers who subscribe to reform but in the crucible of the classroom often embrace more conservative teaching styles (see, e.g., Wilson and Goldenberg [1998], Lloyd [1999], and Skott [2001]). This raises the question of what should constitute reasonable expectations for classroom reform, especially over an extended period of time; most reported studies take place within a calendar year.

The TIMSS 1999 Video Study provides a major opportunity for researchers to consider country differences and similarities and to juxtapose their own research with this study’s findings. It would be interesting to know, for example, the extent to which beginning teachers (those with 1 or 2 years of experience) differ from their more experienced counterparts. Although the median level of teacher experience was quite high across countries, the within-country variance was considerable. For example, the experience of U.S. teachers ranged from 1 to 40 years. The smallest range (from 1 to 33 years) occurred in the Netherlands. What would the analysis have looked like had only beginning teachers been considered?

At various times when I read the report, I wanted to know what mathematics the authors were really talking about. When low-complexity problems were pitted against high-complexity problems, what were examples of each type of problem? Further, I had trouble with the notion of what were called application problems. These problems were apparently defined in terms of the number of steps needed to solve the problems, regardless of whether the problems’ contexts involved real-life situations (the usual definition of application). In countries other than the Netherlands, only 9 to 27 percent of problems per lesson involved real-life connections.

Much research in the United States, particularly that conducted prior to the 1990s, has focused on efficient ways of teaching mathematics. For example, McKinney (1986) and Leinhardt (1988) revealed that expert teachers could cover more problems in a given class period than could novice teachers, and they could do so with greater flexibility. However, the lesson pattern, or “signature,” for Japanese teachers suggests that the number of problems covered in a given lesson is not as important as the depth with which the content is addressed. Whereas U.S. teachers spent an average of 5 minutes per independent problem per lesson, Japanese teachers spent an average of 15 minutes. Although research in the United States over the past decade has become decidedly more interpretive, researchers would be wise to focus more explicitly on the interplay between what gets taught, how it gets taught, and the depth with which mathematics is taught.

In the United States, the length of mathematics lessons varied a great deal, showing a considerably greater range than in the Czech Republic, Japan, and Switzerland. This finding, juxtaposed with the fact that only 22 percent of problems per U.S. lesson focused on geometry, suggests that some U.S. students may not be getting much geometry, including both two- and three-dimensional geometry. The role of school geometry in the United States, particularly at the middle school level, deserves careful consideration in developing teacher education programs for both preservice and inservice teachers.

It was both exhilarating and exasperating to read Teaching Mathematics in Seven Countries: Results From the TIMSS 1999 Video Study. Given the multitude of classrooms observed, there seemed to be a boundless number of strategic research sites. Overall, teachers in the United States appeared more conservative in their teaching styles than did their counterparts in other countries, although not markedly so. One wonders why. We can only conclude that there is much work to be done if the teaching of mathematics in the United States is to bear some family resemblance to the NCTM Standards. The question is, “Where in the world do we begin?”

¹ Hong Kong is a Special Administrative Region (SAR) of the People’s Republic of China. For convenience, this commentary refers to Hong Kong as a country.

² U.S. and Czech scores on the 1999 mathematics assessment did not differ significantly, and Switzerland did not participate in the 1999 mathematics assessment.