Some Characteristics of Eighth Grade Mathematics Classes in the TIMSS Videotape Study

situations involve only abstract concepts such as numbers and geometric shapes-even specific examples such as a rectangle three inches wide and five inches long-in contrast to concrete situations that involve physical or social representations of abstract concepts-such as a 3-by-5 inch index card. One reason reviewers were interested in this dichotomy was the call from some advocates of mathematics education reform in the United States for far more mathematics instruction to be done in "real world" contexts. Reviewers wanted to determine whether the proportion of concrete settings was significantly lower in the United States than in Japan or Germany. The locus of control of the solution of a problem-whether the task in the context of the class essentially included its own method of solution-is another aspect of the extent to which instruction was consistent with some current calls for reform of mathematics education in the United States. An important component of problem solving is the solver's control of the process of solving a problem [3]. A comparison of the extent to which exercises appear to require solvers to find solution methods should provide one indication of the extent to which mathematics instruction is oriented toward "problem solving" and also, perhaps, of the extent to which routine skills are emphasized. Similarly, the complexity of exercises is another indicator of their sophistication and mathematical richness, especially from a problem solving perspective. Determining whether a problem requires a "single-step" or "multi-step" solution depends upon not only the statement of the problem but also the presumed sophistication of the solver. For example, solving a simple linear equation such as 5x + 4 = 19 would be considered a two-step process for students who are first learning how to solve such equations but would constitute one very small step for students who had developed proficiency at solving systems of linear equations. As a result, reviewers had to consider the problem in the context of the class when judging its complexity. The reviewers classified the subject and described the subject matter of each class. Each of the 90 classes was classified according to its likely position in a traditional (1980s) United States college preparatory mathematics curriculum. The three classifications used were "Before Algebra," "Algebra," and "Geometry." Approximately half the classes were classified as Geometry, 30% as Algebra, and the remainder as Before Algebra. The reviewers also rated each class on the basis of their judgment of its potential for helping students understand mathematics. The five categories used 1998] EIGHTH GRADE MATHEMATICS CLASSES 795 This content downloaded from 157.55.39.178 on Sun, 24 Jul 2016 06:17:36 UTC All use subject to http://about.jstor.org/terms were "Weak," "Almost Good," "Good," "Better," and "Strong." While this is quite subjective, the four reviewers were able to reach complete agreement in all 90 cases. Readers of this paper who would like to make their own judgments may send email to timss@ed.gov to inquire about obtaining copies of the tables and other data from the TIMSS video study. The table for each class consisted of a few pages, typically about three, that included a summary description of each segment of the class. Segments were defined and identified by the members of the Stigler laboratory. A new segment started when there was a significant change in activity or in the content being presented. Reviewers categorized each segment by the nature of its mathematical activity and by its role in the structure of the class. The segments became the nodes of the directed graphs used to represent graphically the structure of the