Lessons Learned: What International Assessments Tell Us about Math Achievement

4: Lack of Focus in the Mathematics Curriculum: Symptom or Cause

Two reports, released in conjunction with the original TIMSS (Third International Mathematics and Science Study) in 1997, analyzed the curriculum component of the study and offered one significant explanation for the relatively poor performance of U.S. students in mathematics.1 The data indicated that the U.S. curriculum typically covered more topics at each grade level than did that of any other country participating in TIMSS— leading to the description of the U.S. math curriculum as one that was “a mile wide and [an] inch deep.” This characterization was true of state curriculum standards, the National Council of Teachers of Mathematics (NCTM) standards, district standards, textbooks, and even the curriculum as implemented by the teachers. There was little focus in the U.S. curriculum , given the large number of topics that were intended to be, and actually were, covered. In this paper, we return to the issue of curriculum focus and ask whether such a concept is useful, not just as a hypothesis about the United States but, more generally, in understanding cross-national differences in achievement. Since 1997, when these original observations were made, greater insight has been gained into the cross-national differences in curriculum and their relationship to achievement.2 One such insight has been labeled coherence. It Lack of Focus in the Mathematics Curriculum: Symptom or Cause? william h. schmidt and richard t. houang 4 65 66 william h. schmidt and richard t. houang is applicable to different aspects of the curriculum, but here we examine coherence in curriculum content standards: Content standards, taken together, are coherent if they are articulated over time as a sequence of topics and performances consistent with the logical and, if appropriate, hierarchical nature of the disciplinary content from which the subject matter derives. This is not to suggest that there is only one coherent sequence, but rather that any such coherent sequence reflect[s] the inherent structure of the discipline. This implies that for a set of content standards “to be coherent” they must evolve from particulars (e.g., simple mathematics facts and routine computational procedures associated with whole numbers and fractions) to deeper structures. It is these deeper structures by which the particulars are connected (such as an understanding of the rational number system and its properties). This evolution should occur both over time within a particular grade level and as the student progresses across grades.3 If content standards reflect the structure of a discipline, then the “depth” of those standards should increase as students move across the grades. Failure to increase in depth, sophistication, and complexity across the grades would indicate a lack of coherence. Extensive repetition of virtually the same standards across grade levels is found in the United States. This repetition runs counter to the idea of “coherent” development, is unwarranted, and contributes to a lack of focus. Such repetition can be replaced with standards that form a trajectory by linking coverage of the topics over grades and by reducing the repetition over the same grade levels. Such an approach would represent a “continuing penetration of the discipline moving to a deeper structure that makes things ‘simpler .’”4 That is, coherent development is, in the long run, fundamentally simpler than virtual repetition without development in depth and sophistication and without attaching to fundamental, unifying ideas of the discipline. Once one begins to consider the concept of coherence, one could conclude that focus is merely a symptom or indicator and that its absence suggests that coherence is lacking in the curriculum. In this interpretation, the lack of coherence is the real difficulty. The lack of focus, the large number of topics in a curriculum, is a symptom, like a high temperature, of a problem and not the problem itself. Some have argued that focus is not important at all. For example, Baker and LeTendre contend that the move toward globalization includes globalization of the curriculum. They suggest that most nations now intend the “same knowledge base in mathematics and science, significantly reducing crossnational variation in the intended curriculum.”5 Using the concepts of breadth and depth, which parallel our concept of focus, they find only small differences in both dimensions (as they define them) across the TIMSS countries and no significant relationship between breadth and depth of curricula and crossnational differences in achievement. There is a problematic clash of views about focus and its significance. Is...