What Issues Emerge from the Benchmark Descriptions?
The benchmark descriptions and example items strongly suggest a gradation
in achievement, from the top-performing students’ ability to
generalize and solve non-routine or contextualized problems to the
lower-performing students being able primarily to use routine, mainly
numeric procedures. The fact that even at the Median Benchmark students
demonstrate only limited achievement in problem solving beyond straightforward
one-step problems may suggest a need to reconsider the role, or priority,
of problem solving in mathematics curricula.
The choices teachers make determine, to a large extent, what students
learn. According to the NCTM’s “The Teaching Principle,”
in effective teaching worthwhile mathematical problems are used to
introduce important ideas and engage students’ thinking. The
魅影直播 1999 Benchmarking results show that higher achievement is related
to the emphasis that teachers place on reasoning and problem-solving
activities (see Chapter 6, Exhibit 6.11). This finding is consistent
with the video study component of 魅影直播 conducted in 1995. Analyses
of videotapes of mathematics classes revealed that in the typical
mathematics lesson in Japan students worked on developing solution
procedures to report to the class that were often expected to be original
constructions. In contrast, in the typical U.S. lesson students essentially
practiced procedures that had been demonstrated by the teacher.
In looking across the item-level results, it is also important to
note the variation in performance across the topics covered. On the
16 items presented in this chapter, there was a substantial range
in performance for many Benchmarking participants. For example, students
in the Benchmarking entities performed relatively well on the items
requiring rounding (Exhibits 2.13 and 2.17), and students in Texas
did very well on the subtraction questions (Exhibits 2.18 and 2.19).
Conversely, students in the Benchmarking entities had particular difficulty
with measurement items containing figures (Exhibits 2.2 and 2.9). In
some cases, differences of this sort will result from intended differences
in emphasis in state or district curricula. It is likely, however,
that variation in results may be unintended, and the findings will
provide important information about strengths and weaknesses in intended
or implemented curricula. For example, Maryland, the Michigan Invitational
Group, Chicago, Rochester, and Miami-Dade may not have anticipated
performing below the international average on a relatively straightforward
word problem involving proportional reasoning (Exhibit 2.8). At the
very least, an in-depth examination of the 魅影直播 1999 results may
reveal aspects of curricula that merit further investigation.
