The Ambiguity of Teaching to the Test Standards

2. TEACHING MATHEMATICS AND SCIENCE
solving “open-ended” problems) rather than with fundamental changes in
practice. Simon and Tzur (1999) and Simon, Tzur, Heinz, Kinzel, and
Schwan-Smith (2000) suggested that teachers often have a limited understanding
of what the changes advocated by mathematics education researchers
and the standards really mean. They say that many teachers
interpret the changes that are needed as discouraging telling and showing,
using manipulatives, or having students work in small groups, for example.
Though all of these strategies may well contribute to a better teaching environment,
taken alone, or even in combination, such strategies may not necessarily
lead to greater student learning (Boaler, 2002; Schorr, 2002; Schorr
& Firestone, 2001; Schorr & Koellner-Clark, 2003; Simon & Tzur, 1999;
Spillane & Zeulli, 1999). These strategies may represent a step in the right
direction, but the authors of the Principles and Standards for School Mathematics
caution that many of the pedagogical ideas from the NCTM Standards—such
as the emphasis on discourse, worthwhile mathematical tasks,
or learning through problem solving—have been enacted without deeper
changes in actual classroom practices (NCTM, 2000).
Goldsmith and Shifter (1993) pointed out that developing a better form
of teaching requires more than the acquisition of some new instructional
techniques or strategies. It demands a reconceptualization of the entire process
of teaching and learning, and this entails substantive change in teachers’
knowledge and beliefs about instruction and content (Schorr, 2002;
Schorr & Lesh, 2003; Simon et al., 2000; Simon & Tzur, 1999). But, teaching
practices are not easily modified or revised. Stigler and Hiebert (1999)
argued that teaching is a cultural activity, and cultural activities “evolve
over long periods of time in ways that are consistent with the stable web of
beliefs and assumptions that are part of the culture … and rest on a relatively
small and tacit set of core beliefs about the nature of the subject, about
how students learn, and about the role that a teacher should play in the
classroom” (p. 87). Given the long history of relatively stable instructional
practices in this country (cf. Cuban, 1993), it is not surprising that the
changes that have been reported are more akin to the adoption of new strategies
or techniques rather than fundamental shifts in practices.
In addition to setting forth a vision emphasizing understanding and inquiry-based
learning, state and national standards (AAAS Project 2061,
1993; NCTM, 1989, 2000; NRC, 1996; Rosenstein et al., 1997) also emphasize
new content (e.g., discrete mathematics). Some proponents of testing
consider tests to be the best hope for using the authority of the state to improve
teaching and learning. They believe that standards, and associated
standards-based tests, used effectively, can steer teachers and students to
content areas that are important for modern society but have received inadequate
attention in the past (NCTM, 2000). Others at least hope that tests
may prompt teachers to consider more challenging content and activities
that provide an opportunity for students to develop an understanding of
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