## Appendix B## A Research Base for the Instructional Criteria in Project 2061’s Mathematics Curriculum Materials Analysis ProcedureProject 2061’s instructional criteria represent a set of features that are characteristic of good instructional design. They require analysts to consider how effective a material is in: stating the purpose for learning the content; taking account of what students already know; engaging students in meaningful learning experiences that provide for the development and understanding of, and practice with mathematics ideas; providing opportunities for reflection and assessment of learning; and supplying the important aspects of a quality learning environment that provides for the success of all students. These criteria were derived from research on learning and teaching and from the craft
knowledge of experienced educators. The primary sources for the research background
include: Chapter 13, "Effective Learning and Teaching," of Overall, the research tells us that effective instruction provides students with a carefully developed set of experiences. For example, one summary of research concluded that the five components used by successful teachers to help students develop mathematical ideas are: attending to prerequisites, developing relationships, employing representations, attending to student perceptions, and emphasizing the generality of mathematics concepts (Good, Grouws, & Ebmeier, 1983). In addition, Peterson (1988) found that "students learn efficiently when their teachers first structure new information for them and help them relate it to what they already know, then monitor their performance and provide corrective feedback recitation, drill, practice, or applications activity." The following summary of the research supporting Project 2061’s analysis of mathematics textbooks is intended to provide key examples of relevant findings, rather than a comprehensive review of the literature. (It is organized according to the seven categories of the Project 2061 criteria.)
The positive effects of providing students with a clear idea of the purpose, goals, and content of the lesson ahead of time is most closely associated with the work of Ausubel (1968). In addition to stating the purpose at the beginning, effective lessons include summaries that review the mathematical concept being studied (Madsen-Nason, 1988), helping students to close the loop to formalize what they have learned. The sequence of lessons is also important in accomplishing the stated purpose, since mathematical ideas often build on each other. For example, Mack (1990) determined that prior knowledge can influence the sequence of instruction. Therefore careful design of lesson flow is important. Requiring that such actions be clear and meaningful to students is a common sense reminder that students themselves need to understand the intended purposes, and that how those purposes are communicated is important.
The importance of taking account of students’ ideas is captured in the statement
by Ausubel (1968) that "the most important single factor influencing learning is what
the learner already knows." Helping teachers to understand students’ knowledge and thinking leads to using
improved instructional strategies (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Cobb et al., 1991). Instructional materials should provide opportunities for students to make connections between and among mathematical ideas and skills. Resnick (1987) concluded that without explicit assistance in connecting ideas, people do not usually learn concepts simply by building up pieces of knowledge. Unless materials attend to students’ prior knowledge and teachers are alerted to it, the sequence of activities might be inappropriate (Mack, 1990). Moreover, further misconceptions may develop or achievement will be diminished for many students who are unable to develop more sophisticated ideas, partially due to persistent errors (Brown & VanLehn, 1982; Matz, 1980).
The simple use of hands-on materials is not sufficient to promote learning and understanding. However, an early review of research showed that when concrete materials were used appropriately (which took place in about half the studies), student achievement was better than when more abstract approaches were used (Suydam & Higgins, 1977). The key is the appropriate use of concrete materials, which includes a sufficient number of firsthand experiences that are provided in a variety of contexts. Research has shown that learning activities involving problem situations that vary along critical dimensions do promote connections and understanding (Case & Sandieson, 1988; Vergnaud, 1988). Appropriate use of firsthand experiences also requires that the situations be embedded in problem situations that have meaning for students (Brown, Collins, & Duguid, 1989). Taking this somewhat further, recent research indicates that mathematical learning is enhanced when instruction emphasizes student engagement with tasks that are both meaningful and challenging (Stein & Lane, 1996). Tasks that are challenging are important in producing the kind of higher order thinking necessary for connections and understanding to take place.
The introduction of mathematical terms, symbols, and procedures is critical to developing skill and understanding. Students at all grade levels need to work consistently at developing their understanding of the ideas captured in conventional mathematics representations and symbols (Greeno & Hall, 1997). However, Wagner and Parker (1993) found that extensive work with symbolic manipulation before developing solid understanding results in inability to progress beyond mechanical manipulations. Because representations of mathematical ideas are so important to conceptual development (Ball, 1988; Hiebert & Wearne, 1986), these representations should be carefully developed, and should connect with earlier informal and concrete experiences. According to Resnick (1982), making explicit the connections between concrete representations and their associated symbols helps students construct necessary relationships. Student understanding of concepts leads to the ability to generate new connections (Mayer, 1989) and promotes remembering ideas so concepts and procedures can be applied to solving problems and in learning more advanced concepts (Bruner, 1960). In addition, strong connections between concepts enhances transfer to other contexts (Carpenter & Moser, 1984; Kieren, 1988). For learning to become formalized and ready to use, appropriate and meaningful practice in a variety of contexts and applications is necessary (Peterson, 1988). However, Hiebert et al. (1997) found that excessive practice before attaining understanding can lead to difficulty in making sense of the procedures later.
Difficulties in mathematical problem solving are often caused by students’ ineffective use of what they already know (Schoenfeld, 1992). However, classroom discourse can exploit the use of language as a powerful tool for orienting and focusing attention and is crucial for constructing relationships (Greeno, 1988; Resnick & Omanson, 1987). Specifically, students who are expected to engage in communication about mathematics will have improved conceptions of the nature of mathematics (Lampert, 1989). Work in pairs and small groups can be an effective tool for promoting student communication. For example, both Slavin (1989) and Webb (1989) found that work in small groups can enhance achievement through student interaction if the work is focused carefully on learning mathematical ideas. In addition, guidance of student interpretation and reasoning through classroom discourse and work in small groups can help students construct and formalize their ideas so they are more accessible. Students need the opportunity for self-discovery, and activities that are unstructured enough to allow them to derive generalizations and invent their own procedures (Doyle, 1983). Questions in the lesson summary can also help students reflect on the mathematical concepts and help them establish linkages between mathematical topics (Madsen-Nason, 1988).
The alignment of mathematics learning goals with assessment items and tasks is a critical requirement that is not often accomplished well. Without this alignment, it is unlikely that assessment tasks can be used effectively to monitor learning or make instructional decisions. Research shows that elementary mathematics teachers tend to use more diverse assessment techniques while secondary teachers emphasize tests (Gullickson, 1985) and rely heavily on published paper and pencil tests (Stiggins & Bridgeford, 1986). At both levels, assessment that is integrated or embedded with instruction is important for estimating the effectiveness of lessons and making decisions about individual and group progress. In addition, multiple types of assessments that are integrated with learning mathematics can lead to teachers’ understanding of student thinking and strategies that promote higher order thinking and problem solving (Kulm, 1994). The convergence of assessment evidence from different sources is necessary in order to make instructional decisions (Stiggins, 1997), so a variety of applications and contexts as well as multiple types of assessment strategies should be used. For example, Zehavi, Bruckheimer, and Ben-Zvi (1988) found that the use of projects in and out of class for assessment enhances student mathematical achievement.
The keys to quality instruction are a well-prepared teacher, instructional materials that provide opportunities to learn important mathematics, and an environment with high expectations that encourages thinking. Research has revealed that the motivational climate in classrooms is enhanced through encouraging enthusiasm for learning, reducing anxiety, and inducing curiosity (Brophy, 1983), and that students perform better at problem solving in a supportive atmosphere (Thompson & Thompson, 1989). Knowledge of mathematics content enables the teacher to provide rich and flexible mathematics instruction, but knowledge of mathematics alone is not sufficient—the knowledge must be well-connected with how students learn specific ideas. Some teachers draw the strategy and rationale for their approaches to teaching mathematics from their knowledge of the discipline of mathematics itself (Ball, 1991). Ultimately, teachers whose knowledge of mathematics is more connected and conceptual are also more conceptual in their teaching (Leinhardt, Putnam, Stein, & Baxter, 1991; Steinberg, Haymore, & Marks, 1985). Not all students, even in the same classroom, receive the same quality of instruction. For example, the nature, frequency, and duration of mathematics teachers’ interactions with male and female students sometimes differ significantly (Leder, 1992; Meyer & Koehler, 1990). In addition, national assessments reveal a picture of racial and ethnic disparities in mathematics achievement (Dossey, Mullis, Lindquist, & Chambers, 1988) that has complex roots but can be traced to factors such as socio-economic status and language proficiency (Fernandez & Nielsen, 1986). All children, including those who have been underserved, can and do learn mathematics when they have access to quality mathematics instruction (Campbell, 1995; Silver & Stein, 1996). Project 2061’s mathematics curriculum materials analysis procedure will help educators determine which curriculum materials will provide the quality instruction that research shows students need. ## ReferencesAmerican Association for the Advancement of Science. (1989). American Association for the Advancement of Science. (1993). Ausubel, D. P. (1968). Ball, D. L. (1988). Ball, D. L. (1991). What’s all this talk about discourse? Bell, A., Greer, B., Grimison, L., & Mangan, C. (1989). Children’s performance on
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