Teachers Talk: Pressure Points in the K-8 Mathematics Curriculum

Abstract

Forty K-8 teachers participated in small, in-depth, facilitated discussions about “pressure points” in the curriculum. We define a pressure point as a topic, skill, or concept that is crucial to future mathematics learning but which many or most students do not master to the extent expected at a given grade level. They are issues that persist from one grade level to the next; eventually they impair the ability of students to succeed in technical disciplines. The teachers identified a number of pressure points; we focus on an understanding of place value and ”reasonableness” of answer as two examples that were identified across all grade levels. Our small-scale study represents one approach to integrating teachers into the process of identifying important and relevant research questions in mathematics learning. We argue that the pressure points identified by teachers are areas in which targeted research would have maximum impact on learning and teaching, from teacher preparation to targeted diagnostic tools to student success rates. The Third International Mathematics and Science Study (TIMSS) shows a pattern of declining performance with respect to students from other countries as U.S. students progress through elementary and secondary education (National Center for Educational Statistics 1996, 1997, 1998). Curriculum content, the order of presentation and depth of coverage of topics, and the pedagogy associated with each topic have all been cited as points of difference between the U.S. math curriculum and the curricula of other, higher-scoring countries (Cogan and Schmidt 1999; Schmidt et al. 1996, 1997, 1999). The most recent TIMSS data, from 2003, provide little evidence of improvement: the U.S. ranked 12 at grade four and declined to 15 by grade eight. Indeed, the data from TIMMS indicate no significant overall improvement in U.S. math scores since 1995 (Mullis et al. 2004). Similarly data from the Program for International Student Assessment (PISA) 2003 show the U.S. ranking 24 in both mathematical literacy and problem solving (Lemke et al. 2004). The National Assessment of Educational Progress (NAEP) provided confirmatory evidence, finding that only 36 percent of fourth graders and 30 percent of eighth graders were performing “at or above the proficient” level in 2005 (Perie 2005). According to Sanders (2004), since 1975 the U.S. has dropped from the rank of 3 to that of 17 in terms of the percentage of students pursuing science and engineering, while the number of jobs in these fields is growing at more than three times the rate of other professions. This situation is clearly connected to mathematics performance, with nearly three fourths of U.S. post-secondary mathematics enrollments identified as remedial or introductory (Madison 2003). Given these and other similar statistics and findings, it is crucial to identify what skills prove to be particularly challenging both for children to learn and for teachers to instruct. But mathematics education, even at the lower grade levels, involves a complex learning process including many different concepts, algorithms, and relationships, and it is not entirely clear what sort of mathematics education research will be of most benefit to classroom teachers. Here, we report the results of in-depth discussions with first to ninth grade teachers about what sorts of research would have the most practical value in their classrooms immediately. Because TIMSS and other studies report difficulties nationwide, the teachers were asked to think about how their students learn mathematics and what difficulties their students typically encounter in learning mathematics on a largescale, systemic level. Specifically, teachers were asked to identify “pressure points” in the K-8 curriculum. We defined a pressure point to be a topic, skill, or concept that many or most students do not master to the extent expected in a given grade, and that also is critical to future learning. That is, pressure points are bits of content in the curriculum for which a better understanding of how students learn would, in the teachers’ views, drastically improve the number of students succeeding at that grade level and also have an impact on future learning of 1 Rheinlander et al.: Teachers Talk: Pressure Points in K-8 Mathematics Produced by The Berkeley Electronic Press, 2008 mathematics. Rather than address issues of standards (What should students know and be able to do at a given grade level?) we asked the teachers to draw upon years of experience in the classroom to describe their students’ greatest difficulties. Teachers often possess understanding of student learning that is different from researchers’ understanding; this paper presents one approach to integrating teachers into the process of identifying and defining research questions in mathematics learning that are particularly important to them. In a review of research on teachers’ decision making, Shavelson and Stern (1981) recommend that research on teaching should not merely make recommendations for practice, but should link those research recommendations to actual implementation in practice. With respect to questions of content, this kind of approach to mathematics research is reflected in the current study. Here, the crucial “link” is represented by a two-way conversation between researchers and practitioners, with the eventual goal of producing research findings and establishing new teaching practices that will link the two, reciprocally and iteratively. We also interpret the term “research recommendation” to be context dependent and, in particular, content dependent. Ball (2000) discusses the futility of separating issues of pedagogy from the content that it is supposed to address, citing three problems that need to be solved for any given piece of content: what teachers need to know, how they have to know it, and how they can be helped to learn to use it. To this we would add the importance of designing research that addresses those content issues that will have the most far-reaching impact on student learning. By our definition these pressure points are issues that defy the best intentions of knowledgeable and dedicated teachers, continue from one grade level to the next, and eventually impair the ability of students to succeed in technical disciplines. Clearly, the decision about what content to spend time and energy on does not belong to the teacher alone. External pressures from administrators, parents, colleagues, standardized test scores, and textbook choices all affect how time is allocated in the classroom (Barr 1975; Floden et al. 1981, 1980; Shavelson and Stern 1981). Schwille, Porter, and Gant (1979) compiled an inventory of external pressures that might cause teachers to change the focus of the curriculum and noted that teachers seemed very willing to change the content of instruction, no matter which pressure was applied. A recent comparison of mathematics curriculum standards across states reveals, “a consistent lack of national consensus regarding common learning expectations in mathematics at particular grade levels” (Reys and Lappan 2007, 680). The multiplicity of forces at work here are consistent with Ball and Cohen’s (1996, 6) finding that textbooks alone are “uneven” agents of curricular change. Considering both the central role of teachers in the classroom and their willingness to adjust (Schwille, Porter, and Gant, 1979), it is important to note that teachers will use “reliable information” to 2 Numeracy Vol. 1, Iss. 1 [2008], Art. 4 http://services.bepress.com/numeracy/vol1/iss1/art4 make important content decisions (Shavelson, Cadwell, and Izu, (1977, 95). Research that is targeted to the questions and issues of most concern to teachers of mathematics, as defined by teachers of mathematics, can be just such a reliable source of information. Some researchers do attempt to address directly the kinds of knowledge about teaching that teachers themselves can provide (Cochran-Smith and Lytle 1990, 1993), considering the types of knowledge that teachers bring to their vocation in terms of practical knowledge of content, students, and pedagogical possibilities (Clandinin and Connelly 1987, 1991; Connelly and Clandinin 1985, 1988, 1990; Elbaz 1983, 1991). Some also consider the kind of knowledge arising from reflection on practice (Schon 1983, 1987, 1991). Although the programs of these researchers regard teachers as producing and possessing their own knowledge (Fenstermacher 1994, 13) in neither case do researchers attempt to mine the knowledge teachers might collectively hold on systemic issues. The assumption of many researchers seems to be that educational issues arise, are recognized, and are subsequently solved or not solved, strictly within the boundaries of the classroom and between the teacher and the students. In contrast, the teachers in the present study demonstrated that even with a wealth of knowledge and tools at their disposal, some learning goals—many that were tied to content spanning across grade levels—remained unmet. In this study we attempt to identify the parameters of a new content-based research targeting issues that cannot be addressed with known methods in a single classroom. The point of this paper is to use teachers’ observations to identify pressure points in the curriculum where targeted research would have the most impact.