LITERATURE REVIEW
/ Project Description / Research Objectives & Questions / Concepts / LIterature Review /
School algebra. Since the mid-19 th century algebra has been a standard high school course in the United States, providing a conceptual foundation and a language for studying abstractions and generalizations of numbers, and serving as a prerequisite to other mathematics courses and as a requirement for admission to higher education (Jones & Coxford, 1970). In recent decades many educators have argued that algebra should play a more prominent role in school mathematics, and that it should be taught to all students at an earlier age than has been typical (National Council of Teachers of Mathematics [NCTM], 1989, 2000; Usiskin, 1987). This has led to debate and discussion about what should constitute “school algebra.”
Because of its multiple roles in the school curriculum, algebra serves as a filter for educational opportunities available to students (U.S. Department of Education, 1997). A crucial concern is that students from racial and ethnic minorities have fewer opportunities to learn algebra than students from the majority. In recent years algebra has been called the new civil right (Moses, 1995; Moses & Cobb, 2001), and Algebra for All has become a goal of both professional organizations and local and state educational agencies (Achieve, Inc., 2002; NCTM, 1989, 2000). In a study of 12,500 students, Gamoran and Hannigan (2000) found that all students, including those with very low prior achievement, benefit from studying algebra, where “benefit” is defined by growth in mathematical achievement between grades 8 and 10. In recent years, efforts have been made across the country to ensure that more students have access to algebra. Nationally, the percent of students who graduate from high school having studied algebra had increased to 92% in 1996 (Dossey & Usiskin, 2000).
Despite the increases in enrollment in algebra courses, U.S. student performance on algebra items on both national and international assessments is still disappointing (see Blume & Heckman, 2000; Valverde et al., 2002). The dramatic evidence of the superior algebra performance of students in many other countries suggests that a major factor in students’ performance is the school mathematics curriculum. Recent studies of the relation between curriculum and students’ achievement indicate that high school students using standards-based curricula perform better than students using more traditional curricula on items testing conceptual understanding, use of graphs and other representations, and mathematics set in realistic contexts. (Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; Swafford, 2003; Thompson & Senk, 2001).
In general, little is known about the mathematical or pedagogical demands inherent in teaching either with standards-based instructional materials or with the more typical algebra instructional materials that have emphasized symbolic manipulation and algebraic structure. Some research (e.g., Lloyd,1999; Lloyd & Wilson, 1998; Wilson & Lloyd, 2000) provides insight into the challenges that standards-based instructional materials present to practicing teachers, and in particular, how teachers’ knowledge of algebra influence their instruction. Weiss, Banilower, McMahon, and Smith (2001) report that many secondary school mathematics teachers perceive a need for professional development. In their national sample of high school mathematics teachers, the majority of whom were not using standards-based instructional materials, about a third reported a desire to deepen their mathematics content knowledge and 40% wanted to understand student thinking in mathematics.
Teachers' knowledge. Early studies of teachers’ subject matter knowledge found little empirical evidence of connections between larger "amounts" of teacher subject matter knowledge and greater student achievement. Begle concluded that “There are no experts who can distinguish the effective from the ineffective teacher merely on the basis of easily observable teacher characteristics. Similarly, the effects of a teacher’s subject matter knowledge and attitudes on student learning seem to be far less powerful than most of us realized.” (1979, p. 54). Reviews and studies done in the past decade have begun to uncover connections between teachers' knowledge and students' achievement, as well as to highlight complexity about teacher knowledge and background issues. Monk (1994) and Rowan, Chiang, and Miller (1997) have found that secondary school mathematics teachers’ knowledge has a positive impact on student knowledge. These lines of work also have produced counterintuitive findings: Rowan, Correnti, and Miller (2002) report that “in mathematics… students who were taught by a teacher with an advanced degree in mathematics did worse than those who were taught by a teacher not having a mathematics degree.” (pp. 13-14), and Monk found that advanced mathematical coursework by teachers beyond a set of five courses added little value in terms of student gains.
Wilson, Floden, and Ferrini-Mundy (2001) point out that one serious problem in studying these relationships is the lack of sufficiently sensitive measures for examining teacher characteristics. Rowan et al. (1997), Rowan et al. (2002), and Monk (1994) all allude to the same issue. Relationships between teacher knowledge and student performance might become more visible if the indicators of teacher knowledge were sufficiently sensitive to measure the kind of knowledge that is most likely, from a theoretical standpoint, to impact student learning. This might include pedagogical content knowledge (Shulman, 1986) or mathematical knowledge for teaching (Ball & Bass, 2000a). Some small-scale studies have demonstrated these connections, but we lack appropriate theoretical frames, and instruments to measure that kind of knowledge in large-scale ways at the secondary school level.
Various scholars have focused on issues of the use of knowledge in teaching (e.g., Ball, 1991; Kennedy, 1997; Ma, 1999; Wilson, Shulman, & Richert, 1987). Ball and her colleagues (Ball, 1988, 1990a, 1990b; Ball & Bass, 2000a, 2000b; Ball & Hill, 2003) have been pioneers in conceptualizing and framing questions about mathematical knowledge for teaching at the elementary level. They contend that there is a special kind of mathematical work entailed in teaching, and that teachers are more likely to engage in this kind of mathematical work than others who work in mathematically intensive careers. A few years ago, Ball and her collaborators began to design frameworks, instruments, mathematical tasks and items to measure the knowledge of teachers of elementary and middle grades in ways closely keyed to the actual work of instruction (e.g. Hill & Ball, n.d.). Complementary work is needed at the secondary school level (see Ferrini-Mundy & Findell, 2001).
Teacher knowledge at the secondary level is now measured primarily by instruments such PRAXIS tests and locally designed state assessments, most of which have focused on pure content knowledge; few have incorporated ideas about the mathematical knowledge needed for teaching. We argue that efforts to find connections and relationships between teachers’ knowledge and student performance will be constrained until more nuanced and valid instruments exist.
The preparation and professional development of teachers of secondary school mathematics. There is currently a strong national focus on the mathematical issues in the preparation and continued professional development of teachers of mathematics, primarily at the elementary school level. However, with the No Child Left Behind legislation requiring high quality teachers who have “demonstrated competence in subject knowledge and teaching skills” (U.S. Department of Education, 2002, p. 38), new issues are arising for those who prepare teachers and provide professional development, at all levels.
The dominant approach to the mathematical preparation of secondary school mathematics teachers in the United States for nearly four decades has been to require that they complete an undergraduate major (or a near-equivalent) in mathematics (Ferrini-Mundy & Graham, in press). Prospective teachers typically take courses in calculus, linear algebra, abstract algebra and analysis – courses offered widely for mathematics majors -- under the assumption that such mathematical background is useful in teaching. Our national investment in mathematics courses at the undergraduate level is substantial (see Conference Board of the Mathematical Sciences [CBMS], 2002, for information about undergraduate course enrollments), and a large proportion of the students in these courses are preparing to be teachers of secondary school mathematics. Yet we know little about the effectiveness of such preparation for teaching.
There is considerable discussion today about what additional, or different, mathematical experiences or knowledge might be crucial for secondary school teachers (e.g., CBMS, 2001; Cuoco, 2001; Usiskin, 2000). Efforts are being made to connect the traditional mathematics of the undergraduate curriculum to the curricular emphases of the secondary school, and to prepare teachers with the special mathematical knowledge needed for teaching (e.g., Cooney, Brown, Dossey, Schrage, & Wittmann, 1996; Graham, Portnoy, & Grundmeier, 2002; Usiskin, Peressini, Marchisotto, & Stanley, 2003). Researchers have raised theoretical questions about the impossibility of separating knowledge and the contexts and activities in which it develops (Borko et al., 2000), and about the promise of “opportunities to learn content that either simulate or are situated in the contexts in which subject matter is used – core activities of teaching.” ( Ball & Bass, 2000a, p. 99). Few such examples exist at the secondary preservice mathematics level, and we lack research tools for studying the impact of these approaches.
Similar issues prevail in discussion about the appropriate content and emphasis in professional learning opportunities for experienced teachers, with increased calls for professional development experiences and teacher institutes that are strongly focused on mathematical content. Indeed, with an emphasis on earlier introduction of algebra ideas in the elementary and middle grades, and a wider range of algebra content being introduced in secondary school curricula, it seems logical that teachers will need to know more algebra than ever needed before. Yet a key question that has not been well addressed through research is: What is the algebra that teachers need to know to help them be effective in their secondary school classrooms, and how can we measure it?
The secondary school algebra curriculum is growing and shifting as national and local policy expectations change. Preparing teachers adequately for this moving curricular target means identifying the key ideas and themes that transcend various “perspectives” on school algebra, and then determining how teacher need to know mathematics in those areas so it is useful to them in their teaching. The proposed study takes up these questions.
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