High Expectations: A "How^" of Achieving Equitable Mathematics Classrooms

Dramatic changes are occurring in the teaching and learning of mathematics in the nation's classrooms. It is no longer sufficient to be proficient in basic skills; students are now expected to develop deep conceptual understanding of a broad range of mathematical content areas. While there has been an increased focus on the need for equal opportunities for all students, African American students continue to score below their white counterparts, particularly in these critical areas. This paper suggests that low expectations play a significant role in the level of achievement of African American students in mathematics. It then examines the teaching of an effective mathematics teacher in an urban middle school and finds that high expectations are at the core of his teaching. Examples are given to illustrate three facets of his high expectations. As the vision of what it means to be mathematically literate in our society has shifted from a basic skills curriculum/or some to a more demanding standard/or all, the limits of past pedagogical practice have become increasingly apparent. In response to this, the National Council of Teachers of Mathematics (NCTM, 1989) published its Standards for mathematics education. These Standards reoriented conventional sequencing of mathematics concepts, encouraging the introduction of geometric and algebraic concepts as early as the first grade. They also emphasized the development of conceptual understanding through a range of exploratory strategies including manipulatives, word problems, and open-ended questions. Basic skills continue to have an important role, however they are greatly de-emphasized relative to past practice, and they are envisioned as emerging through the application of conceptual understanding, rather than through direct instruction. Over the past two decades, the underachievement of minority students in mathematics has been well-documented (Secada, 1992; Tate, 1997). Tate (1997) examined the mathematics achievement of diverse groups from national trend studies and found that African American and Hispanic students continue to score at significantly lower levels than white and Asian American students. He further reported that while minority students have made achievement gains in recent years, these gains were only on low-level and basic mathematics skills. This is problematic since basic skill proficiency is not enough for "true knowledge and mastery of mathematics" (Secada, 1992, p. 630). Recognizing the disparity between minority students and their white counterparts, NCTM (2000) published a revised version of the standards of mathematics education 1. This work was completed when the first author was a Post-Doctoral Fellow at the University of Pittsburgh and the second author was a Post Doctoral Fellow ofthe DiME Project, University of California Berkeley. Vol. 56, Nos. 2 & 3, July, 2005 127 The Continuing Spirit of Brown entitled Principles and Standards for School Mathematics which attempts to include one of the important omissions of the previous standards document (i.e., NCTM, 1989). This omission includes the importance of high expectations for all students in order to implement the "new" visions of mathematics curriculum and teaching as suggested by the previous document. NCTM (2000) provides the mathematics education and mathematics education research communities with an Equity Principle which suggests that "equity [in mathematics education] requires high expectations and worthwhile opportunities for all" (p. 12). The issues here are: what constitutes worthwhile opportunities at the secondary level and what can be done to promote high expectations of minority students among the mathematics education community? Worthwhile Opportunities for All NCTM (2000) asserts that schools have an obligation to ensure that all students participate in a strong instructional program that supports their mathematics learning. Such programs are built around worthwhile opportunities that elicit mathematics learning among all students. These opportunities include, but are not limited to, problem solving, reasoning, and communication. Moreover, such opportunities should embody aspects of abstraction, invention, proof and application (Romberg, 1987)). In accordance with this view, Schoenfeld (1994) posits that "when one focuses on mathematical thinking the ability to do or use mathematics then one needs to pay attention to: content, problem-solving strategies or heuristics, control (how well one uses resources), beliefs, and how well one functions as a member of a mathematical community" (p. 58). This shift from computational "basics" to a "body of knowledge" to be incorporated within mathematical tasks reflects a shift of seeing mathematics as ideas and not just procedures. The important issue here is how does this transition in what constitutes mathematics learning alter the current makeup ofthe mathematics curriculum of minority students? In particular, what kinds of curriculum and instruction uphold the idea of worthwhile opportunities? Worthwhile Teaching The National Council of Teachers of Mathematics (NCTM, 1991) encourages mathematics teachers to focus on new and innovative techniques and strategies to improve the quality of mathematics instruction. By developing means other than the "empty vessel" pedagogy, diverse populations can be supported and accommodated in their mathematics learning. The standards (i.e., NCTM 1989 and 1991) call for a curriculum that prepares students to be mathematically literate (i.e., to use mathematics to communicate, problem solve, and reason) and instructional practices that are student-teacher interactive, tapping higher-order cognitive thinking, problem solving, and discovery learning. Mathematics teaching with a new vision requires teachers to view both the mathematics classroom and their roles in teaching differently. In order to allow students to construct knowledge, mathematics classes should be viewed as places where students rediscover and re-invent concepts and methods of solution. Viewing mathematics classrooms in this manner requires a different perspective ofthe roles of mathematics teachers in such settings. What are some ofthe "new" roles of mathematics teachers in such environments? Lappan (1998) notes the following four new roles: (1) engaging students in the task, (2) pushing student thinking while the exploration is proceeding, (3) helping students to make the mathematics more explicit during whole-class and group interaction and synthesis, and 128 The Negro Educational Review