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InterMath: Technology and the Teaching and Learning of Middle Grades Mathematics

The pedagogical shifts embodied in a series of documents published by the National Council of Teachers of Mathematics (NCTM) emphasize vastly different mathematics teaching-learning approaches than are typical in today's classrooms (NCTM, 1989, 1991, 1995). Rather than static knowledge and skills detached from both other domains and everyday events, mathematics is viewed as problem solving, reasoning, and communicating that empowers students to confidently "explore, conjecture, and reason logically [about the world around them]" (NCTM, 1989, p.5).

"How can teachers teach a mathematics that they never have learned, in ways that they never experienced?"

- From Policy and Practice: An Overview, by David Cohen and Deborah Ball

Teachers are not merely keepers and transmitters of mathematical knowledge; they facilitate student engagement by posing relevant problems that encourage mathematical thinking and communication. Contemporary perspectives transcend rote memorization of procedures, computational algorithms, paper-and-pencil drills, and manipulations of symbols. Mathematics teachers and students are now encouraged to engage in deep mathematical thinking involving analysis, problem finding and problem solving, and rich conceptual understanding.

According to the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), conceptual understanding "enables children to acquire clear and stable concepts by constructing meanings in the context of physical situations and allows mathematical abstractions to emerge from empirical experience" (p. 17). Conceptual understanding is specifically reinforced in the content and emphasis changes in K-8 mathematics proposed in the Standards. The Standards suggest decreased attention in areas such as complex paper-and-pencil computations; practicing routine, one-step problems; learning isolated topics; developing skills out of context; memorizing procedures, such as cross-multiplication, without understanding; and relying on outside authority (teacher or answer key) (pp. 21, 71). Conversely, they stress increased attention in several key areas such as pursuing open-ended problems and extended problem-solving projects; creating algorithms and procedures; representing situations verbally, numerically, graphically, geometrically, or symbolically; exploring relationships among different representations; estimation and the reasonableness of answers; connecting topics within and outside of mathematics; questioning; and justification of thinking (pp. 20, 70). Collectively, NCTM standards advocate methods that emphasize developing mathematical power: conceptual understanding, problem solving, reasoning, connection building, communicating, and self-confidence.

Concurrent with NCTM Standards refinements, numerous efforts were advanced by publishers and researchers to improve mathematics curriculum. NSF, for example, funded five comprehensive middle-school mathematics curricula that are now widely available (Connected Mathematics Project, Seeing and Thinking Mathematically, Mathematics in Context, Six Through Eight Mathematics, Middle-school Mathematics Through Applications Project). These materials (and efforts such as Vanderbilt's Jasper Woodbury) promote reform in applications, problem solving, conceptual understanding, building connections across mathematics domains and uses, alternative assessment strategies, and mathematics communications skills.

Reform, however, does not occur simply because new standards or approaches emerge. Several barriers have hampered reform efforts. One barrier appears to be linked to resilient and pervasive beliefs among preservice and inservice teachers as to what constitutes mathematics (Anderson & Piazza, 1996; Ball, 1988; Dossey, 1992; Thompson, 1984; 1992). Even and Lappan (1994) identified several widely held teacher beliefs: (1) computational proficiency is the major mathematics curriculum goal; (2) mathematical knowledge is ruled bound and unconnected; (3) teaching is telling and learning is memorizing (p. 129). Howson, Keitel, and Kilpatrick (1981) noted that many curriculum projects fail because teachers tend to proceduralize methods in ways that are often inconsistent with the curriculum's underlying epistemological and pedagogical assumptions. This has been particularly evident in the use of widely available drill-and-practice programs that could be used to support emerging pedagogies, but rarely are. David Cohen (1990), for example, documents the activities of a well-intentioned teacher who, based on lectures about reform mathematics, believed her methods were consistent with the current reform movement. However, she never actually experienced "doing mathematics" or learning mathematics in these new ways herself. While her intent and motives were admirable, the lack of experience in participating as a learner inherently limited her understanding and insight in implementing the approaches. In order to promote conceptual change teachers must themselves experience mathematics as we want our students to: as conjecturing, reasoning, communicating, and problem solving. Such experiences should prompt teachers to examine their fundamental beliefs about such questions as, "What is mathematics?" "What does it mean to know mathematics?" "How do students learn mathematics?" "What is the role of the teacher in the mathematics classroom?"

Research in mathematics education suggests that a teacher's conception of mathematics has a strong impact on how mathematics is approached in the classroom (Cooney, 1985; Thompson, 1984; Thompson, 1992). The nature of the classroom environment in which mathematics is done strongly affects how students view the subject and how it should be taught and learned. A common theme found throughout the reform documents is "What students learn is fundamentally tied to how they learn it" (NCTM, 1989, p. 5; NCTM, 1991, p. 21). Thus, if we want our students to view mathematics not as a static body of rules and procedures, but as a meaningful and dynamic, yet connected body of knowledge, we must make an impact on their teachers' view of mathematics.

In short, if we want our teachers to meaningfully teach mathematics, they must experience meaningful mathematics. In the words of Cohen and Ball (1990), "How can teachers teach a mathematics that they never have learned, in ways that they never experienced?" We cannot expect teachers to teach in a manner consistent with reform advocates simply because they have been told what to do or how to do it. To help our teachers meaningfully teach and model mathematical thinking, they must experience relevant mathematics as students, benefiting from both the discovery processes as well as guidance and modeling of capable peers. Better teacher models are needed at all levels, K-12 through university, if we hope to break the cycle.

InterMath proposes to provide middle school teachers with this type of experience in a 45-hour workshop environment. The content of the workshops will be centered around the middle-school mathematics curriculum as identified by Georgia's Quality Core Curriculum (QCC) and the Curriculum and Evaluations Standards for School Mathematics (National Council of Teachers of Mathematics, 1989).

Furthermore, InterMath develops a mechanism that provides and sustains assistance to middle-grades mathematics teachers in their use of technology to implement mathematics education reforms. Technology enables mathematics education reform, but it is not reform per se (cf. Kaput, 1992). Teachers are provided extended opportunities to experience and do mathematics in an environment supported by diverse technologies (Dreyfus & Eisenberg, 1996). The approach focuses on developing mathematical power--understanding, using, and appreciating mathematics. Our interest is in empowering teachers through the use of technology in mathematics exploration, open-ended problem solving, interpreting mathematics, developing understanding, and communicating about mathematics (see Bransford, et al, 1996; Schoenfeld, 1982, 1989, 1992; Silver, 1987).

Rationale of Workshop Activities

In designing the workshops, we have kept in mind the work of Malone and Lepper (1987) concerning the design of instructional environments that are intrinsically motivating. They have identified four sources of intrinsic motivation in learning activities: (1) an appropriate level of challenge, (2) appealing to the sense of curiosity, (3) provides the learner with a sense of control, and (4) encourages the learner to be involved in a world of fantasy in which students can experience vicariously rewards and satisfactions that might not be available to them otherwise. While a teacher may not be able to incorporate all of these sources of intrinsic motivation into every learning activity, incorporating at least one appears to increase the likelihood that the activity will be intrinsically motivating.

Pertaining to the first source of intrinsic motivation, we have included a variety of problems that strike a balance between being a challenge and being do-able. By posing challenging problems within a familiar context, teachers will develop confidence (self-efficacy) in problem solving, so they will be more likely to engage the activities. The context of the problems enables teachers to safely sample and reflect on their own approaches to problem solving. The second source of intrinsic motivation is appealing to the sense of curiosity. Activities can stimulate curiosity by introducing ideas that are surprising or discrepant from the learner's existing beliefs and ideas. While the mathematical problems posed in the laboratories will center on middle-school curriculum, they are more open-ended and generative than is typically seen in the middle-school curriculum. Problems can be used as a springboard for ideas and investigations that participants find personally intriguing. Furthermore, teachers will be able to choose among several activities in which to actually engage. They can choose activities that are most applicable to their classroom needs and relevant to their mathematical understanding. Since participants can choose activities based on their preferences, the second and third sources of intrinsic motivation (appealing to the sense of curiosity and providing the learner with a sense of control) will be reflected throughout the laboratory. The fourth source of intrinsic motivation is encouraging engagement through fantasy. As an example of a task using fantasy, consider the following problem requiring the use of the Pythagorean theorem:

The student needs to calculate the distance from point a to point b in order to inform Captain James T. Kirk about how to set the transformer beam on the Federation Starship Enterprise so they can pick up the necessary dilithium crystals directly below on the planet's surface. Kirk only knows the distances of the ship and the crystals from a third point where his scouting party has stopped (Lepper & Hodell, 1980).

Fantasies are more intrinsically motivating when they employ characters and situations with which the student can identify. Faced with either this fantasy-like problem or a series of abstract problems in which students are asked to find the length of one side of a triangle, one can imagine which type of problem students would prefer.

The philosophy permeating InterMath is that teachers first must relearn mathematics in ways most educators have not, that is in a more open-ended, generative manner so they may come to understand what reform documents mean by "meaningful learning." Furthermore, by encouraging teachers to create/modify their own curriculum units, InterMath attempts to avoid what Howson, et al. (1981) warn may be a cause for failed reform -- teachers failing to assume ownership of reforms.