Concrete Representation and Number Line Models: Connecting and Extending. (2000)

There is no question that appropriate manipulative use in developmental mathematics enhances students' ability to construct mathematical knowledge. The National Council of Teachers of Mathematics (NCTM) Principles and Standards 2000 echoes the original NCTM Curriculum and Evaluation Standards for School Mathematics' (1989) suggestion: From their concrete experiences students can begin to abstract the more complex meanings and ideas that provide them with access to higher mathematical thought. Although the American Mathematical Association of Two-Year Colleges (1995) standards document Crossroads in Mathematics reemphasizes that idea-pointing to the need for models, manipulatives, and visual representation of concepts in developmental programs that lead students to abstract symbolic forms-some developmental mathematics instructors may not be familiar or comfortable with their use. It is important for us, as instructors, to be aware of ways to help students make the connection to the more abstract symbolic and graphical representations. One area of research that can help inform the understanding of how students best, how they make such connections, is the literature on learning styles. Bernice McCarthy (1997) has identified four styles of learning, experiencing (Type 1), conceptualizing (Type 2), applying (Type 3), and creating (Type 4) (p. 49 ), and also has stressed the importance of structuring instruction to not just accommodate learning styles but to strengthen students' ability to learn in different ways. Excellence and higher-order thinking demand that we honor both sides of the brain, teaching interactively with hands-on, real-life messy problem solving...Unfortunately, however, teachers persist in lecturing and using logical, sequential problem solving most of the time (p. 50). Although McCarthy's work has not included college students, many postsecondary developmental students may be primarily Type 1, or experiential, and may need hands-on activities to help them conceptualize and apply. In an effort to identify the learning characteristics of the new student, for example, the Myers-Briggs Type Indicator (MBTI) was administered to 4,000 entering college students over a 15-year period from approximately 1978 to 1993 (Schroeder, 1993). The results indicate that approximately 60 percent of entering students prefer the sensing mode of perceiving compared to 40 percent who prefer the intuitive mode. The learning styles of those who prefer sensing are characterized by a preference for direct, concrete experiences. (p. 22) These data applied to a sample of all entering freshmen, not at-risk students only. Another study (Susabda, 1992)-using Kolb's Learning Style Inventory and McCarthy's Teaching Style Inventory-investigated the relationship between students whose learning style matched or mismatched their faculty's preferred teaching style and their related academic performance at a Christian high school. Findings indicated that average and below average students' learning styles tended to be more concretely dimensional. The study also found that mismatched students with average or below average academic performance were more dependent on their preferred teaching style than students demonstrating superior academic performance. Although research findings are inconsistent, and studies directly measuring developmental college students' preferred learning approaches were not found, related research leads in the direction of seeking alternatives to link the abstract and concrete for at-risk learners in college settings. Here we will investigate ways to do just that. How can we connect the work we do with integer chips to the continuous number line? How can we connect and extend that work to the development of rational number representations, operations, and percent problems? How can we lead students from concrete representations to graphical and symbolic representations? …