Word problems provide a meaningful context for developing number concepts and the meanings of operations (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). This idea is supported by a theory that describes the situated nature of learning (Brown, Collins, & Duguid, 1989; Lave & Wenger, 1991). Ultimately, students must also learn to work meaningfully and skillfully with mathematics in its abstract, or symbolic, form, especially if they are to be successful in higher mathematics (Sfard, 2000).
Math Pathways & Pitfalls® helps students make sense of—and solve—both word problems and problems in their purely symbolic forms. Math Pathways & Pitfalls encourages mental, visual, and paper-and-pencil solution methods. Regardless of whether or not the problem is in context, students are expected to explain why their solution process makes sense.
Findings From Studies
Math Pathways & Pitfalls Classroom Observation Transfer Study, 2006–2007
This qualitative study was conducted by Heller Research Associates and funded by the U.S. Department of Education’s Institute of Education Sciences (R305K050050). It was designed to describe ways in which teachers may transfer practices from the Math Pathways & Pitfalls curricula to their regular, non–Math Pathways & Pitfalls lessons in math or other subjects. The pedagogical practices of interest in this study focused on three dimensions: mathematics; language and discourse; and fairness among learners. The study documents examples of ways in which teachers borrow practices from the Math Pathways & Pitfalls lessons and the reasons for those choices. Descriptive data on transfer can be drawn on in future Math Pathways & Pitfalls professional development as a source of ideas that teachers might find useful.
Key Findings:
Observations: Over the 6-month period between observations, rubric scores increased, indicating that teachers’ instruction generally developed in all three areas (mathematics; language and discourse; and fairness) after implementation. Average scores show classrooms in the beginning-developing range during the fall observation and in the developing-advanced range during the spring observation. All teachers improved in at least one area, and no teacher scored lower during the second observation than they did on the first. These results indicate that teaching practices during regular, non–Math Pathways & Pitfalls math lessons developed in the direction of general and specific aspects of the Math Pathways & Pitfalls approach.
Mean total scores on the 9-point rubric for the post–Math Pathways & Pitfalls implementation data were significantly greater than mean total scores before Math Pathways & Pitfalls was implemented (Wilcoxon T+ = 36, p < .005, n = 8). Math Pathways & Pitfalls practices appear to transfer from Math Pathways & Pitfalls lessons to non–Math Pathways & Pitfalls lessons, but without a comparison group we cannot tell for sure.
Interviews: Teachers were better able to articulate the causes and effects of increased participation during the second interview and were generally more specific about effective strategies and the strengths and weaknesses of their students. During the second interview, the interviewer also asked specifically how the Math Pathways & Pitfalls course was affecting their non–Math Pathways & Pitfalls lessons, and teachers cited a wide range of Math Pathways & Pitfalls teaching strategies and practices. In addition, teachers were generally very positive about their Math Pathways & Pitfalls experience.
Conclusions: In short, these results indicate that teaching practices during regular, non–Math Pathways & Pitfalls math lessons developed in the direction of general and specific aspects of the Math Pathways & Pitfalls approach that was implemented over the intervening 6-month period. This is an important finding because it raises the possibility that a relatively small “dose” of Math Pathways & Pitfalls lessons (about 15 hours during the school year) may have a powerful effect on mathematical learning. If so, studies of this phenomenon may lead to understandings about which practices are more likely to transfer and may provide clues about how Math Pathways & Pitfalls supports the adoption of new practices. Ultimately, this information will be valuable in the design of new instructional materials and professional development for teachers.