Scientific Basis

What Is the Scientific Foundation for the Design of Math Pathways & Pitfalls?

Math Pathways & Pitfalls draws upon a wide range of theory and research on children’s mathematical thinking, academic language development, and effective and equitable instructional practices. The following describes the key research-based features of the program, followed by a selected list of references.

Math Pathways & Pitfalls Turns Pitfalls Into Pathways for Learning

As one of our students has pointed out, “My father says you can learn from your mistakes.” The pitfalls highlighted in Math Pathways & Pitfalls lessons address some of the most common misconceptions cited in the research literature on student thinking and used as distractors on state and national assessments. During each lesson, students contrast correct and incorrect ways to solve a problem. They talk explicitly about why a particular pitfall occurs, how to avoid the pitfall, and how to think correctly about the mathematics in the problem.

Experimental studies by cognitive scientists Durkin and Rittle-Johnson (2009) suggest that comparing examples of common mathematical errors to examples that are correct may prevent such errors from being made in the future. They also report that the contrast between correct and incorrect examples prompts students to recognize correct concepts. Other research supports the idea that pitfalls serve as strong motivators for inquiry and sense making (Festinger, 1957; Borasi, 1994).

Math Pathways & Pitfalls Addresses Key Mathematics Standards in Number and Algebraic Reasoning

The mathematical content of Math Pathways & Pitfalls focuses on key concepts and skills that are found in the National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points and in state and local standards. The lessons for grades K–3 focus on building concepts related to number sense, operations, and equality. The lessons for grades 4–8 target concepts related to fractions, decimals, ratios, proportions, percents, algebraic expressions, and equations. For help selecting lessons that support standards in which students need stronger development or extra reinforcement, see our NCTM Curriculum Focal Points / Math Pathways & Pitfalls Correlation Table.

Math Pathways & Pitfalls Provides Tools That Develop a Community of Learners

Math Pathways & Pitfalls draws from research about how to build communities in which students not only learn mathematics but also learn how to participate in the discourse practices of mathematics (Lampert, 1990; Ball, 1997; Cobb, Wood, & Yackel, 1993). Math Pathways & Pitfalls provides opportunities for students to participate safely while also inviting intellectual risk-taking (Beghetto, 2004). Each Math Pathways & Pitfalls lesson includes opportunities for individual, partner, and whole class participation, allowing students to take on riskier roles as they feel ready.

The accompanying Discussion Builders poster contains sentence stems that embed the academic language used in mathematical discourse and supports risk-taking and respectful discussion. A Video for Students also models students taking risks by presenting their ideas publicly and agreeing and disagreeing respectfully.

Math Pathways & Pitfalls Addresses Multiple Modalities

Math Pathways & Pitfalls tasks mediate learning in multiple ways, using oral, print, and other visual modes of communication. For example, mathematical concepts in Math Pathways & Pitfalls lessons are initially developed by discussing the thinking of two fictional students. Their thinking, presented in print and supported by labeled drawings, purposefully embeds important conceptual ideas to prime the discussion.

Having both text and drawings to refer to allows students to examine — and reexamine — the mathematical ideas. After the start of the Math Pathways & Pitfalls lesson lays a conceptual foundation, students’ own solutions serve as the basis for discussion and learning. The lesson encourages students to use multiple modalities to explain their solutions, such as explaining their thinking orally and using drawings, symbols, and text labels to make their explanations clear. Learning mediated through multiple modalities is particularly important in providing access for visual learners and English learners (Echevarria, Vogt, & Short, 2004).

Math Pathways & Pitfalls Provides Word Problems and Symbolic Problems

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 purely symbolic form. Also, 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.

Math Pathways & Pitfalls Develops Mathematical Language

The meaning that students develop for a mathematical idea is closely entwined with the way they use language to reason about the idea and learn to translate among words, symbols, and meanings of that idea (Vygotsky, 1962; Schleppegrell, 2004; Cummins, 1980; Pimm, 1995; Solomon & Rhodes, 1995). Discussion-based lessons, such as Math Pathways & Pitfalls, support language development. As Khisty (1992, 1995) points out, students develop language by talking.

Math Pathways & Pitfalls teaching guides help teachers anticipate language confusion and support communication to ensure that English learners have access to the discussion. In addition to fostering oral communication, Math Pathways & Pitfalls lessons help students learn to comprehend complex mathematical text and write mathematical explanations. To support vocabulary development, lessons open with a quick review of important mathematical words, including examples of how they are used. Finally, the Discussion Builders elicit increasingly more sophisticated use of academic language and reasoning as students progress through the grades.

Math Pathways & Pitfalls Prompts a Metacognitive and Proactive Stance Toward Learning

Instructional approaches that encourage self-monitoring — or metacognition — have been shown to support learning with understanding (Donovan & Bransford, 2005). Math Pathways & Pitfalls lessons include several structures and reminders for students to become more proactively aware of their thinking and learning processes. For example, Math Pathways & Pitfalls lessons encourage students to consider where someone might make a pitfall as they solve a problem. Students learn to ask themselves whether or not a solution makes sense. These opportunities for reflection help students become more conscious of how they learn and ways they can monitor their own learning and problem solving.

Math Pathways & Pitfalls Incorporates Tools for Professional Learning and Lesson Study

Math Pathways & Pitfalls provides practical professional development options that respond to different needs, budgetary constraints, and time allocations. Teachers can learn how to use Math Pathways & Pitfalls on their own or with colleagues by viewing the Math Pathways & Pitfalls Video for Teachers (on the DVD included with each book) while completing the Professional Development Tasks (also found in each book).

The Math Pathways & Pitfalls teaching guides, which provide mathematical insights and teaching tips for each lesson, have served as a resource for lesson study groups. Finally, each Math Pathways & Pitfalls lesson embeds structures and prompts that support the development of effective and equitable teaching habits. Additional professional development opportunities are available:


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Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531.

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Durkin, K., & Rittle-Johnson, B. (2009, June 7). Comparing incorrect and correct examples when learning about decimals and the effects on explanation quality. Poster session presented at the Institute of Education Sciences Research Conference.

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