Why does this model work?

The section provides an explanation of how deliberate pedagogy serves as a scaffold for assisting students in learning how to take notes and then use these notes to solve mathematical problems. Most important, the reform-oriented curriculum places unusual demands on the students than that of a more traditional curriculum. It is imperative that these demands be taught to students who are compelled to meet them. Manouchehri and Goodman (2000) suggest that when using problem solving based curriculum, teachers need to include approaches that facilitate “guiding students’ inquiry, mapping gradual development of both the content and

Mathematical Problem Solving 84

learner’s thinking, and creating a balance between fostering students’ conceptual understanding while assisting them in acquisition of basic skills” (p. 29). According to the model described in this chapter, the teachers went beyond simply requiring students to take notes. Mr. Orland and Miss Lipan modeled their own thought processes through both writing and oral statements. These two modalities generate a quasi- interaction with students, in which students focus on capturing their teacher’s application of an effective strategy and then restating in their own words what they observed and understood about the problem launched and modeled. This model allows both the teacher and students to make metacognitive advances; the teacher and students think about the way they are learning the mathematics at hand, as opposed to merely thinking about the mathematics content itself. This occurs because the teacher provides deliberate and explicit modeling of note taking, which helps the students to develop the metacognitive skills necessary for learning how to take notes that will subsequently help them to think about the concepts they are learning.

Metacognition refers to the abstract thought process through which an individual thinks about and reflects upon one’s own thinking. In Miss Lipan’s modeling, for example, she states, “I am going to name my strategy and prove how I can use it to find the area. My strategy is called

surround-and-subtract.” Then she both says and writes, “Step 1:

Surround the original shape. Draw a square/rectangle around the borders of the original shape. Step 2: Find the total area of the square/rectangle you drew around the original shape. Step 3: Subtract the area of the outside shape from the total area of the square/rectangle you drew. This is Total Area – Outside shape = Area of Original Shape.” In addition, Miss Lipan demonstrates how to solve the problem, making her implicit thinking and knowledge explicit for her students. Maccini, Mulcahy, and Wilson (2007) assert that, given the difficulties many students with learning disabilities in mathematics have accessing reform-based curriculum, it is essential to integrate pedagogical practices that are both deliberate and explicit.

Another reason that this model is an improvement over current use of note taking in mathematics instruction is that the teacher allows time for students to review and reflect on their notes, which are scaffolded

Note Taking as Deliberate Pedagogy 85

by the written reflection questions. Giving crucial scaffolding is fundamental in the learning phase of note taking as learners progress through the zone of proximal development. Mr. Orland and Miss Lipan engage students in dialogue about the problem to provide any clarification that they might need about the task set before them. With respect to this approach, Farmer (1995) offers the explanation, “the ability to solve problems through dialogue with [teachers] or peers is a harbinger of competencies that will later become internalized” (p. 305). In this instance, the teachers also used many of the techniques that Rogoff (1990) describes as “scaffolded learning supports.” These include the following: elaborating, linking, prompting, simplifying, and providing affective support.

An important observation is that because of organizing and applying a deliberate instructional approach, the actual situation may also be described as nominative. In other words, Mr. Orland and Miss Lipan set criteria pertaining to the most effective way of taking notes and, at the same time, state the requirements for meeting those criteria. The central point here is that the mathematics learning community, as in these two classrooms, exists as a means for appropriating deliberate discourse that is within reach for all learners. Students’ note taking skills improve over time as students grow in their understanding of how their notes help them to organize, understand, and shape their ideas in a meaningful way. In addition to note taking as a deliberate part of instruction, teachers need to explicitly communicate their expectations regarding the use of written notes for completing a problem-solving task. These expectations must be generalized to classroom norms and procedures that are applied to all classroom activities; students must become accustomed to their teacher setting high expectations to prevent negative fallout that could result in low problem solving performance.

A final explanation of why this model works involves the collaborative nature of the model. As noted earlier, the teacher educator served as the more knowledgeable other in assisting Mr. Orland and Miss Lipan throughout the planning phase, observing and taking notes during the launching and modeling phase, and meeting with the teachers after the lesson to share observation notes, which included critical reflection and questions about the recent mathematics instruction. We

Mathematical Problem Solving 86

worked together to achieve a shared understanding of the teaching and learning of mathematics. As Vygotsky (1978, 1981) theorizes, when teachers are challenged to work on activities collectively and are encouraged to achieve what they are not capable of doing individually, they are likely to move forward in their development as mathematics teachers, especially when the dialogue between the teacher and the other is sustained long enough to become a deliberate process. This, in turn, benefits students because it provides them with teachers that are more knowledgeable about how to make reform-oriented curricula accessible to all learners.

4 Conclusion

Bruner’s conception of knowledge representation and Vygotsky’s construct of the zone of proximal development have a place in contemporary discussion of the importance of creating effective learning communities that support all learners, especially when using reform- oriented curricula. Teachers need to understand how to assist all learners in their zone of proximal development, providing opportunities for problem-solving and inquiry-based activities that encourage the development of complex thinking and logical reasoning. Professional development activities that coalesce with Bruner and Vygotsky’s research can assist teachers in the development of deliberate pedagogical practices. These practices must be designed to enhance mathematical learning experiences, such as supporting development of the students as effective note takers. Such a framework calls for a reconceptualization of the traditional role of teacher and learner. The emphasis is on processes and strategies rather than products and solutions. In other words, teachers must call attention to the why and how of mathematics, instead of merely focusing on the what—the final answer. This method of deliberate pedagogy allows teachers to move beyond the function of imparting knowledge and organizational skills. What discerns the note taking instructional approach described in this chapter is that it reflects the needs of the learner, the demands and purposes of the mathematical content, and the attributes of the context in which the launching and

Note Taking as Deliberate Pedagogy 87

modeling transpired. As practicing teachers strive to work with diverse learners using reform-oriented mathematics curricula, the learner, the content, and the instructional context must be given considerable attention to be effective and successful (Albert, 2003; Draper, 2002; Maccini, Mulcahy, & Wilson 2007; White, 2003). Note taking can provide students with a structure for organization; further, when teachers deliberately use it to model the thinking and learning processes, note taking can provide much more—a metacognitive tool to enhance students’ mathematical understanding.

References

Albert, L. R. (2003). Creating caring learning communities that support academic success. In C. Cimino, R. M. Haney, & J. M. O’Keefe, (Eds.). Educating young adolescents: Conversations in excellence (pp. 51-65). Washington, DC: National Catholic Education Association.

Albert, L. R. (2000). Outside in, inside out: Seventh grade students’ mathematical thought process. Educational Studies in Mathematics, 41, 109-142.

Boch, F. & Piolat, A. (2005). Note taking and learning: A summary of research. The WAC Journal, 16, 101-113.

Boaler, J. (2002). Learning from teaching: Exploring the relationship between reform curriculum and equity. Journal for Research in Mathematics Education, 33(4), 239-258.

Bruner, J. S. (1996). The culture of education. Cambridge, MA: Harvard University Press.

Bruner, J. S. (1973). The relevance of education. New York: W. W. Norton. Bruner, J. S. (1966). Toward a theory of instruction. New York: W. W. Norton.

Bruner, J. S. (1963). On knowledge: Essays for the left hand. Cambridge, MA: Harvard University Press.

Draper, R. J. (2002). School mathematics reform, constructivism, and literacy: A case for literacy instruction in the reform-oriented math classroom. Journal of Adolescent and Adult Literacy, 45(6), 520-529.

Driscoll, M. (1994). Psychology of learning for instruction. Boston: Allyn and Bacon. Farmer, F. (1995). Voice reprised: Three etudes for a dialogic understanding. Rhetoric

Review, 13, 304-320.

Greenfield, P. M. (1984). Theory of the teacher in the learning activities of everyday life. In B. Rogoff & J. Lave (Eds.), Everyday cognition (pp. 117-138). Cambridge, MA: Harvard University Press.

Mathematical Problem Solving 88

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2004). Connected mathematics; Growing, growing, growing. Needham, MA: Pearson Prentice Hall.

Maccini, P., Mulcahy, C. A., & Wilson, M. S. (2007). A follow-up of mathematics interventions for secondary students with learning disabilities. Learning Disabilities Research and Practice, 22(1), 58-74.

Manouchehri, A. & Goodman, T. (2000). Implementing mathematics reform: Challenge within. Educational Studies in Mathematics, 42, 1-34.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Paul, W. (1974). How to study in college. Boston: Houghton Mifflin.

Piolat, A. Olive, T., & Kellogg, R. (2005). Cognitive effort during note taking. Applied Cognitive Psychology, 19, 291-312.

Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social context. New York: Oxford University Press.

Saye, J. W. & Brush, T. (2002). Scaffolding critical reasoning about history and social issues in multimedia-supported learning environments. Educational Technology Research and Development, 50(3), 77-96.

Simons, K. D. & Klein, J. D. (2007). The impact of scaffolding and student achievement levels in a problem-based learning environment. Instructional Science, 35, 41-72. Vygotsky, L. S. (1994). The problem of cultural development of the child. In R. Van

Der Veer and J. Valsiner (Eds.), The Vygotsky reader (pp. 57-72). Cambridge, Massachusetts: Blackwell.

Vygotsky, L. S. (1986). Thought and language. Cambridge, MA: MIT Press.

Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 144-188). Armonk, NY: Sharpe. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological

processes. Cambridge, MA: Harvard University.

Wertsch, J. V. (1980). The significance of dialogue in Vygotsky’s account of social, egocentric, and inner speech. Contemporary Educational Psychology, 5, 120-162. Wertsch, J. V. (1979). From social interaction to higher psychological processes: A

classification and application of Vygotsky’s theory. Human Development, 22, 1-22. White, D. Y. (2003). Promoting productive mathematical classroom discourse with

89 Chapter 5

Japanese Approach to Teaching