under the principle of epistemology, which encompasses the study of the origin, nature, limits, and methods of knowledge. How might teachers encourage students to use their notes to discern what they know about the mathematical knowledge learned through their problem solving
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reform-oriented textbooks? How does the deliberate nature of note taking scaffold students in their learning of new mathematical concepts presented in these texts? Implicated in the epistemology of mathematical learning via note taking is the work and research of Bruner (1996, 1966, & 1963) and Vygotsky (1994, 1986, 1981, & 1978). Their work is useful in explaining what processes trigger student performance when participating in practices directed by deliberate pedagogy and when identifying what resources or experiences may form the basis for student learning of mathematics. Both Bruner and Vygotsky position their notion of learning and development within a practical view in which the authenticity of learning is deduced, negotiated, and consensual. Such a view suggests that mathematical learning calls attention to the ways in which learning is changed and continuously renewed as learners interact in contexts that scaffold sense-making of the content (Driscoll, 1994). Note taking then, as a tool, allows students to demonstrate their conceptual understanding of mathematics on an abstract level in that writing is a concrete representation of thought (Albert, 2000).
Bruner (1973) asserts that learners move from a concrete understanding to an abstract understanding of the mathematical concepts they encounter. When used deliberately and purposefully, note taking acts as a bridge that connects the concrete domain to the abstract domain presented in reform-oriented texts. This can be seen when mathematics teachers apply deliberate pedagogical methods for taking notes in their classrooms; metacognition fosters students’ awareness of what they need to learn, when and how they need to learn it, and self-knowledge of personal and intellectual qualities. Thus, “[k] nowing is a process, not a product” (Bruner, 1973, p. 72). From Bruner’s perspective, students can best operate at a high metacognitive level when pedagogical processes are deliberate and intentional.
Like Bruner, Vygotsky (1994, 1978) believed that learning in the classroom must coalesce with deliberate pedagogy. Vygotsky introduced the notion of the zone of proximal development to explain how students make the transition from interpsychological functioning to intrapsychological functioning; deliberate pedagogical practices that include note taking may assist students in making this transition. The zone of proximal development is the distance between a student’s “actual
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developmental level as determined by independent problem solving” and the higher level of “potential development as determined through problem solving under adult guidance or in collaboration with more capable peers” (1978, p. 86). This idea is useful in explaining, at least in part, why the phenomenon of using note taking as a tool for scaffolding mathematical learning makes sense. Greenfield (1984) suggests that the metaphorical nature of a scaffold, as it is known in building construction, has five attributes. “It provides a support; it functions as a tool; it extends the range of the workers; it allows the worker to accomplish a task not otherwise possible; and it is used selectively to aid the worker where needed…a scaffold would not be used for example, when a carpenter is working five feet from the ground” (p. 118). The attributes of a scaffold also make clear how note taking can be a tool that assists students in solving problems that may be difficult or unfamiliar. For example, as students apply strategies and techniques to solve difficult or unfamiliar problems, they use their written notes to help them begin to connect their thinking to mathematical ideas. Note taking provides students with opportunities to learn through writing while extending their understanding of concepts and content; the inner dialogue with self unequivocally offers students opportunities to write, practice, and make their thinking visual and concrete (Albert, 2000).
Note taking can be viewed as a “conceptual hard scaffold” that guides students in the problem solving process (Saye and Brush, 2002). Hard scaffolds are fixed supports or guides based on teachers’ prior expectations and knowledge of difficulties students might encounter as they engage in problem solving tasks. Therefore, conceptual hard scaffolds can provide directions that help students seek relevant information to use when problem solving. Conceptual hard scaffolds may be models of approaches or processes (Simons and Klein, 2007). For example, a teacher may provide a model of a simpler problem so students can apply it to solve a complex problem. Then, the students can use their written notes as a model of how to start a difficult problem or as a hint and support in the problem solving process. Such an approach may promote higher-order thinking as well as a way for students to make connections between simple and complex problems.
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Students need proper scaffolding to be introduced to the deliberate process of note taking; similarly, their teachers need proper scaffolding to learn how to teach this new approach to note taking. Professional development provides the opportunity for teachers to develop an understanding of and to improve upon the pedagogical practices needed to effectively practice deliberate note taking. The implementation of this approach to note taking is successful in the practical teaching episodes presented largely because the teachers received instruction and feedback. Their learning, just as their students’ learning, is scaffolded so as to prompt a progression of cognitive functioning that results in an improved, more equitable approach to mathematics instruction.