Learning and Mathematics

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Math Horizon - Ball (1987)

Deborah Ball of Michigan State University teaches math education in the College of Education and as part of her research teaches a math class in a local elementary school. In this article she examines the challenge of creating classroom practices for third graders of diverse racial, ethnic, and socioeconomic backgrounds in the spirit of current reform, with ideals involving student engagement in authentic tasks. Using her own elementary school mathematics classroom, the author presents three dilemmas -- of content, discourse, and community-- that arise in trying to teach in ways that are "intellectually honest." Ball frames and responds to these dilemmas, providing a view of underlying pedagogical complexities and the conditions needed in order to work toward current educational visions.
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Article:

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93 (4). pp. 373-397.

Quotes:

Bruner's (1960) "notion of 'intellectual honesty'... has most captured my imagination. Writing on the topic of readiness for learning, he argued: We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. No evidence exists to contradict it; considerable evidence is being amassed that supports it' (Bruner, 1960, p. 33). My undergraduate students sometimes squirm a bit and make weak, nervous jokes. 'Calculus? Can a first grader learn calculus?' But I, more experienced with young children, am quite convinced. The things that children wonder about, think, and invent are deep and tough. Learning to hear them is, I think, at the heart of being a teacher" (p. 374)

"The idea of intellectual honesty makes sense. Somehow, what I do with children should be honest, both to who they are and to what I am responsible to help them learn. Intellectual honesty implies twin imperatives of responsiveness and responsiblity. But I wonder: How do I create experiences for my students that connect with what they now know and care about but that also transcend their present? How do I value their interests and also connect them to ideas and traditions growing out of centuries of mathematical exploration and invention?" (p. 374-375).

"Because mathematical knowledge is socially constructed and validated, sense making is both individual and consensual. Drawing mathematically reasonable conclusions involves the capacity to make mathematically sound arguments to convince oneself and others of the plausibility of a conjecture or solution. It also entails the capacity to appraise and react to others' reasoning and to be willing to change one's mind for good reasons. Thus, community is a crucial part of making connections between mathematical and pedagogical practice" (p. 376).

"Bruner (1960) argues that children should encounter 'rudimentary versions' of the subject matter that can be refined as they move through school. This position, he acknowledges, is predicated on the assumption that 'there is continuity between what a scholar does on the forefront of his discipline and what a child does in approaching it for the first time...' But what constitutes a defensible and effective 'rudimentary version'? And what distinguishes intellectually honest [fragments] from distortions of the subject matter?" (p. 376).

"... I note three problems inherent in attempting to model classrooms on ideas about authentic mathematical practice... First, constructing a classroom pedagogy on the discipline of mathematics would be in some ways inappropriate, even irresponsible. Mathematicians focus on a small range of problems, working out their ideas largely alone. Teachers, in contrast, are charged with helping all students learn mathematics, in the same room at the same time. The required curriculum must be covered and skills developed. With 180 days to spend and a lot of content to visit, teachers cannot afford to allow students to spend months developing one idea or learning to solve a certain class of problems... Moreover, certain aspects of the discipline would be unattractive to replicate in mathematics classrooms. For instance, the competitiveness among research mathematicians..." (p. 377) is hardly a desirable model for an elementary classroom.

Ball reports frustrations with attempting to employ real-world materials: "Money [was used] as a representation context for exploring negative numbers" (p. 381). However, as one of her students said, "There is no such thing as below-zero dollars!" (p. 382). "With money, many children never used negative numbers to represent debt: They were inclined to report that someone had "$6" and "also owes so-and-so eight dollars" rather than using "-8" to represent the debt. They were also inclined to leave positive values (money) and negative ones (debt) unresolved" (p. 383).

-- Summarized by Mia Ong Wenbourne

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