Ball (1993)

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With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics


Ideas like "understanding," "authenticity," and "community" are central in current debates about curriculum, instruction, and assessment. Many believe that teaching and learning would be improved if classrooms were organized to engage students in authentic tasks, guided by teachers with deep disciplinary understandings. Students would conjecture, experiment, and make arguments; they would frame and solve problems; and they would read, write, and create things that mattered to them. This article examines the challenge of creating classroom practices in the spirit of these ideals. With a window on her own teaching of elementary school mathematics, the author presents three dilemmas-of content, discourse, and community-that arise in trying to teach in ways that are, in Bruner's terms, "intellectually honest." These dilemmas arise reasonably from competing and worth-while aims and from the uncertainties inherent in striving to attain them. The article traces and explores the author's framing of and response to these dilemmas, providing a view of the pedagogical complexities that underlie current educational visions and the conditions needed to work toward them.

Outline of Headings

  • A Restatement of the Pedagogical Challenge
  • Dilemma No. 1: Representing the Content: The Case of Teaching Negative Numbers
    • Rationale for Teaching Integers
    • Analyzing the Content and Thinking about Learners
    • Trying Money
    • Students' Learning
    • Dilemmas of Content and Representation
  • Dilemma No. 2: Respecting Children as Mathematical Thinkers: The Case of "Sean Numbers"
    • Rationale for Teaching Invention
    • Appreciating the Mathematics in the Child
    • Students' Learning
    • Dilemmas of Respecting Children as Thinkers
  • Dilemma No. 3: Creating and Using Community
    • Community Learning
    • Students' Learning
    • Dilemmas of Creating and Using Community
  • Dilemmas of Trying to Be "Intellectually Honest" in Teaching Mathematics


Video of "Sean's Numbers" and related materials are available at

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397. doi:10.1086/461730
author = {Ball, Deborah Loewenberg},
doi = {10.1086/461730},
journal = {The Elementary School Journal},
number = {4},
pages = {373--397},
title = {{With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics}},
url = {},
volume = {93},
year = {1993}