Obstacles and Affordances for Integer Reasoning: An Analysis of Children's Thinking and the History of Mathematics

Author:
Jessica Pierson Bishop The University of Georgia

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Lisa L. Lamb San Diego State University

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Randolph A. Philipp San Diego State University

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Ian Whitacre Florida State University

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Bonnie P. Schappelle San Diego State University

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Melinda L. Lewis San Diego State University

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Volume/Issue:
Volume 45: Issue 1
Page(s):
19–61
DOI:
https://doi.org/10.5951/jresematheduc.45.1.0019
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Subjects:
Math Topic, Other Topic, Reasoning/Sense Making/Proof, Research, Research Study
Keywords:
Children's thinking; Cognitive affordance; Cognitive obstacle; Integers; Mathematical history; Negative numbers

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or knowledge that may, initially, interfere with one's learning integers (which we call cognitive obstacles) and understanding or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affordances). We analyzed historical mathematical writings related to integers as well as clinical interviews with children ages 6-10 to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles.

Contributor Notes

Jessica Pierson Bishop, Department of Mathematics and Science Education, The University of Georgia, 105 Aderhold Hall, Athens, GA 30602; jpbishop@uga.edu

Lisa L. Lamb, School of Teacher Education and Center for Research in Mathematics and Science Education, San Diego State University, CRMSE, 6475 Alvarado Rd, Suite 206, San Diego, CA 92120; lisa.lamb@sdsu.edu

Randolph A. Philipp, School of Teacher Education and Center for Research in Mathematics and Science Education, San Diego State University, CRMSE, 6475 Alvarado Rd, Suite 206, San Diego, CA 92120; rphilipp@mail.sdsu.edu

Ian Whitacre, School of Teacher Education, Florida State University, 1114 West Call Street, G123 Stone Building, Tallahassee, FL, 32306; iwhitacre@fsu.edu

Bonnie P. Schappelle, Center for Research in Mathematics and Science Education, San Diego State University, CRMSE, 6475 Alvarado Rd, Suite 206, San Diego, CA, 92120; bschappelle@projects.sdsu.edu

Melinda L. Lewis, Center for Research in Mathematics and Science Education, San Diego State University, CRMSE, 6475 Alvarado Rd, Suite 206, San Diego, CA, 92120; mindydanenhauer@hotmail.com

(Corresponding author is Bishop jpbishop@uga.edu)
(Corresponding author is Lamb lisa.lamb@sdsu.edu)
(Corresponding author is Philipp rphilipp@mail.sdsu.edu)
(Corresponding author is Whitacre iwhitacre@fsu.edu)
(Corresponding author is Schappelle bschappelle@projects.sdsu.edu)
(Corresponding author is Lewis mindydanenhauer@hotmail.com)
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Journal for Research in Mathematics Education
Cover Journal for Research in Mathematics Education
Copyright:
The National Council of Teachers of Mathematics, Inc. All rights reserved. 2014
Page Count:
43
Article Category:
Research Article
Print Publication Date:
01 Jan 2014
Online Publication Date:
Jan 2014
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