Students’ Thinking About Integer Open Number Sentences

We share a subset of the 41 underlying strategies that comprise five ways of reasoning about integer addition and subtraction: formal, order-based, analogy-based, computational, and emergent. The examples of the strategies are designed to provide clear comparisons and contrasts to support both teachers and researchers in understanding specific strategies within the ways of reasoning. The ability to categorize strategies into one of five ways of reasoning may enable teachers to organize knowledge of student thinking in ways that are useable and accessible for them and provide researchers with sufficient information about the strategies and ways of reasoning such that they can reliably build on this work.

[1]  Lisa L. C. Lamb,et al.  Leveraging Structure: Logical Necessity in the Context of Integer Arithmetic , 2016 .

[2]  L. Bofferding.  Negative Integer Understanding: Characterizing First Graders' Mental Models , 2014 .

[3]  M. Chiu.  Using Metaphors to Understand and Solve Arithmetic Problems: Novices and Experts Working With Negative Numbers , 2001 .

[4]  Lisa L. C. Lamb,et al.  Using order to reason about negative numbers: the case of Violet , 2014 .

[5]  Amy J. Hackenberg.  Students’ Reasoning With Reversible Multiplicative Relationships , 2010 .

[6]  Herbert P. Ginsburg,et al.  Entering the child's mind , 1997 .

[7]  Thomas P. Carpenter,et al.  Children's Mathematics: Cognitively Guided Instruction , 1999 .

[8]  Aurora Gallardo.  Negative Numbers in the Teaching of Arithmetic. Repercussions in Elementary Algebra. , 1995 .

[9]  T. P. Carpenter,et al.  Using Knowledge of Children’s Mathematics Thinking in Classroom Teaching: An Experimental Study , 1989 .

[10]  Lisa L. C. Lamb,et al.  Developing Symbol Sense for the Minus Sign , 2012 .

[11]  Jean-Claude Bringuier,et al.  Conversations with jean piaget , 1982 .

[12]  Joëlle Vlassis.  The balance model: Hindrance or support for the solving of linear equations with one unknown , 2002 .

[13]  David Tall,et al.  Concept image and concept definition in mathematics with particular reference to limits and continuity , 1981 .

[14]  T. P. Carpenter,et al.  Teachers' pedagogical content knowledge of students' problem solving in elementary arithmetic , 1988 .

[15]  A. Arcavi.  Symbol Sense: Informal Sense-Making in Formal Mathematics. , 1994 .

[16]  Karen C. Fuson,et al.  Research on whole number addition and subtraction. , 1992 .

[17]  Michelle Stephan,et al.  A Proposed Instructional Theory for Integer Addition and Subtraction , 2012 .

[18]  Marilyn P. Carlson,et al.  Applying Covariational Reasoning While Modeling Dynamic Events: A Framework and a Study. , 2002 .

[19]  Lisa L. C. Lamb,et al.  Obstacles and Affordances for Integer Reasoning: An Analysis of Children's Thinking and the History of Mathematics , 2014 .

[20]  Konstantinos P. Christou,et al.  What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra , 2012 .

[21]  Steven R. Williams.  Models of Limit Held by College Calculus Students. , 1991 .

[22]  Juliet M. Corbin,et al.  Basics of Qualitative Research (3rd ed.): Techniques and Procedures for Developing Grounded Theory , 2008 .

[23]  Sean Larsen,et al.  Coming to Understand the Formal Definition of Limit: Insights Gained From Engaging Students in Reinvention , 2012 .

[24]  Irit Peled,et al.  Signed Numbers and Algebraic Thinking , 2017 .