Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition

Subtraction can be understood by two basic models—taking away (ta) and determining the difference (dd)—and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed...

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Bibliographic Details
Published in:Educational studies in mathematics Vol. 79; no. 3; pp. 389 - 408
Main Authors: Selter, Christoph, Prediger, Susanne, Nührenbörger, Marcus, Hußmann, Stephan
Format: Journal Article
Language:English
Published: Dordrecht Springer 01.03.2012
Springer Netherlands
Subjects:
Addition
Algebra
Algorithms
Analysis
Arithmetic
Case Studies
Children
Education
Equations (Mathematics)
Learning
Learning Processes
Mathematical Logic
Mathematical procedures
Mathematical relations
Mathematics
Mathematics Education
Mathematics Instruction
Mathematics Skills
Mental Computation
Modeling
Subtraction
Arithmetic and algebra
Inverse relation
Determining the difference
Algebraic equations
Models of subtraction
Standard algorithms
Mental arithmetic
Negative numbers
Taking away
ISSN:0013-1954, 1573-0816
Online Access:Get full text
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Summary:Subtraction can be understood by two basic models—taking away (ta) and determining the difference (dd)—and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed in a longitudinal perspective on mathematics learning processes, the article highlights some exemplary steps in which the inverse relation is discussed in light of the two models, namely mental subtraction, the standard algorithms for subtraction, negative numbers and manipulations for solving algebraic equations. For each step, the article presents educational considerations for fostering a flexible use of the two models as well as of the inverse relation between subtraction and addition. In each section, a mathedidactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched.
ISSN:0013-1954
1573-0816
DOI:10.1007/s10649-011-9305-6