Didaktische Probleme der elementaren Algebra

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. Two models of subtractionMany adults as well as school children understand subtraction solely as taking away. In this paper, we shall show the importance of the second model of subtraction (determining the difference) and the relevance of the inverse relation between addition and subtraction by adopting a longitudinal perspective. 1 Beforehand, some remarks are necessary with respect to the notions that we use. 1 Note that we do not present an empirical longitudinal study where we followed a set of students or a programme over a longer period of time. We adopt a longitudinal perspective for conducting a mathedidactical analysis (van den Heuvel-Panhuizen and Treffers, 2009).C. Selter (*) : S. Prediger : M. Nührenbörger : S. Hußmann

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“…Usiskin, 1988;Malle, 1993;Mason, 1996) and for the equal sign (e.g. Kieran, 1988) for more than 20 years.…”

Section: Understanding Manipulations For Solving Algebraic Equationsmentioning

confidence: 99%

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

et al. 2011

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“…These differences are exacerbated by the fact that relational meaning has different interpretations (e.g., Malle, 1993;Cortes et al, 1990);Prediger (2010) summarized six different meanings for equality (p.81):…”

Section: Understandings Of Equality and The Equal Signmentioning

confidence: 99%

Rethinking Algebra Teaching in the Light of ‘Orchestration of Signs’ – Exploring the “Equal Sign” in a Norwegian Mathematics Classroom

Pepin

Bergem

Klette

2014

Algebra Teaching Around the World

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“…Gemäß Kieran (1989) Hoch und Dreyfus (2004, Linchevski und Livneh (1999), Malle (1993), Sfard und Linchevski (1994), Star et al (2005).…”

Section: Introductionunclassified

Strukturieren eines algebraischen Ausdrucks als Herstellen von Bezügen

Rüede

2012

J Math Didakt

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The structuring of an algebraic expression as the production of relations Abstract Structuring algebraic expressions is an individual process in which a person sets different parts of the expression in relation to each other. A description of these relations shows two things: one, into which parts the individual sorted the algebraic expression and how he or she interprets them, and two, in which way the individual uses these relations. Studying these relations therefore allows a reconstruction of the personal meaning which an individual ascribes to the algebraic expressions. In this article, this conception is used to analyse interviews in which individuals analysed algebraic expressions. Four types of relations were identified:Producing relations within an algebraic expression can mean to simplify it visually, to change it, to re-interpret parts of it or to classify the expression. These insights are important for the teaching of algebra, particularly for meaning negotiation within individual structures.2

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Didaktische Probleme der elementaren Algebra

Cited by 94 publications

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

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

Rethinking Algebra Teaching in the Light of ‘Orchestration of Signs’ – Exploring the “Equal Sign” in a Norwegian Mathematics Classroom

Strukturieren eines algebraischen Ausdrucks als Herstellen von Bezügen

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