Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

Nührenbörger, Marcus; Steinbring, Heinz

doi:10.1007/s10857-009-9100-9

Cited by 28 publications

(16 citation statements)

References 8 publications

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“…Only a few of them investigate how the "ambiguity" (e.g. Nührenbörger & Steinbring, 2009) of interpreting an illustration as well as a symbolic representation of a subtraction problem can be fostered. Here, we see a clear indication of the need for carefully designed teaching experiments.…”

Section: Model Of Subtractionmentioning

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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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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Section: Model Of Subtractionmentioning

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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“…In the revision of our experiments, we then try to understand the social conditions that support the emergence of collective argumentation in order to learn how to initiate argumentation in subsequent experiments in a more effective way. Thereby we are influenced by learning and teaching theories from scientific neighbour-disciplines of mathematics education, mainly using approaches of symbolic interactionism and ethnomethodology (Voigt 1994;Krummheuer 1995;Yackel and Cobb 1996), theories of argumentation (Schwarzkopf 2003), and epistemological theories (Steinbring 2005;Nührenbörger and Steinbring 2009).…”

Section: Example: the Realisation Of Argumentationmentioning

confidence: 99%

Design Science and Its Importance in the German Mathematics Educational Discussion

Nührenbörger

Rösken-Winter

Fung

et al. 2016

ICME-13 Topical Surveys

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1 from design research but also pursue particular accentuations: The first contribution (Sect. 2.2) reports on a project that positions design science between normative and descriptive theories. That is, mathematical learning opportunities have been designed from a normative viewpoint while learning processes have been explained on the basis of descriptive theories. Examples are provided that illustrate such a proceeding. The second contribution (Sect. 2.3) also reports on a developmental research project. Here, studying carefully designed teaching units is in the focus while paying explicit attention to problems that teachers face in the classroom. One main goal of the research is to help teachers to develop their mathematical knowledge and to ultimately enhance their teaching. Subsequently, in Sect. 2.4 Wittmann elaborates on the conception of empirical studies within design research that start from the mathematics involved in terms of structure-genetic didactical analyses. Particularly, bridging didactical theories and practice is pursued by collective teaching experiments. Finally, Sect. 2.5 takes up the strands presented so far and sheds light on developments of design science from a national and international perspective. Thereby, particular emphasis is assigned to design research from a learning perspective as extending characteristics from design research. The chapter ends with a summary and an outlook on further developments.

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“…For example, children were asked to find all the number combinations to complete sums such as: [?+?=7], [?+?=12]. A similar type of task was used by Nührenbörger and Steinbring (2009) but in a study that analysed forms of mathematical interaction and communication of mathematical knowledge between children at the beginning of primary school and their teachers.…”

Section: Procedural Skill and Conceptual Knowledge: The Iterative Modelmentioning

confidence: 99%

Procedural and conceptual changes in young children’s problem solving

Voutsina

2011

Educ Stud Math

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This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5-6-year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task and how children's application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children's pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children's overt behaviour reveals a dynamic interplay between children's developing representation of the task, their improved procedures and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the "iterative model" hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children's arithmetic problem solving.

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Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

Cited by 28 publications

References 8 publications

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

Design Science and Its Importance in the German Mathematics Educational Discussion

Procedural and conceptual changes in young children’s problem solving

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