Abstraction and embodiment: exploring the process of grasping a general

This paper reports from a case study which explores kindergarten children’s mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes “a point of view from which the concrete can be seen as meaningfully related” (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children’s mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys’ awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys’ gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children’s concrete embodied activities.

Beyond categories: dynamic qualitative analysis of visuospatial representation in arithmetic

Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad categorizations of representations and argues the benefits of dynamic qualitative analysis of arithmetical-representational strategy across multiple semi-independent aspects of display, calculation, and interaction. It proposes an alternative methodological approach combining the structured organization of classification with the detailed nuance of description and describes a systematic but flexible framework for analysing nonstandard visuospatial representations of early arithmetic. This approach is intended for use by researchers or practitioners, for interpretation of multimodal and nonstandard visuospatial representations, and for identification of small differences in learners’ developing arithmetical-representational strategies, including changes over time. Application is illustrated using selected data from a microanalytic study of struggling students’ multiplication and division in scenario tasks.

The process of problem posing: development of a descriptive phase model of problem posing

The aim of this study is to develop a descriptive phase model for problem-posing activities based on structured situations. For this purpose, 36 task-based interviews with pre-service primary and secondary mathematics teachers working in pairs who were given two structured problem-posing situations were conducted. Through an inductive-deductive category development, five types of activities (situation analysis, variation, generation, problem-solving, evaluation) were identified. These activities were coded in so-called episodes, allowing time-covering analyses of the observed processes. Recurring transitions between these episodes were observed, through which a descriptive phase model was derived. In addition, coding of the developed episode types was validated for its interrater agreement.

Teacher noticing and its growth toward expertise: an expert–novice comparison with pre-service and in-service secondary mathematics teachers

Although strong references to expertise in different theoretical approaches to teacher noticing have been made in the last decades, empirical knowledge about the development of teacher noticing from novice to expert level is scarce. The present study aims to close this research gap by comparing three different groups of mathematics teachers with different degrees of professional teaching experience—pre-service teachers at the master’s level, early career teachers, and experienced teachers—using data sampled in the frame of the research program from the Teacher Education and Development Study in Mathematics (TEDS-M). Furthermore, the construct of teacher noticing is assessed in a differentiated way by analyzing different noticing facets. Findings confirm that three facets of teacher noticing can be empirically distinguished—perception of important classroom events, their interpretation, and decisions regarding further developments. The results reveal a considerable increase in professional noticing between master’s students and practicing teachers. However, in contrast to other studies, among examples from East Asia, a stagnation or decrease in professional noticing between early career teachers and experienced teachers could be observed. Overall, the study highlights the cultural dependency of expertise development regarding teachers’ noticing.