Subject-specific vocabulary, particularly in mathematics is acknowledged as essential for learning. “Without an understanding of the vocabulary that is used routinely in mathematics instruction, textbooks, and word problems, students are handicapped in their efforts to learn mathematics” (Miller, 1999, p. 312). While Miller’s observation is largely uncontested there is need for research on how pupils deal with potential ambiguities that may arise with subject-specific vocabulary attached to number sense. This is especially relevant given some educators' emphasis on definitions as a measure of cognition (cf. Vinner, 1991). It has been established that definitions, particularly concerning number sense, are not without difficulties; Griffin (2004) argues that while aspects of number sense are easily recognizable, defining it precisely is challenging. The present research is a case study of pre-school pupils endeavour to explain the mathematics concepts associated with number sense namely: numerals and digits (in Swedish tal & siffror). Unlike studies focusing solely on mathematics vocabulary as a prerequisite for engaging in mathematical activities (e.g., Miller, 1999, Griffin, 2004) the present study seeks to explore how pupils connects previous knowledge and available recourses in a sense making process. The guiding questions are: i) how do the pupils explain the concepts in question? ii) what are some of the emerging thoughts in their sense making process? Data was collected through an interview based on aspects related to the pupils written answer to a task. Using a semiotic framework (Olande, 2014), the pupils’ sense making process and emerging ideas connected to number sense were identified. Preliminary results highlight the fluid nature of the concepts and also demonstrates the pupils’ use of what is at hand - the apparent to approach some profound aspects of number sense.

References

Griffin, S. (2004) Building number sense with Number Worlds: A mathematics program for you children. Early Childhood Research Quarterly 19(1), 173-180

Miller, D. L. (1993). Making the connection with language. Arithmetic Teacher, 40(6) 311-316

Olande, O (2014). Graphical artefacts: Taxonomy of students’ response to test items. Educational Studies in Mathematics, 85 (1) 53-74

Vinner, S. (1991). ‘The role of definitions in the teaching and learning of mathematics’, in D. Tall (ed.), Advanced Mathematical Thinking, Mathematics Education Library, Kluwer Academic Publishers, Dordrecht, pp. 65–81

The present case study explores the assessment practice of two examiners from different academic backgrounds on an undergraduate thesis work in mathematics education. Reflection notes and feedback from two instances of a thesis-writing process are interrogated using a framework based on a semiotics perspective to meaning-making. It is shown that the examiners utilize aspects that are ‘immediate and non-contested’ to successively make accessible that which is ‘withdrawn yet to be revealed.’ In this process, varied interventional approaches are observed to support desirable aspects of teacher professional knowledge including critical-analytical disposition. The study also highlights knowledge integration for teaching, scientific disposition, collaboration, and knowledge transformation as some of the desirable aspects of teacher professional knowledge. The findings indicate that knowledge integration and transformation remain an issue of concern for pre-service teachers. This raises the question of how desirable traits for a teacher-as-a researcher can be promoted within the context of thesis work.

The present study explores a task containing graphical artefacts from the Swedish national test (Nationella provet – NP) for a sub-sample of grade 9 students’ solutions. The sub-sample comprising of 115 students’ solutions to the task is closely analysed, using an analytical construct founded on “identification” as well as the “critical-analytical” approach to problem solving. Based on this construct it is observed that a sizable number of students’ solutions follow a visual strategy with strong reliance on everyday forms of expression. Given the nature and purpose of NP we posit that students use the methods and tools that reflect general school practice. The analysis used in this study is perceived as problematising the assessment of competency from high stake tests, and the educational setting in general.

This study reports from a pre-service teacher’s online learning and assessment activity on determining variability of two graphical artefacts. Using a critical-analytical perspective to data, the present study indicate that the prospective teachers surveyed showed awareness of relevant subject specific operators and methods however, these seem not be well coordinated and were submerged in forms of expressions characterized by intuitive methods and everyday language. Significantly the prospective teachers seemed to substitute statistical and mathematical methods with explanatory metaphors which while providing room for deeper subject specific engagement were however, only used superficially. Their reliance on everyday forms of expression and visual perception is perceive as a factor that might have hampered their effective choice and application of relevant subject specific tools and forms of expression. This observation puts to task the role of informal methods in statistics education.

The present study, carried out in the Nordic countries, examines the characteristics of students' scholastic performance on items containing graphical artefacts, that is, bar graphs, pie charts and line graphs, selected from the Programme for International Student Assessment (PISA) survey test. Graphical analysis of statistical data resulted in the observation of two major categories of performance by the students. The results of cluster analysis also confirmed the two approaches. One approach consists of items perceived as requiring identification, that is, focusing primarily on perceptual elements. The other consisting of items requiring a critical-analytical approach, that is, involving evaluation of the graphical system, active interaction with subject specific operators and forms of expression. The general observation is that the pattern of response is similar for all these countries, with items demanding an identification approach showing comparatively higher scores than for items perceived as demanding a critical-analytical approach.

Given the importance of a critical-analytical disposition in the case of graphical artefacts, this paper explores graphicacy based on students’ answers to an item from PISA survey test. Primarily, results from the written test were analyzed using PISA’s doubledigit rubrics or coding. In evaluating these categories, it is observed that just a small percentage of students are able to produce answers that reflect a critical-analytical approach with respect to the use of statistical/mathematical operators and forms of expressions. Secondly, video observation shows that students tend to employ what is perceived as an ”identification approach” while discussing the task. Whereas elements of mathematical and statistical ideas can be identified in the students’ discussion, these are not explicitly stated and are largely submerged in everyday concerns and forms of expression. 

The research concerns presented in this dissertation addresses aspects of students’ interaction with commonly occurring graphical artefacts in teaching and learning environments. In particular, focus is on how the students make sense of these artefacts in relation to subject specific tools and forms of expression.

The dissertation comprises of four studies guided by a semiotics cultural- historical perspective to cognition. The first study which is largely quantitative, analyses the percentage scores from students’ responses to selected items from OECD PISA surveys for items containing graphical elements. The second and the third studies in keeping with a more sociocultural perspective to learning as point of departure, examine the students’ collaborative interaction around tasks containing graphical elements. The fourth study explores the nature of students’ solutions from the Swedish national test in mathematics based on a tools and forms of expression sensitive empirically derived construct of Identification contra Critical-analytical approach to graphicacy.

The main outcomes of these studies can be summarised as follows: first there is justification for re-examining the predominant characterisation of students’ interaction with graphical artefacts. Secondly, while it is not uncommon for students to take a more visual-perceptive and intuitive approach to graphicacy, results from task items interactions indicate that a Critical-analytical approach seems to be more reliable and capable of yielding desirable outcomes. The outcomes of these studies call for vigilance on the type of tasks used in relation to graphicacy and how these can be used to foster students’ Critical-analytical disposition.

This paper reports findings from study on students discourse in a graphical milieu from PISA 2003. This was a follow up to analysis of some items containing graphics done for 8 countries where it was found that response rate for questions requiring explanatory communication was relatively low for a paper and pen environment as well as in an environment allowing collaboration and cooperation among students.