Investigating teachers’ appraisal of unexpected moments and underlying values: an exploratory case in the context of changing mathematics classroom discourse

This article provides an exploratory case study that examines what one teacher indicated as unexpected as she worked to become more purposeful about her classroom discourse practices. We found that she highlighted three areas as being unexpected: (1) aspects of lesson enactment; (2) characteristics of student learning and (3) her own intentionality or purposefulness. We interpret these instances through the lens of the systemic functional linguistics (SFL) appraisal framework in order to understand how she evaluates the instances she highlights and connect these instances to the literature on values in the teaching and learning of mathematics.     

A teacher’s learning process in dual design research: learning to scaffold language in a multilingual mathematics classroom

In this paper, we argue that dual design research (DDR) is a fruitful way to promote and trace the development of a mathematics teacher’s expertise. We address the question of how a teacher participating in dual design research can learn to scaffold students’ development of the language required for mathematical learning in multilingual classrooms. Empirical data were collected from two teaching experiments (each with 8 lessons, and 21 and 22 students, aged 11–12 years), for which lesson series about line graphs were co-designed by the researchers and the teacher. The teacher’s learning process was promoted (e.g. by conducting stimulated recall interviews and providing feedback) and traced (e.g. by carrying out 5 pre- and post-interviews before and after the teaching experiments). An analytic framework for teachers’ reported and derived learning outcomes was used to analyse pre- and post-interviews. The teacher’s learning process was analysed in terms of changes in knowledge and beliefs, changes in practice and intentions for practice. Further analysis showed that this learning process could be attributed to the characteristics of dual design research, for instance the cyclic and interventionist character, the continuous process of prediction and reflection that lies at its heart, and the process of co-designing complemented with stimulated recall interviews.     

Poverty,inequality and mathematics performance: the case of South Africa’s post-apartheid context

South Africa’s recent history of apartheid, its resultant high levels of poverty and extreme social and economic distance between rich and poor continue to play-out in education in complex ways. The country provides a somewhat different context for exploring the relationship between SES and education than other countries. The apartheid era only ended in 1994, after which education became the vehicle for transforming society and a political rhetoric of equity and quality education for all was prioritized. Thus education focused on redressing inequalities; and major curriculum change, with on-going revisions, was attempted. In this sense engagement with SES and education became foregrounded in policy, political discourse and research literature. Yet for all the political will and rhetoric little has been achieved and indicators are that inequality has worsened in mathematics education, where it is particularly pronounced. This paper proposes that continued research confirming poverty–underachievement links, which suggest an inevitability of positive correlations, is unhelpful. Instead we should explore issues of disempowerment and agency, constraints and possibilities, and the complex interplay of factors that create these widely established national statistics while simultaneously defying them in particular local contexts. Such research could shift the focus from a discourse of deficit and helplessness towards a discourse of possibilities in the struggle for equity and quality education for all.     

Pre-service science teachers’ perceptions of mathematics courses in a science teacher education programme1

Most science departments offer compulsory mathematics courses to their students with the expectation that students can apply their experience from the mathematics courses to other fields of study, including science. The current study first aims to investigate the views of pre-service science teachers of science-teaching preparation degrees and their expectations regarding the difficulty level of mathematics courses in science-teaching education programmes. Second, the study investigates changes and the reasons behind the changes in their interest regarding mathematics after completing these courses. Third, the current study seeks to reveal undergraduate science teachers’ opinions regarding the contribution of undergraduate mathematics courses to their professional development. Being qualitative in nature, this study was a case study. According to the results, almost all of the students considered that undergraduate mathematics courses were ‘difficult’ because of the complex and intensive content of the courses and their poor background mathematical knowledge. Moreover, the majority of science undergraduates mentioned that mathematics would contribute to their professional development as a science teacher. On the other hand, they declared a negative change in their attitude towards mathematics after completing the mathematics courses due to continuous failure at mathematics and their teachers’ lack of knowledge in terms of teaching mathematics.     

Exploring main obstacles of inflexibility in mathematics teachers’ behaviour in accepting new ideas: the case of equivalence between infinite sets

This article seeks to explore the most important obstacles of inflexibility in mathematics teachers’ behaviour in accepting new ideas, especially, while encountering a new situation. In so doing, the concept of equivalence between infinite sets, which is one of the challenging issues in mathematics teaching and learning, was selected as the subject matter of the study. The participants of the study were 64 high-school mathematics teachers. To collect the data, they were exposed to a challenging new situation involving equivalence between infinite sets. Then they were interviewed regarding the decisions they had made. The analysis of the data revealed that the main obstacles to accepting new ideas by mathematics teachers were preference to well-known procedures and tendency to work with methods with a more general extensive domain of the function. In addition, while the spirit of the article seems to imply that teachers are not flexible, in terms of accepting a new method, the data on teacher choices highlighted the positive results of teacher flexibility.     

Elementary teachers’ beliefs on the role of struggle in the mathematics classroom

Reform-oriented approaches to mathematics instruction view struggle as critical to learning; however, research suggests many teachers resist providing opportunities for students to struggle. Ninety-three early-years Australian elementary teachers completed a questionnaire about their understanding of the role of struggle in the mathematics classroom. Thematic analysis of data revealed that most teachers (75 %) held positive beliefs about struggle, with four overlapping themes emerging: building resilience, central to learning mathematics, developing problem solving skills and facilitating peer-to-peer learning. Many of the remaining teachers (16 %) held what constituted conditionally positive beliefs about struggle, emphasising that the level of challenge provided needed to be suitable for a given student and adequately scaffolded. The overwhelmingly positive characterisation of student struggle was surprising given prior research but consistent with our contention that an emphasis on growth mindsets in educational contexts over the last decade has seen a shift in teachers’ willingness to embrace struggle.     

Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops

In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.     

The Mathematics in Society Project: a new conception of mathematics(1) †

The Mathematics in Society Project (MISP) began in 1980 as an international association of mathematics educators in three continents. The main purpose of MISP is the writing of innovative secondary school mathematics courses based on a new conception of mathematics itself. The starting point for MISP was the fact that mathematics in school is found to be difficult and unpleasant by the great majority of pupils, but mathematics in society is widely diffused and used implicitly by most people. MISP therefore sees mathematics as a ‘living body’ representing all its uses (implicit and explicit) in society, in contrast to the ‘skeleton’ concept of mathematics which has led to such failure in schools. The gradual development of this new conception is informally analysed in the style of Kuhn and Lakatos.

     

Disciplinary literacy in mathematics: One mathematician’s reading practices

Recent scholarship on disciplinary literacy calls for an emphasis on teaching discipline-specific language/literacy practices. An understanding of these practices is, therefore, essential to literacy instruction in secondary content areas such as mathematics. This case study examined one mathematician’s reading practices, with a focus on the strategies he used in text comprehension. Data collected include the mathematician’s think-alouds during reading, discussion of his reading think-alouds, and semi-structured interviews. These data were analyzed qualitatively through an iterative process involving multiple readings and identification and refinement of codes. The analysis revealed that the mathematician engaged in extensive reading and employed an array of strategies—rereading, close reading, monitoring and questioning, summarizing and paraphrasing, storying, drawing on prior knowledge and experience, evaluating and verifying, and note-taking and visualizing—to help him make sense of what he read. These findings provide important insights that can inform mathematics teachers’ efforts to support students’ mathematics reading/learning.     

Exploiting the potential of primary historical sources in primary school: a focus on teacher’s actions

Integrating history of mathematics in classes could be a hard task with young pupils. Indeed, original historical sources have a language that is far from the modern one. Such texts represent cultural artefacts that can give access to mathematical knowledge. The teacher can exploit such potential acting as a mediator between the mathematical signs of the source and those signs that are accessible to students. Through a case study, we investigate the role of the teacher in the process of semiotic mediation during a collective discussion. The analysed intervention is made of two phases: firstly, students work collaboratively and secondly, the teacher mediates a discussion aimed at institutionalizing the knowledge. During the discussion, working on a text from Tartaglia’s translation of Euclid’s Elements, a group of fifth graders constructs a definition of prime numbers. Referring to the Theory of Semiotic Mediation, we analyse the role of the teacher in building up semiotic chains linking students’ productions to an institutionalized knowledge emerging from the collective discussion. We highlight how teacher’s focalization on students’ words allows the progress of the discussion: the potential of the historical text is exploited fostering a definition that is close to culturally shared mathematics.     

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.     

Factors affecting student success in a first-year mathematics course: a South African experience

In spite of sustained efforts tertiary institutions implement to try and improve student academic performance, the number of students succeeding in first-year mathematics courses remains disturbingly low. For most students, the gap between their mathematical capability and the competencies they are expected and need to develop to function effectively in these courses persists even after course instruction. In this study, an instrument for identifying and examining factors affecting student performance and success in a first-year Mathematics university course was developed and administered to 86 students. The overall Cronbach's Alpha coefficient for the questionnaire was found to be 0.916. Having identified variables from prior research known to affect student performance, factor analysis was used to identify variables exhibiting the greatest impact on student performance. The variables included prior academic knowledge, workload, student approaches to learning, assessment, student support teaching quality, methods and resources. From the analysis, students' perceptions of their workload emerged as the factor having the greatest impact on student's performance, followed by the matriculation examination score. The findings are discussed and strategies that can be used to improve teaching and contribute to student success in a first-year mathematics course in a South African context are presented.     

A new proof of a generalization of Gerzon’s bound

Periodica Mathematica Hungarica - In this paper we give a short, new proof of a natural generalization of Gerzon’s bound. This bound improves the Delsarte, Goethals and Seidel’s upper...