Mediated action in teachers’ discussions about mathematics tasks

This paper presents analyses of teachers?? discussions within mathematics teaching developmental research projects, taking mediation as the central construct. The relations in the so-called ??didactic triangle?? form the basic framework for the analysis of two episodes in which upper secondary school teachers discuss and prepare tasks for classroom use. The analysis leads to the suggestion that the focus on tasks places an emphasis on the task as object and its resolution as goal; mathematics has the role of a mediating artefact. Subject content in the didactic triangle is thus displaced by the task and learning mathematics may be relegated to a subordinate position.     

From design-based research to re-sourcing ‘in the wild’: reflections on studies of the co-evolution of mathematics teaching resources and practices

This commentary paper looks across the studies of the design and use of mathematics teaching resources included in this issue. It analyses everyday and educational notions of resource; particularly how usage of that term varies across the papers. Key characteristics of each study are identified and the studies are organized into five broad groups representing different lines of investigation of resource design and use. Across the studies, the broad notion of appropriation is influential in conceptualizing user/tool relations, with the instrumental/documentational approach particularly prominent, although close inspection shows that researchers draw on this in different ways. A majority of the studies relate to professional development interventions in a context of large-scale reform efforts, offering insights into a range of approaches to providing practical support for teacher appropriation of resources. One study investigates the resource systems established by ordinary teachers: alongside other research, this highlights the challenges of a re-sourcing approach in which teachers collaborate to curate a localized resource system, and points to conditions which are conducive to the success of such efforts.     

The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: practitioners’ views

This study investigates the different ways by which secondary school mathematics teachers view how advanced mathematics studies are relevant to expertise in classroom instruction. Data sources for this study included position papers and written notes from a group interview of 15 Israeli teachers who studied in a special master’s program, of which advanced mathematics courses comprise a sizeable share. Data analysis was iterative and comparative, aiming at identifying and characterizing teachers’ different perspectives. Overall, all participating teachers thought that the advanced mathematics studies in the program were relevant to their teaching of secondary school mathematics. Moreover, teachers specifically mentioned the importance of studying contemporary mathematics from research mathematicians. All teachers pointed out at least one specific feature that they viewed as relevant to their work: advanced mathematics courses (1) as a resource for teaching secondary school mathematics, (2) for improving understanding about what mathematics is, and (3) for reminding teachers what learning mathematics feels like.     

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.     

Students’ experiences with mathematics teaching and learning: listening to unheard voices

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.     

The relationship between mathematicians’ pedagogical goals,orientations, and common teaching practices in advanced mathematics

Despite mathematics educators’ research into more effective modes of teaching, lecture is still the dominant mode of instruction in undergraduate mathematics courses. Surveys suggest this is because most mathematicians believe this is the best way to teach. This paper answers a call by mathematics education researchers to explore mathematicians’ needs and goals concerning teaching. We interviewed eight mathematicians about findings in the mathematics education research literature concerning common pedagogical practices of instructors of advanced mathematics classes: “chalk talk,” the presentation of formal and informal content, and teacher questioning. We then analyzed the responses for resources, orientations, and goals that might influence the participants to engage in these practices. We describe how participants believed common lecturing practices allowed them to achieve their goals and aligned with their orientations. We discuss these findings in depth and consider what implications they may have for researchers that aim to change mathematicians’ teaching practices.     

Factors shaping mathematics lecturers’ service teaching in different departments

In this article we focus on university lecturers’ approaches to the service teaching and factors that influence their approaches. We present data obtained from the interviews with 19 mathematics and three physics lecturers along with the observations of two mathematics lecturers’ calculus courses. The findings show that lecturers’ approaches to teaching the same topic vary across departments; that is, they consciously privilege different aspects of mathematics, set different questions on examinations and follow different textbooks while teaching in different departments. We discuss factors influencing lecturers’ decision of what (mathematics) to teach in different departments and offer educational implications for service mathematics teaching in terms of students’ mathematical needs and the role of mathematics for client students.     

Elementary school teachers’ growth in inquiry-based teaching of mathematics

This article reports on a self-directed, school-based, practice-based professional development (PD) experience aimed at helping elementary school teachers to develop knowledge and expertise in inquiry-based teaching of mathematics. It discusses the characteristics of the self-directed orientation of this PD that supported the teachers’ learning, the nature of the inquiry-based knowledge they constructed, and the impact on their teaching. It highlights the centrality of agency, practical knowledge, and situated learning in this PD approach. The findings suggest that this approach can help mathematics teachers who want to be the architect of their own learning to transform their classrooms in meaningful and desirable ways.     

Instructors’ use of technology in post-secondary undergraduate mathematics teaching: a local study

In this study, instructors of undergraduate mathematics from post-secondary institutions in Newfoundland were surveyed (N = 13) and interviewed (N = 8) about their use of, experiences with, and views on, technologically assisted teaching. It was found that the majority of them regularly use technologies for organizational and communication purposes. However, the use of math-specific technology such as computer algebra systems, or dynamic geometry software for instructional, exploratory, and creative activities with students takes place mostly on an individual basis, only occasionally, and is very much topic specific. This was even the case for those instructors who use technology proficiently in their research. The data also suggested that familiarity with and discussions of examples of technology implementation in teaching at regular and field-oriented professional development seminars within mathematics departments could potentially increase the use of math-specific technology by instructors.     

Changing students’ beliefs about the relevance of mathematics in an advanced secondary mathematics class

ABSTRACT

This study shows that using authentic contexts for learning differential equations in a differentiation-by-interest setting can enhance students’ beliefs about the relevance of mathematics. The students in this study were studying advanced mathematics (wiskunde D) at upper secondary school in the Netherlands. These students are often not aware of the relevance of the mathematics they have to learn in school. More insights into the application of mathematics in other sciences can be beneficial for these students in terms of preparation for their future study and career. A course differentiating by student interest with new context-rich curriculum materials was developed in order to enhance students’ beliefs about the relevance of mathematics. The intervention aimed at teaching differential equations through guided small-group tasks in scientific, medical or economical contexts. The results show that students’ beliefs about the relevance of mathematics improved, and they appreciated experiencing how the mathematics was applied in real-life situations.     

Examining preservice teachers’ noticing of equity-based teaching practices to empower students engaging in productive struggle

Our study examined ways preservice teachers (PSTs) make connections between teaching practices and use of student resources that support productive struggle and promote equity. Our research questions are: (1) How do PSTs notice and describe the equity-based mathematics teaching practice of leveraging student resources to support student struggles? and (2) In what ways do PSTs make connections to and interpret the role student resources play in the resolution of students’ mathematical struggles? The qualitative study examined 39 PSTs in a mathematics content course for PSTs. Data come from a video analysis assignment where PSTs described their mathematical interpretations of the student struggle(s) and teacher’s use of student resources to support the struggle resolution. Findings show that PSTs noticed teacher moves that leveraged student’s mathematical thinking and linguistic funds of knowledge and based the productive level of the struggle on actions built upon peers, linguistic knowledge and prior mathematical knowledge.     

Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations

Digital resources offer opportunities to improve mathematics teaching and learning, but meanwhile may question teachers’ practices. This process of changing teaching practices is challenging for teachers who are not familiar with digital resources. The issue, therefore, is what teaching practices such so-called ‘mid-adopting’ mathematics teachers develop in their teaching with digital resources, and what skills and knowledge they need for this. To address this question, a theoretical framework including notions of instrumental orchestration and the TPACK model for teachers’ technological pedagogical content knowledge underpins the setting-up of a project with twelve mathematics teachers, novice in the field of integrating technology in teaching. Technology-rich teaching resources are provided, as well as support through face-to-face group meetings and virtual communication. Data include lesson observations and questionnaires. The results include a taxonomy of orchestrations, an inventory of skills and knowledge needed, and an overview of the relationships between them. During the project, teachers do change their orchestrations and acquire skills. On a theoretical level, the articulation of the instrumental orchestration model and the TPACK model seems promising.     

Approach to mathematics in textbooks at tertiary level – exploring authors’ views about their texts

The aim of this article is to present and discuss some results from an inquiry into mathematics textbooks authors’ visions about their texts and approaches they choose when new concepts are introduced. Authors’ responses are discussed in relation to results about students’ difficulties with approaching calculus reported by previous research. A questionnaire has been designed and sent to seven authors of the most used calculus textbooks in Norway and four authors have responded. The responses show that the authors mainly view teaching in terms of transmission so they focus mainly on getting the mathematical content correct and ‘clear’. The dominant view is that the textbook is intended to help the students to learn by explaining and clarifying. The authors prefer the approach to introduce new concepts based on the traditional way of perceiving mathematics as a system of definitions, examples and exercises. The results of this study may enhance our understanding of the role of the textbook at tertiary level. They may also form a foundation for further research.     

Developing teachers’ and teaching researchers’ professional competence in mathematics through Chinese Lesson Study

This study examines co-learning of mathematics practicing teachers and mathematics teaching researchers through parallel lesson study in China. Two cases are illustrated and compared to highlight what practicing teachers and teaching researchers learned. The practicing teachers developed their competence in identifying instructional objectives, improving instructional process, selecting and sequencing mathematical tasks, and developing professional vision. The mathematics teaching researchers developed their professional competence in effectively carrying out teaching research activities, effectively mentoring teachers, and deepening the understanding of teaching.     

Values in preservice mathematics teachers’ discussions of the Body Mass Index - A critical perspective

This article explores the values that come to the fore when preservice mathematics teachers (PTs) ¹ engage in critical discussions about the role of mathematical models in society. The specific model that was discussed was the Body Mass Index (BMI) ². From the analysis of the PTs’ discussions of the BMI from a mathematical and societal point of view several mathematical and mathematics educational values were identified such as openness, rationalism, progress, reasoning, evaluating, and problematizing the instrumental understanding of mathematics. In addition, critical thinking about mathematics in society as emphasized in curricula in the three countries involved in the study, was identified with four categories of complementary pairs. Knowing the mathematical and mathematics educational values underpinning PTs’ discussions and their connection to critical thinking is important for successfully engaging with the role of mathematics in society.