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.     

A revised theorization of the relationship between teachers’ conceptions of mathematics and its teaching

We assembled the ideas about mathematics and about its teaching which were expressed by mathematicians and mathematics educators into two pairs of ‘official’ (collective) conceptions: mathematics is either static or dynamic, and mathematics teaching is either closed or open. These polar conceptions produce a 4-pair relationship between the conceptions of mathematics and its teaching. The adherence to official conceptions was tapped by a questionnaire encompassing 176 Israeli high school mathematics teachers, aimed at examining the relationship between their conceptions of mathematics and its teaching. The majority of these teachers either hold a single conception in one of the domains or do not adhere to any conception, and a quarter of them hold either the static-closed or dynamic-open pairs of conceptions that prevail among teachers in other countries. Consequently, we define a conception of an entity as a comprehensive and homogenous set of ideas about a particular characteristic or feature of that entity. Reality is that teachers practice their profession without adhering to any official conception, and perhaps are (/to be?/) praised for their reluctance to blindly adopt the clear-cut rigid official conceptions of mathematics and its teaching while maintaining their individual and independent blends of ideas.     

‘New mathematics’ and teaching

Inside the scientific world it is not always understood that the mood of mathematics, which is a product and a part of culture, can change with time. This is partly why many have been surprised by the coming of the so‐called new mathematics.

In the truly creative mathematical mind two opposite tendencies coexist: the logical and the imaginative. Apparently it seems that new mathematics can be reduced to a purely logical machinery. In fact it contains as much imaginative contributions as classical mathematics. But it is difficult to show simultaneously the logical sequence of propositions and the clumsy progression of research itself. Mathematical exposition does not always follow the ‘ most natural slopes’ of the mind. Unfamiliar presentations often give an impression of ‘ abstraction ‘, more familiar ones an impression of concreteness ‘.

So it appears that difficulties with new mathematics are mostly of psychological origin. Misuses of it can easily raise up intolerance reactions and emotional blocks. Perhaps insisting upon the fact that, here as elsewhere, it is important to be able to guess, to realize that intuition and imagination are essential, could help to make new mathematics better understood, more useful and more able to be considered as a unifing element among sciences.     

Teachers’ views on dynamically linked multiple representations, pedagogical practices and students’ understanding of mathematics using TI-Nspire in Scottish secondary schools

Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons and what evidence do they provide to support their findings? Throughout session 2008–2009, this empirical research project involved six Scottish secondary schools, two mathematics teachers from each school and students from different ages and stages. Teachers used TI-Nspire PC software and students the TI-Nspire handheld technology. This technology is specifically designed to allow dynamically linked multiple representations of mathematical concepts such that pupils can observe links between cause and effect in different representations such as dynamic geometry, graphs, lists and spreadsheets. The teachers were convinced that the use of multiple representations of mathematical concepts enhanced their students’ relational understanding of these concepts, provided evidence to support their argument and described changes in their classroom pedagogy.     

Novice facilitators’ changes in practices: From launching to managing discussions about mathematics teaching

Novice facilitators of professional development (PD) programs for mathematics teachers often face challenges in leading productive discussions and achieving the goals of these programs. Although research in this area is gradually accumulating, not much is known about how novice facilitators address these challenges and change their practices accordingly. This paper presents case studies of two novice facilitators of PD programs in two different countries. The analyses look at their work over one year, to illustrate the changes in their practices while managing discussions. The results show that although the facilitators operated in different contexts, their practices and their processes of change resembled, suggesting that these processes are not idiosyncratic. We argue that novice facilitators’ changes in practices correspond to changes in their resources, orientations, goals, and identities and that PD program teams can support these changes.     

The relationship between preservice elementary teachers’ solutions to a pattern generalization problem and difficulties they anticipate in teaching it

To explore the relationship between elementary preservice teachers’ (PTs’) solutions to a pattern generalization problem and the difficulties they expected to encounter when teaching the same problem to students, we administered a task-based questionnaire to 154 participants at a large Southwestern university in the US. Employing inductive content analysis, we identified possible links between PTs' solutions and their anticipated difficulties. PTs who solved the problem using figurative reasoning tended to anticipate difficulties related to pedagogical moves to support students’ mathematical understanding. In contrast, PTs who solved the problem using algebraic formulations were likely to anticipate difficulties related to teaching algebraic knowledge and supporting procedural fluency. Also, only PTs who solved the problem using figurative reasoning anticipated difficulties associated with eliciting and evaluating student thinking, whereas PTs who used formulas to solve the problem expected difficulties related to their own self-efficacy and confidence. We discuss three implications for mathematics teacher education.     

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.     

Teachers’ pedagogical content knowledge in teaching word problem solving strategies

ZDM – Mathematics Education - This research addressed Hungarian pre-service and in-service (both elementary and lower secondary) teachers’ pedagogical content knowledge concerning the...     

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.     

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.     

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.     

Student teachers’ types of probing questions in inquiry-based mathematics teaching with and without GeoGebra

Previous studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims of this study are to elaborate on different ways of asking students to give explanations in inquiry-based mathematics teaching and to investigate whether questioning in GeoGebra lessons differs from questioning in other lessons. Data was collected by video recording 29 Finnish mathematics student teachers’ lessons in secondary and upper secondary schools. The lesson videos were coded for the student teachers’ probing questions. After this, categories for the types of probing questions were created, which is elaborated in this paper. It was found that the student teachers who used GeoGebra emphasized conceptual probing questions during the explore phase of a lesson slightly more than the other student teachers.     

The concept ‘model’ and its potential role in mathematics education

Under the general term ‘mathematical models’ is now subsumed a sufficiently diversified collection of distinct types of models to merit categorization and an attempt at a unified definition. In this study a broad ‘working definition’ for the concept ‘model’ is proposed; basic types of models are identified and illustrated, and the impact of conscious instruction of models on mathematics education is analysed.     

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 relationship between the Fermi–Dirac distribution and statistical distributions in languages

Mathematical Notes - In this article, we study, from the mathematical point of view, the analogies between language and multi-particle systems in thermodynamics. We attempt to introduce an...     

Teachers’ gestures and speech in mathematics lessons: forging common ground by resolving trouble spots

This research focused on how teachers establish and maintain shared understanding with students during classroom mathematics instruction. We studied the micro-level interventions that teachers implement spontaneously as a lesson unfolds, which we call micro-interventions. In particular, we focused on teachers’ micro-interventions around trouble spots, defined as points during the lesson when students display lack of understanding. We investigated how teachers use gestures along with speech in responding to such trouble spots in a corpus of six middle-school mathematics lessons. Trouble spots were a regular occurrence in the lessons (M = 10.2 per lesson). We hypothesized that, in the face of trouble spots, teachers might increase their use of gestures in an effort to re-establish shared understanding with students. Thus, we predicted that teachers would gesture more in turns immediately following trouble spots than in turns immediately preceding trouble spots. This hypothesis was supported with quantitative analyses of teachers’ gesture frequency and gesture rates, and with qualitative analyses of representative cases. Thus, teachers use gestures adaptively in micro-interventions in order to foster common ground when instructional communication breaks down.