Mathematics Teaching Practices Research Articles

Demystifying Proofs Through Structured Interaction: A Case Study of One Instructor’s Teaching in an Undergraduate Analysis Course

The concept of proof is central to every undergraduate mathematics curriculum, but it is also a difficult concept for university students to understand and for university instructors to teach. Prior studies have contributed useful research knowledge about the nature and sources of undergraduate students’ difficulties with proof, but they have paid less attention to the development of effective instructional practices to enhance student learning of proof. As a step towards this direction, this paper makes a contribution to the limited body of case studies of promising or potentially effective teaching practices in undergraduate mathematics by reporting a case study of an instructor’s teaching, locally characterized as “effective,” in an undergraduate analysis course at a leading Indian university. The instructor did not deviate from the so called “Definition-Theorem-Proof” (DTP) format that is followed in most proof-oriented university mathematics courses, but her teaching presented a set of features that, collectively, form a rather innovative teaching practice at the undergraduate level. Specifically, our case study shows that, even in the context of a rather crowded university classroom, proof can be demystified for students through structured interaction between the instructor and the students, that is, an interactive, conversational style of proof instruction invoking the participation of students. This is based on a solid foundation in symbolic logic at the very outset and on a significant time investment in definitions being explained in depth using informal language, visual aids, and real-life analogies. In addition to contributing to existing images of potentially effective teaching practices in undergraduate mathematics, this paper draws attention to the Indian educational context that has had little representation in international forums of mathematics education research thus far.

Preface

H Retnawati1, Kismiantini1 and O A Montesinos-López2 1 Mathematics Education Department, Universitas Negeri Yogyakarta, Indonesia 2 Faculty of Telematic, Universidad de Colima, MexicoWe are honored to present a collection of articles from the 3nd International Seminar on Innovation in Mathematics and Mathematics Education (ISIMMED 2019) which was conducted in Universitas Negeri Yogyakarta, Indonesia from 3 to 4 October 2019. In accordance with the development of science and technology, the practice of research activities and learning activities needs to be adjusted. This adjustment needs to be linked to Science, Technology, Engineering and Mathematics (STEM). All of us know that the course on STEM is inseparable from mathematical content such as algebra, geometry, analysis, applied mathematics, statistics, and computing, and also teaching and learning.With the theme of “Innovation in STEM Education: Current Research Trends and Practices”, this seminar discussed the growth of technology that gave many impacts in all part of life, particularly in mathematics and mathematics education. The sub theme is research in STEM Education, teaching and instruction in stem education, assessment in stem education, ethnomathematics, statistics and its applications, algebra and its applications, geometry and its applications, analysis and its applications, and computer science and its applications. We hope that this proceeding will generate innovative concepts and practices in mathematics and mathematics education, and as an attempt to build competitive human resource for researchers, educators, practitioners and policy makers to share their experiences during the ISIMMED 2019. After a series of review process, 72 articles are selected to be published in this Scopus-indexed proceeding. The remaining articles are published in the regular proceeding.The editors and the committees of ISIMMED 2019 would to thank the participants who have contributed and shared their scientific works in this proceeding. We also would to express our gratitude to every committee member for organizing the conference and to Universitas Negeri Yogyakarta for the financial support.2 March 2020EditorsHeri RETNAWATIKISMIANTINIOsval Antonio MONTESINOS-LÓPEZList of Committees are available in the pdf.

Climber Prospective Teacher: Relationship Beliefs and Mathematics Teaching Practice

This research explained the description of the beliefs of prospective teacher who had an adversity quotient type climber and the relationship between beliefs and mathematics teaching practices in the classroom. One student of climber prospective teacher who had taken an internship course and had good communication skill was willing to become a research subject. Semi-structured interview and observation of learning in class were used to obtain the research data. The data that had been obtained were coded, reduced, presented, triangulated to obtain credible data, and then conclusions were drawn. The results show that the climber math prospective teacher believes that mathematics is as something dynamic, namely the space of creation, human invention that develops continuously. Teaching mathematics is student-centered through problem providing or problem solving, and learning is as a construction of student’s active understanding to build knowledge. A more detailed explanation of beliefs description and the relationship with the mathematics teaching practice is described in this research.

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.

Banality of mathematical expertise

Practices within research mathematics can and do serve as models for mathematics education. However, typically such inspirations impose a devastating narrowness in relation to reflections on mathematics. This narrowness I refer to as the “banality of mathematical expertise”. Reflections on mathematics can be expressed through a philosophy of mathematics that goes beyond the traditional emphasis on ontological and epistemological dimensions, to become four-dimensional by also addressing social and ethical issues. Many working philosophies of mathematics operate within a narrow scope of reflections, seemingly located within an ethical vacuum. The consequence is a cultivation of a banality, manifest in many university studies in mathematics as well as in dominant research paradigms in mathematics. This constitutes a serious limitation in providing models for mathematics education. By contrast, there exist examples of practices of mathematics education that demonstrate a richness of reflections on mathematics. Accordingly, I address the extent to which such practices of critical mathematics education could serve as models for research mathematics and mathematics education at the university level.

Teacher Professional Learning with/in Place: Storying the Work of Decolonizing Mathematics Education from within a Colonial Structure

We come together, five mathematics teacher educators from varied cultural backgrounds and diverse academic pathways, interested in teacher professional learning, and interested in exploring our understanding of colonial practices in mathematics education specifically and in education more generally. In this paper, we share our stories, drawing upon our own experiences in conversation with each other and in dialogue with academic readings. Our paper studies tensions encountered as we explore decolonizing educational practices within colonial structures, paying close attention to place/land-based pedagogies. In recognizing education as a colonial act, we examine some of the literature in decolonizing education and research before introducing arguments for the role of mathematics education in colonizing educational practices. As a collaborative research group, we met regularly over a period of 3 months, collecting audio recordings of our meetings, transcriptions, shared individual writings, and written responses to each other’s writing and academic readings. We draw on narrative inquiry to structure our experiences of decolonizing mathematics teacher professional learning through living and telling stories and then through re-telling and re-living stories. We argue that this process is rewarding and challenging, and requires individual and collective ongoing dialogue. With our stories, we challenge the places where mathematics is performed, for example from classrooms to learning gardens, and what counts as mathematics. We conclude with questions to frame future directions and dialogues with an invitation to others to respond through critical dialogue and practice.

Transformar Exercícios em Cenários para Investigação

O texto descreve e analisa atividades sobre educação matemática crítica realizadas em uma disciplina de um curso de licenciatura em matemática. O referencial teórico adotado consistiu especialmente dos conceitos de paradigma do exercício e cenários para investigação, de Ole Skovsmose. A principal atividade consistiu em os licenciandos abrirem um exercício de um livro didático a fim de transformá-lo em uma atividade investigativa. O objetivo geral de ensino foi refletir sobre possibilidades de modificar as práticas docentes tradicionais de ensino de matemática. A pesquisa qualitativa teve como objetivo compreender os aspectos considerados pelos estudantes nesse processo de abrir um exercício e as características atribuídas por eles para uma atividade investigativa. Os primeiros resultados indicam que uma atividade investigativa precisa ter um apelo à realidade dos alunos da educação básica. A realização da atividade e sua discussão foram consideradas um primeiro passo dos licenciandos em direção à educação matemática crítica.

EXPLORING METACOGNITIVE REGULATION AND STUDENTS’ INTERACTION IN MATHEMATICS LEARNING: AN ANALYSIS OF NEEDS TO ENHANCE STUDENTS’ MASTERY

Purpose of the study: This study is a survey of teachers’ and students' perceptions of mathematics teaching and learning practices. The purpose of this study was to determine the level of learning practice based on metacognitive regulation strategies and student interactions to increase student achievement. Methodology: A quantitative survey of 45 mathematics teachers and 52 Form 2 students using questionnaires was conducted to look at the level of mathematics teaching and learning practices and to obtain agreement on the need for learning metacognitive regulation strategies and student interactions. Findings: The findings show that teachers' level of knowledge about metacognitive regulation strategies, level of teaching and learning based on metacognitive regulation and student interaction strategies, and level of mastery and application of student mathematics skills are moderate. Implications: Furthermore, teachers and students agree that teaching and learning practices based on metacognitive regulation strategies and student interaction need to be implemented as a mathematics learning intervention to enhance students’ mastery in mathematics. Implications of this study led to the suggestion that metacognitive regulation strategies be used as a best practice and guide to transform teaching and learning methods. Novelty/Originality of this study: This study is an extension of the study of aspects of metacognitive skills by looking at both the view of the teacher and the student. Results show that students' metacognitive skills and interactions are related to and influence learning.

Using video-Based on Tasks for Improving Mathematical Practice and supporting the productive struggle in Learning Math among Student Teachers in the Faculty of Education

The main objective of the current research is to identify the use of video-Based on Tasks to improve Mathematical Practice and support the productive struggle in learning mathematics among teachers of the faculty of Education. The research tools were prepared and adjusted that included a test, score card, and support for productive struggle in mathematics learning. Basically, it was applied to a sample consisting of (20 male and female students) from the third-year students at the Faculty of Education, Tanta University. The study adopted the quasi-experimental curriculum with experimental design (pre-design after one group). In effect, the findings of the research manifested the effectiveness of using video-Based on Tasks to improve mathematical practices and support the productive struggle of student teachers in the faculty of Education. Accordingly, it was recommended to pay attention to international standards for mathematical practices and effective mathematical teaching practices. In addition, it is vital to pay attention to employing video-based on tasks and activating them within the school curriculum.

Early Mathematics Teaching Practices for Students with Autism Spectrum Disorders

Early Mathematics Teaching Practices for Students with Autism Spectrum Disorders

An experience of cultural transposition of the El’konin-Davydov curriculum

We will present cultural transposition as a particular perspective to frame the use of foreign mathematics education practices as an opportunity for questioning the didactic practices of one’s own cultural context. This requires a process activated by researchers, who deconstruct the cultural layers underpinning the foreign education practice before proposing it to teachers. We discuss the theoretical premises of this approach and, in accordance with them, we propose a transposition process of the El’konin-Davydov curriculum. In particular, we will show how our deconstruction process has affected the design and implementations of particular Professional Development courses (PDs) in Italy. Finally, we will present a case study of a teacher involved in one of these PDs to observe her new educational awareness.

Beyond quality metrics: defying journal rankings as the philosopher’s stone of mathematics education research

In recent years, the field of mathematics education has witnessed the emergence of several journal rankings. Within these rankings, Ibero-American journals have had little—to almost no—presence. This raises awareness on the current state of journal indexes and on what these indexes do and how they have been used in constituting sites of exclusion within the mathematics education community. We contend that ranking systems are perceived as the philosopher’s stone of academia, in the sense that they have the ability to convert any material (an article, a journal, the academic production of a scholar) into a precious material within the academic world. This alchemic move not only exacerbates exclusion, but also configures a point system that constitutes places for enjoyment and fetishistic disavowal within researchers. Thus, we want to challenge the assumption that mainstream journal rankings are able to accurately reflect the quality, impact, and reputation of mathematics education journals by taking as “empirical data” the Ibero-American journals in the field of mathematics education. In problematizing journal ranking effects in the production, dissemination, and socialization of knowledge, we seek to open a discussion regarding the economic-political dynamics that govern publishing practices in mathematics education and that are entangled in the production of knowledge in our discipline.

Identifying Effective Mathematics Teaching: Some Questions for Research

This paper argues that there is a gap in the research on teaching and learning mathematics, and thus a need for research into effective teaching practices. From a selective review of literature, reasons are examined as to why less emphasis appears to have been placed on generating knowledge about teaching and effective mathematical teaching practices than on mathematics teachers and their knowledge. Possible reasons for this lack of attention to teaching include assumptions that, as researchers, we already know what effective teaching is or that if teachers embrace certain principles about teaching or learning, then effective teaching will necessarily ensue. Arguing that both of these positions neglect to address important aspects of developing knowledge about effective teaching, the paper raises a number of questions that research might address in moving the balance more towards examining the actual practices that might characterise effective teaching.

The Problem-Based Threshold: Shifting Pre-Service Teachers’ Thinking About Mathematics Instruction

The purpose of the mathematics methods course where this study took place was to support pre-service teachers (PSTs) as their thinking about instruction transitioned from a traditional, teacher-centered approach to a more problem-based, student-centered approach, and to guide them as they developed strategies for teaching through problem solving. We used writing assignments to prompt PSTs to think critically about mathematics instruction. We then analyzed their writing to determine whether PSTs’ thinking about instruction changed throughout the course. The results show that at the beginning of the course, PSTs’ thinking was deeply rooted in traditional views, but that as the course progressed, their thinking shifted toward more problem-based views. Additionally, we examined PSTs’ writing to look for instances where their traditional thinking interfered with their emerging problem-based ideas about instruction. Determining the ways PSTs’ struggled to develop problem-based views has significant implications for mathematics methods course design and teaching practices.

An Exploratory Survey on Teaching Practices Integrating Nutrition and Mathematics in Australian Primary Schools

Overweight and obesity prevalence in children has increased worldwide. One of the stated reasons for this rise is the increase in portion sizes, perhaps due to individuals having difficulties with portion size estimation. Portion size estimation could be improved with portion size education involving the mathematical concepts of volume and capacity. The current study aims to explore mathematical teaching practices focussing on volume and capacity measurement in Australian primary schools. A convenience sample of 101 teachers completed an online survey reporting information on their teaching background, experience, use of resources, digital games and curricular integration. We found that teachers tend to use Smartboards and cubes to teach volume and capacity, and that the majority use nutrition-related examples such as household measures, recipes and grocery shopping. Teachers also expressed having positive beliefs and interest in using digital games for these concepts. Although mathematics is most commonly integrated with science and technology, teachers believe a digital game integrating nutrition and mathematics could be very useful. Research on the development and implementation of an educational resource to combine these subjects using technology is needed.

An Examination of the Relationships of Mathematics Self-Efficacy and Teaching Practices among Elementary, Secondary, and Special Education Educators

ABSTRACTTeacher self-efficacy and teaching practices are both related to effective teaching and learning of mathematics. Given the theoretical and empirical basis for the importance of teacher self-efficacy, we hypothesized that teacher reported self-efficacy would be positively related to teacher self-reported effective teaching practices. This study examined the self-reported construct data across in-service educator grade level (i.e., elementary and secondary) and educator position (i.e., general education or special education) to determine the associations of constructs and the potential for differences between groups. Correlations between self-efficacy and teacher practices in-service teachers were analyzed using Kendall’s Tau. A Kruskal-Wallis one-way analysis of variance was used to analyze the three participant groups (general education: elementary, general education: secondary, and special education) for the teacher self-efficacy and self-reported teaching practices constructs. Similarities and differences among the relationships between educator grade level, educator position, self-efficacy, and teaching practices are discussed. In addition, implications for future research and professional development with educators are outlined.

Observing Mathematics Teaching Practices to Promote Professional Development: An Analysis of Approaches to Probability

This study aims to analyse the approaches to probability that were carried out by the participants of the teaching practice of Primary Education teachers and, more specifically, their teaching trajectories. To do this we analysed 23 video-recorded classes of all levels, using a previously validated instrument. Results show trajectories characterised by a strong presence of intuitive approach at initial levels, in conjunction with a much lower presence of frequency approach and an absence of other approaches. As the level increases, the presence of intuitive approach decreases, being replaced by subjective, frequency and classical ones, with the latter two having the greatest presence in the upper levels. We conclude that results will be useful for strengthening the professional development of Primary Education teachers in relation to the teaching of probability.

Preservice teacher learning experiences of entrepreneurial thinking in a STEM investigation

Connections between innovation policy and science, technology, engineering, and mathematics (STEM) education practices are poorly defined across a variety of national and international contexts. This disconnect is particularly evident in countries such as Australia that are not traditionally associated with an entrepreneurial culture. This study explores connections between entrepreneurial thinking and STEM Education in the context of a pre-service teaching course where learning is experienced through an integrated STEM investigation project. An entrepreneurial problem-validation technique is applied by preservice teachers as the starting point for their investigation, and it is the performance and experience of learning through that technique, which is the focus of this paper. My research question is: How do pre-service teachers perform and experience entrepreneurial validated learning as part of a STEM investigation? To address this question the study is informed by a phenomenological orientation drawing upon preservice teachers’ investigation reports and reflective journals. Data were analyzed using a phenomenological form of inductive analysis to develop key themes that describe learning experiences when generating and validating ideas and problems for a STEM investigation. Findings from this study point to the emotional experiences of learning through entrepreneurial thinking, how these experiences challenge learner values, and the need to explore further the connections between entrepreneurial thinking and STEM education in naturalistic contexts.

Abolition by the Numbers Mathematics as a Tool to Dismantle the Carceral State (and Build Alternatives)

Galvanized by organized communities, movements against mass incarceration have steadily coalesced and crept into schools over the last decade and a half. This political moment creates new opportunities for those working for justice in communities and in schools, and demands that subject-matter educators engage in this work. While there is a robust “reform” discourse within mathematics education, the field has been constructed as neatly outside the carceral sphere and remains relatively isolated from this burgeoning movement. This article draws on the authors’ practices and bodies of research to explore mathematics as an agent of the carceral state and to imagine mathematics education practices may stop the state from harming—including killing—Black, Brown, queer, migrant, poor, and other marginalized people. The authors also consider how mathematics could be a core tool for an abolitionist movement.

Mathematical competencies revisited

This article deals with what it means to possess competence in mathematics. It takes its point of departure in the fact that the notions of mathematical competence and mathematical competencies have gained a foothold as well as momentum in mathematics education research, development and practice throughout the last two decades. The Danish so-called KOM Project (KOM: Competencies and the Learning of Mathematics), the report from which was published in 2002, has played an instrumental role in that development. Since then, a host of new developments has taken place, and we—as the authors of the original report—have felt the need to take stock of this development and revisit the conceptualisation of the basic notions in order to provide an updated version of the original conceptual framework and terminology. Whilst the fundamentals of this framework have been preserved in this article, the version presented here in addition to an up-to-date terminology offers greater clarity and sharpness and richer explanations than found in the original.