Mathematical Practice Research Articles

The MTSK model as a tool for designing tasks for teacher education

This paper studies the knowledge developed by a group of sixty-two pre-service primary teachers during a training session on the definition of a polygon and the mathematical practice of defining. We explore the knowledge developed by pre-service teachers when they carry out a series of training tasks oriented around the practice of defining with primary pupils, within the theoretical framework of the Mathematics Teachers’ Specialised Knowledge model, and using videos of an authentic lesson as prompt. Data were collected by means of video recordings of the implementation of the training tasks, in conjunction with the pre-service teachers’ own written observations as they watched the recorded lesson. A content analysis was carried out on the collected data using the Mathematics Teachers’ Specialised Knowledge model. The results highlight the way in which knowledge of the mathematical practice of defining is constructed, along with the mathematical objects involved in this practice, and pedagogical aspects of defining with young pupils. They also illustrate how a sequence of tasks oriented around a model of teachers’ knowledge can contribute to the development of this knowledge in initial training programmes, and more especially provide orientation about the training of pre-service teachers in the mathematical practice of defining.

A Review of Brahmagupta's Contributions

The life, accomplishments, and legacy of prominent Indian mathematician and astronomer Brahmagupta from the 7th century are examined in this review. Brahmagupta, who is well-known for his contributions to algebra, arithmetic, and astronomy, provided ground-breaking ideas in his "Brāhmasphuṭasiddhānta,"a foundational work that included the rules for mathematical operations involving zero and negative integers. His efforts established the groundwork for contemporary mathematics, inspiring other researchers in mediaeval Europe as well as the Islamic world. In order to emphasize Brahmagupta's methods, his contribution to the advancement of mathematical thought, and the ongoing influence of his work on current mathematical practices, this review synthesizes historical accounts and contemporary interpretations.

Aksiologi Sains pada Matematika

This research investigates the relationship between the axiology of science and mathematics, focusing on values, ethics, and norms in the context of scientific research. The study underscores the importance of integrating the axiology of science into mathematics, highlighting that values and ethics should be integral considerations in the development of mathematical knowledge. The research methodology employed is a comprehensive literature review, exploring the contributions of ethical values to the development of mathematics and the ethical implications in mathematical practice. The results of the study reveal that mathematics is not a separate entity from values and ethics; rather, it is closely intertwined with them. The research details how personal values, scientific values, and social values influence the mathematical research process and its application in various scientific disciplines. Ethical implications in the development of mathematical theory are also a significant focus, considering how ethics affect axiomatic considerations, definitions, and theorems in mathematics. This research suggests that integrating the axiology of science into mathematics is highly relevant for the advancement of mathematical knowledge. It helps mitigate biases, enhances understanding of virtues, raises ethical awareness, and improves the abilities of mathematicians to adhere to values, ethics, and norms in their research. The research findings hold significance in the contexts of mathematical education and practice and in promoting ethical considerations in the use of mathematics in science and society.

Denying Infinity: Pragmatism in Abraham Robinson’s Philosophy of Mathematics

Abraham Robinson is well-known as the inventor of nonstandard analysis, which uses nonstandard models to give the notions of infinitesimal and infinitely large magnitudes a precise interpretation. Less discussed, although subtle and original–if ultimately flawed–is Robinson's work in the philosophy of mathematics. The foundational position he inherited from David Hilbert undermines not only the use of nonstandard analysis, but also Robinson's considerable corpus of pre-logic contributions to the field in such diverse areas as differential equations and aeronautics. This tension emerges from Robinson's disbelief in the existence of infinite totalities (any mention of them is ‘literally meaningless’) and the fact that much of his work involves them. I argue that he treats infinitary avenues of mathematics as useful tools to avoid this difficulty, but that this is not successful to the extent that these tools must be justified by a conservative extension from finitary mathematics. While Robinson provides a compelling and unorthodox pragmatic justification for the role of formal systems in mathematical practice despite their apparent infinitary presuppositions, he deflates mainstream mathematics to a collection of games that occasionally produces meaningful results. This amounts to giving up on a commitment to reconciling his finitism with his mathematical practice.

Disrupting racial storylines about black girls in mathematics through teaching content and building relationships

This study is a qualitative case study that focuses on the instructional practice of a white woman teacher with three Black girls. We draw upon literature that describes racial storylines for Black girls to focus on how a white Woman teacher navigates racial storylines to enact instructional practices that are humanzing for Black girls. Specifically, we show how interactions the teacher has with three Black girls disrupt racial storylines about invisibility/hypervisibility, behavior, and ability. Implications of this work pushes back against the idea that Black girls are monolithic and illustrate practices in mathematics that support the success of Black girls.

Bridging the gap: Examining parental involvement strategies and their impact on homework completion rates in mathematics

This research examines the influence of parental involvement tactics on students' mathematics homework completion rates, aimed at pinpointing crucial factors and evaluating the efficacy of such strategies. Utilizing a qualitative-quantitative approach, including with students, parents, and teachers, the study identifies distractions, workload, family obligations, and insufficient teacher support as significant barriers to mathematics homework completion. Time management, comprehension, and fatigue also play minor roles. The findings indicate that targeted parental involvement strategies can significantly enhance completion rates, with effective methods including accessible support, designated study spaces, breaking down assignments, fostering student autonomy, and leveraging technology. Additionally, parental involvement fosters stronger parent-child relationships, boosts student confidence, and promotes a positive attitude toward mathematics. To create a conducive learning environment, schools, teachers, and parents should collaborate, implementing recommendations such as creating quiet areas, promoting digital detox initiatives, managing workload, offering targeted support, establishing study environments, and setting clear expectations for mathematics practice.

Problem-Based Learning in Blended Classrooms: An Approach in Covid 19 Emergency Situation

This research paper, reports the findings of using problem-based learning approach in Blended Learning (BL) environment, while practicing grade 9 mathematical problems by fifty grade 9 students. This three week program was framed in contrast of traditional schooling, intending as enrichment and not remediation, for science group students’ practices in mathematics. The study was framed through formative experiment research method and practiced at public sector Higher Secondary School Jehangira district Swabi, aiming to explore the advantages and hazards of PBL model when applied in BL environment. The data was collected through an open ended self-developed questionnaire and was analyzed by using simple frequency distribution. The results achieved were mostly positive such as students found productive PBL approach through WhatsApp application like solving examples helped them in solving other problems, course work was completed timely, and using smart phone with peers in groups helped in better understanding the problems. But showed some concerns such as poor availability of network/internet services or non-affordability of having smart phone or internet packages, and parents showed unwillingness for using smart phone/internet. On the basis conclusion, it was suggested that these issues can be resolved by teaming up with parents alongside convincing head of the institute.

Teachers’ mathematical work based on examples presented in the teaching of algebra in secondary education

Both international examination results and scientific evidence related to the teaching and learning of school algebra point to the necessity for analyzing mathematical practice in the classroom. In this context, we have formulated the research objective of characterizing the mathematical work that teachers favor based on the examples that they implement in the teaching of topics within the domain of secondary school algebra. To this end, this study is situated within the interpretive paradigm utilizing an instrumental collective case study approach. The study design entails the selection of two representative cases of mathematics teachers in Chilean schools, with the objective of understanding and describing their mathematical work in the secondary school classroom based on the examples they present. The results show that the first teacher utilizes varied examples that activate distinct forms of mathematical thinking, promoting the activation of both instrumental and discursive genesis. In contrast, the second teacher follows a more traditional approach, with the use of examples to illustrate and motivate, and a direct presentation of knowledge, activating primarily semiotic genesis, in which a transition between the numerical and the algebraic is lacking. Ultimately, the results highlight the importance of deeper consideration of the choice of examples in the teaching of algebra and how these examples can influence students’ learning. Furthermore, additional ideas are put forward for future work in this line of research associated with the use of examples in the classroom.

Unlocking mathematical marvels: Exploring number patterns in the Rangku Alu traditional game

Ethnomathematics, an interdisciplinary field at the intersection of cultural studies and mathematics, examines the intricate mathematical practices found within diverse cultural settings. This research focuses on Rangku Alu, a traditional game from Indonesia, to explore its embedded mathematical elements. Using a qualitative ethnographic methodology, the study uncovers the mathematical concepts within the game's cultural context. It traces the origins and cultural importance of Rangku Alu in Manggarai, East Nusa Tenggara, through combining participant observation, interviews, and literature review. The game, which prominently features bamboo, is a staple activity in harvest celebrations and serves as a way to express gratitude. During the game, players engage in synchronized movements that create rhythmic patterns and interactive sequences. The analysis reveals seven distinct numerical patterns, derived from the strikes of bamboo, jumping sequences, and player formations. These patterns include both odd and even sequences, as well as basic arithmetic operations. Rangku Alu not only entertains but also provides a unique setting for exploring numerical concepts. This study highlights the potential of integrating traditional practices like Rangku Alu into mathematics education to promote inclusivity and enhance mathematical understanding in multicultural environments.

STUDENTS’ PRACTICE OF MATHEMATICS AND ITS EFFECT ON THEIR NUMERICAL SKILLS IN PROBLEM-SOLVING IN SECONDARY SCHOOLS IN FAKO DIVISION, SOUTH WEST REGION OF CAMEROON

For several decades mathematics has been called the “critical filter”, because students who are inadequately prepared in mathematics during secondary school lose many career choices that would otherwise be available to them. Also, for those students who pursue post-secondary education in Cameroon, success in O-Level mathematics is a pre-requisite for most degree programmes in business, computing and the sciences. This makes mathematics very pivotal in reshaping the future and students’ attitude towards mathematics an area of concern. Negative attitudes towards mathematics are damaging, leading to disengagement, increased anxiety and lack of confidence, and reluctance to try to improve skills. Cameroon needs graduates with advanced mathematics skills to promote innovation, data synthesis, and technology if it is to solve challenging problems and be competitive in the global scenario. This study examined students’ practice of mathematics and its effect on their numerical skills in problem- solving in secondary schools in Fako Division, South West Region of Cameroon. The objective is to find out whether students’ practice of mathematics has any influence on their numerical skills in problem-solving. The mixed research methodology with convergent parallel design was used. The target population comprised 6350 form five students from secondary schools in Fako Division. The accessible population was made up of 1036 students and the sample size of the study was made of 512 form five students from six colleges using the random sampling technique. There were also 26 teachers and 6 Heads of Mathematics Department. Data was collected using questionnaire and interview guide. Instruments were validated and the reliability coefficients gave satisfactory values of 0.75 and 0.743 for students’ and teachers’ questionnaire respectively. The statistical analyses and findings reveal that students’ practice of mathematics has a positive correlation with their numerical skills in problem-solving. It was recommended that: (1) problem-solving should be part of each curricular unit and begin in kindergarten. For this to be effective, the teaching of problem-solving should not be isolated, instead, it can serve to support and enrich the learning of mathematics concepts and notation. (2) teachers should cultivate students’ interest in mathematics as early as possible. Varying classroom instruction practices could be a remedy to enhance students’ understanding, achievement, and motivation in learning mathematics.

Design and practice of secondary mathematics teaching based on PAD class model

Design and practice of secondary mathematics teaching based on PAD class model

The Critical Roles of a Mathematics Specialist in Establishing Effective, Coordinated Professional Development Systems

ABSTRACT For nearly three decades, mathematics research has indicated the need for high-quality mathematics instruction including both ambitious and equitable practices, as well as continuous, innovative learning opportunities for educators. Teaching practices and standards for mathematical practice have been identified to provide guidance. While research connects effective professional development (PD) to improved instructional practices, there is a need for additional research on the role mathematics specialists play in developing and sustaining PDs. This qualitative case study examined the multiple roles of a specialist while laying the foundational components for a coordinated PD system to improve and strengthen mathematics teaching and learning in elementary classrooms. The PD efforts included the implementation of a teacher subsystem where a specialist led all components consisting of pull-out PD, mathematics coaching, collaborative time, and teacher networks. This study took place at a rural, pre-K through fourth-grade school in a southeastern state. Analysis of the data identified the specialist’s key roles in each component of the teacher subsystem. The results indicated that the development and implementation of a teacher subsystem impacts the overall effectiveness of PD. Findings serve as a foundation for specialists to design and implement coordinated efforts that can have positive impacts on the teaching and learning of mathematics.

Equitable Pedagogical Practices in Learning Mathematics at Secondary Schools in Nepal

The central focus of this study is to explore equitable pedagogical practices in learning mathematics in diversified learners’ mathematics classrooms. This study focused on investigating the challenges faced by mathematics teachers to conduct equitable pedagogical practices in learning mathematics in mathematics classrooms and exploring the possible ways to promote equitable pedagogical practices in learning mathematics. An interpretive critical theory paradigm was employed for collecting, analysing, and interpreting data in an iterative process. Qualitative narratives were collected through in-depth interviews, focus group discussions (FGD), and classroom observations of which five secondary-level mathematics teachers of Dhading were observed. Through multi-layered theme analysis and interpretation, the study produced five emergent challenges: linguistic issues, classroom diversity, the dominance of banking pedagogy, student absenteeism, school environment, and socioeconomic position. Moreover, the study showed eight different ways to promote equitable pedagogical practices in learning mathematics: providing additional support for marginalized students, using multi-methods of mathematics teaching, promoting peer and group work, individual treatment, motivational counseling, teaching mathematics relating to daily life, promoting collaborative learning and cooperative teaching, and promoting culturally responsive teaching.

A Formative Study Towards the Inclusion of Indigenous Technologies and Knowledge Practices in Science, Technology, Engineering, Arts, and Mathematics (STEAM) Curriculum Settings

A Formative Study Towards the Inclusion of Indigenous Technologies and Knowledge Practices in Science, Technology, Engineering, Arts, and Mathematics (STEAM) Curriculum Settings

A Formative Study Towards the Inclusion of Indigenous Technologies and Knowledge Practices in Science, Technology, Engineering, Arts, and Mathematics (STEAM) Curriculum Settings

This work on STEAM and education for sustainable development was undertaken across a series of collaborative case studies as formative research on the inclusion of Indigenous technologies and knowledge practices in teacher education. We noted that, despite the current academic imperative to decolonise southern African education, one seldom finds the inclusion of Indigenous technologies and Indigenous heritage practices in the contemporary classroom. Teachers we worked with were highly interested in their Indigenous knowledge heritage. Yet, they reported that they, and the young teachers they work with, simply do not have the heritage knowledge capital to include Indigenous knowledge systems in their teaching. Other challenges they face are the time it takes to engage community knowledge holders, and to find knowledge relating to school subject disciplines. The teachers observed that students prioritise modernity over Indigenous heritage and technologies, often undervaluing the latter as a forgotten past. Three exploratory cases delved into teacher education's response to challenges through co-engaged work, and this paper synthesises the emerging evidence-aiming to refine pedagogical tools for integrating Indigenous knowledge into STEAM education. A cultural-historical approach was used to frame the study and to derive insights and inferences in co-engaged lesson design research with teachers.

On metaphors of mathematics: Between Blumenberg’s nonconceptuality and Grothendieck’s waves

How can metaphors account for the formation of mathematical concepts, for changes in mathematical practices, or for the handling of mathematical problems? Following Hans Blumenberg’s thought, this paper aims to unfold a possible answer to these questions by viewing the metaphorical frameworks accompanying these changes as essential for an understanding of how changes in mathematical practices have been accounted for. I will focus especially on cases in which these changes were caused by encounters with a mathematical object which did not yet have a well-defined concept, but also show that such indeterminacy remains with the mathematical concept even after it is considered ‘well-defined’. As the paper will show, this ‘forefield’ [Vorfeld] of the concept is addressed by Blumenberg’s account of metaphorology on the one hand, and accompanied by his later account of nonconceptuality [Unbegrifflichkeit] on the other hand. While Blumenberg himself did not develop a full-fledged philosophy of mathematics or of mathematical practices, I aim to show that one can nevertheless extract from his writings a unique position concerning the role metaphors play in mathematics. To this end, Blumenberg’s account of nautical and oceanic metaphors and Alexandre Grothendieck’s philosophy of mathematical practice provide fruitful starting points.

Problem solving in mathematics education: tracing its foundations and current research-practice trends

In tracing recent research trends and directions in mathematical problem-solving, it is argued that advances in mathematics practices occur and take place around two intertwined activities, mathematics problem formulation and ways to approach and solve those problems. In this context, a problematizing principle emerges as central activity to organize mathematics curriculum proposals and ways to structure problem-solving learning environments. Subjects’ use of concrete, abstract, symbolic, or digital tools not only influences the ways to pose and pursue mathematical problems; but also shapes the type of representation, exploration, and reasoning they engage to work and solve problems. Problem-solving foundations that privilege learners’ development of habits of mathematical practices that involve an inquiry method to formulate conjectures, to look for different ways to represent and approach problems, and to support and communicate results shed light on directions of current research trends and the relevance of rethinking curriculum proposals and extending problem-solving environments in terms of teachers/students’ consistent use of digital tools and online developments.

Defining “Ethical Mathematical Practice” Through Engagement with Discipline-Adjacent Practice Standards and the Mathematical Community

This project explored what constitutes “ethical practice of mathematics”. Thematic analysis of ethical practice standards from mathematics-adjacent disciplines (statistics and computing), were combined with two organizational codes of conduct and community input resulting in over 100 items. These analyses identified 29 of the 52 items in the 2018 American Statistical Association Ethical Guidelines for Statistical Practice, and 15 of the 24 additional (unique) items from the 2018 Association of Computing Machinery Code of Ethics for inclusion. Three of the 29 items synthesized from the 2019 American Mathematical Society Code of Ethics, and zero of the Mathematical Association of America Code of Ethics, were identified as reflective of “ethical mathematical practice” beyond items already identified from the other two codes. The community contributed six unique items. Item stems were standardized to, “The ethical mathematics practitioner…”. Invitations to complete the 30-min online survey were shared nationally (US) via Mathematics organization listservs and other widespread emails and announcements. We received 142 individual responses to the national survey, 75% of whom endorsed 41/52 items, with 90–100% endorsing 20/52 items on the survey. Items from different sources were endorsed at both high and low rates. A final thematic analysis yielded 44 items, grouped into “General” (12 items), “Profession” (10 items) and “Scholarship” (11 items). Moreover, for the practitioner in a leader/mentor/supervisor/instructor role, there are an additional 11 items (4 General/7 Professional). These results suggest that the community perceives a much wider range of behaviors by mathematicians to be subject to ethical practice standards than had been previously included in professional organization codes. The results provide evidence against the argument that mathematics practitioners engaged in “pure” or “theoretical” work have minimal, small, or no ethical obligations.

Meaning-making systems: A multimodal analysis of a Latinx student’s mathematical learning

This article discusses multimodal practices in the context of teaching and learning and how this idea might inform and facilitate mathematical learning, especially for Latinx students. We discuss qualitative data drawn from a study of an elementary bilingual classroom (age 10 and age 11) in a Midwestern city (USA) that is exceptional because the students successfully do high-level mathematics. We describe one class episode and one student’s use of multiple resources to create meaning. Through this we highlight the nature and relevance of multimodal practices for learning mathematics. This case highlights the necessity of creating environments, where students, especially those who have been historically excluded, use resources to make meaning and gain greater access to mathematics.

Mathematical practices are everywhere: The intersections of pre‐service teacher claims, non‐mathematics‐education faculty claims, and observable actions

AbstractMathematics traditionally has been taught in a way that reifies the cultural belief that it is discrete in nature and, therefore, irrelevant to other content areas and the real world. National Council of Teachers of Mathematics, inclusive of mathematics teacher educators, views this belief as unproductive to learning and has stressed the need for students to make connections between mathematical content areas, to other disciplines, and to the real world. Unfortunately, many teacher‐preparation programs (TPPs) structurally are organized with overt divisions between content areas, which likely perpetuates the belief that mathematics is disconnected and irrelevant. Nonetheless, this project was motivated by the hope that at the individual classroom level within such TPPs, non‐mathematics teacher educators (TEs) were integrating mathematical practices and, thus, assisting in changing this unproductive belief. An analysis was completed of a three‐part, data‐collection process of (a) 180 pre‐service teachers' assignments requiring that they align activities from non‐mathematics education courses with mathematical practices; (b) 16 TEs' questionnaire data on how they use mathematical practices in their instruction; and (c) 11 full‐class, video‐taped observations of TEs teaching. Findings revealed that TEs do implement mathematical practices but do not claim always that this is what they are doing.