Mathematical Practice Research Articles

Beyond content: exploring the neglected dimensions of mathematics literacy

Abstract Becoming mathematically literate means not only knowledge of the content of mathematics but also understanding of the nature of mathematics and the literate practices involved in the creation, communication, and consumption of its content. This case study examined one mathematician’s view of the nature of mathematics and his literate practices. Data collected include semi-structured interviews with the mathematician, observations of his daily work routines, and his think-alouds during the reading of a disciplinary text. These data were analyzed qualitatively through an iterative process involving multiple readings and identification and refinement of codes. The analysis revealed that the mathematician (a) viewed mathematics as rigorous and demanding, both theoretical and practical, relatively stable but highly rewarding, interconnected with other disciplines, and involving discipline-legitimated discursive practices; (b) engaged in extensive reading/viewing and writing, valued learning from repeated trials and errors, and collaborated with an international network of peers in research; (c) used a range of reading strategies (e.g. close reading, summarizing, questioning, storying, evaluating, annotating) to help him make sense of the disciplinary text; and (d) marshaled both verbal and visual resources to create specialized knowledge, engage in rigorous reasoning, develop logical argument, and construct professional identity. These findings provide important insights that can help teachers design activities that are authentic to mathematics practices and effective for promoting mathematics literacy.

Mathematics Teacher Educators’ Practices to Support Teachers in the Design of Mathematical Tasks

Since teachers have the greatest impact on student learning, it is crucial to consider how professional development programs (PDP) for teachers can enhance their contribution, especially in designing mathematical tasks for teaching. This paper focuses on identifying patterns of practices of mathematics teacher educators related to crucial aspects of two teacher PDPs: one conducted face-to-face and the other using a Massive Open Online Course (MOOC). The Meta-Didactical Transposition is employed as the theoretical framework for comparing the two PDPs and for identifying patterns of practices. The findings suggest that educators, both in face-to-face and online settings, consider certain practices to guide teachers in designing mathematical tasks. This paper aims to share experiences of good practices that can be implemented by other researchers seeking to guide teachers in task design, either alone or in small groups.

Integration of ICT’s into assessment practices of school mathematics

Este artículo pretende divulgar un análisis de revisión de literatura sobre la integración de las Tecnologías de la Información y Comunicación (TIC) a las prácticas evaluativas para el aprendizaje de las matemáticas escolares. Para tal fin, se lleva a cabo la respectiva revisión en tres fases: fase de búsqueda, fase de selección y fase de sistematización-análisis. La última etapa recibió especial énfasis, debido a que se logra desarrollar una investigación de carácter cualitativo con enfoque descriptivo. Además, en la etapa de sistematización, la información fue triangulada a través de una bitácora de búsqueda y una matriz de revisión de literatura. Estos dos instrumentos permitieron establecer 39 categorías iniciales de los textos estudiados. Luego, estas categorías pasaron por un proceso de análisis y síntesis, especificando tres categorías generales relacionadas con la pregunta. Este proceso de indagación viene esclareciendo, entre otros aspectos, que sin importar el tipo de evaluación que se articule a las TIC, los medios de recolección de información más usados son: test diagnósticos, exámenes y cuestionarios pensados para la medición y estandarización de los aprendizajes, dejando las técnicas de evaluación en las que el estudiante tiene un rol activo en lugares secundarios.

What is the area of a pig? Problem posing and problem solving in early childhood education

Problem posing and problem solving serve as a crucial element of classroom instruction in early mathematics education, and have long been a topic of study of many practitioners and researchers. Used as a powerful tool for differentiation, they affect the ways in which the practice of mathematics is perceived by students and also help teachers gauge children’s understanding of concepts. In the student-centered approach, problem posing and problem solving can successfully engage students in creative educational situations. This research has been part of a broader design research project, the aim of which was to investigate the relationship between a classroom environment that allows for dialogue and young children’s propensity to design and solve their own tasks. Methodology included taking field notes and photographs, followed by reflection sessions. Turning young mathematicians into independent inquirers helped them gain authentic ownership of their knowledge. Additionally, it aided in the development of the young children’s competencies in effective engagement in problem posing activities. The toolbox of instructional techniques for problem posing in the classroom evolved, transforming mathematical classrooms into inquiry polygons for all learners.

A Conceptual-Methodological Framework to Investigate the Mathematical Practices with DGS in Secondary Students

This contribution presents a conceptual-methodological framework to study mathematical practices from the instrumental aspect inherent to the activity of learning instrumented with Dynamic Geometry Systems (DGS) of a student at a secondary school [In Argentina, secondary education is compulsory training for young people from 12 to 17 years old]. This framework has a conceptualization of the mathematical practice with Dynamic Geometry Systems and a method to favour its operationalization. The framework has been built from diverse theoretical perspectives, in the environment of ergonomic approach of instrumental approximation in a didactical perspective and a qualitative methodology. The method to define the practice indicators that allow to identify it in the field is illustrated with parts of the results of the research in which it originated. One important implication of the framework is that it values, from a didactical perspective, the instrumental character of the activity of using Dynamic Geometry Systems in students that learn geometry, through a concrete definition of mathematical practices with Dynamic Geometry Systems and the operationalization of its analysis for a geometry content. This framework is an original contribution that refers to the knowledge about the existence of mathematical practices with Dynamic Geometry Systems about the content of school geometry.

Characterizing Secondary Teachers’ Structural Reasoning

The Common Core State Standards for Mathematical Practice asks students to look for and make use of structure. Hence, mathematics teacher educators need to prepare teachers to support students’ structural reasoning. In this article, we present tasks and rubrics designed and validated to characterize teachers’ structural reasoning for the purposes of professional development. Initially, tasks were designed and improved using interviews and small pilot studies. Next, we gave written structure tasks to over 600 teachers in two countries and developed and validated rubrics to categorize responses. Our work contributes to the preparation and support of mathematics teachers as they develop their own structural reasoning and their ability to help students develop structural reasoning.

A Mathematics Educator Walks into a Physics Class: Identifying Math Skills in Students’ Physics Problem-Solving Practices

AbstractOne of the main goals of lower division “service” mathematics courses is to provide STEM-intending students with opportunities to engage in activities and contexts that can support their efforts to apply the mathematical ideas they are learning to successive major courses. The Mathematics Association of America has supported many mathematicians’ efforts to ask partner discipline faculty what topics and habits of mind they feel should be covered in mathematics classes to prepare them for their subsequent classes. We add to this work with a twist: Instead of asking physics faculty what they want students to know, we analyzed videos that students in an introductory physics class created so that we could ask ourselves what mathematical practices were most and least prevalent in the students’ physics problem-solving efforts. A qualitative analysis of the results, which we present here, indicated that most students were proficient in math practices involving problem setup and that the majority were able to apply mathematical concepts such as trigonometry and the solving of algebraic equations. However, only 44% of the student groups concluded their explanations by discussing answer reasonability and only 18% conducted a unit analysis to determine if their answers were applicable to the context of the problems assigned, even though both of these elements are important components of the overall sensemaking process. This report presents examples that illustrate these results and concludes with implications for teaching both entry-level mathematics and physics courses by modeling productive problem-solving and sensemaking practices.

Using visual representations to enhance isiXhosa home language learners’ mathematical understanding

Background: Several isiXhosa home language (HL) learners are excluded from meaningful mathematics learning because they are taught in English. Not only do teachers lack epistemological and pedagogical confidence in using multiple languages when teaching mathematics, but there are no mathematical registers for African languages that allow for adequate mathematical teaching and learning. There is a scarcity of research on what constitutes effective mathematics instruction for isiXhosa HL learners in South African language of learning and teaching (LoLT) Grade 1 classrooms.Aim: The purpose of this study was to explore the experiences of Grade 1 teachers using visual representations to enhance isiXhosa HL learners’ understanding of mathematics in the English- LoLT in Grade 1 classrooms.Setting: This study was conducted at four primary schools in the Western Cape’s Metro East Education District.Methods: This study employs a qualitative research approach in conjunction with an adapted interactive qualitative analysis (IQA) systems method to collect in-depth data about current mathematics practices in English LoLT in Grade 1 classrooms. The data were analysed using John Stuart Mill’s analytical comparison technique.Results: This study found that semiotics such as visual (and concrete) representations assist isiXhosa HL learners to grasp and understand mathematical concepts easily.Conclusion: This study emphasises the significance of using sufficient visual representation strategies to enhance isiXhosa HL learners’ mathematical understanding in the English LoLT in Grade 1 classrooms.Contribution: The outcomes of this study can make a positive contribution to current mathematics practice in terms of supporting isiXhosa HL learners in English LoLT in Grade 1 classrooms.

Exploring early mathematics through picturebooks: A case study in the context of Head Start

The purpose of the study was to better understand the phenomenon of exploring early mathematics through book reading. The study centers on Head Start and lays on Bronfenbrenner’s bioecological framework). Two sub-questions guided the qualitative single case study of six Head Start adult participants (teachers, parents, administrators): (1) What are Head Start participants’ experiences in exploring early mathematics through picturebooks? and (2) What do Head Start participants say about exploring early mathematics through picturebooks? Findings showed that participants expressed interest toward exploring early mathematics through picturebooks. Picturebooks were commonly used in the classroom but also accessible for families. Participants provided evidence of mathematics practices and discussions around mathematics in the school and in the home. The participants’ sayings and experiences in exploring early mathematics through picturebooks aligned with child development and contexts of learning, two cornerstones of Developmentally Appropriate Practice (DAP), but potential obstacles emerged. Implications for researchers and practitioners are discussed.

Mathematical knowledge for teaching as a didactic praxeology

It is widely acknowledged that teachers' work has both practical and theoretical dimensions, both praxis and logos, and that teacher education should address both dimensions. However, as argued in this article, existing discourses for describing the mathematical knowledge of teachers and what it takes to become competent in it are not always clear to what extent these discourses address knowledge about the practices of mathematics teaching or theories that provide arguments for the practices. Specifically, I argue for the coordination of the framework of mathematical knowledge for teaching (MKT) with the analytical distinction of logos and praxis combined in teaching practices. Moreover, I have backed this argument with examples of the proposed categories of the adjusted framework from the existing body of work in mathematics education. Each category of the proposed framework, whether concerning a teaching method or a theoretical consideration, was indeed exemplified by some concept, approach, or perspective from the field of mathematics education. The framework, I believe, would prove useful in engaging mathematics teacher education discursively, for instance, in analyzing teacher education materials to determine the extent to which they offer mere methods, theoretical arguments, or a combination of both.

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.

Classroom interactions that shape the nature of students’ appropriation of a complex mathematical practice

Scholars continue to emphasize the importance of fostering proficiency with mathematical practices as an educational outcome. As teachers attempt to support students in developing these practices, they communicate subtle messages about their nature. However, researchers lack a detailed understanding of the classroom interactions that communicate these messages. To begin to address this gap in the literature, we investigated the relationship between the types of classroom interactions around the mathematical practice of imposing structure and the ways students subsequently engaged in that practice. This led to the identification of three types of classroom interactions that shaped the nature of students’ appropriation of imposing structure: (a) engaging students in the practice, (b) providing different representations of the practice, and (c) reflecting on different instantiations of the practice. Our examination of the nature of these interactions suggests teachers must attend to details as they support students to appropriate mathematical practices in formal learning environments.

Exploring affordances and limitations of between-group movement in elementary mathematics classrooms

In this exploratory study, we examine how between-group movement, as an autonomy-promoting practice, might incentivize or disincentivize sixth-grade students’ engagement in two mathematical practices: (1) making sense of problems and persevering in solving them; and (2) constructing viable arguments and critiquing the reasoning of others. Between-group movement refers to a pedagogical strategy wherein teachers allow groups to physically move within the classroom while problem-solving to discuss strategies, ask for help, or check their work with other groups. Exploring both the affordances and limitations of between-group movement, we found that between-group movement supported groups to construct viable justifications, among other sense-making mathematical practices. However, we also found that some groups over-scaffolded during between-group conversations which disincentivized meaningful engagement in mathematical practices. Furthermore, between-group movement revealed some equity concerns in relation to status-based privileges. The findings imply that between-group movement can be a constructive pedagogical practice under specific conditions.

Elementary Teachers’ Perceptions and Practices of Real-life Application of Mathematics: A Case of a Private School in Karachi

Mathematics is a practical subject and is very important for developing problem-solving skills in learners that will help them in their everyday lives. Nonetheless, it is an unpopular subject among the school learners due to its complexity and existing teaching pedagogies. Also, it is believed that teachers’ practices are influenced by their perceptions. Therefore, this study endeavoured to explore the perceptions and practices of elementary mathematics teachers regarding the real-life applications of mathematics. Using a case study approach, mathematics teachers from one private sector school were participants in this research. Data collection tools included individual semi-structured interviews with elementary mathematics teachers to uncover their beliefs and classroom observations to gauge their practices. Thematic analysis was employed to analyse the qualitative data collected. Prominent findings illustrated that teachers’ perceptions and understanding of real-life applications of mathematics are limited and are applied minimally in their instructions, while traditional teaching methods dominate the classroom environment. Importantly, participating teachers emphasized the need for teacher training programs focusing on real-life applications in teaching of mathematics. The findings also highlighted that the teachers, school administration, and teacher trainers should work together to revise and adapt pedagogical approaches to effectively integrate real life applications of mathematics into the lessons, enhancing students’ mathematical understanding.

Paul Cohen’s philosophy of mathematics and its reflection in his mathematical practice

This paper studies Paul Cohen’s philosophy of mathematics and mathematical practice as expressed in his writing on set-theoretic consistency proofs using his method of forcing. Since Cohen did not consider himself a philosopher and was somewhat reluctant about philosophy, the analysis uses semiotic and literary textual methodologies rather than mainstream philosophical ones. Specifically, I follow some ideas of Lévi-Strauss’s structural semiotics and some literary narratological methodologies. I show how Cohen’s reflections and rhetoric attempt to bridge what he experiences as an uncomfortable tension between reality and the formal by means of his notion of intuition.

Teachers’ perceptions of less successfully organized professional development practices in mathematics: A study of nine secondary schools in Shanghai, China

Professional development (PD) for mathematics teachers in China, especially in Shanghai, has received growing international attention. However, most of available research concerning Chinese PD has focused on successful practices, and far too little attention has been paid to less successfully organized PD practices, particularly for mathematics teachers in Shanghai. This study aims to examine key aspects and underlying reasons for less successfully organized PD practices in Shanghai from teachers’ perspectives. The data were collected from 132 mathematics teachers in 9 randomly selected secondary schools in Shanghai through a questionnaire survey and follow-up interviews. The results show that Shanghai mathematics teachers perceived “time, duration and frequency”, “assessment and management” and “objective” as the three most unsatisfactory aspects in less successfully organized PD they attended, and they considered that organized PD practices were less successful mainly due to lack of assessment for the PD organizers and of necessary coordination between PD organizers at different levels. In addition, there were statistically significant differences in teachers’ perceptions of various specific problems in different aspects about less successfully organized PD between teachers with different demographic features, such as length of teaching experience and gender. Implications of the findings of the study to Chinese educational settings and beyond are discussed at the end of the paper.

Etnomatematika Melalui Simbol dan Komunikasi Nonverbal dalam Tradisi Marosok di Minangkabau

Studies that examine mathematical ideas or practices in various cultural activities are known as ethnomathematics. Ethnomathematics studies mathematical ideas in various cultural activities carried out by ethnic, social or professional groups. This research was conducted as part of the exploration in the Marosok trade tradition by the Minangkabau tribe in West Sumatra. This study aims to reveal the practice of mathematics through symbols and gestures contained in buying and selling transactions through the Marosok tradition in Minangkabau. The Marosok buying and selling tradition is the tradition of shaking hands "Marosok" or touching a finger covered with cloth (cover), between the seller and the buyer with the aim of getting an agreement on the price in the sale and purchase of livestock using nonverbal communication. This nonverbal communication activity can be used as a context to support language diversity in mathematics. Language diversity can facilitate all students with diverse learning needs. This research uses qualitative research with literature study method. The results obtained from this study indicate that symbols and finger gestures in the Marosok tradition contain the basic numbers one, two, three, four, five and two and a half as well as two types of operations in mathematics, namely addition and subtraction. These finger symbols and gestures can be used as a math practice.

Inquiry-based mathematics education and attitudes towards mathematics: tracking profiles for teaching

Research on the relationships between the main constructs underlying inquiry-based learning is rarely reported in mathematics education research. Considering this as a complex problem which is worth to be investigated, the present study aims to provide some empirical evidences that might serve as an insight to support further investigations on the relationships between attitudes towards mathematics and inquiry-based learning approaches. Thus, this study adopts a descriptive research design where no variables are manipulated but observed and measured in order to identify changes depicted in data collection. An instructional design focusing on the nature of mathematical inquiry is carried out with the participation of 304 secondary and high school students, and a clustering approach is used to look at how participants are grouped around certain attitudinal profiles before and after such mathematical practice. The results show how the heterogeneity of attitudinal profiles present in the classroom evolves positively in terms of perceived usefulness of mathematics and mathematical self-concept as perception of competence in mathematics. This fact provides some basis that might be used for further research on the idea that certain forms of development in inquiry-based mathematics education (IBME) based on greater immersion in the nature and culture of mathematics can help students to improve their attitudes towards mathematics.

Inclusive mathematics education: special education paradigms and possibilities of pedagogical practices

The challenge of inclusion is constant and is in historical development with progress, setbacks, conflicts. Therefore, the objective of this article is to analyze the paradigms present in the history of Special Education and their possible impacts and presence in current times regarding the pedagogical practices of teachers who teach Mathematics.The study has a qualitative, exploratory approach; data production procedures are bibliographic and documental research, followed by descriptive analysis. The research reveals that there are social, cultural and historical factors, present in the history of Brazil and of school education, which is marked by segregation principles, dichotomous vision between the so-called "normal" and "abnormal", ableism, which constitute constant obstacles in the search for Inclusive Education and differentiated pedagogical practices in Mathematics that consider diversity as part of the human constitution. In this context, the pedagogical practice of the teacher who teaches Mathematics, often, to be based on the inclusive perspective, needs transformation, awareness of the historical aspects and critical reflection on the history of exclusion to enable resistance and permanent search for human rights and inclusion.

Definitions in practice: An interview study

In the philosophy of mathematical practice, the aim is to understand the various aspects of this practice. Even though definitions are a central element of mathematical practice, the study of this aspect of mathematical practice is still in its infancy. In particular, there is little empirical evidence to substantiate claims about definitions in practice. In this article, we address this gap by reporting on an empirical investigation on how mathematicians create definitions and which roles and properties they attribute to them. On the basis of interviews with thirteen research mathematicians, we provide a broad range of relevant aspects of definitions. In particular, we address various roles of definitions and show that definitions are not just a product of mathematical factors, but also of social and contingent factors. Furthermore, we provide concrete examples of how mathematicians interact and think about definition. This broad empirical basis with a variety of examples provides an optimal starting point for future investigations into definitions in mathematical practice.