Mathematical Practice Research Articles

Differences in Ethnomathematical Characteristics between Buton Traditional Houses

The mathematical practice of the Buton people can be seen from the ethnomathematics of the Buton traditional house. This qualitative study aims to explore the differences in ethnomathematical characteristics between Buton traditional houses. The subjects of this study were 3 traditional leaders and 3 builders of Buton traditional houses. Data were collected through measurements, interviews, and observations of the traditional Buton house in Baubau city, the capital of the Buton kingdom, Indonesia. The credibility of the data was tested by using triangulation of sources. Data were analyzed descriptively qualitatively from Miles, Huberman, and Saldana in the form of data condensation, data presentation, and conclusions. The results of data analysis showed that the differences in the ethnomathematical characteristics of the three Buton traditional houses can be seen from the size and area of the house, the angle of the roof and stairs, the arrangement of the roof, the number of poles, the number of rooms, the model of the house, and the type, model, and the number of windows used. Teachers need to know the differences in ethnomathematical characteristics so that they can be used to increase students’ active participation in learning mathematics in class.

Addressing Dehumanizing Mathematical Practices: Using Supervisory Leaders’ Experiential Knowledge to Transform the Mathematics Classroom

Deficit language concerning historically marginalized students pervades much of education today. Black, Brown, and Indigenous children experience marginalization and dehumanizing practices in classrooms instead of participating in a safe space to learn and grow. For this paper we employ a crucial component from Critical Race Theory to address systemic racism in schools: we listen to the lived experiences of professionals of color. These personal narratives open avenues for social justice through critiquing current and historical political, economic, and sociocultural practices and policies. This study examined how four Black collaborators – one high school principal, one middle school principal, one elementary principal, and one special education teacher – each with decades of instructional experience, address four key dehumanizing practices students of color experience in classrooms across the country in their own supervision practices.

Ethnomathematics values in Temple of Heaven: An Imperial Sacrificial Altar in Beijing, China

Many studies are proving that learning mathematics with an ethnomathematical approach can improve students’ mathematical skills. Developing and using ethnomathematics concepts are important to raise history and cultural awareness of mathematics. This study aims to analyse the ethnomathematics values of the Temple of Heaven. Temple of Heaven is one of the famous heritage sites in Beijing, China, which bears many ethnomathematics concepts. The researchers applied a qualitative method in this study. The subject of this research is the Temple of Heaven building that is located in Beijing, China. Researchers identified the geometrical concept present in the exterior, interior design, and building structure of the Temple of Heaven building. This research shows the existence of mathematical concepts in the architecture of the Temple of Heaven. This research result can help teachers in making mathematical practice questions with ethnomathematics concepts.

Prospective Primary School Teachers’ Competence for the Cognitive Analysis of Students’ Solutions to Proportionality Tasks

In order to foster the learning of mathematics, the teacher must be able to analyse and assess the students’ mathematical activity. The explicit recognition of objects and processes involved in mathematical practices is a competence that the teacher should develop. This cognitive analysis competence allows the teacher to understand the processes of mathematical learning, to foresee conflicts of meanings and to establish different possibilities for institutionalising the mathematical knowledge involved.In this article we present the results of the evaluation phase of a training intervention with eighty-eight prospective primary school teachers, which aims to promote and assess the competence for the cognitive analysis of students’ solutions to proportionality tasks. To this end, we proposed the prospective teachers to interpret different students’ solution strategies for a problem, recognise the mathematical elements (languages, concepts, propositions, procedures and arguments) put at stake in each strategy, and to analyse the algebraic character of the mathematical practices involved in them. The results reveal the prospective teachers’ limitations for the analysis and assessment of non-usual resolution strategies, the identification of key mathematical objects and the discrimination of arithmetic and algebraic activity in the students’ solutions. The improvement of the results requires the following actions: to allow prospective teachers to become acquainted with different forms of reasoning that can be applied in proportionality situations, delve more deeply into the algebraic character of mathematical activity, and extend the number and variety of situation problems that prospective teachers can analyse and discuss.

The community mathematics project: Using a parent tutoring program to develop sense-making skills in novice mathematics educators

Programs that provide underserved populations with access to high-quality mathematics through workshops, clubs, and classroom observations (Civil, Stoehr, & Salazar, 2019; Civil & Bernier, 2006; Diaz-Palomar, Menendez, & Civil, 2011; Jimenez-Castellanos, Ochoa, & Olivos, 2016; Muir, 2011, 2012; Quintos, Civil, & Bratton, 2019) are vital. The community mathematics project (CMP) is one such program that takes a holistic approach to transforming mathematics education. Through partnerships with local institutions, the CMP pairs a prospective teacher (PT) with a parent for online mathematics tutoring sessions. This study describes how PTs in the CMP supported parents’ mathematics development of sense-making (Odden & Russ, 2017) in mathematics by providing relevant tasks, employing multiple strategies, using virtual manipulatives, and questioning in one–one-one tutoring sessions. Findings suggest that this type of field experience with an explicit focus on a sense-making approach supported PTs in learning mathematics, engaging in partnerships with parents and members of the community, and attending broader issues of social contexts of learning. Further, PTs attended to individual considerations such as sequencing and scaffolding, relevancy of the content, task development, and building towards common mathematical practices. Finally, onsite coaching and tutoring experiences were formative in developing essential skills for teaching mathematics in elementary school in the following areas: content and pedagogical content knowledge, development of core pedagogical practices, funds of knowledge approaches and collaboration with families, and dispositions toward (teaching) mathematics such as reframing mistakes, emphasis on sense-making, and co-construction of knowledge.

Mathematics Assessment Practices of Primary School Teachers in France

We investigated French primary school teachers’ assessment practice in mathematics. Using an online questionnaire on teachers’ background, teaching, and grading practice, we were able to determine assessment profiles of 604 primary school teachers. As evidenced by the teachers’ scores on the latent factors Assessment purposes, Assessment practices, and Differentiation, teachers with the profile of Enthusiastic assessors view assessment as more useful and use it more often to adapt their instruction than teachers with the profile of Unenthusiastic assessors. This can be useful for practice and sheds more light on French teachers’ assessment practices in mathematics. It is also interesting to compare the results of this survey with those from China and the Netherlands, as the differences reflect different assessment cultures and may shed light on some of the results of international large-scale assessments such as PISA.

Analysis tasks based on a theorem in Nonlinear Approximation theory

The systematic design and analysis of tasks which can be implemented in first-year university courses and point to advanced inner- and extra-mathematically rich issues and their rationales is an open problem in the didactics of mathematics in higher education. Potentials of such tasks can be seen with regard to learning processes in the first year of study, which are less compartmentalized and allow for an extended acquisition of rationales of advanced mathematical practices and concepts. Against this background, subject-specific potentials of advanced mathematics are examined in this contribution. The Anthropological Theory of the Didactic (ATD) serves as the theoretical framework and in the analyses notions from its 4T-model are applied. Structural observations in praxeological terms are illustrated by examples chosen from a presentation of a classical result in Nonlinear Approximation. At the specific focus of the praxeological analyses are aspects for bridging and extending concepts within and across Analysis. On this basis, tasks designed for first-year university Analysis courses are analysed in detail. Moreover, general characteristics of the methodological approach realized in this contribution are discussed.

Towards a historical approach to physics education

Modern physics is a complex multiplicity of practices: theoretical, mathematical, experimental and simulation practices. Experimental and simulation practices are related to the pragmatic dimension of a physical theory. Mathematical practices are related to the syntactic dimension of a physical theory, but theoretical practices involve an often neglected semantic dimension. Physics and consequently teaching physics are usually reduced to the syntactic and pragmatic dimensions. Semantic dimension is linked to the conceptualization of the physical reality, to the conception of Nature. By neglecting the semantic dimension, physics is reduced to a pure mathematical game and to technological manipulations. Thus, the cultural aspect of science is lost and physics education is reduced to a mere technical training. This process of de-culturalization of science had its roots in the Enlightenment’s turn in physics to free it from theology and metaphysics and had its completion in the post-second-world-war era. I believe we have to recover the cultural aspects of physics to understand it more deeply in its whole complexity. We have to recover all its relationships with other disciplines as philosophy, mathematics, psychology, sociology and other sciences, even theology, which are fundamental to constitute its semantic dimension. Historical approach to physics and physics education is the only way to recover this interdisciplinarity at the roots of the various physical conceptions of Nature. In this way, for example, we can understand that beyond mechanics there is a mechanist conception of Nature, beyond thermodynamics there is a thermodynamical conception of Nature, and beyond electromagnetism there is an electromagnetic conception of Nature. These different conceptions of Nature are not compatible and their historical fight has produced the relativistic, quantum and chaos revolutions in XX century physics. These different conceptions of Nature imply different existential self-understanding of the meaning of mankind in the universe and different ethical perspectives.

Subjectivities in curricular practices in Mathematics in Youth and Adult Education

Background: The curriculum practices, especially those related to Mathematics, developed in EJA, can exert a strong influence on the students' sentiment of self-efficacy and, in turn, influence the cognitive processing of mathematical knowledge. Objective: To investigate which meanings of curricular practices in Mathematics are subjectived by EJA teachers. Design: Narratives of 5 teachers who work at EJA, whose data collection was through interviews. Environmente and participants: The research took place in a rural municipal public school located in a city in the interior of Ceará. Data collection and analysis: Textual Discursive Analysis. Results: The curricular practices developed in EJA, especially those related to mathematics, can exert a strong influence on the student's sentiment of self-efficacy and influence their cognitive processing of mathematical knowledge. The moments of collective pedagogical planning and training influence the selection of curriculum content and teaching materials, guiding the construction of the teachers' curricular practices, influencing the construction of their subjectivities, in a permanent relationship between subjects who interact and confront each other from different ways for their pedagogical work, including the support they seek from their peers. Conclusions: Thinking about the subjectivities of the actors involved in EJA should mean turning the attention to the teaching and learning processes, which encompass a range of diversities of social groups with very different cultures, expectations, ages and interests, meaning greater attention to production and implementation of appropriate curricula and teaching for these groups of students, considering their cultures, experiences and previous experiences.

Ameta‐analysison the effects of technology's functions and roles on students' mathematics achievement in K‐12 classrooms

AbstractBackgroundWith the recent pivot to online instruction and/or to mixed online/face‐to‐face (i.e., hybrid) models of teaching necessitated by the novel coronavirus pandemic, the uses of technology to support instruction would seem to have great importance. Whereas effective integration of technology in mathematics classrooms requires recognition of students’ needs and instructional goals that technology could fulfill, very few comprehensive meta‐analytical studies have focused on the roles of instructional technology in support of specific learning goals.ObjectivesIn this meta‐analysis, we investigated and reviewed the roles and affordances of technology in mathematics instruction and its effects on students' mathematics achievement.MethodsCross‐searching multiple bibliographic databases generated a total of 3276 potentially relevant full‐text studies. After applying a series of inclusion/exclusion criteria, we kept 77 studies with 151 effect sizes. Two researchers then coded for seven sets of study features and student characteristics: technology roles, publication‐related features (years and resources), research design, sample size, grade levels, duration of interventions, and types of the testing instruments. The coding achieved acceptable levels of inter‐rater reliabilities. Since effect sizes varied across studies and student populations, random‐effects models were used for overall effect estimates and moderator analyses.Results and conclusionsOverall, this meta‐analysis revealed a small, positive, and statistically significant effect of technology on students' mathematics achievement with an average Hedges' s g (ḡ) of 0.23. The largest significant moderator effects were found when technology was used to design and support collaborative and communicative environments (ḡ= 0.49); the next‐largest moderator effects were found when using technology as supports for problem‐solving (ḡ= 0.39), followed by the roles of as supports for conceptual development (ḡ= 0.31) and as adaptive mathematics practices (ḡ= 0.28). This study also found that the effects of technology were differentiated as a function of publication resources, research design, duration of intervention, and the testing instrument.Major takeawaysFocusing on the roles and functions of technology, this study can contribute to guide potential synthetical studies as well as the future design of educational technology. In particular, technology is used effectively when it is used (a) to create and design a collaborative and communicative learning environment where students have chances to collaborate and interact with each other and (b) as supports for mathematics problem‐solving and for mathematics conceptual development. On the other hand, technology is less effective when it is used for monitoring and assessing students' learning processes without follow‐up instructions.The findings suggested that technology intervention lasting for a shorter time is the better choice. When technology is used for longer periods of instruction, maintaining its novelty and students’ curiosity should be considered. This study also suggested that incorporating computer‐adaptive testing in technology‐based intervention programs is more likely to detect technology effects as adaptive testing could continuously adjust to the abilities of individual students.

Norms That Regulate the Theorem Construction Process in an Inquiry Classroom of 3D Geometry: Teacher’s Management to Promote Them

This paper aims to illustrate how a teacher instilled norms that regulate the theorem construction process in a three-dimensional geometry course. The course was part of a preservice mathematics teacher program, and it was characterized by promoting inquiry and argumentation. We analyze class excerpts in which students address tasks that require formulating conjectures, that emerge as a solution to a problem and proving such conjectures, and the teacher leads whole-class activities where students’ productions are exposed. For this, we used elements of the didactical analysis proposed by the onto-semiotic approach and Toulmin’s model for argumentation. The teacher’s professional actions that promoted reiterative actions in students’ mathematical practices were identified; we illustrate how these professional actions impelled students’ actions to become norms concerning issues about the legitimacy of different types of arguments (e.g., analogical and abductive) in the theorem construction process.

Formative Assessment in Online Distance Learning Modality through the Lens of Mathematics and Science Teachers

Formative assessment provides ongoing feedback to monitor student learning and improve teachers’ instruction. But the COVID-19 pandemic has changed the mode of learning delivery and has affected formative assessment. Hence, the researchers explored the practices, challenges, and adaptive strategies of Mathematics and Science Senior High school teachers on formative assessment of students’ learning in an online distance learning modality. Narrative inquiry research design was used in which eight teachers and six students who were chosen through convenience sampling were subjected to in-depth interview. The audio-recorded interviews were transcribed and underwent coding, categorizing, and thematizing. Results revealed that teachers employ practices that allow them to provide immediate feedback, utilize online assessment tools, probe questions, encourage students’ active participation, and align learning competencies with formative assessment. However, they encounter challenges that affect students’ learning such as lack of motivation, unavailability of internet/data connection and gadgets, low digital literacy skills, minimal peer assistance, and authenticity of formative assessment results. But they adapt strategies to address these challenges through designing differentiated assessment, showing empathy, reinventing pedagogical goals through meaningful online tools, promoting student-centered learning, and providing consistent means of communication. Findings implied that teachers use various online formative assessment practices similar to what they do in face-to-face classes. Challenges are evident in the entirety of instruction. However, teachers are dynamic in improving their practices. An action plan consisting of activities, objectives, and description is hereby proposed to improve the online formative assessment practices of mathematics and science teachers to enhance students’ learning.

The Development and Validation of Mathematical Thinking Beliefs (MTB) Instrument

Mathematical thinking beliefs are a significant element in supporting teachers’ practices of learning and teaching school mathematics. Most prior researches on mathematics beliefs have focused on teachers’ beliefs about learning and teaching mathematics in general. This research aimed to validate a Mathematical Thinking Belief instruments (MTB). The instrument was distributed to 523 mathematics teachers in Omani public schools. These teachers were asked to write their opinion in an instrument with a 5-point Likert scale. Exploratory factor analysis and confirmatory factor analysis have been applied to test the MTB using AMOS 25.0. All constructs show acceptable internal consistency reliability. The model meet a good model fit for the mathematical thinking beliefs measurement model through different fit index tests such as CMINDF, CFI, GFI, SRMR, RMSEA, and TLI. The results show that all indices of fit criteria were achieved. It is also indicated that acceptable validity, construct reliability and variance extracted values.

Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning

Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers’ role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students’ work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice.

Ethnomodelling as a glocalization process of mathematical practices through cultural dynamism

Ethnomodelling can be considered as the association between ethnomathematics and mathematical modelling that enables members of distinct cultural groups to perceive a different reality in relation to the nature of mathematical knowledge. It also provides insights into many diverse forms of mathematics developed locally. Thus, ethnomodelling is defined as the study of mathematical phenomena that adds cultural components to the modelling process. The development of this connection is conducted through three cultural approaches: local, global, and glocal, which are used in the conduction of ethnomodelling investigations that aim to work against colonialism in order to value and respect sociocultural diversity of members of distinct cultural groups. Because ethnomodelling seeks to promote the development of understanding of differences through dialogue; it is important to argue for its inclusion as a translational process for systems taken from the reality of the members of diverse cultures. In this article we argue that ethnomodelling creates a firm foundation that allows for the integration of these three cultural approaches in exploring mathematical knowledge developed in distinct cultural groups through cultural dynamism.

Connecting expectations of Te Whāriki and The New Zealand Curriculum: Examples of mathematical practices

The first year of primary school aims to be closely connected with early childhood education, yet this is often invisible in the curriculum of specific subjects. This paper sets out an approach that uses mathematical practices as a curriculum tool that reconceptualises school mathematics. Using the early childhood mathematics framework of Te Kākano, the strands of mathematical practices are important descriptors of mathematical activity for children. We describe examples of mathematical learning from both early childhood and the first year of school, and make a case for using mathematical practices as a conceptual tool for designing a mathematics curriculum in the first years of school.

Mathematics in the middle: The relationship between measurement and metamorphic matter

This paper revisits philosophical questions regarding the relationship between mathematics and matter. I briefly present four contrary and contemporary perspectives on the speculative force of mathematics, as a provocation for further discussion on the subject of sciento-metrics. I first consider the ideas of the philosopher Quentin Meillassoux, as a way of setting the stage for various kinds of materialist philosophies of mathematics. I then turn to the ideas of two mathematicians - Fernando Zalamea and Giuseppe Longo - and a computer scientist - Gregory Chaitin - and explore how their discussions of contemporary mathematical practice offer important insight (and twist) regarding the relationship between mathematics and matter.

Comparing the Mathematical Practices Pre-Service Teachers and Mathematics Teacher Educators Identified as Relevant to Problems and Tasks

In the U.S., state adopted or developed college- and career-ready mathematics standards, including the Common Core State Standards for Mathematics, not only impact districts, students, and their teachers, but also university teacher preparation programs. In order to attain and sustain Common Core’s vision of developing mathematically competent citizens, teacher preparation programs must support pre-service teachers’ development of practical conceptions of the Standards for Mathematical Practice. In this article, we examine the mathematical practices middle grades pre-service teachers (grades 4-9 licensure) and mathematics teacher educators identified as playing a role in attempts to make sense of and work toward solutions to mathematics problems. In addition, we compare the mathematical practices indicated both within and across pre-service teachers and mathematics teacher educators. Results identify pre-service teachers’ potential difficulties operationalizing six specific mathematical habits of mind. Finally, we describe how such comparisons can guide the design of future teacher education and professional learning by describing a process for identifying problems and tasks with the greatest potential to support pre-service teachers’ development of practical conceptions of mathematics or other content-specific habits of mind.

Relevant Arithmetic and Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture.

Social constructivism in mathematics? The promise and shortcomings of Julian Cole\u2019s institutional account

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed.