Mathematical Practice Research Articles

Automating Induction by Reflection

Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires an infinite number of axioms, which is not a feasible input to a computer-aided theorem prover requiring a finite input. Mathematical practice is to specify these infinite sets of axioms as axiom schemes. Unfortunately these schematic definitions cannot be formalized in first-order logic, and therefore not supported as inputs for first-order theorem provers. In this work we introduce a new method, inspired by the field of axiomatic theories of truth, that allows to express schematic inductive definitions, in the standard syntax of multi-sorted first-order logic. Further we test the practical feasibility of the method with state-of-the-art theorem provers, comparing it to solvers' native techniques for handling induction.

What Is a Theorem (in Practice)? The Role of Metamathematics in the Making of Mathematics

This article advocates the benefits of a sociological perspective for the philosophy of mathematical practice. Drawing from the literature of the sociology of sciences, it defends a community-centered approach of the study of mathematical practice and assesses the role of the notion of metamathematics in mathematical change and in stabilized mathematical practices. It relies on the case study of the emergence of geometric control theory at the beginning of the 1970s and of the citational practices associated to the community of control theory since the mid-1990s. The case study shows that the introduction of geometric tools in control theory at the end of the 1960s induced a change in the metamathematical views that control theorists had on their objects. It is then demonstrated how membership to the community of control theory shapes the production and the reception of the theorems of Stefan, Sussmann and Nagano. Interpreting the historical development and citational practices of this community through the perspective of metamathematics, this paper concludes by discussing the role of the orbit theorem in control theory, both as a cognitive label and as a social marker of membership to this community.

Rules, understanding and language games in mathematics

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games.

APPLICATION OF HISTORICAL DATA IN STUDYING THE VALUE «TIME» IN THE ELEMENTARY CLASSES

This article examines the features of the use of historical data in the study of the value of «time» in primary school. Shown is their educational, upbringing and developmental role in the formation of temporal representations in younger students, the expansion and deepening of knowledge, the development of cognitive interest, in the implementation of the connection between mathematical theory and practice.

Dual-Language Teachers’ Use of Technology to Facilitate Mathematical Discourse

Teachers’ beliefs, professional development, access to hardware/software, and instructional contexts all influence teachers’ use of technology (Ertmer et al., 2012; González-Carriedo & Esprívalo Harrell, 2018; Ottenbreit-Leftwich et al., 2010). The first three factors have been studied extensively, but additional research on the instructional contexts of teachers, especially those working in less common settings, may enhance our knowledge of how technology shapes teachers’ instruction and, subsequently, student learning. This case study highlights three teachers’ beliefs and practices of incorporating technology to enhance mathematics learning and mathematical discourse in public dual-language elementary schools. Through interviews, video-recorded lessons, and workshop discussions, researchers examined why and how teachers leveraged technology to enhance mathematics learning in a bilingual program. The analysis indicates that teachers’ reasoning and technology use sought to increase opportunities for discourse, a significant component of the mathematical practices (National Governors Association Center for Best Practices, 2010), strengthening their bilingual students’ understanding and application of mathematical concepts, processes, and proficiencies.

Embedding STEM in the mathematics classroom: A case study of vocational high school teacher’s practice

Mathematics teachers often question the practice of Science, Technology, Engineering, and Mathematics (STEM) education without diminishing mathematics’s potential as a conceptual tool to solve problems. This qualitative case study research explores the STEM teaching practices carried out by two mathematics vocational high school teachers. Further exploration seeks to investigate how is mathematics playing a role in the STEM project they are carrying. Data were collected through documents and media analysis (lesson plans, students’ work, photos, and video) and interview with teachers. The findings showed that both teachers chose to implement the embedded-STEM approach in their first time STEM teaching experience. They used mathematics to facilitate students’ problem-solving, mathematics modelling, and reasoning in the STEM project. The teachers had different practices and perspectives in designing, implementing, and assessing embedded-STEM mathematics classrooms. The contribution of this research is to provide practical cases of embedded-STEM mathematics teaching, which may be relevant as a solution for countries that do not offer a specific STEM curriculum.

Mathematics Heritage Project: An Exploration Empowering Students' Mathematical Identities

The International Study Group on Ethnomathematics (ISGEm) supports incorporating cultural diversity of mathematical practices to promote the teaching and learning of school mathematics. Through The Mathematics Heritage Project, students at a middle school in the southeastern United States developed unique creations to connect with the mathematics connected to their identities and self-identified cultural group. Upon reflection, students reported an increased awareness of the relevance of mathematics in their lives and a sense of ownership that is both meaningful and modern.

Growing mathematical learners

Results in the International Trends in Mathematics and Sciences Study, and the National Monitoring Study of Student Achievement, have meant there has been recent focus on mathematics education in New Zealand. As with previous iterations of assessment result reports, the quality of teachers, teaching, teacher education, professional learning, mathematics programmes and curriculum have been questioned by the media. It seems that change in mathematics education in New Zealand may, once again, be imminent. In this paper, there is a plea that any change needs to retain what is positive about mathematics education in New Zealand. Teachers need to be supported to grow their own mathematics content knowledge and pedagogical content knowledge so they can make autonomous and sound decisions for their own students’ needs, while retaining the ethic of care they already foster in their classrooms. Support needs to be in the form of rich, centrally managed yet locally adapted, non-divisive resources and professional learning.
 
 Furthermore, any change to mathematics education in New Zealand needs to be centred around students and teachers engaging in authentic mathematics practice. A challenging task approach is described that supports this practice, which has been successfully trialled with New Zealand teachers. In this approach, content is not pre-taught. Rather, teachers support the students to solve challenging tasks by moving through the phases of Launch, Explore, Summarise, and Reflect.

Opportunities to Learn From (Advanced) Mathematical Coursework: A Teacher Perspective on Observed Classroom Practice

This article explores secondary teachers’ opportunities to learn from an innovative real analysis course, as reflected in their actual classroom teaching. The course used cases of teaching as a site for applying mathematics and developing pedagogical mathematical practices. This article explores particular teaching moments in (N= 6) secondary teachers’ classrooms, and the attributions they gave for why they engaged in those teaching practices. Teachers engaged in instructional practices that exemplified course objectives, and their attributions for their actions contribute a teacher perspective on opportunities to learn in teacher education from (advanced) mathematical coursework. Results highlight cases of teaching and modeled instruction as catalysts of change and as opportunities to develop pedagogy from mathematical activity, and vice versa.

Developmental Change of Approximate Number System Acuity (Keenness) Reveals Delay

Purpose: Major aim of the study was to investigate the development of mathematical thinking and processing in Pakistani sample. Particular focus of the study was to figure out whether Pakistani people across various age groups process numbers with similar sophistication as their western counter parts from developed countries. Mathematics plays huge role in the development of the society and research evidence in this context from Pakistan is scarce.
 Design/Methodology/Approach: 261 participants ranging from 5 to 72 years of age participated in the study. Panamath task being the robust measure of ANS acuity was administered.
 Findings: Results revealed that numerical acuity got precise with an increase in age. However, most sophisticated acuity has been shown around age 46-50 as compared to the western population showing its peak around 30 years of age. Delay in developing most sophisticated approximate number system acuity across the groups as compared to the trend reported in the western population raises many questions in terms of cultural variations and practices contributing to the development of number sense.
 Implications/Originality/Value: We need to improve our mathematical learning and teaching practices so that it could be helpful in economic growth in turn by better mathematical performance across various age groups. On a general note, economical practices, math related curriculum policies, lack in math related games, math practices at home and at educational institutions with varying level of curriculum and pedagogical practices might be a contributor for this trend. The study has important implications for understanding the development of number sense cross-culturally keeping in view the evidence from various cultures.

Research on Mathematics Classroom Teaching Based on the Model of Hierarchical Teaching Management

With the implementation of nine-year compulsory education, and our school is located at the border of urban and rural areas, there are many children in rural areas and the quality of students varies. The problems of mathematics learning in our school have come to the fore. Some do not learn at all, and some cannot learn. Coupled with the traditional thinking that mathematics is a “big problem” in students’ minds, students’ learning initiative and self-consciousness are generally higher. Poor, it was difficult to improve at one time. On the other hand, limited by students’ large classes, number of class hours and 45 minutes of class time, classroom teaching can provide students of different levels with limited participation opportunities. Therefore, as a mathematics teacher in actual work, they will constantly try to use better methods. To change the problem of student differences in class teaching mode. By analyzing the status quo of mathematics teaching in the middle school, studying the reform practice of mathematics teaching at home and abroad, it is decided to conduct experimental research of layered teaching in mathematics classroom teaching of our school. Hierarchical teaching conforms to the dialectical theory of materialism, the theory of pedagogy, the learning psychology of students, and the principles of teaching students in accordance with their aptitude in the teaching theory. It has a broad theoretical foundation.

Examining Preservice Teachers' Familiarity and Experiences with Mathematical Modeling Practices

ABSTRACT Mathematical modeling has been recognized and highlighted in most standards and reports as an important concept to the teaching and learning of mathematics. The study investigated 208 PK–8 preservice teachers’ (PSTs’) familiarity and experiences with mathematical modeling. The study adopted a survey-design perspective to collect data from 10 universities in a mid-western state in the United States of America (U.S.). Data collected were analyzed using both quantitative and qualitative methods. Results from the study indicated that PSTs were familiar with some outcomes related to mathematical modeling practices. However, their familiarity and experiences seem limited, and their perception of mathematical modeling practices were traditional in nature. Most of the PSTs had misconceived definitions regarding mathematical modeling and confused mathematical modeling with model mathematics. Furthermore, most PSTs reported they had no formal exposure or training in practices associated with mathematical modeling during their teacher preparation program. The findings suggest that infusing modeling courses or modules in existing methods or content courses in teacher education programs can be an effective way to elevate PSTs from unfamiliarity of mathematical modeling practices and standards These results along with a supplemental discussion have implications for teacher preparation programs and researchers centered on mathematical modeling education.

La práctica matemática en la Guatemala colonial del siglo XVIII

En este artículo discuto elementos de la práctica matemática en la Guatemala colonial del siglo XVIII. La discusión de dicha práctica gira en particular alrededor del libro de matemáticas más antiguo que existe publicado en el período colonial en América Central: la Arithmetica practica del padre Juan José de Padilla, publicado en 1732. En la primera parte del artículo, describo brevemente mi encuentro con el libro de Padilla y los problemas contextuales que encontré para ofrecer una edición facsímil del mismo. En la segunda parte, defino la aproximación historiográfica que utilizo. Argumento que investigar acerca de las prácticas matemáticas históricas es indagar acerca de las matemáticas tal como fueron imaginadas, pensadas y utilizadas dentro de un cierto contexto histórico cultural. También es indagar sobre las maneras en que las matemáticas, los matemáticos, los libros de matemáticas y otros artefactos encajan y responden ideológicamente a las estructuras económicas, políticas y educativas de su tiempo.

Virtue theory of mathematical practices: an introduction

Virtue theory of mathematical practices: an introduction

A curriculum of mathematical practices

ABSTRACT This paper argues for reconceptualising an educational curriculum that locates its primacy in practices. The argument is framed around the core purpose of education: to help people ‘live well in a world worth living in’. Living well and learning about what this means is typically guided by epistemologically based curricula, and conversely, school curricula determine the substance of education. We argue that this understanding of education is too narrow, and as a consequence, it severs the relationship between knowing and practising. We propose that a curriculum of mathematical practices is required for human flourishing, where the focus is on mathematical practices rather than predominantly on knowledge. To demonstrate our position, we consider different kinds of mathematical practices needed during the Covid-19 crisis. We examine how a practice-approach forms the basis for a future-oriented curriculum which might better equip individuals and societies to respond to conditions which disrupt their everyday circumstances.

Commentary on Paul Ernest’s Theory about Teachers’ Beliefs and Practice

In this short communication, the author analyzed Paul Ernest’s theory on relationships between teachers’ beliefs, and their impact on teachers’ practice of mathematics. The author considered the teachers’ espoused and enacted models of mathematics assessment in addition to the teachers' views of the nature of mathematics, teaching, and learning models. The author also considered three purposes of mathematics assessment.

The instructional situations in which mathematics teachers provide substantive feedback

Feedback provided by mathematics teachers usually addresses procedural skills and, to a much lesser extent, other competencies such as conceptual understanding or engagement in mathematical practices. As most previous literature has studied feedback provided on homework or video prompts, how teachers provide such feedback in the classroom is poorly understood. Here, sixteen lessons taught by five teachers were purposefully sampled from a larger video study (172 lessons) as lessons with high-quality feedback according to a standardized observation instrument. The analysis focused on the instructional situations in which teachers provided feedback. When teachers provided procedural feedback, the situations were orderly and effective. Feedback on conceptual understanding and mathematical practices was provided in situations when students were especially challenged, and entailed a series of complex decisions, thereby placing demands on the teachers to manage both the students’ understanding and behavior. We argue that researchers should focus on how teachers and students negotiate the instructional situation to allow for feedback on conceptual understanding or mathematical practices.

Inside and Out: Factors That Support and Hinder the Self-Advocacy of Undergraduates with ADHD and/or Specific Learning Disabilities in STEM.

Self-advocacy is linked to the success and retention of students with disabilities in college. Self-advocacy is defined as communicating individual wants, needs, and rights to determine and pursue required accommodations. While self-advocacy is linked to academic success, little is known about how students with disabilities in science, technology, engineering, and mathematics (STEM) practice self-advocacy. We previously developed a model of self-advocacy for STEM students with attention-deficit/hyperactivity disorder (ADHD) and/or specific learning disabilities (SLD). Here, we use this model to examine what factors support or hinder self-advocacy in undergraduate STEM courses. We conducted semistructured interviews with 25 STEM majors with ADHD and/or SLD and used qualitative approaches to analyze our data. We found internal factors, or factors within a participant, and external factors, the situations and people, described by our participants, that influenced self-advocacy. These factors often interacted and functioned as a support or barrier, depending on the individuals and their unique experiences. We developed a model to understand how factors supported or hindered self-advocacy in STEM. Supporting factors contributed to a sense of comfort and security for our participants and informed their perceptions that accommodation use was accepted in a STEM course. We share implications for research and teaching based on our results.

Mathematical beliefs and self-reported practices of Chinese early childhood teachers in the context of teaching mathematics during block play

ABSTRACT In China, a shift in early math teaching from a teacher-directed approach to a child-centered approach has been recommended. This study explored Chinese teachers’ beliefs about early math teaching and learning, math teaching efficacy, and math-related practices in the context of unit block play. Early childhood teachers (N = 391) from Ningbo, Zhejiang province, participated in this study. The results of a questionnaire-based survey indicated that teachers’ beliefs about math teaching and learning and math teaching efficacy were significant predictors of their math teaching practices during unit block play. Math teaching efficacy mediated the positive relationship between constructivist beliefs and child-centered practices, but not between traditional beliefs and teacher-directed practices. The results have implications for early childhood teacher education in China and future studies on early math teaching and learning.

Theorizing STEM Leadership: Agency, Identity and Community

Abstract STEM education, when perceived as integrated learning that encompasses knowledge, skills and practices of Science, Technology, Engineering and Mathematics, points to a need to re-examine ways of classification of school subjects and learning. Consequently, dilemmas related to integrated STEM education arise. School leaders are faced with the task to organize teams to address issues such as the ownership of STEM, identity issues such as STEM teacher or teacher of STEM subjects, evaluation of STEM programs and resources to support STEM education. The unique characteristics of integrated disciplines demand leaders who understand the unique characteristics and demands of each discipline and to apply them to build a synergistic platform to magnify the similarities and harness the differences for learning. In this paper, we present an argument for STEM leadership to focus on building STEM teachers’ agency, identity and sense of belonging to a community. These three aspects are important for meaningful planning, enactment and sustainability of STEM programs since teachers’ beliefs, intentions, actions and empowerment are known to be instrumental in the success of many educational reforms.