The Roles of the Aesthetic in Mathematical Inquiry

Mathematicians have long claimed that the aesthetic plays a fundamental role in the development and appreciation of mathematical knowledge. To date, however, it has been unclear how the aesthetic might contribute to the teaching and learning of school mathematics. This is due in part to the fact that mathematicians' aesthetic claims have been inadequately analyzed, making it difficult for mathematics educators to discern any potential pedagogical benefits. This article provides a pragmatic analysis of the roles of the aesthetic in mathematical inquiry. It then probes some of the beliefs and values that underlie mathematical aesthetic responses and reveals the important interplay between the aesthetic, cognitive, and affective processes involved in mathematical inquiry.     

The Roles of the Aesthetic in Mathematical Inquiry

Mathematicians have long claimed that the aesthetic plays a fundamental role in the development and appreciation of mathematical knowledge. To date, however, it has been unclear how the aesthetic might contribute to the teaching and learning of school mathematics. This is due in part to the fact that mathematicians' aesthetic claims have been inadequately analyzed, making it difficult for mathematics educators to discern any potential pedagogical benefits. This article provides a pragmatic analysis of the roles of the aesthetic in mathematical inquiry. It then probes some of the beliefs and values that underlie mathematical aesthetic responses and reveals the important interplay between the aesthetic, cognitive, and affective processes involved in mathematical inquiry.     

Mathematics Learning and Participation as Racialized Forms of Experience: African American Parents Speak on the Struggle for Mathematics Literacy

This article draws on 3 ethnographic and participant observation studies of African American parents and adults from 3 northern California communities. Although studies have shown that African American parents hold the same folk theories about mathematics as other parents, stressing it as an important school subject, few studies have sought to directly examine their beliefs about constraints and opportunities associated with mathematics learning for both themselves and their children. I argue that, as they situate the struggle for mathematical literacy within the larger contexts of African American, political, socioeconomic, and educational struggle, these parents help reveal that mathematics learning and participation can be conceptualized as racialized forms of experience. As they attempt to become doers of mathematics and advocates for their children's mathematics learning, discriminatory experiences have continued to subjugate some of these parents, whereas others—as demonstrated in their oppositional voices and behaviors—resisted their continued subjugation based on a belief that mathematics knowledge, beyond its role in schools, can be used to change the conditions of their lives. The characterization of mathematics learning as racialized experience put forth in this article contrasts with culture-free and situated perspectives of mathematics learning often found in the literature. As a result of their experiences with oppression in this society, the concept of race has historically played a major role in the lives of African Americans. Although race has dubious value as a scientific classification system, it has had real consequences for the life experiences and life opportunities of African Americans in the United States. Race is a socially constructed concept which is [a] defining characteristic for African American group membership. (Sellers, Smith, Shelton, Rowley, & Chavous, 1998, p. 18)     

Pre-service mathematics teachers become full participants in inquiry investigations

Research is described concerning the effectiveness of inquiry-based laboratory environments created in US mathematics/science education programme courses. Laboratory projects were conducted using a framework that allowed pre-service teachers to explore, analyse, and communicate ‘investigable’ realms of physical phenomena. Goals were for pre-service teachers to experience the value of learning in an inquiry-enhanced environment and to engage in contextualized mathematics so they would utilize this instruction in their future classrooms. It is proposed that inquiry-based laboratories are needed within the mathematics classroom in order to allow students the opportunity to contextualize, to connect to other disciplines, and to experience mathematical concepts. Pre-service teachers were expected to pursue conjectures, collect data, think critically, and communicate findings. This qualitative research shows how the use of inquiry can complement the learning of mathematical content and educational strategies for pre-service teachers. Results provide detailed information for teacher educators regarding instructional design of contextualized mathematical inquiry.     

Inquiry as an entry point to equity in the classroom

ABSTRACT

Although many policy documents include equity as part of mathematics education standards and principles, researchers continue to explore means by which equity might be supported in classrooms and at the institutional level. Teaching practices that include opportunities for students to engage in active learning have been proposed to address equity. In this paper, through aligning some characteristics of inquiry put forth by Cook, Murphy and Fukawa-Connelly with Gutiérrez's dimensions of equity, we theoretically explore the ways in which active learning teaching practices that focus on inquiry could support equity in the classroom.     

Developing mathematical modelling skills: The role of CAS

Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters This is an important part of validating a mathematical model formulated to describe, a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.     

Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR research programs

By continuing a contrast with the DNR research program, begun in Harel and Koichu (2010), I discuss several important issues with respect to teaching and learning mathematics that have emerged from our research program which studies learning that occurs through students’ mathematical activity and indicate issues of complementarity between DNR and our research program. I make distinctions about what we mean by inquiring into the mechanisms of conceptual learning and how it differs from work that elucidates steps in the development of a mathematical concept. I argue that the construct of disequilibrium is neither necessary nor sufficient to explain mathematics conceptual learning. I describe an emerging approach to instruction aimed at particular mathematical understandings that fosters reinvention of mathematical concepts without depending on students’ success solving novel problems.     

Aesthetics as a liberating force in mathematics education?

This article investigates different meanings associated with contemporary scholarship on the aesthetic dimension of inquiry and experience, and uses them to suggest possibilities for challenging widely held beliefs about the elitist and/or frivolous nature of aesthetic concerns in mathematics education. By relating aesthetics to emerging areas of interest in mathematics education such as affect, embodiment and enculturation, as well as to issues of power and discourse, this article argues for aesthetic awareness as a liberating, and also connective force in mathematics education.     

Music as Embodied Mathematics: A Study of a Mutually Informing Affinity

The argument examined in this paper is that music – when approached through making and responding to coherent musical structures,facilitated by multiple, intuitively accessible representations – can become a learning context in which basic mathematical ideas can be elicited and perceived as relevant and important. Students' inquiry into the bases for their perceptions of musical coherence provides a path into the mathematics of ratio,proportion, fractions, and common multiples. Ina similar manner, we conjecture that other topics in mathematics – patterns of change,transformations and invariants – might also expose, illuminate and account for more general organizing structures in music. Drawing on experience with 11–12 year old students working in a software music/math environment, we illustrate the role of multiple representations, multi-media, and the use of multiple sensory modalities in eliciting and developing students' initially implicit knowledge of music and its inherent mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Karen in motion: The role of physical enactment in developing an understanding of distance,time, and speed

The role of direct kinesthetic experience in mathematics education remains relatively unexamined. What role can physical enactment play in mathematics learning? What, if any, implications does it carry for classroom teaching? In this article I explore the role that a third grader's kinesthetic experience plays in supporting her learning of the mathematics of motion, a content area typically for older students. Based on analyses of two individual interviews and classroom participation, I argue that Karen's ability to use physical enactment to inhabit motion trips, along with a thoughtfully emergent curriculum design, created a learning environment that enabled Karen to develop a deep, conceptual understanding of distance, time, and speed.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals

Although mathematics educators seem to agree on the importance of teaching mathematics for understanding, what they mean by understanding varies greatly. In this article, I elaborate and exemplify the construct of key developmental understanding to emphasize a particular aspect of teaching for understanding and to offer a construct that could be used to frame the identification of conceptual learning goals in mathematics. The key developmental understanding construct is based on extant empirical and theoretical work. The construct can be used in the context of research and curriculum development. Using a classroom example involving fractions, I illustrate how focusing on key developmental understandings leads to particular, potentially useful types of pedagogical thinking and directions for inquiry.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.     

利用数学建模为载体 培养医科学生研究性学习能力的探索

分析了在医学数学教学中开展研究性学习的意义.介绍了数学建模与研究性学习的关系.探索了开展研究性学习活动的途径.总结了开展研究性学习活动的效果.     

The relationship between mathematical problem-solving skills and self-regulated learning through homework behaviours,motivation, and metacognition

Studies highlight that using appropriate strategies during problem solving is important to improve problem-solving skills and draw attention to the fact that using these skills is an important part of students’ self-regulated learning ability. Studies on this matter view the self-regulated learning ability as key to improving problem-solving skills. The aim of this study is to investigate the relationship between mathematical problem-solving skills and the three dimensions of self-regulated learning (motivation, metacognition, and behaviour), and whether this relationship is of a predictive nature. The sample of this study consists of 323 students from two public secondary schools in Istanbul. In this study, the mathematics homework behaviour scale was administered to measure students’ homework behaviours. For metacognition measurements, the mathematics metacognition skills test for students was administered to measure offline mathematical metacognitive skills, and the metacognitive experience scale was used to measure the online mathematical metacognitive experience. The internal and external motivational scales used in the Programme for International Student Assessment (PISA) test were administered to measure motivation. A hierarchic regression analysis was conducted to determine the relationship between the dependent and independent variables in the study. Based on the findings, a model was formed in which 24% of the total variance in students’ mathematical problem-solving skills is explained by the three sub-dimensions of the self-regulated learning model: internal motivation (13%), willingness to do homework (7%), and post-problem retrospective metacognitive experience (4%).     

Using narrative inquiry for investigating the becoming of a mathematics teacher

This article presents narrative inquiry as a method for research in mathematics education, in particular the study of how pre-service teachers’ views of mathematics develop during elementary teacher education. I describe two different, complementary approaches to applying narrative analysis, one focusing on the content of a narrative, the other focusing on the form. The examples discussed are taken from interviews with and teaching portfolios compiled by four pre-service teachers. In analysing the content of the students’ narratives, I use emplotment to construct a retrospective explanation of how one pre-service teacher’s own experiences at school were reflected in the development of her mathematical identity. In analysing the form of the narratives, I also look at how the students told their stories, using linguistic features, for example, to identify core events in the accounts. This particular focus seems to be promising in locating turning points in the trainees’ views of mathematics.     

The role of prediction in the teaching and learning of mathematics

The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.     

Simon’s team’s contributions to scientific progress in mathematics education: A commentary on the Learning Through Activity (LTA) research program

I discuss two ways in which the Learning Through Activity (LTA) research program contributes to scientific progress in mathematics education: (a) providing general and content-specific constructs to explain conceptual learning and instructional design that corroborate and/or elaborate on previous work and (b) raising new questions/issues. The general constructs include using instructional design as testable models of learning and using theoretical constructs to guide real-time, instructional adaptations. In this sense, the general constructs promote understanding of linkages between conceptual learning and instruction in mathematics. The concept-specific constructs consist of empirically-grounded, hypothetical learning trajectories (HLTs) for fractional and multiplicative reasoning. Each HLT consists of specific, intended conceptual changes and tasks that can bring them forth. Questions raised for me by the LTA work involve inconsistencies between the stance on learning and reported teaching-learning interactions that effectively led to students’ abstraction of the intended mathematical concepts.     

THE NATURE OF PARTICIPATION AFFORDED BY TASKS,QUESTIONS AND PROMPTS IN MATHEMATICS CLASSROOMS

This paper reports on the development of an analytical instrument which identifies mathematical affordances in the public tasks, questions and prompts of mathematics classrooms. The aim is to become more articulate about mathematical activity. I have explored the use of several frameworks which identify learning outcomes, structures of knowledge, mental actions, teaching actions and intentions and found that none of them give me access to the detail of what makes one mathematics lesson different from another for learners. From the experience of using these I devised a new analytical tool which unfolds patterns of participation afforded in mathematics lessons. This tool has been tested on several videos of lessons, and has been used by pre-service teaching students to analyse their own lessons.     

Perspectives on technology mediated learning in secondary school mathematics classrooms

The introduction of technology resources into mathematics classrooms promises to create opportunities for enhancing students’ learning through active engagement with mathematical ideas; however, little consideration has been given to the pedagogical implications of technology as a mediator of mathematics learning. This paper draws on data from a 3-year longitudinal study of senior secondary school classrooms to examine pedagogical issues in using technology in mathematics teaching — where “technology” includes not only computers and graphics calculators but also projection devices that allow screen output to be viewed by the whole class. We theorise and illustrate four roles for technology in relation to such teaching and learning interactions — master, servant, partner, and extension of self. Our research shows how technology can facilitate collaborative inquiry, during both small group interactions and whole class discussions where students use the computer or calculator and screen projection to share and test their mathematical understanding.