Making abstract mathematics concrete in and out of school

We adopt a neo-Vygotskian view that a fully concrete scientific concept can only emerge from engaging in practice with systems of theoretical concepts, such as when mathematics is used to make sense of outside school or vocational practices. From this perspective, the literature on mathematics outside school tends to dichotomise in- and out-of-school practice and glamorises the latter as more authentic and situated than academic mathematics. We then examine case study ethnographies of mathematics in which this picture seemed to break down in moments of mathematical problem solving and modelling in practice: (1) when amateur or professional players decided to investigate the mathematics of darts scoring to develop their “outing” strategies and (2) when a prevocational mathematics course task challenged would-be mathematics teachers’ concept of fractions. These examples are used to develop the Vygotskian framework in relation to vocational and workplace mathematics. Finally, we propose that a unified view of mathematics, outside and inside school, on the basis of Vygotsky’s approach to everyday and scientific thought, can usefully orientate further research in vocational mathematics education.     

REJOINDER

In this paper, we explore the influence of the teacher on the language games being played in a Grade 3 mathematics classroom. The kinds of listening strategies used by the teacher seemed to have an impact on the kind of language game being played as a small group of students engaged in a problem‐solving activity. Understanding classroom conversation from a language‐games perspective is used to explore the kinds of sense‐making, student engagement, and new understandings evolving through classroom discourse. Language games may also be important for encouraging students to extend and transform their understandings. We describe how learning to ‘teach etcetera’ as a way of helping students ‘see‐as’ and move beyond narrowly defined predetermined content may entail playing more open‐ended language games and learning how to ‘listen’.     

Improving the Odds: Identifying Language Activities that Support the Language Development of Preschoolers with Poorer Vocabulary Skills

The aim of the current study is to determine what language activities Norwegian preschool children took part in, and to examine whether these language activities predict children’s language comprehension. We tested children (n?=?134) with language measures at age 4/5 and age 5/6 and interviewed their teachers (n?=?71) about the kinds of language activities the children engaged in during that school year. Teachers reported a variety of classroom language activities, ranging from informal language stimulation of everyday situations to more explicit language activities such as book reading, language games, vocabulary training and school preparation groups. Book reading every day significantly predicted children’s language comprehension.     

Changes in school mathematics knowledge in Greece: a Bernsteinian analysis

The purpose of this paper is to critically examine the changes that took place in school mathematics knowledge for pupils aged six and seven (first grade of Greek elementary school) and its pedagogical approach, which took shape following the reforms of the mathematics curricula in 1982 and 2003. Our analysis is based on Bernstein’s theoretical framework on pedagogic discourse and Dowling’s theoretical framework on textual domains. The results showed that there are differences between the 1982 and 2003 reforms, in terms both of the content of mathematics knowledge, as well as of the forms of pedagogic interaction between the teacher and the pupil.     

Language Games in the Ivory Tower: Comparing the Philosophical Investigations with Hermann Hesse's The Glass Bead Game

Wittgenstein explores learning through practice in the Philosophical Investigations by means of an extended analogy with games. However, does this concern with learning also necessarily extend to education, in our institutional understanding of the word? While Wittgenstein's examples of language learning and use are always shared or social, he does not discuss formal educational institutions as such. He does not wish to found a ‘school of thought’, and is suspicious of philosophy acting as a theory that can be applied to other areas of life. While Wittgenstein's focus on developing independent thinking was neither individualistic nor anti‐institutional, it did, however, focus on developing the thinking of his students rather than theorising about how this could be applied on a large scale. An analysis of Hermann Hesse's novel, The Glass Bead Game will help us to pick up where Wittgenstein deliberately left off—thinking about how (or if) one can institutionalise learning methods that encourage thinking for oneself. These differences in the writers’ treatment of education will become evident in the differences between their game analogies. While language‐games combat our ‘craving for generality’ in Philosophical Investigations, the Glass Bead Game represents this craving, and how it manifests itself throughout history in disciplines other than logic and philosophy of language. It also represents the potential for institutions to become insular, exclusive communities.     

Perceived autonomy support and autonomous motivation toward mathematics activities in educational and out-of-school contexts is related to mathematics homework behavior and attainment

We adopted a trans-contextual model of motivation to examine the processes by which school students' perceived autonomy support (defined as students' perceptions that their teachers' support their autonomous or self-determined motivation) and autonomous forms of motivation (defined as motivation to act out of a sense of choice, ownership, and personal agency) toward mathematics activities in an educational context predict autonomous motivation and intentions toward mathematics homework, and actual mathematics homework behavior and attainment, as measured by homework grades, in an out-of-school context. A three-wave prospective study design was adopted. High-school students (N = 216) completed self-report measures of perceived autonomy support and autonomous forms of motivation toward mathematics activities in school in the first wave of data collection. One-week later, participants completed measures of autonomous forms of motivation, attitudes, subjective norms, perceived behavioral control, and intentions with respect to mathematics homework outside school. Students' self-reported homework behavior and homework grades from students' class teachers were collected 5-weeks later. A structural equation model supported model hypotheses. Perceived autonomy support and autonomous forms of motivation toward mathematics activities in school were related to autonomous forms of motivation toward mathematics homework outside of school. Autonomous forms of motivation toward mathematics homework predicted intentions to do mathematics homework mediated by attitudes, subjective norms and perceived behavioral control. Intentions predicted self-reported mathematics homework behavior and mathematics homework grades. Perceived autonomy support and autonomous forms of motivation toward mathematics in school had statistically significant indirect effects on mathematics homework intentions mediated by the motivational sequence of the model. Results provide preliminary support for the model and evidence that autonomous motivation toward mathematics activities in the classroom is linked with autonomous motivation, intention, behavior and actual attainment in mathematics homework outside of school.     

Towards a theory of mathematics homework as a social practice

This article presents a theoretical conceptualization of mathematics homework as a social practice. Rather than considering homework as a task or an artifact, this approach frames homework in terms of the social contexts in which students participate and how students participate in those contexts. This perspective has long been suggested by homework researchers but has not been developed as a framework for understanding homework. Drawing from Wenger’s (1998) social theory of learning and research grounded in sociocultural theory, this conceptualization makes central meaning making and identity development, and puts forth meaning and identity as lenses for understanding students’ participation in the practice of mathematics homework. To further develop this conceptualization of homework, I draw on data from an ethnographic study of the role and meaning of mathematics homework in the lives of middle school students. Case studies of two students are presented to demonstrate the relationships among the meaning of homework, students’ identities, and their participation in the practice of homework. One student’s experiences support him in identifying as a capable mathematics student who is bound for continued academic success through high school to college. Thus, he comes to take on mathematics homework as a means to learn and succeed. The other student’s experiences support him in building an identity that leads him to reject homework. This conceptualization of homework and the case study data have implications for the practice of homework and for theories of students’ motivational dispositions in the context of mathematics homework and learning in general.     

生活情境教学法用于小学数学教学的实践分析

生活处处有数学。无论是从新时期数学课程改革的要求来看,还是从教育教学理论分析或者小学生的全面发展来看,将生活情境教学法用于小学数学教学都具有重要意义。本文首先从数学学科的特点出发,简要分析生活情境教学法应用研究的必要性;然后从新课程标准、教育学理论依据、学生全面发展三个方面分析生活情境教学法用于小学数学教学实践的依据,进而探究相关的实践方法。     

Elementary-age children’s conceptions about mathematics utility and their home-based mathematics engagement

Integrating multiple theoretical frameworks, the authors examined rising first- to fourth-grade students’ mathematics utility conceptions—their knowledge and beliefs about the usefulness of mathematics, home-based mathematics engagement, and grade-level differences in mathematics utility conceptions and home engagement. Most children viewed mathematics as heavily focused on low-level mathematics operations and as learned and used primarily in school. Older children showed more awareness of mathematics as part of daily living, but still viewed mathematics as mostly school-based—more so than their younger counterparts. Results suggest that awareness of mathematics in daily life may be associated with children’s mathematics utility value (perceived usefulness of mathematics). Although children engaged in activities at home with the potential to foster mathematics development, the frequency of engagement was not related to their awareness of mathematics in daily activities. Thus, there may be untapped opportunities for young children to connect the mathematics they learn in school to their daily life.     

Effects of playing mathematics computer games on primary school students’ multiplicative reasoning ability

This study used a large-scale cluster randomized longitudinal experiment (N = 719; 35 schools) to investigate the effects of online mathematics mini-games on primary school students’ multiplicative reasoning ability. The experiment included four conditions: playing at school, integrated in a lesson (E_school), playing at home without attention at school (E_home), playing at home with debriefing at school (E_home-school) and, in the control group, playing at school mini-games on other mathematics topics (C). The mini-games were played in Grade 2 and Grade 3 (32 mini-games in total). Using tests at the end of each grade, effects on three aspects of multiplicative reasoning ability were measured: knowledge of multiplicative number facts, skills in multiplicative operations, and insight in multiplicative number relations and properties of multiplicative operations. Through path analyses it was found that the mini-games were most effective in the E_home-school condition, where both students’ skills and their insight were positively affected as compared to the control group (significant ds ranging from 0.22 to 0.29). In the E_school condition, an effect was only found for insight in Grade 2 (d = 0.35), while in the E_home condition no significant effects were found. Students’ gameplay behavior (time and effort put in the games) was in some cases, but not always, related to their learning outcomes.     

The impacts of an early mathematics curriculum on oral language and literacy

Competence in early mathematics is crucial for later school success. Although research indicates that early mathematics curricula improve children's mathematics skill, such curricula's impacts on oral language and early literacy skills are not known. This project is the first to investigate the effects of an intensive pre-kindergarten mathematics curriculum, Building Blocks, on the oral language and letter recognition of children participating in a large-scale cluster randomized trial project. Results showed no evidence that children who were taught mathematics using the curriculum performed differently than control children who received the typical district mathematics instruction on measures of letter recognition, and on two of the oral language (story retell) subtests, sentence length and inferential reasoning (emotive content). However, children in the Building Blocks group outperformed children in the control group on four oral language subtests: ability to recall key words, use of complex utterances, willingness to reproduce narratives independently, and inferential reasoning (practical content).     

Professional development for practicing mathematics teachers: a critical connection to English language learner students in mainstream USA classrooms

This article examines the effects of changing teacher self-efficacy on student achievement. Results from a quantitative study of mathematics teachers in a USA mid-Atlantic state showed that teachers experienced reduced self-efficacy when working with English language learner students (ELLs) relative to non-ELLs. Further, the article explores professional development opportunities for strengthening teachers’ effectiveness with ELLs. The study utilized a four-part survey instrument based on the Teacher Sense of Efficacy Scale (TSES). All participants (n = 181) were mathematics educators from the highlighted state, and 94 % of them were preK-12 public school practicing classroom teachers. Study results indicate that teachers’ participation in professional development on ELL instruction is positively correlated to their heightened sense of self-efficacy. This finding points to continuing education as a potentially viable approach to reducing academic challenges for ELL students in mainstream mathematics classrooms by developing ever more effective practicing teachers.     

MATHEMATICS AS AN IM/PURE KNOWLEDGE SYSTEM: SYMBIOSIS, (W)HOLISM AND SYNERGY IN MATHEMATICS EDUCATION

The problem of culturally decontextualised mathematics education faced by Nepali students, teachers and teacher educators has often been oriented by the view of the nature of mathematics as a body of pure knowledge, which gives rise to an exclusive emphasis on an ideology of singularity, epistemology of objectivism, language of universality and logic of certainty whilst developing curriculum, conceiving pedagogies and implementing assessment strategies in school mathematics education and mathematics teacher education programmes. With epistemic referents of dialectical logics and performative imagination, an alternative view of the nature of mathematics as an impure knowledge is discussed with its possible disempowering features, such as essentialism, hegemony and dualisms. Finally, an inclusive view of the nature of mathematics as im/pure knowledge system is articulated with the help of various forms of dialectics.     

试论新课改下小学数学教学的创新

随着新课程改革的不断深入,小学数学教师也应转变传统的教学观念,创新数学教学方法,以达到新课程改革对小学阶段数学教学目标的新要求。小学阶段的数学教学能为学生未来更好地学习数学知识打好理论基础,在这一段时间内培养学生的良好学习习惯能使学生的学习达到事半功倍的效果。因此抓住这一关键期对学生进行教育,对帮助学生形成数学思维、提高学生运用数学知识解决实际问题的能力有重要作用。基于此,本文将研究在新课改背景下小学数学教学方法的几种创新形式。     

A cognitive analysis of Cauchy’s conceptions of function,continuity, limit and infinitesimal,with implications for teaching the calculus

In this paper, we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints.     

Critical alignment in inquiry-based practice in developing mathematics teaching

This paper reports a case study from a mathematics teaching developmental research project. The theoretical foundation for the research comprises communities of inquiry and critical alignment, with which the developmental methodology has a particular synergy. This synergy is the main focus of the paper. The paper elaborates theoretical and methodological antecedents of the project and traces these through a case study of developments in the practices of one upper secondary school team and a group of university didacticians (mathematics teacher educators and researchers) during the first year of the project. The case study reveals that critical alignment and inquiry (necessarily) bring uncertainty and risk, and foster tensions within the teachers’ practice and between the practices of teachers and didacticians. In exposing these uncertainties, risks and tensions, the paper points to their value for the learning and knowledge gained by participants.     

Taiwanese high school students’ perspectives on effective mathematics teaching behaviors

This study developed a questionnaire composed of teaching-behavior items obtained from pioneering empirical studies, for measuring students’ perspectives on effective mathematics teaching. The questionnaire was used to survey 4514 high school students in Taiwan, where Western innovative mathematics instruction has been promoted since a curriculum reform in 1993. Exploratory factor analysis revealed factors contributing to how teachers should handle teaching materials, use representations and teaching methods, and guide problem solving. Ten factors were identified. Seven are rooted in traditional Chinese educational culture (e.g., detailed illustration in teaching problem solving), and three were influenced by Western culture (e.g., student active-learning activities in teaching methods). The findings show that teaching behaviors that either require a considerable amount of time or minimize use of time are relatively less favored, including learning in small groups or through games in the student active-learning activities factor and only providing important steps in the speedy lecture factor.     

Multiple Paths to Mathematics Practice in Al-Kashi’s Key to Arithmetic

In this paper, I discuss one of the most distinguishing features of Jamshid al-Kashi’s pedagogy from his Key to Arithmetic, a well-known Arabic mathematics textbook from the fifteenth century. This feature is the multiple paths that he includes to find a desired result. In the first section light is shed on al-Kashi’s life and his contributions to mathematics and astronomy. Section 2 starts with a brief discussion of the contents and pedagogy of the Key to Arithmetic. Al-Kashi’s multiple approaches are discussed through four different examples of his versatility in presenting a topic from multiple perspectives. These examples are multiple definitions, multiple algorithms, multiple formulas, and multiple methods for solving word problems. Section 3 is devoted to some benefits that can be gained by implementing al-Kashi’s multiple paths approach in modern curricula. For this discussion, examples from two teaching modules taken from the Key to Arithmetic and implemented in Pre-Calculus and mathematics courses for preservice teachers are discussed. Also, the conclusions are supported by some aspects of these modules. This paper is an attempt to help mathematics educators explore more benefits from reading from original sources.