Modelling in Mathematics Classrooms: reflections on past developments and the future

This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments. After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future—for what remains to be done to enable modelling to make its essential contribution to the «functional mathematics», the mathematical literacy, of future citizens and professionals. What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them? These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging.     

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.     

How do we understand mathematical practices in non-mathematical fields? Reflections inspired by cases from 12th and 13th century China

Recent studies of the history of mathematics have shed light on the diversity of mathematical practices in the ancient world. In this article, I address this issue from two perspectives. First, I analyze different mathematical instruments used in different domains in 12^th and 13^th century China. Second, by analyzing how the text is related to the problems written down in the same period, I argue that there existed two categories of mathematical problems. As a result, I suggest that problems, textual procedures, and material operations should be considered together when regarding various mathematical practices in mathematical and non-mathematical fields.     

高师本科数学分析教学改革的研究与实践

以信息社会为特征的新世纪开始提出扫“数学盲”的历史任务 ,而脱盲的标志则是要懂得微积分基础 [1 ] ,然微分与积分是数学分析的主体的结构 .因此 ,2 1世纪的大学数学教育应该搞好数学分析课程的改革与建设 ,开展教学研究 .本文将给出我们在数学分析教学改革的研究与实践 [1 - 3] 的一系列的做法 ,以便请教于同行     

A quote a day educates

Conclusion  I often ponder on my duties as a teacher of the subject I love. I feel I am responsible for more than simply transmitting knowledge. I wish I could help my students see mathematics from various vantage points. One of these should be from a point high enough to afford a full, sweeping view of the mathematical valley below—maybe missing the details we strive to convey in class-but seeing thelandscape of mathematics. Claude Bragdon said, “Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.” I want my students to consider that such a bold statement might actually be true.     

Equity in mathematics education: unions and intersections of feminist and social justice literature

Traditional models of gender equity incorporating deficit frameworks and creating norms based on male experiences have been challenged by models emphasizing the social construction of gender and positing that women may come to know things in different ways from men. This paper draws on the latter form of feminist theory while treating gender equity in mathematics as intimately interconnected with equity issues by social class and ethnicity. I integrate feminist and social justice literature in mathematics education and argue that to secure a transformative, sustainable impact on equity, we must treat mathematics as an integral component of a larger system producing educated citizens. I argue the need for a mathematics education with tri-fold support for mathematical literacy, critical literacy, and community literacy. Respectively, emphases are on mathematics, social critique, and community relations and actions. Currently, the integration of these three literacies is extremely limited in mathematics.     

Cross-curricular activities within one subject?

The structural organization, of the Danish Gymnasium greatly hinders cross-curricular activities. However, it is possible to integrate other subjects in the mathematics curriculum, not the least due to the existence of the so-called “aspects” I will discuss a particular course on modeling ozone depletion which was framed by the “model aspect”. The organization and outcome of the course are linked to three types of competencies mathematical. technological and reflective. I will focus on the reflective competency, in particular the criticla evaluation of mathematical models and their use. One conclusion is that modeling furthers all three competencies, and thus should be given more emphasis in mathematics instruction. However, if the reflective competency is to be furthered, the topic must be seen in a broader societal context, and this would be better supported by cross-curricular activities.     

The onto-semiotic approach to research in mathematics education

In this paper we synthesize the theoretical model about mathematical cognition and instruction that we have been developing in the past years, which provides conceptual and methodological tools to pose and deal with research problems in mathematics education. Following Steiner’s Theory of Mathematics Education Programme, this theoretical framework is based on elements taken from diverse disciplines such as anthropology, semiotics and ecology. We also assume complementary elements from different theoretical models used in mathematics education to develop a unified approach to didactic phenomena that takes into account their epistemological, cognitive, socio cultural and instructional dimensions.     

Toward the problem-centered classroom: trends in mathematical problem solving in Japan

In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks

We explore the conjecture that engaging teachers with activities which feature mathematical practices from the past (history-focused tasks) and in today’s mathematics classrooms (mathtasks) can promote teachers’ problematizing of mathematics and its pedagogy. Here, we sample evidence of discursive shifts observed as twelve mathematics teachers engage with a set of problematizing activities (PA) – three rounds of history-focused and mathtask combinations – during a four–month postgraduate course. We trace how the commognitive conflicts orchestrated in the PA triggered changes in the teachers’ narratives about: mathematical objects (such as what a function is); how mathematical objects come to be (such as what led to the emergence of the function object); and, pedagogy (such as what value may lie in listening to students or in trialing innovative assessment practices). Our study explores a hitherto under-researched capacity of the commognitive framework to steer the design, evidence identification and impact evaluation of pedagogical interventions.     

国内数学学科论著产出的结构以及与国际数学热门研究领域的比较

本文以统计数据为依据对国内数学学科近五年的研究状况、发展趋势以及国内学者数学论著的产出结构做了分析，并与国际数学领域的研究动向和发展态势进行了比较．     

An Investigation of Relationships between Seventh-Grade Students' Beliefs and Their Participation during Mathematics Discussions in Two Classrooms

As mathematics teachers attempt to promote classroom discourse that emphasizes reasoning about mathematical concepts and supports students' development of mathematical autonomy, not all students will participate similarly. For the purposes of this research report, I examined how 15 seventh-grade students participated during whole-class discussions in two mathematics classrooms. Additionally, I interpreted the nature of students' participation in relation to their beliefs about participating in whole-class discussions, extending results reported previously (Jansen, 2006) about a wider range of students' beliefs and goals in discussion-oriented mathematics classrooms. Students who believed mathematics discussions were threatening avoided talking about mathematics conceptually across both classrooms, yet these students participated by talking about mathematics procedurally. In addition, students' beliefs about appropriate behavior during mathematics class appeared to constrain whether they critiqued solutions of their classmates in both classrooms. Results suggest that coordinating analyses of students' beliefs and participation, particularly focusing on students who participate outside of typical interaction patterns in a classroom, can provide insights for engaging more students in mathematics classroom discussions.     

Mathematical Sociology: Some Educational and Organizational Problems of an Emergent Sub‐Discipline

Three major areas of mathematical sociology are critically reviewed: analysis of measurement, statistical analysis and model building. Next, some social problems, created by the introduction of mathematics into sociology, are discussed. These include the emergence of inflated expectations for mathematical sociology which are subsequently disappointed, and the potential status threat which mathematical sociology poses for non‐mathematical sociologists. Examples of mathematical applications in the construction of causal models, population projection and the analysis of stability in social groups are discussed. Following this, the role of mathematics in the education of undergraduate sociology majors is considered. Neither mathematics nor statistics should be required of such persons, but they should be encouraged to acquire a mathematical background if interested. Statistics should, however, be required of sociology graduate students. The graduate training of mathematical sociologists should emphasize research over course work. An apprenticeship relationship with a faculty member working in mathematical sociology is highly desirable for these students. A substantive specialty is also useful since it enables mathematical sociologists to stay in contact with mainstream sociology. Emphasis is placed on the function of present research in legitimating future expanded mathematical education of sociologists.     

Discrete mathematics in primary and secondary schools in the United States

This article provides a status report on discrete mathematics in America's schools, including an overview of publications and programs that have had major impact. It discusses why discrete mathematics should be introduced in the schools and the authors' efforts to advocate, facilitate, and support the adoption of discrete mathematics topics in the schools. Their perspective is that discrete mathematics should be viewed not only as a collection of new and interesting mathematical topics, but, more importantly, as a vehicle for providing teachers with a new way to think about traditional mathematical topics and new strategies for engaging their students in the study of mathematics.     

Students Explaining Solutions in Student-Directed Groups: Cooperative Learning and Reform in Mathematics Education

Cooperative learning experiences can contribute to mathematics education reform by stimulating student communication. Sixth grade student conversations were recorded on four occasions over a four month period when they were working in cooperative groups. The results indicated that routine compliance with the requirement to “explain” superseded authentic dialogues about mathematical ideas. Student conversations were influenced by the model of explanation exchanges emerging from the teacher's visits to groups. Teacher influence was mediated by students' past experiences. The findings suggest that teachers implementing reform should help students develop criteria for judging mathematical arguments and confront student conceptions directly to deepen debates.     

Mathematics standards and curricula in the Netherlands

This paper addresses the question of what mathematics Dutch students should learn according to the standards as established by the Dutch Ministry of Education. The focus is on primary school and the foundation phase of secondary school. This means that the paper covers the range from kindergarten to grade 8 (4~14 years olds). Apart from giving an overview of the standards, we also discuss the standards' nature and history Furthermore, we look at textbooks and examination programs that in the Netherlands both have a key role in determining the intended mathematics curriculum. In addition to addressing the mathematical content, we also pay attention to the way mathematics is taught. The domain-specific education theory that forms the basis for the Dutch approach to teaching mathematics is called “Realistic Mathematics Education” Achievement scores of Dutch students from national and international tests complete this paper. These scores reveal what the standards bring us in terms of students' mathematical understanding. In addition to informing an international audience about the Dutch standards and curricula, we include some critical reflections on them.     

A Situated and Sociocultural Perspective on Bilingual Mathematics Learners

My aim in this article is to explore 3 perspectives on bilingual mathematics learners and to consider how a situated and sociocultural perspective can inform work in this area. The 1st perspective focuses on acquisition of vocabulary, the 2nd focuses on the construction of multiple meanings across registers, and the 3rd focuses on participation in mathematical practices. The 3rd perspective is based on sociocultural and situated views of both language and mathematics learning. In 2 mathematical discussions, I illustrate how a situated and sociocultural perspective can complicate our understanding of bilingual mathematics learners and expand our view of what counts as competence in mathematical communication.     

A Situated and Sociocultural Perspective on Bilingual Mathematics Learners

My aim in this article is to explore 3 perspectives on bilingual mathematics learners and to consider how a situated and sociocultural perspective can inform work in this area. The 1st perspective focuses on acquisition of vocabulary, the 2nd focuses on the construction of multiple meanings across registers, and the 3rd focuses on participation in mathematical practices. The 3rd perspective is based on sociocultural and situated views of both language and mathematics learning. In 2 mathematical discussions, I illustrate how a situated and sociocultural perspective can complicate our understanding of bilingual mathematics learners and expand our view of what counts as competence in mathematical communication.