A Brief Comparison of the U.S. and Chinese Middle School Mathematics Programs

The U.S. generally has a less intense mathematics curriculum in the middle school grades than China. Some factors contributing to the lower intensity in the U.S. mathematics curriculum are textbooks with extensive drill, repetition of content, lack of challenging problem solving, lower curricular and cultural expectations, and ability grouping. In comparison, China utilizes challenging problem solving, sequential development of content without repetition, expectations of hard work, high values for mathematics by the curriculum and culture, and a common curriculum for all as aspects of mathematics instruction. The U.S. is taking a positive direction in its mathematics curriculum with the use of technology and reform while compulsory education is mandating that the theoretical depth of middle school curriculums in China be lowered for all of its students in grades 1–9.     

The graphics calculator in mathematics education

This article covers a project conducted by the Freudenthal Institute from August 1991 to September 1994 entitled “The graphics calculator in mathematics education.” The theory of realistic mathematics education was taken as the point of departure for formulating the hypotheses. The developmental research design was used. Observation of the students' behavior during the experimental lessons supports the premise that the graphics calculator can stimulate the use of realistic contexts, the exploratory and dynamic approach to mathematics, a more integrated view of mathematics, and a more flexible behavior in problem solving.     

Aims in Mathematical Education and their Implications for the Training of Mathematics Teachers

The extent to which aims in mathematical education are common to different spheres of education and industry is considered, and an approach to mathematics in terms of problem‐solving is illustrated by two examples of investigations. It is suggested that greater emphasis should be given to problem‐solving as opposed to content‐‐this view is related to specified aims in mathematics teaching. After a discussion of the provisional status of some current educational assumptions, implications for the training of teachers are briefly examined with an indication of some mathematics education courses offered at the University of Keele. In a short conclusion some points on transfer of abilities, the balance between discovery and technique, public and private mathematics, and in‐service training are considered.     

Connections and confusion: teaching perimeter and area with a problem-solving oriented unit

Problem-solving-oriented mathematics curricula are viewed as important vehicles to help achieve K-12 mathematics education reform goals. Although mathematics curriculum projects are currently underway to develop such materials, little is known about how teachers actually use problem-solving-oriented curricula in their classrooms. This article profiles a middle-school mathematics teacher and examines her use of two problems from a pilot version of a sixth-grade unit developed by a mathematics curriculum project. The teacher's use of the two problems reveals that although problem-solving-oriented curricula can be used to yield rich opportunities for problem solving and making mathematical connections, such materials can also provide sites for student confusion and uncertainty. Examination of this variance suggests that further attention should be devoted to learning about teachers' use of problem-solving-oriented mathematics curricula. Such inquiry could inform the increasing development and use of problem-solving-oriented curricula.     

Problem solving in the mathematics classroom: the German perspective

In Germany, problem solving has important roots that date back at least to the beginning of the twentieth century. However, problem solving was not primarily an aspect of mathematics education but was particularly influenced by cognitive psychologists. Above all, the Gestalt psychology developed by researchers such as Köhler (Intelligenzprüfungen an Anthropoiden. Verlag der Königlichen Akademie des Wissens, Berlin, 1917; English translation: The mentality of apes. Harcourt, Brace, New York, 1925), Duncker (Zur Psychologie des produktiven Denkens. Springer, Berlin, 1935), Wertheimer (Productive thinking. Harper, New York, 1945), and Metzger (Schöpferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this research had hardly any influence on mathematics education—neither as a scientific discipline nor as a foundation for mathematics instruction. In the German mathematics classroom, problem solving, which is according to Halmos (in Am Math Mon 87:519–524, 1980) the “heart of mathematics,” did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situation may change. In the past few years, nationwide standards for school mathematics have been introduced in Germany. In these standards, problem solving is specifically addressed as a process-oriented standard that should be part of the mathematics classroom through all grades. This article provides an overview on problem solving in Germany with reference to psychology, mathematics, and mathematics education. It starts with a presentation of the historical roots but gives also insights into contemporary developments and the classroom practice.     

Mathematical problem solving in didactic institutions as a complex system: The case of elementary calculus

The question of problem-solving activities in didactic institutions is critical in mathematics education for two important reasons. It is a main factor of learning according to Piaget, and it is a means for students to try to align their behaviors to expected institutional references. Mathematical reasoning during problem solving in didactic institutions is studied in the present work as a complex system of interfering constraints. Results tend to show that this system may be understood as being ruled by ternary interactions between three poles: the student, the teacher, and the knowledge itself. Simultaneously, theoretical and pragmatic considerations are focused on problem solving in mathematics: the specific epistemological difficulties of each domain of knowledge to be studied, the computational asymmetry between mathematical concepts and procedures, and the influence of implicit teacher expectations through students' decoding of local “didactic contracts.”     

Students’ and Teachers’ Conceptual Metaphors for Mathematical Problem Solving

Metaphors are regularly used by mathematics teachers to relate difficult or complex concepts in classrooms. A complex topic of concern in mathematics education, and most STEM‐based education classes, is problem solving. This study identified how students and teachers contextualize mathematical problem solving through their choice of metaphors. Twenty‐two high‐school student and six teacher interviews demonstrated a rich foundation for these shared experiences by identifying the conceptual metaphors. This mixed‐methods approach qualitatively identified conceptual metaphors via interpretive phenomenology and then quantitatively analyzed the frequency and popularity of the metaphors to explore whether a coherent metaphorical system exists with teachers and students. This study identified the existence of a set of metaphors that describe how multiple classrooms of geometry students and teachers make sense of mathematical problem solving. Moreover, this study determined that the most popular metaphors for problem solving were shared by both students and teachers. The existence of a coherent set of metaphors for problem solving creates a discursive space for teachers to converse with students about problem solving concretely. Moreover, the methodology provides a means to address other complex concepts in STEM education fields that revolve around experiential understanding.     

Effective collaboration in the productive failure process

Productive failure is a learning design that encompasses problem solving prior to instruction and the learning that occurs during and after this process. In the mathematics education literature, there is a need for analyses of students’ interactions that occur as they collaborate during the productive failure process. In this paper, we contribute to this area by taking a closer look at students’ interactions that characterize an effective productive failure process. In analyzing video footage of two different groups of students working on invention tasks in a flipped mathematics classroom, we observed that the productive failure process seemed to work best in groups of students among whom the instructional design evoked students’ intellectual need and curiosity. These students also developed a set routine for solving problems whose solutions are difficult to find without prior direct instruction on the topic, which proved valuable on follow-up in-class and posttest problems.     

Online education for in-service secondary teachers and the incorporation of mathematics technology in the classroom

This paper reviews existing research on how in-service high school teachers have learned about, worked on or thought about the incorporation of mathematics technology into their teaching practices. The paper reviews different scenarios of instruction issuing from important research related to teacher professional development. Specifically, we will deal with contributions to online in-service mathematics teacher education that refer to the use of digital technologies in classroom teaching practices. The different articles reviewed belong to a range of teams of researchers from several universities and countries, and who have implemented distinct online education approaches. That work has allowed the gaining of knowledge on the specificities of using Web 2.0 tools for mathematics professional development (MPD), the function that online teacher interaction has in teacher learning, and the actual classroom conditions in which mathematics technology is incorporated into instructional practice. This paper describes and discusses the design features of those approaches emphasizing the main concepts and their underpinning theoretical frames, noting important design elements, and specific results. Finally, the paper discusses how some of these research findings are connected with emergent issues in the field of MPD.     

Curriculum developers and problem solving: the case of Israeli elementary school projects

Problem solving has been a main focus in mathematics education for several decades, yet it seems that its definition and classroom implementation are far from being consensual. We explore the views and approaches of a small community: the project leaders of five elementary mathematics curriculum development projects in Israel, working within a centralized system, which dictates the syllabus. We describe and analyze their views along six categories: What are problems? What are not problems? Classification of problems, problem solving and individual differences, the ratio of problem solving tasks to other tasks in the project, and the role of heuristics and metacognition in teaching problem solving. We describe, exemplify, interpret and discuss the (few) points of convergence and the many different approaches. Finally, we reflect on the possible role of research in settling those differences. We speculate that our analysis and results go beyond the local and the idiosyncratic.     

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments.     

Handheld technology for mathematics education: flashback into the future

In the 1990s, handheld technology allowed overcoming infrastructural limitations that had hindered until then the integration of ICT in mathematics education. In this paper, we reflect on this integration of handheld technology from a personal perspective, as well as on the lessons to be learnt from it. The main lesson in our opinion concerns the growing awareness that students’ mathematical thinking is deeply affected by their work with technology in a complex and subtle way. Theories on instrumentation and orchestration make explicit this subtlety and help to design and realise technology-rich mathematics education. As a conclusion, extrapolation of these lessons to a future with mobile multi-functional handheld technology leads to the issues of connectivity and in- and out-of-school collaborative work as major issues for future research.     

Humans-with-media and continuing education for mathematics teachers in online environments

This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25?years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that ??blends?? aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers?? classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.     

Calculator Use on NAEP: A Look at Fourth‐ and Eighth‐Grade Mathematics Achievement

This article summarizes research conducted on calculator block items from the 2007 fourth‐ and eighth‐grade National Assessment of Educational Progress Main Mathematics. Calculator items from the assessment were categorized into two categories: problem‐solving items and noncomputational mathematics concept items. A calculator has the potential to be used as a problem‐solving tool for items categorized in the first category. On the other hand, there are no practical uses for calculators for noncomputational mathematics concept items. Item‐level performance data were disaggregated by student‐reported calculator use to investigate the differences in achievement of those fourth‐ and eighth‐grade students who chose to use calculators versus those who did not, and whether or not the nation's fourth and eighth graders are able to identify items where calculator use serves as an aide for solving a given mathematical problem. Results from the analysis show that eighth graders, in particular, benefit most from the use of calculators on problem‐solving items. A small percentage of students at both grade levels attempted to use a calculator to solve problems in the noncomputational mathematics concept category (items in which the use of a calculator does not serve as a tool to solve the problem).     

Mathematical problem solving: an evolving research and practice domain

Research programs in mathematical problem solving have evolved with the development and availability of computational tools. I review and discuss research programs that have influenced and shaped the development of mathematical education in Mexico and elsewhere. An overarching principle that distinguishes the problem solving approach to develop and learn mathematics is to conceptualize the discipline as a set of dilemmas or problems that need to be explored and solved in terms of mathematical resources and strategies. In this context, relevant questions that help structure and organize this paper include: What does it mean to learn mathematics in terms of problem solving? To what extent do research programs in problem solving orient curricular proposals? What types of instructional scenarios promote the students’ development of mathematical thinking based on problem solving? What type of reasoning do students develop as a result of using distinct computational tools in mathematical problem solving?     

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.     

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.     

Problem solving in Chinese mathematics education: research and practice

This paper is an attempt to paint a picture of problem solving in Chinese mathematics education, where problem solving has been viewed both as an instructional goal and as an instructional approach. In discussing problem-solving research from four perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based. We also describe several problem-solving activities in the Chinese classroom, including “one problem multiple solutions,” “multiple problems one solution,” and “one problem multiple changes.” Unfortunately, there are no empirical investigations that document the actual effectiveness and reasons for the effectiveness of those problem-solving activities. Nevertheless, these problem-solving activities should be useful references for helping students make sense of mathematics.     

In the books there are golden houses: mathematics assessment in East Asia

In this paper, some fundamental issues on mathematics assessment and how they are related to the underlying cultural values in East Asia are discussed. Features of the East Asian culture that impact on mathematics assessment include the pragmatic nature of the culture, the social orientation of East Asian people, and the lop-sided stress on the utilitarian function of education. East Asians stress the algorithmic side of mathematics, and mathematics is viewed more as a set of techniques for calculation and problem solving. The notion of fairness in assessment is of paramount importance, and there is a great trust in examination as a fair method of differentiating between the able and the less able. The selection function of education and assessment has great impact on how mathematics is taught, and assessment constitutes an extrinsic motivation which directs student learning. Finally, the strengths and weaknesses of these East Asian values are discussed.     

Students’ Activity in Computer-Supported Collaborative Problem Solving in Mathematics

The purpose of this study was to analyse secondary school students’ (N = 16) computer-supported collaborative mathematical problem solving. The problem addressed in the study was: What kinds of metacognitive processes appear during computer-supported collaborative learning in mathematics? Another aim of the study was to consider the applicability of networked learning in mathematics. The network-based learning environment Knowledge Forum (KF) was used to support students’ collaborative problem solving. The data consist of 188 posted computer notes, portfolio material such as notebooks, and observations. The computer notes were analysed through three stages of qualitative content analysis. The three stages were content analysis of computer notesin mathematical problem solving, content analysis of mathematical problem solving activity and content analysis of the students’ metacognitive activity. The results of the content analysis illustrate how networked discussions mediated mathematical knowledge and students’ questions, while the mathematical problem solving activity shows that the students co-regulate their thinking. The results of the content analysis of the students’ metacognitive activity revealed that the students use metacognitive knowledge and make metacognitive judgments and perform monitoring during networked discussions. In conclusion, the results of this study demonstrate that working with the networked technology contributes to the students’ use of their mathematical knowledge and stimulates them into making their thinking visible. The findings also show some metacognitive activity in the students’ computer-supported collaborative problem solving in mathematics.