Mathematics Education as a Design Science

We propose re-conceptualizing the field of mathematics education research as that of a design science akin to engineering and other emerging interdisciplinary fields which involve the interaction of “subjects”, conceptual systems and technology influenced by social constraints and affordances. Numerous examples from the history and philosophy of science and mathematics and ongoing findings of M&M research are drawn to illustrate our notion of mathematics education research as a design sicence. Our ideas are intended as a framework and do not constitute a, “grand” theory (see Lester. 2005, this issue). That is, we provide a framework (a system of thinking together with accompanying concepts, language, methodologies, tools, and so on) that provides structure to help mathematics education researchers develop both models and theories, which encourage diversity and emphasize Darwinian processes such as: (a) selection (rigorous testing), (b) communication (so that productive ways of thinking spread throughout relevant communities), and (c) accumulation (so that productive ways of thinking are not lost and get integrated into future developments).     

Methods of evaluation and scale of functional spaces

Linear equations of mathematical physics with constant coefficients have fed calculational mathematics since the 18th century. The area of nonlinear equations with variable coefficients arose due to gas-hydrodynamic problems in the 20th century. Now, one of the methods of research for properties of solutions of such equations and, accordingly, applied problems is the use of calculations on modern computers. The capacities of computers and their efficiency have increased in the 21st century and allowed progress to be made in solving applied problems, except for cases of methodical errors in calculations. One of the basic sources of such methodical errors is the “uncontrollable” machine accuracy of calculations. One of the methods of solving such numerical problems is a suitable localization of the problem and the choice of an adequate basis of the necessary functional space. Below, we state new results in this area of mathematical and applied research.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Connecting multiplicative thinking and mathematical reasoning in the middle years

Mathematical reasoning and problem solving are recognised as essential 21st century skills. However, international assessments of mathematical literacy suggest these are areas of difficulty for many students. Evidenced-based learning trajectories that identify the key ideas and strategies needed to teach mathematics for understanding and support these important capacities over time are needed to support teachers and curriculum developers so that they do not have to rely solely on mathematics content knowledge. Given this goal and recent evidence to suggest a relationship between the development of multiplicative thinking and mathematical reasoning, this paper explores the processes involved in developing a single, integrated scale for multiplicative thinking and mathematical reasoning using data from a four-year design-based project to establish learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years of schooling.     

Pre-service Teachers’ Developing Conceptions about the Nature and Pedagogy of Mathematical Modeling in the Context of a Mathematical Modeling Course

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers’ developing conceptions about (a) the nature of mathematical modeling in simulations of “real life” problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students’ ways of thinking. The researchers’ conceptual development, on the other hand, involved realizing the critical aspect of the “teacher role” played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.     

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.     

From convergence to divergence: the development of mathematical problem solving in research, curriculum, and classroom practice in Singapore

Following the movement of problem solving in the US and other parts of the world in the 1980s, problem solving became the central focus of Singapore’s national school mathematics curriculum in 1990 and thereafter the key theme in research and practice. Different from some other countries, this situation has largely not changed in Singapore mathematics education since then. However, within the domain of problem solving, mathematics educators in Singapore focused more on the fundamental knowledge, basic skills, and heuristics for problem solving till the mid 1990s. In particular, problem solving heuristics, especially the so-called “model method”, a term most widely used for problem solving, received much attention in syllabus, research, and classroom instruction. Since the late 1990s, following the national vision of “Thinking Schools, Learning Nation” and nurturing modern citizens with independent, critical, and creative thinking, Singapore mathematics educators’ attention has greatly expanded to the development of students’ higher-order thinking, self-reflection and self-regulation, alternative ways of assessment and instruction, among other aspects concerning problem solving. Researchers have also looked into the advantages and disadvantages of Singapore’s textbooks in representing problem solving, and the findings of these investigations have influenced the development of the latest school mathematics textbooks.     

Prospective teachers anticipate challenges fostering the mathematical practice of making sense

Problem solving has long been a priority in mathematics education, and the first Common Core mathematical practice (SMP1) focuses on this priority through the language of “Make sense of problems and persevere in solving them.” We present findings from a survey about how prospective elementary teachers' (PTs) make sense of potential difficulties with fostering SMP1. Findings suggested that PTs' common anticipated difficulties relate to planning a solution pathway and self monitoring whether the solution makes sense. Moreover, a third of PTs disclosed that their anticipated difficulties are linked to their own personal struggles with aspects of SMP1. An alternative interpretation of SMP1 surfaced in which a small number of PTs described SMP1 as necessitating that a teacher teach multiple solution methods to students, instead of engaging students in productive struggle to develop their own strategies. We present a framework illustrating the connections between SMP 1 and Pólya's problem solving phases, and we discuss how these findings connect to and build on previous research of PTs' experiences with problem solving. We offer implications for the targeted support needed in teacher preparation programs to address these struggles, to prevent them from being replicated in their students.     

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.     

Problem solving in the United States, 1970–2008: research and theory, practice and politics

Problem solving was a major focus of mathematics education research in the US from the mid-1970s though the late 1980s. By the mid-1990s research under the banner of “problem solving” was seen less frequently as the field’s attention turned to other areas. However, research in those areas did incorporate some ideas from the problem solving research, and that work continues to evolve in important ways. In curricular terms, the problem solving research of the 1970s and 1980s (see, e.g., Lester in J Res Math Educ, 25(6), 660–675, 1994, and Schoenfeld in Handbook for research on mathematics teaching and learning, MacMillan, New York, pp 334–370, 1992, for reviews) gave birth to the “reform” or “standards-based” curriculum movement. New curricula embodying ideas from the research were created in the 1990s and began to enter the marketplace. These curricula were controversial. Despite evidence that they tend to produce positive results, they may well fall victim to the “math wars” as the “back to basics” movement in the US is revitalized.     

Mathematical creativity and giftedness: a commentary on and review of theory, new operational views, and ways forward

In this commentary we synthesize and critique three papers in this special issue of ZDM (Leikin and Lev; Kattou, Kontoyianni, Pitta-Pantazi, and Christou; Pitta-Pantazi, Sophocleous, and Christou). In particular we address the theory that bridges the constructs of “mathematical creativity” and “mathematical giftedness” by reviewing the related literature. Finally, we discuss the need for a reliable metric to assess problem difficulty and problem sequencing in instruments that purport to measure mathematical creativity, as well as the need to situate mathematics education research within an existing canon of work in mainstream psychology.     

Problem solving and Working Mathematically: an Australian perspective

This article reviews “problem solving” in mathematics in Australia and how it has evolved in recent years. In particular, problem solving is examined from the perspectives of research, curricula and instructional practice, and assessment. We identify three key themes underlying observed changes in the research agenda in Australia in relation to problem solving: Obliteration, Maturation and Generalisation. Within state mathematics curricula in Australia, changes in the language and construction of the curriculum and in related policy documents have subsumed problem solving within the broader category of Working Mathematically. In relation to assessment, research in Australia has demonstrated the need for alignment of curriculum, instruction and assessment, particularly in the case of complex performances such as mathematical problem solving. Within the category of Working Mathematically, recent Australian curriculum documents appear to accept an obligation to provide both standards for mathematical problem solving and student work samples that illustrate such complex performances and how they might be assessed.     

问题驱动 多元表征的概念教学探究——对偶线性规划教学案例设计

数学与应用数学(师范)专业中的《运筹学》具有跨学科、实践性的课程特点,目标在于培养职前教师用数学方法解决实际问题的能力.结合义务教育阶段新课程标准中"四基"的提出这一背景,本文将以线性规划部分(运筹数学)对偶线性规划概念的引入这一知识模块为例,探讨通过问题串形式进行问题驱动、多元表征的概念教学过程.即遵循问题驱动—兴趣驱动—问题意识发展—提出和解决新问题,依据数学与外部联系、数学内部联系两条主线设计教学和学习,探索如何通过问题驱动、多元表征的结构化教学过程引导学生的学习方式发生改变,增强探究学习的动机,发展问题解决能力.课堂教学实践证明效果优于以往单一的讲授式教学法,一定程度上提高了学生的学业成绩、应用问题的兴趣和问题解决意识.     

Minimalism as a Guiding Principle: Linking Mathematical Learning to Everyday Knowledge

Studies report that students often fail to consider familiar aspects of reality in solving mathematical word problems. This study explored how different features of mathematical problems influence the way that undergraduate students employ realistic considerations in mathematical problem solving. Incorporating familiar contents in the word problems was found to have only a limited impact. Instead, removing contextual constraints from the problem goal was found to motivate students to validate their problem solving in terms of their everyday experiences. Based on these findings, what determines the authenticity and relevance of a mathematical problem seems to be whether the problem allows students to freely reconstruct the problem situation by making use of their imagination and everyday experiences. In short, the basic principle seems to be “less is more”; that is, fewer constraints in problem goals could function to help students personally associate problem solving with their everyday experiences.     

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?     

Learning the Algebraic Method of Solving Problems

This study of students' attempts to formulate and solve algebra word problems shows that the logic underlying algebraic problem solving methods is little understood. Students' prior experiences with solving problems in arithmetic gives them a compulsion to calculate which is manifested in the meaning they give to “the unknown” and how they use letters, their interpretation of what an equation is, and the methods they choose to solve equations. At every stage of the process of solving problems by algebra, students were deflected from the algebraic path by reverting to thinking grounded in arithmetic problem solving methods.     

Basic formulas and languages: PART II.Applications to E0L systems and forms

This paper (the second of two parts) settles the decidability status of several properties of derivations in E0L systems (forms). In particular we show that the so called “one-to-many simulation” among E0L forms is decidable, solving in this way an open problem from [6]. We use a more general mathematical framework, based on the theory of well-quasi orders, developed in Part I of this paper [2].     

What Is the Object of the Encapsulation of a Process?

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?     

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.