This edited volume is part of the ICME - 13 Monographs series, which assembles edited versions of papers originally presented at the Thirteenth International Congress on Mathematical Education in Hamburg, Germany. The volume represents research from a multinational set of mathematics education researchers who have studied problem solving, problem posing, and relationships of problem solving to technology and to teaching. The primary audience for the book is the community of education researchers who want to understand the nature of problem solving in mathematics and the environments and tools that support students to engage in problem solving. The volume will be a useful addition to the libraries of education researchers and to mathematics teacher educators who are charged with supporting preservice or inservice teachers’ development of capacities to incorporate problem solving into their instruction. Instructors of mathematics are not the primary audience for this volume; however, instructors who incorporate problem solving into their teaching and curriculum might also find the implications of this volume useful in framing their planning and instruction.

The chapters are clustered into five themed parts. The first part focuses on research and theory on problem solving heuristics. In the Chapter 1, Tjoe presents the findings of a qualitative inquiry into U.S. high school students’ familiarity with the idea of using multiple solution methods, their ability to understand solutions that were presented to them, and their capacity to connect solutions to each other and to navigate between solution methods. In Chapter 2, Maciejewski introduces a theoretical model of problem solving anchored in mathematical foresight. Using interviews with students in an undergraduate mathematics course, Maciejewski illustrates the utility of the model for describing students’ problem solving processes.

The second part (Chapter 3 through Chapter 5) focuses on the theme of problem solving with technology. In Chapter 3, Carreira and Jacinto illustrate, through a case study, how one student from Portugal uses digital technology in unexpected but productive ways to support his problem solving. Santos - Trigo, in Chapter 4, synthesizes research and theory to illustrate how digital technologies offer tools to teachers and to students for representing mathematical objects and for using those representations to solve problems. Amado, Carreira, and Nobre conclude the second part of the volume in Chapter 5 by sharing how middle grades mathematics students in Portugal used spreadsheet software successfully to investigate and to solve a linear programming problem prior to their introduction to formal algebraic techniques for representing and solving such problems.

The third part, Chapter 6 and Chapter 7, shifts to focus on problem posing. In Chapter 6, Hersant and Choquet use data from elementary classrooms in France to investigate how characteristics of problems posed to students create different opportunities for those students to engage in authentic inquiry-based learning. Chapter 7 wraps up the third part with a report by Malaspina, Torres, and Rubio of a multiple case study of 15 high school mathematics teachers from Peru and their construction of problem sequences intended to scaffold students’ problem solving.

Part four (Chapter 8 through Chapter 11) examines questions about how to assess problem solving. In Chapter 8, Loh and Lee report the findings of a mixed methods study of the phases of metacognition and the levels of metacognition observed in self-reports of problem solving from 783 Singaporean secondary students. Chanudet, in Chapter 9, uses data from a problem-solving centered course in Geneva, Switzerland to examine tools for summative assessment of problem solving and to explore whether using such tools impacts teachers’ formative assessment practices. In Chapter 10, Di Martino and Signorini share results of an empirical investigation into the potential for a standardized problem-solving assessment to give insights into Italian students’ problem-solving processes and the aspects of the process that might be obscured by the time constraints associated with standardized assessment. This part concludes with Chapter 11, in which Mendoza Álvarez, Rhoads, and Campbell report on the development and validation of Likert scale items to assess students’ uses of sense-making strategies, representational strategies, self-monitoring, justifying of methods, and perception of challenge during the problem-solving process.

The volume concludes with a final section of four chapters that examine various aspects of the environment in which problem solving occurs. In Chapter 12, Koichu and Keller describe the successes and challenges of creating and sustaining a productive online problem-solving forum intended to support 10th grade Israeli students’ out - of - class problem solving. Liljedahl, in Chapter 13, documents findings suggesting that the orientation of writing surfaces (vertical versus horizontal), relative permanence of writing on those surfaces (e.g., erase-able versus permanent) have nontrivial effects on how quickly Canadian high school students, working in small groups, got started on a problem solving task, the ways that they used work surfaces to record their thinking, the eagerness with which they engaged in problem solving, the extent to which they interacted with each other and with other groups, and the extent to which they persisted during their problem solving. Liljedahl offers an argument that an environment in which students work at vertical whiteboards is most likely to promote greater amounts of student engagement and student thinking. In Chapter 14, Felmer, Perdomo-Díaz, and Reyes investigate the outcomes of a professional development program designed to support Chilean teachers’ abilities to create problem-based learning environments by developing their knowledge of the importance of problem solving for students’ learning, by deepening their own knowledge of mathematics by engaging the teachers in problem solving, by asking them to reflect on the role of the teacher during classroom episodes of student problem solving, and by engaging them in introducing problem solving in their own classrooms. The authors share their interpretation of findings regarding the relationship between the program and teachers’ beliefs, teachers’ practices, and their students’ learning. Chapter 15 closes the fifth part, and the volume, with an examination by Kin, Yap, Guan, Hoong, Lam, Seng, Choon, and Dindyal of factors that supported the scale-up and sustainability of a program designed to infuse problem solving activity into the curriculum, assessment, and instructional practices across a variety of types of secondary schools in Singapore. Using the perspective that an innovation operates within a system of interactions among policy, school-level factors, and program-level factors, the authors examine the interactions among school - level and program - level factors and the effects of those interactions on the sustainability of the program at various institutions.

The chapters of this volume offer interesting insights into the phenomenon of problem solving and the technological tools and learning environments that support students to engage in problem solving. Although the audience is primarily mathematics education researchers, the various chapters also have significant and potentially useful implications for those who are interested in incorporating more problem-solving activity into mathematics instruction.

Duane Gaysay is an Assistant Professor of Mathematics Education at Syracuse University. His scholarly interests are in students’ mathematical thinking and in the pedagogy of preparing secondary mathematics teachers.