Word problems in mathematics education: a survey

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

     

Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics

The current study examines whether the engagement of mathematics teachers in modelling activities and subsequent changes in their conceptions about these activities affect their beliefs about mathematics. The sample comprised 52 mathematics teachers working in small groups in four modelling activities. The data were collected from teachers' Reports about features of each activity, interviews and questionnaires on teachers' beliefs about mathematics. The findings indicated changes in teachers' conceptions about the modelling activities. Most teachers referred to the first activity as a mathematical problem but emphasized only the mathematical notions or the mathematical operations in the modelling process; changes in their conceptions were gradual. Most of the teachers referred to the fourth activity as a mathematical problem and emphasized features of the whole modelling process. The results of the interviews indicated that changes in the teachers' conceptions can be attributed to structure of the activities, group discussions, solution paths and elicited models. These changes about modelling activities were reflected in teachers' beliefs about mathematics. The quantitative findings indicated that the teachers developed more constructive beliefs about mathematics after engagement in the modelling activities and that the difference was significant, however there was no significant difference regarding changes in their traditional beliefs.     

Developing teachers' models for assessing students' competence in mathematical modelling through lesson study

Applications and modelling have gained a prominent role in mathematics education reform documents and curricula. Thus, there is a growing need for studies focusing on the effective use of mathematical modelling in classrooms. Assessment is an integral part of using modelling activities in classrooms, since it allows teachers to identify and manage problems that arise in various stages of the modelling process. However, teachers’ difficulties in assessing student modelling work are a challenge to be considered when implementing modelling in the classroom. Thus, the purpose of this study was to investigate how teachers’ knowledge on generating assessment criteria for assessing student competence in mathematical modelling evolved through a professional development programme, which is based on a lesson study approach and modelling perspective. The data was collected with four teachers from two public high schools over a five-month period. The professional development programme included a cyclical process, with each cycle consisting of an introductory meeting, the implementation of a model-eliciting activity with students, and a follow-up meeting. The results showed that the professional development programme contributed to teachers’ knowledge for generating assessment criteria on the products, and the observable actions that affect the modelling cycle.     

Trends in the evolution of models & modeling perspectives on mathematical learning and problem solving

In this paper we briefly outline the models and modelling (M&M) perspective of mathematical thinking and learning relevant for the 21st century. Models and modeling (M&M) research often investigates the nature of understandings and abilities that are needed in order for students to be able to use what they have (presumably) learned in the classroom in “real life” situations beyond school Nonetheless, M&M perspectives evolved out of research on concept development more than research on problem solving; and, rather than being preoccupied with the kind of word problems emphasized in textbooks and standardized tests, we focus on (simulations of) problem solving “in the wild.” Also, we give special attention to the fact that, in a technology-basedage of information, significant changes are occurring in the kinds of “mathematical thinking” that is coming to be needed in the everyday lives of ordinary people in the 21^st century—as well as in the lives of productive people in future-oriented fields that are heavy users of mathematics, science, and technology.     

Promoting interdisciplinarity through mathematical modelling

This article presents one approach to addressing the important issue of interdisciplinarity in the primary school mathematics curriculum, namely, through realistic mathematical modelling problems. Such problems draw upon other disciplines for their contexts and data. The article initially considers the nature of modelling with complex systems and discusses how such experiences differ from existing problem-solving activities in the primary mathematics curriculum. Principles for designing interdisciplinary modelling problems are then addressed, with reference to two mathematical modelling problems—one based in the scientific domain and the other in the literary domain. Examples of the models children have created in solving these problems follow. A reflection on the differences in the diversity and sophistication of these models raises issues regarding the design of interdisciplinary modelling problems. The article concludes with suggested opportunities for generating multidisciplinary projects within the regular mathematics curriculum.     

The role of ethnomathematics in mathematics education cases from the horn of Africa

The aim of this paper is to highlight the role that ethnomathematics may have in the mathematical curriculum in the Horn of Africa. It is also a first attempt to document some social pratices and native procedures that people living in this region use to manage their “daily mathematical problems”. Examples from the local culture, which could be used to introduce mathematical arguments in the classroom, are described. The paper finally deals with the possible ways these cultural events may be included in the mathematical syllabi, such as coining new mathematics terms in the local languages, or as preparing mathematics textbooks and classroom activities.     

Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms

Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed.     

How to connect school mathematics with students' out-of-school knowledge

In this paper we present an explorative study for which special cultural artifacts have been used, i.e. supermarket receipts, to try to construct with 9-year old pupils (fourth class of primary school) a new mathematical knowledge, i.e. the algorithm for multiplication of decimal numbers. Furthermore also estimation and approximation processes have been introduced, procedures that are not commonly used in ordinary teaching activity. In our study the receipts, through some modifications, have become more explicitly tools of mediation and integration between in and out-of school knowledge, so they can be utilized to create new mathematical goals, thus becoming real mathematizing tools and constituting a didactic interface between in and out-of-school mathematics. In agreement with ethnomathematical perspective we deem that it is a task for the teacher to know, in order to be able to profitably take account of the teaching, the life experienced by the pupil. Future mathematics teachers should be prepared a) to see mathematics incorporated into real world, b) to investigate mathematical ideas and practices of their pupils, and c) to look for ways to incorporate into the curriculum elements belonging to the sociocultural environment of the pupils, as a starting point for mathematical activities in the classroom. In this way the motivation, interest and curiosity of the pupils will be increased and the attitude towards mathematics of both pupils and teachers will be changed.     

The ostensive dimension through the lenses of two didactic approaches

The paper presents how two different theories—the APC-space and the ATD—can frame in a complementary way the semiotic (or ostensive) dimension of mathematical activity in the way they approach teaching and learning phenomena. The two perspectives coincide in the same subject: the importance given to ostensive objects (gestures, discourses, written symbols, etc.) not only as signs but also as essential tools of mathematical practices. On the one hand, APC-space starts from a general semiotic analysis in terms of “semiotic bundles” that is to be integrated into a more specific epistemological analysis of mathematical activity. On the other hand, ATD proposes a general model of mathematical knowledge and practice in terms of “praxeologies” that has to include a more specific analysis of the role of ostensive objects in the development of mathematical activities in the classroom. The articulation of both theoretical perspectives is proposed as a contribution to the development of suitable frames for Networking Theories in mathematics education.     

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.     

Seeking research-grounded solutions to problems of practice: classroom-based interventions in mathematics education

Research on classroom-based interventions in mathematics education has two core aims: (a) to improve classroom practice by engineering ways to act upon problems of practice; and (b) to deepen theoretical understanding of classroom phenomena that relate to these problems. Although there are notable examples of classroom-based intervention studies in mathematics education research since at least the 1930s, the number of such studies is small and acutely disproportionate to the number of studies that have documented problems of classroom practice for which solutions are sorely needed. In this paper we first make a case for the importance of research on classroom-based interventions and identify three important features of this research, which we then use to review the papers in this special issue. We also consider the issue of ‘scaling up’ promising classroom-based interventions in mathematics education, and we discuss a major obstacle that most such interventions find on the way to scaling up. This obstacle relates to their long duration, which means that possible adoption of these interventions would require practitioners to do major reorganizations of the mathematics curricula they follow in order to accommodate the time demands of the interventions. We argue that it is important, and conjecture that it is possible, to design interventions of short duration in mathematics education to alleviate major problems of classroom practice. Such interventions would be more amenable to scaling up, for they would allow more control over confounding variables and would make more practicable their incorporation into existing curriculum structures.     

Technology-active mathematical modelling

Problems in mathematical modelling and data analysis are discussed from a constructivist perspective. This approach provides students with realistic opportunities to connect mathematics to significant social and environmental problems while incorporating recent advances made possible by today's mathematically powerful calculators. Also included are methods for enhancing students' abilities to shift among a wide range of representations using the modelling capabilities in graphing utilities. Consideration is further given to the changes that technology imposes on the classroom culture, including changes in students' attitudes about modelling techniques and difficulties in locating appropriate problems. The article concludes by discussing the integration of teaching and assessment with mathematical modelling.     

How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving?

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers’ self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the ‘what I know, what I look for’ method. These results led us to hypothesize that the observed gap may have an impact on students’ comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students’ literal and inferential understanding of word problems. A total of 278 fourth graders (9–10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students’ understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.     

The Impact of Enacted Mathematics Curriculum Models on Prospective Elementary Teachers' Course Perceptions and Beliefs

This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.     

A tale of two sets of norms: Comparing opportunities for student agency in mathematics lessons with and without interactive simulations

Previous research has documented the importance of setting up productive norms in mathematics classrooms. Studies have also shown the potential for activities involving interactive simulations (sims) to support student engagement and learning. In this study, we investigated the relationship between norms and sim-based activities. In particular, we examined the social and sociomathematical norms in lessons taught with and without the use of PhET sims in the same teacher’s middle-school mathematics classroom. There were statistically significant differences in indicators of social norms between the two types of lessons. In sim lessons, the teacher more frequently took the role of a facilitator of mathematical ideas, and students exhibited conceptual agency more often than they did in non-sim lessons. On the other hand, there was substantial overlap: the teacher usually acted as an evaluator, and the students usually exhibited disciplinary agency in both types of lessons. However, there was a stark contrast in sociomathematical norms between the two types of lessons. Students’ specifically mathematical obligations in non-sim lessons consistently included practicing procedures in isolation and appealing to rules. Obligations in sim lessons included developing and sharing strategies, making conjectures and providing justifications. In both types of lessons, students were obligated to recall mathematical facts and vocabulary. Thus, the social norms were broadly consistent except for important differences in frequency, whereas we found substantial qualitative contrasts in the sociomathematical norms in the two types of lessons. This case provides evidence that contrasting norms can exist within the same classroom. We argue from our data that these differences may be mediated by curricular choices—in this case, the use of sims.     

Motion, speed, and other ideas that “should be put in books”

This paper focuses on a portion of a research project involving a group of inner-city middle school students who used SimCalc simulation software over the course of an entire school year to investigate ideas relating to graphical representations of motion and speed. The classroom environment was one in which students openly defended and justified their thinking as they actively explored and solved rich mathematical problems. The activities, generally speaking, involved functions that were intended to tap students’ real world intuitions as well as prior mathematical skills and understandings about speed, motion, and other graphical representations that underlie the mathematics of motion. Results indicate that these students did build ideas related to those concepts. This paper will provide documentation of the ways in which these students interpreted graphical representations involving linear and quadratic functions that are associated with constant and linearly changing velocities, respectively.     

With a focus on ‘Grundvorstellungen’ Part 1: a theoretical integration into current concepts

Current comparative studies such as PISA assess individual achievement in an attempt to grasp the concept of competence. Working with mathematics is then put into concrete terms in the area of application. Thereby, mathematical work is understood as a process of modelling: At first, mathematical models are taken from a real problem; then the mathematical model is solved; finally the mathematical solution is interpreted with a view to reality and the original problem is validated by the solution. During this cycle the main focus is on the transition between reality and the mathematical level. Mental objects are necessary for this transition. These mental objects are described in the German didactic with the concept of Grundvorstellungen'. In the delimitation to related educational constructs, ‘Grundvorstellungen’ can be described as mental models of a mathematical concept.     

Arguments constructed within the mathematical modelling cycle

Socio-cultural theories in mathematics education field recently emphasize the importance of the collective argumentation within small-group work. Since mathematical modelling tasks require a process in which students search for a solution for real life problems through small-group work, the arguments in this process become an issue of concern. This study examines the arguments constructed within the mathematical modelling cycle by considering the participants’ modelling processes. In this context, four primary pre-service mathematics teachers worked on a modelling task and their arguments were explained through the components of Toulmin’s argumentation schema. Findings revealed that the data and the claims of most of the arguments corresponded to the starting and ending points of the modelling transition in which the current arguments constructed. The existence of the arguments corresponded through warrant-claim originated from inquiring the assumptions in the modelling cycle. In addition, the participants made assumptions as warrants to support their arguments and as rebuttals to show the degree of certainty of claims in intra-group challenging situations. Both the warrants and the backings depended on modelling context as well as mathematics context.     

Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis

In the United States and elsewhere, prospective teachers of secondary mathematics are usually required to complete numerous advanced mathematics courses before obtaining certification. However, several research studies suggest that teachers’ experiences in these advanced mathematics courses have little influence on their pedagogical practice and efficacy. To understand this phenomenon, we presented 14 secondary mathematics teachers with four statements and proofs in real analysis that related to secondary content and asked the participants to discuss whether these proofs could inform their teaching of secondary mathematics. In analyzing participants’ remarks, we propose that many teachers view the utility of real analysis in secondary school mathematics teaching using a transport model, where the perceived importance of a real analysis explanation is dependent upon the teacher’s ability to transport that explanation directly into their instruction in a secondary mathematics classroom. Consequently, their perceived value of a real analysis course in their teacher preparation is inherently limited. We discuss implications of the transport model on secondary mathematics teacher education.     

A Relational Perspective on Issues of Cultural Diversity and Equity as They Play Out in the Mathematics Classroom

In this article, we present a relational perspective in which cultural diversity is viewed as a relation between people's participation in the practices of different communities. In the case at hand, the relevant practices were those of students' local, home communities, and the broader communities to which they belonged in wider society on the one hand and the specifically mathematical practices established by the classroom community on the other hand. In the 1st part of the article, we discuss how we might characterize the practices of these various communities by drawing on Wenger's (1998) notion of a community of practice and on Gee's (1997) notion of a Discourse. In doing so, we question the manner in which students are frequently classified exclusively in terms of the standard categories of race and ethnicity in investigations of equity in mathematics education. Later in the article, we clarify that in addition to focusing on the continuities and contrasts between the practices of different communities, the relational perspective also encompasses issues of both power and identity. As we illustrate, the gatekeeping role that mathematics plays in students' access to educational and economic opportunities is not limited to differences in the ways of knowing associated with participation in the practices of different communities. Instead, it also includes difficulties that students experience in reconciling their views of themselves and who they want to become with the identities that they are invited to construct in the mathematics classroom.