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Influence of the Type of Mathematical Problems on Students’ and Pre-service Teachers’ Interest and Performance. A Replication and Elaboration Study
Abstract
Several studies confirm the importance of the role of students’ interest in learning mathematics. This article describes the process of conceptual replication of Rellensmann and Schukajlow’s (2017) research on how the connection to the reality of a mathematical problem affects the interest in solving it. Our study distinguishes between intramathematical problems, word problems and modelling problems. It was implemented with 80 Spanish ninth-grade students and 80 pre-service teachers. The results show that Spanish students are more interested in intramathematical problems and less interested in modelling problems, while pre-service teachers are more interested in problems connected to reality, especially word problems. We also provide data regarding the performance of students and prospective teachers, which is higher in word problems. In addition, we find that there are significant relationships between performance and task-specific interest. These results complement the original study, as they allow us to contrast whether there are differences with German students and to explain the German pre-service teachers’ judgements of students’ interest in problems with and without a connection to reality.
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Although influences of interest on learning are well documented, mediating processes have not been clarified. The authors investigated how individual and situational interest factors contribute to topic interest and text learning. Traditional self-report measures were combined with novel interactive computerized methods of recording cognitive and affective reactions to science and popular culture texts, monitoring their development in real time. Australian and Canadian students read 4 expository texts. Both individual interest variables and specific text titles influenced topic interest. Examination of processes predictive of text learning indicated that topic interest was related to affective response, affect to persistence, and persistence to learning. Combining self-rating scales with dynamic measures of student activities provided new insight into how interest influences learning.
Students’ emotions (e.g., enjoyment, boredom) while doing math and their situational interest in mathematics are important for their learning of mathematics, but examinations of teachers’ judgments of students’ emotions and interest while solving tasks are rare. Moreover, we do not know much about the predictors of teachers’ judgments of students’ emotions and interest. In this study, we addressed preservice teachers’ judgments of students’ task-specific enjoyment, boredom, and situational interest and analyzed whether such judgments are related to preservice teachers’ own enjoyment, boredom, and interest. Furthermore, we aimed to analyze whether preservice teachers’ judgments differ between the two types of tasks under investigation (tasks with and without a connection to reality). To achieve these aims, 182 preservice teachers were randomly assigned to one of two study conditions. In one condition, preservice teachers' judgments of task-specific emotions and situational interest for fictitious 9th-grade students were measured, and in the second condition, preservice teachers' own task-specific emotions and situational interest were measured. The results show that preservice teachers' judgments of students’ task-specific emotions and situational interest differ for tasks with and without a connection to reality. Further, our findings indicate that preservice teachers’ judgments of students’ task-specific emotions and interest differ from preservice teachers' own task-specific emotions and interest. Implications for theoretical models of teachers’ judgments and consequences for teacher education are discussed.
Fermi problems are real-context estimation tasks that are suitable for introducing open-ended problems in primary school education. To ensure their effective introduction in the classroom, teachers must have adequate proficiency to deal with them. One of the key aspects of problem-solving proficiency is flexibility, but there are few studies on flexibility in solving real-context problems. This study, based on an analysis of the errors made by 224 prospective teachers when solving a Fermi problem sequence, establishes performance levels. In addition, we define levels of flexibility in using multiple solutions across the sequence, which allows us to address the main objective: to study the relationship between performance and flexibility. We found that there are significant relationships between flexibility levels and the number and severity of errors made. Encouraging flexibility in prospective teachers may be an efficient way to improve their performance in solving real-context problems.
Although it is a challenge for primary school teachers, real-context estimation problems can be used as an introduction to mathematical modeling. With this aim, we designed a two-phase activity: in the first phase, 224 prospective teachers developed individual action plans to solve a sequence of real-context estimation problems in the classroom; in the second phase, they completed the solution of the same problems working in groups in the real location where the four problems were contextualized. A comparative study showed that, in the second phase, prospective teachers were able to adapt their solutions to contextual features detected in situ that had not been anticipated in the action plans developed during the first phase. Two-phase modeling activities, which permit a comparison of different perspectives on problems, facilitate the experience of collaborative work. These activities could be incorporated into prospective teachers' initial training as a useful resource for improving their problem-solving expertise.
Emotions and motivation are important for learning and achievement in mathematics. In this paper, we present an overview of research on students’ emotions and motivation in mathematics. First, we briefly review how early research has developed into the current state-of-the-art and outline the following key characteristics of emotions and motivation: objects, valence, temporal stability (vs. variability), and situational specificity (vs. generality). Second, we summarize major theories in the field (the control-value theory of achievement emotions, expectancy-value theory of achievement-related motivation, self-determination theory of human motivation, and social-cognitive theory of self-efficacy). Third, we present an overview of instructional characteristics that have been shown to foster emotions and motivation. Fourth, we provide an overview of the contributions to the special issue on “Emotions and Motivation in Mathematics Education and Educational Psychology.” Finally, we suggest directions for future research in the field with respect to advancing theory, improving measurement, and considering diversity and inclusion.
According to expectancy-value models, two important motivational constructs are task value and self-efficacy expectations. As students work intensively on mathematical problems in the classroom, teachers need the competence to judge whether students value solving the problem and how confident they are that they will succeed. However, not much is known about the mechanisms behind teachers’ judgments concerning students’ motivation. The goal of the present study was to answer the following research questions: (a) How do preservice teachers rate their own task value and self-efficacy for modelling, word, and intramathematical problems? (b) Are there differences in preservice teachers’ diagnostic judgments concerning school students’ task value and self-efficacy for modelling, word, and intramathematical problems? (c) Do preservice teachers’ own task value and self-efficacy for modelling, word, and intramathematical problems differ from preservice teachers’ judgments of school students’ task value and self-efficacy? A total of 182 preservice teachers were asked about the value of 12 problems, their confidence in solving the problems, and how hypothetical ninth graders would judge these constructs. Preservice teachers’ ratings of their own perceptions regarding both constructs were lower for modelling problems than for other problems. When judging school students’ motivation, ratings of both constructs were higher for word problems. Preservice teachers’ ratings of their own motivation were significantly higher across all types of problems than their judgments of school students’ perspectives. A positive correlation between preservice teachers’ own motivation and their judgments of school students’ motivation indicates the importance of teachers’ perceptions of their own motivation to solve mathematical problems for diagnostic judgments.
The success or failure of education systems in promoting the problem-solving skills of students depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary math textbooks of two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on: 1) the quantity of arithmetic word problems; 2) a variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures; and 3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in math textbooks of Singapore and Spain. Furthermore, in both countries, math textbooks emphasized (additive) combine 1 and multiplication-simple rate problems. Notably, Singaporean textbooks contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching-learning process in the classroom.
A textbook constitutes the hegemonic material of the educational institution. It acts as a mediator between the official curriculum and the educational practice. Given its potential influence in the classroom, this study analyzes the treatment of word problems included in the mathematics textbooks published by the publishing houses with the greatest diffusion in Spain at every primary education grade. Three variables were analyzed: their semantic structure, their degree of challenge, and their situational context. The results indicate that most of the problems included in textbooks are characterized by low complexity and variability regarding their semantic structure. They are also characterized by a limited degree of challenge and by being presented in highly standardized situational contexts. Likewise, it is found that there is no evolution in the treatment of these problems with respect to previous studies carried out in the Spanish context. Therefore, it is concluded that the mathematics textbooks currently used in schools are not effective tools to address the process of teaching-learning problem solving.
This open access book presents a structural model and an associated test instrument designed to provide a detailed analysis of professional competences for teaching mathematical modelling. The conceptualisation is based on the COACTIV model, which describes aspects, areas and facets of professional competences of teachers. The manual provides an overview of the essential teaching skills in application-related contexts and offers the tools needed to capture these aspects. It discusses the objectives and application areas of the instrument, as well as the development of the test. In addition, it describes the implementation and evaluates the quality and results of the structural equation analysis of the model.
Teaching mathematical modelling is a cognitively challenging activity for (prospective) teachers. Thus, teacher education requires a detailed analysis of professional competence for teaching mathematical modelling. Measuring this competence requires theoretical models that accurately describe requirements placed upon teachers, as well as appropriate evaluation tools that adequately capture skills and abilities in this field. This book presents an instrument that measures the professional competences in a sample of 349 prospective teachers.
Análisis de los factores de complejidad en planes de resolución individuales y resoluciones grupales de problemas de estimación de contexto real Analysis of the complexity factors in individual solution plans and group developed solutions for real context estimation problems Resumo. Los problemas de estimación en contexto real pueden utilizarse como iniciación en la elaboración de modelos matemáticos. En este estudio se recogieron los planes de resolución individuales de estudiantes (N=224) del grado de Maestro/a en Educación Primaria que se enfrentaron a una secuencia de problemas de estimación contextualizados. Posteriormente, esos mismos estudiantes trabajando en grupos (N=63), resolvieron los mismos problemas realizando mediciones y estimaciones in situ. Así, el estudio se centra en los denominados factores de complejidad, referidos a los aspectos de la resolución con los que se pretende obtener una estimación más precisa. El objetivo es determinar qué factores de complejidad enriquecen los planes de resolución individuales y cuáles son los que enriquecen las resoluciones grupales. Además, se iden-tifica qué características del contexto real promueven que los estudiantes incluyan determinados factores de complejidad, tanto en los planes de resolución individual como en las resoluciones grupales. Los resultados permiten identificar el impacto del trabajo de campo en el proceso de resolución de tareas de estimación formuladas en contextos reales cercanos; en relación al conocimiento del profesor en formación, los resultados permiten identificar carencias en el conocimiento de las tareas matemáticas para la enseñanza.
Prior work on teachers’ mathematical knowledge has contributed to our understanding of the important role of teachers’ knowledge in teaching and learning. However, one aspect of teachers’ mathematical knowledge has received little attention: strategic competence for word problems. Adapting from one of the most comprehensive characterizations of mathematics learning (NRC, 2001), we argue that teachers’ mathematical knowledge also includes strategic competence, which consists of devising a valid solution strategy, mathematizing the problem (i.e., choosing particular strategies and presentations to translate the word problem into mathematical expressions), and arriving at a correct answer (executing a solution) for a word problem. By examining the responses of 350 fourth- and fifth-grade teachers in the USA to four multistep fraction word problems, we were able to explore manifestations of teachers’ strategic competence for word problems. Findings indicate that teachers’ strategic competence was closely related to whether they devised a valid strategy. Further, how teachers dealt with known and unknown quantities in their mathematization of word problems was an important indicator of their strategic competence. Teachers with strong strategic competence used algebraic notations or pictorial representations and dealt with unknown quantities more frequently in their solution methods than did teachers with weak strategic competence. The results of this study provide evidence for the critical nature of strategic competence as another dimension needed to understand and describe teachers’ mathematical knowledge.
In this study, we examined how situational interest, self-efficacy, and performance predict each other during task engagement, and how they, in turn, contribute to continued interest. Finnish fourth-graders (N = 263) did a computerized inductive reasoning task consisting of two sections. Before and after each section, the students reported their situational interest and self-efficacy, and at the end of the task, students stated whether they would like to do similar tasks again (i.e., continued interest). Students’ domain-specific interest and self-concept in mathematics, and gender differences were controlled. A cross-lagged reciprocal effects model with repeated measures, control variables, and outcomes within the structural equation modeling framework was estimated. The results showed situational interest to have a stronger effect on self-efficacy than vice versa, and that they both partly contributed to task performance. Continued interest was influenced only by situational interest at the end of the task. The patterning of predictions across the different stages of the task suggests these effects to be somewhat sensitive to task characteristics.
In recent years, considerable attention has been given to the knowledge teachers ought to hold for teaching mathematics. Teachers need to hold knowledge of mathematical problem solving for themselves as problem solvers and to help students to become better problem solvers. Thus, a teacher’s knowledge of and for teaching problem solving must be broader than general ability in problem solving. In this article a category-based perspective is used to discuss the types of knowledge that should be included in mathematical problem-solving knowledge for teaching. In particular, what do teachers need to know to teach for problem-solving proficiency? This question is addressed based on a review of the research literature on problem solving in mathematics education. The article discusses the perspective of problem-solving proficiency that framed the review and the findings regarding six categories of knowledge that teachers ought to hold to support students’ development of problem-solving proficiency. It concludes that mathematics problem-solving knowledge for teaching is a complex network of interdependent knowledge. Understanding this interdependence is important to help teachers to hold mathematical problem-solving knowledge for teaching so that it is usable in a meaningful and effective way in supporting problem-solving proficiency in their teaching. The perspective of mathematical problem-solving knowledge for teaching presented in this article can be built on to provide a framework of key knowledge mathematics teachers ought to hold to inform practice-based investigation of it and the design and investigation of learning experiences to help teachers to understand and develop the mathematics knowledge they need to teach for problem-solving proficiency.
The study presented in this article takes a closer look at how French and German high-school students deal with a mathematical modeling problem, what blockages they encounter and how differences in the modeling processes between students from both nations can be explained by differences between the teaching and learning of mathematical modeling in France and Germany. To better understand these differences, firstly, a brief overview is provided on the historical development of mathematics education in both countries, with a focus on mathematical modeling, followed by a qualitative empirical study in both France and Germany. Two main differences can be identified: students’ handling of the real-world situation and their striving for accuracy. Possible reasons for these differences are discussed in relation to national teaching traditions.
Teachers try to help their students learn. But why do they make the particular teaching choices they do? What resources do they draw upon? What accounts for the success or failure of their efforts? In How We Think, esteemed scholar and mathematician, Alan H. Schoenfeld, proposes a groundbreaking theory and model for how we think and act in the classroom and beyond. Based on thirty years of research on problem solving and teaching, Schoenfeld provides compelling evidence for a concrete approach that describes how teachers, and individuals more generally, navigate their way through in-the-moment decision-making in well-practiced domains. Applying his theoretical model to detailed representations and analyses of teachers at work as well as of professionals outside education, Schoenfeld argues that understanding and recognizing the goal-oriented patterns of our day to day decisions can help identify what makes effective or ineffective behavior in the classroom and beyond.
Students’ interest in mathematics is important for their learning of mathematics, and the ability to accurately judge students’ motivational orientation is important for mathematics teachers. The aim of this study was to answer the following research questions: (1) Is there a difference in students’ interest in solving problems with and without a connection to reality? (2) Is there a difference in pre-service teachers’ judgments of students’ interest in problems with and without a connection to reality? (3) Can pre-service teachers accurately judge students’ interest in solving problems with and without a connection to reality? To answer these research questions, we administered a questionnaire in which we asked 100 ninth graders about their task-specific interest after they solved problems with and without a connection to reality. We additionally asked 163 pre-service teachers to judge fictitious ninth graders’ interest in solving the same problems. Contrary to our expectations, students’ interest in real-world problems was lower than their interest in problems without a connection to reality when task difficulty was controlled for. Further, our findings indicate an important discrepancy between students’ interest and pre-service teachers’ judgments of students’ interest. Pre-service teachers overrated students’ interest in solving real-world problems, and they underrated students’ interest in solving intra-mathematical problems. Moreover, the accuracy of pre-service teachers’ judgments of students’ interest was low and ranged widely across pre-service teachers. Implications for teacher education and classroom practice are discussed.
This paper reviews factors that contribute to the development of middle school students’ interest in statistical literacy and its motivational influence on learning. To date very little research has specifically examined the influence of positive affect such as interest on learning in the middle-school statistics context. Two bodies of associated research are available: interest research in a mathematics education context and attitudinal research in a tertiary statistics context. A content analysis of this literature suggests that interest development in middle school statistics will be the result of a complex interplay of classroom influences and individual factors such as: students’ knowledge of statistics, their enjoyment of statistics and their perceptions of competency in relation to the learning of statistics. First published May 2009 at Statistics Education Research Journal: Archives
The topic of this paper is mathematical modelling or—as it is often, more broadly, called—applications and modelling. This has been an important topic in mathematics education during the last few decades, beginning with Pollak’s survey lecture (New Trends in Mathematics Teaching IV, Paris, pp. 232–248, 1979) at ICME-3, Karlsruhe 1976. By using the term “applications and modelling”, both the products and the processes in the interplay between the real world and mathematics are addressed. In this paper, I will try to summarize some important aspects, in particular, concerning the teaching of applications and modelling.
In this study, we investigated teachers’ teaching practices and their underlying beliefs regarding context-based tasks to find a possible explanation for students’ difficulties with these tasks. The research started by surveying 27 Junior High School teachers from seven schools in Indonesia through a written questionnaire. Then, to further examine teachers’ teaching practices related to context-based tasks, four teachers were observed and video recorded in two mathematics lessons in which they were asked to deal with context-based tasks. The questionnaire data revealed that the teachers had a tendency toward a view on teaching and learning mathematics which includes encouraging students to be actively involved in solving problems in various contexts. Although this finding suggests that the teachers may offer opportunities to learn context-based tasks to students, the questionnaire data also revealed that the teachers saw context-based tasks as plain word problems. Furthermore, the observations disclosed that their teaching was mainly teacher-centered and directive, which is not considered to be supportive for learning to solve context-based tasks. Combining the findings of this study with the results from our earlier study on Indonesian students’ errors when solving context-based tasks, we found a relationship between how Indonesian teachers teach context-based tasks and the errors Indonesian students make in solving these tasks. These findings support the conclusion that insufficient opportunity-to-learn to solve context-based tasks offered by teachers is a possible explanation for students’ difficulties in solving these tasks. © 2015, Mathematics Education Research Group of Australasia, Inc.
Este artículo muestra un instrumento para analizar el papel de la resolución de problemas en los libros de texto. Partimos de la descripción del papel otorgado a los libros de texto en los procesos de enseñanza y aprendizaje de las matemáticas, para pasar después a mostrar algunas investigaciones que, desde la educación matemática, se han realizado sobre libros de texto, entre ellas las llevadas a cabo en nuestro grupo. De estas últimas mostraremos ejemplos de cómo el instrumento de análisis señalado permite analizar los tipos de problemas que aparecen en los libros de texto.
The choice and context of authentic problems—such as designing a staircase or a soda can—illustrate the modeling process in several stages.
In this study 192 ninth and tenth graders from 8 German classes were asked about their interest concerning tasks with and without connection to reality. The students were randomly assigned to two experimental groups. The first group was asked about their task-specific interest after and the second group before task processing. The study aimed to answer the following questions: (1) Does students' task-specific interest differ according to the type of problem (intra-mathematical, " dressed up " word problems and modelling problems)? (2) Does task processing influence students' task-specific interest? The analysis showed that there are differences in students' interest regarding tasks with and without connection to reality and that task-specific interest across all types of problems decreases after task processing.
This article presents and discusses three paper-and-pencil tests of students' performance on nonroutine mathematics problems at the college level. These tests assess the following: 1. the frequency with which students apply various problem-solving heuristics, and how far and with what success they pursue them; 2. students' subjective assessments of their own problem-solving behavior; and 3. transfer of heuristic behavior on problems related in various degrees to ones that students have studied. These measures were used to assess students' performance before and after a month-long intensive problem-solving course. They provide evidence that such a course can produce strong changes in students' problem-solving behavior.
Problem solving in mathematics has played an important role in educational research in recent decades and has become an essential content in the teaching and learning of mathematics in different countries. In this work, we analyze how these ideas are reflected in the most sold textbook in Spain. Our focus was the topic of plane geometry in the ninth grade. We conclude that this textbook does not fit in this theoretical evolution, as most of the problems analyzed were exercises of concepts and algorithms applications.
A growing interest to teach mathematics closely connected to its use in daily life has taken place in Indonesia for over a decade (Sembiring, Hadi, and Dolk 2008). This chapter reports an exploratory case study of the building of an awareness of mathematical modelling in teacher education in Indonesia. A modelling task, re-designing a parking lot (Ang 2009), was assigned to groups of pre-service secondary mathematics teachers. All groups undertook the stages of collecting data on a parking lot, identifying limitations in the current design of the parking lot, and proposing a new design based on their observations and analyses. The nature of the mathematical models elicited by pre-service teachers during various stages of completing the modelling task will be examined. Implications of this study suggest the need to encourage pre-service teachers to state the assumptions and real-world considerations and link them to the mathematical model in order to validate their models .
Modelling is an important part of mathematical learning. One characteristic feature of modelling problems is their openness. In this study, we investigated the relationship between interest and performance in solving open modelling problems and closed real-
world problems. We used questionnaires and tests to assess the interest and performance of 143 ninth- and 10th-grade students at different achievement levels. We found that low-achieving students were more interested in solving open modelling problems than closed real-world problems. Also, prior individual interest in mathematics and performance were positively related to situational (task-specific) interest. These results contribute to interest theories by underlining the importance of types of real-world problems and achievement levels for situational interest.
This article presents and discusses three paper-and-pencil tests of students' performance on nonroutine mathematics problems at the college level. These tests assess the following: 1. the frequency with which students apply various problem-solving heuristics, and how far and with what success they pursue them; 2. students' subjective assessments of their own problem-solving behavior; and 3. transfer of heuristic behavior on problems related in various degrees to ones that students have studied. These measures were used to assess students' performance before and after a month-long intensive problem-solving course. They provide evidence that such a course can produce strong changes in students' problem-solving behavior.
This study investigated the relationship between mathematical performance and mathematics self-efficacy, attitudes toward mathematics, and the choice of mathematics-related majors by 153 college women and 109 college men. Mathematics performance was correlated moderately with mathematics self-efficacy. No support was found for Hackett and Betz's (1981) hypothesis that women's mathematics self-efficacy expectations are unrealistically low compared to men's. Both mathematics performance and mathematics self-efficacy were significantly and positively correlated with attitudes toward mathematics, masculine sex-role orientation, and a mathematics-related major. Regression analyses supported the superiority of mathematics self-efficacy over mathematics performance and achievement variables in predicting the choice of a mathematics-related major.
The purpose of this note is to bring to the readers' attention the lack of replication studies appearing in the JRME . We are certainly all aware that one empirical study does not by itself answer a general question about the teaching or learning of mathematics. This is supported by the fact that in numerous journal articles the authors will conclude their reports by suggesting that “further research is needed” in their particular area. Hence, one would expect alJ researchers (particularly those who claim important results) to do follow-up work on their own research, and one would expect to find researchers conducting exact replication studies to lend more credence to their results.
This paper presents a reflection on the importance of replication studies in the development of the field of mathematics education. Based on a literature review, it explains what replication studies are, and why they are important for our field. It also addresses the issue of why, despite their apparent importance, replication studies remain scarce. Some open discussions about replication studies in mathematics education are pointed out, and a couple of issues related to replication studies whose empirical exploration could help to advance our research field are identified. Additionally, a typology of fundamental questions and modes of exploration that could guide a replication study in mathematics education is proposed. The paper closes with a discussion about the necessary conditions to promote and further develop replication studies in mathematics education.
Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.
The present issue of JRME features three articles—Melhuish (2018); Jamil, Larsen, and Hamre (2018); and Thanheiser (2018)—that involve, at least to some degree, replication of prior published studies. In each of these articles, the authors provide a clear rationale for the importance of the work, and in all three cases, the importance of replication of prior work is discussed. The authors of these three articles point out that the scientific community writ large values replication, and they also note that replication is quite rare in educational research generally and in mathematics education research more specifically.
Many studies in mathematics education research occur with a nonrepresentative sample and are never replicated. To challenge this paradigm, the author designed a large-scale study evaluating student conceptions in group theory that surveyed a national, representative sample of students. The author illustrates the potential of replication studies to refine theory and theoretical propositions in 3 ways: by offering alternate interpretations of student responses, by challenging previous pedagogical implications, and by reevaluating connections between theories.
This book is a product of love and respect. If that sounds rather odd I initially apologise, but let me explain why I use those words. The original manuscript was of course Freudenthal’s, but his colleagues have carried the project through to its conclusion with love for the man, and his ideas, and with a respect developed over years of communal effort. Their invitation to me to write this Preface e- bles me to pay my respects to the great man, although I am probably incurring his wrath for writing a Preface for his book without his permission! I just hope he understands the feelings of all colleagues engaged in this particular project. Hans Freudenthal died on October 13th, 1990 when this book project was well in hand. In fact he wrote to me in April 1988, saying “I am thinking about a new book. I have got the sub-title (China Lectures) though I still lack a title”. I was astonished. He had retired in 1975, but of course he kept working. Then in 1985 we had been helping him celebrate his 80th birthday, and although I said in an Editorial Statement in Educational Studies in Mathematics (ESM) at the time “we look forward to him enjoying many more years of non-retirement” I did not expect to see another lengthy manuscript.
Posing modelling problems in class is supposed to increase students’ motivation. As motivation is assumed
to emerge from task value and self-efficacy expectations, the present study considered both constructs with the aims to examine (1) whether students have different values and self-efficacy expectations concerning modelling problems versus dressed up word problems and intra-mathematical problems and (2) whether mathematical content influences task value and self-efficacy concerning different types of problems.
We asked 90 high- and middle-track students (ninth- and tenth-graders) how much they valued modelling problems, dressed up word problems, and intra-mathematical problems and if they were confident they could solve these types of problems. All of the problems that we used could be solved by applying mathematical procedures from two different mathematical content areas (Pythagorean theorem or linear
functions). The results indicated that there were significant differences in students’ task values and self-efficacy depending on the type of problem. Students reported the lowest task values and self-efficacy expectations for modelling problems compared with the other types of problems. Moreover, the differences between students’ task values (but not between students’ self-efficacy expectations) within the three types
of problems seemed to depend on the mathematical content area. Intra-mathematical problems that could be solved by applying the Pythagorean theorem were valued higher than problems involving linear functions, whereas for modelling and dressed up word problems, it was the other way around.
Emotions and motivation are important prerequisites, mediators, and outcomes of learning and achievement. In this article, we first review major theoretical approaches and empirical findings in research on students’ emotions and motivation in mathematics, including a discussion of how classroom instruction can support emotions and motivation. Based on this review, we encourage researchers from mathematics education and other disciplines of educational research to combine their efforts. Second, we provide an overview of the contributions in this special issue, most of which reflect such a combination of efforts by considering perspectives from both mathematics education and other fields of educational research. Finally, we consider the neglect of intervention studies and outline directions for future research. We identify intervention studies that target emotions and motivation as one promising but so far underrepresented line of research in mathematics education and review results from existing intervention studies. For future research, we suggest that researchers should implement fine-grained concepts, assessment instruments, theoretical hypotheses, and methods of analysis tailored to the specific features of the mathematical domain to adequately investigate students’ emotions and motivation in this domain.
We report on a study of mathematical modelling skills of final year mathematics undergraduates, studying a course in mathematical models and modelling where the students' introduction to modelling is through exemplar models. These students are found to be at an intermediate stage between expert and novice, whose modelling capabilities are affected by limitations of their domain knowledge both in organisation and in content. We describe expert-novice behaviours, in terms of novice, intermediate and expert classifications. We consider outcomes from multiple-choice coursework questions, developed in a similar style to those used in more elementary contexts. We link these outcomes, and those from other assessments, to the descriptors of behaviour. Further, we raise questions as to how these outcomes might change perceptions of applying mathematics in real situations and consider implications for education more generally.
In this part of the volume, we shall give an introduction both to the field of applications and modelling in mathematics education and to the present volume. In section 1, we present the field of applications and modelling to the mathematics educator who is not a specialist in the field. In Section 2, we explain the basic terms, notions and distinctions in applications and modelling. On this basis, we provide, in Section 3, the conceptualisation of the field adopted in this ICMI Study. This conceptualisation is centred on a number of issues which will be the subject of Section 4. In Section 5, we briefly outline the historical development of applications and modelling in mathematics education. Finally, in Section 6, the structure and organisation of the present book will be described and explained.
All of the chapters in this book discuss research and theory related to the role that affect plays in mathematical problem solving. This chapter begins with a reanalysis of the affective domain, describing affect in terms of beliefs, attitudes, and emotions. The discussion of these three topics indicates the broad impact of affective factors in mathematics learning and relates beliefs, attitudes, and emotions to key chapters in this book. The chapter continues with a discussion of several broad themes that recur with some regularity in this book, including the central role of affect in problem solving, the need to integrate research on cognition and affect, and the importance of the social context in the study of affective factors in mathematics learning. The chapter concludes with comments on some methodological issues and their implications for future research on affective factors in the learning and teaching of mathematics.
During the last decade several major shifts have occurred in the conceptualisation of mathematics as a domain, of mathematical competence as a goal for instruction, and of the way in which this competence should be acquired through schooling. This chapter begins with a summary of the general characteristics and principles underlying the ongoing world-wide reform of mathematics education. Afterwards it documents and illustrates how these general characteristics and principles permeate a major domain of the mathematics curricula for the elementary school, called ‘Number and Arithmetic’. Five related topics within this domain are discussed, namely: number concepts and number sense, the meaning of arithmetic operations, mastery of basic arithmetic facts, mental and written computation, and word problems as applications of the numerical and arithmetical knowledge and skills. The final section lists some remaining issues and tasks for further curriculum research and development in the domain of number and arithmetic.
Although COACTIV did not gather data directly by means of classroom observation, it nevertheless provided concrete insights into the mathematics instruction provided, by reconstructing learning situations at the task level. Specifically, the tasks actually assigned by the COACTIV teachers were analyzed as documents of mathematics instruction, being classified according to a newly developed classification system with a focus on the potential for cognitive activation. The structure and scope of this classification system are presented in this chapter. As a complement to the self-report measures discussed in Chap. 6, the tasks submitted by the COACTIV teachers provide real, “objective” evidence of the content of mathematics instruction at the end of lower secondary education in Germany. The results presented in Chap. 6 and in this chapter support each other in indicating that mathematics instruction in Germany tends to offer little potential for cognitive activation. Yet, as Chap. 9 shows, exposure to more cognitively demanding tasks has positive effects on student learning gains. On the meta-level, our findings provide evidence for the theoretical argument that the tasks administered play a key role in promoting students’ mathematical learning—and that tasks are thus indeed suitable indicators of cognitive activation in the classroom.
This chapter is the third in a series intended to prompt discussion and debate in the Problem Solving vs. Modeling Theme Group. The chapter addresses distinctions between problem solving and modeling as a means to understand and conduct research by considering three main issues: What constitutes a problem-solving vs. modeling task?: What constitutes problem-solving vs. modeling processes?: and What are some implications for research?
Reflecting important new research developments of the past eight years as well as classic theories of problem solving, this book provides a balanced survey of the higher cognitive processes, human thinking, problem solving, and learning. Divided into four parts, the book discusses associationism and Gestalt theory before introducing current research and theories of induction and deduction in part two. Part three considers recent cognitive theories and discusses information-processing analysis, outlining techniques for analyzing cognition into processes, strategies, and knowledge structures. Finally, the implications and applications of findings in cognitive psychology are examined, with emphasis on applications such as teaching creative problem solving and intelligence measurement. The Second Edition updates research and includes new chapters on everyday thinking, expert problem solving, and analogical reasoning to reflect the growing interest in problem solving within specific domains and in real situations. New and classic problem solving exercises are used liberally throughout the text to illustrate concepts and to enhance the reader's active involvement in the process of problem solving.
Building on and extending existing research, this article proposes a 4-phase model of interest development. The model describes 4 phases in the development and deepening of learner interest: triggered situational interest, maintained situational interest, emerging (less-developed) individual interest, and well-developed individual interest. Affective as well as cognitive factors are considered. Educational implications of the proposed model are identified.
Applying mathematics to real problems is increasingly emphasized in school education;
however, it is often complained that many students are not able to solve mathematical
problems embedded in contexts. In order to solve story problems, a transition from a textual
description to a mathematical notation has to be found, intra-mathematical calculations have to
be performed, and the results have to be interpreted with respect to the described situation. On
the one hand, it is often suggested to consider problems which are embedded in a context from
the very beginning; on the other hand, step-by-step procedures at the beginning of learning
processes are widely proposed. In the present work, it was tested experimentally whether
starting a learning process in a “pure” intra-mathematical way (thus, without a textual description
of a context) is more beneficial than starting a learning process with problems providing a
very short context or with problems providing a detailed context, both with respect to objective
measures and with respect to subjective measures. The results indicate that starting with intramathematical
problems and starting with detailed story problems can both be very effective;
however, interaction effects with prior knowledge have to be taken into account.With respect to
motivational aspects, the results indicate that intra-mathematical problems and focused story
problems are substantially more appreciated by the learners than detailed story problems.