High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

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.     

How Students “Unpack” the Structure of a Word Problem: Graphic Representations and Problem Solving

This research investigated how fourth and fifth grade students spontaneously ‘unpacked’ a word problem when generating a graphic representation to aid in problem solution. Relationships among the type of graphic representation produced, spatial visualization, drawing ability, gender, and problem solving also were examined and described. Instrumentation developed for the study included several math challenge tasks, a spatial visualization task, and a drawing task. For one of the math challenge tasks, students were instructed to draw a picture to assist them with problem solution. These graphic representations generated by students were rated as pictorial or as displaying some level of schematic representation. Schematic representations included germane information from the problem supportive of problem solution. Pictorial representations included expressive and extraneous elements not necessary for problem solution, with no schematic elements. Findings indicated that the majority of students rendered schematic representations, with girls more likely than boys to use schematic representations at a statistically significant level. Students who used schematic visual representations were more successful problem solvers than those pictorially representing problem elements. The more “schematic‐like” the visual representation, the more successful students were at problem solution. Drawing a pictorial representation in the math challenge task also was negatively correlated to drawing skill.     

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.

     

Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study

This study explored Singaporean fourth, fifth, and sixth grade students' mathematical thinking in problem solving and problem posing. The results of this study showed that the majority of Singaporean fourth, fifth, and sixth graders are able to select appropriate solution strategies to solve these problems, and choose appropriate solution representations to clearly communicate their solution processes. Most Singaporean students are able to pose problems beyond the initial figures in the pattern. The results of this study also showed that across the four tasks, as the grade level advances, a higher percentage of students in that grade level show evidence of having correct answers. Surprisingly, the overall statistically significant differences across the three grade levels are mainly due to statistically significant differences between fourth and fifth grade students. Between fifth and sixth grade students, there are no statistically significant differences in most of the analyses. Compared to the findings concerning US and Chinese students' mathematical thinking, Singaporean students seem to be much more similar to Chinese students than to US students.     

The initial treatment of the concept of function in the selected secondary school mathematics textbooks in the US and China

In order to provide insight into cross-national differences in students’ achievement, this study compares the initial treatment of the concept of function sections of Chinese and US textbooks. The number of lessons, contents, and mathematical problems were analyzed. The results show that the US curricula introduce the concept of function one year earlier than the Chinese curriculum and provide strikingly more problems for students to work on. However, the Chinese curriculum emphasizes developing both concepts and procedures and includes more problems that require explanations, visual representations, and problem solving in worked-out examples that may help students formulate multiple solution methods. This result could indicate that instead of the number of problems and early introduction of the concept, the cognitive demands of textbook problems required for student thinking could be one reason for differences in American and Chinese students’ performances in international comparative studies. Implications of these findings for curriculum developers, teachers, and researchers are discussed.     

Teachers’ metacognitive and heuristic approaches to word problem solving: analysis and impact on students’ beliefs and performance

We conducted a 7-month video-based study in two sixth-grade classrooms focusing on teachers’ metacognitive and heuristic approaches to problem solving. All problem-solving lessons were analysed regarding the extent to which teachers implemented a metacognitive model and addressed a set of eight heuristics. We observed clear differences between both teachers’ instructional approaches. Besides, we examined teachers’ and students’ beliefs about the degree to which metacognitive and heuristic skills were addressed in their classrooms and observed that participants’ beliefs were overall in line with our observations of teachers’ instructional approaches. In addition, we investigated how students’ problem-solving skills developed as a result of teachers’ instructional approaches. A positive relationship between students’ spontaneous application of heuristics to solve non-routine word problems and teachers’ references to these skills in their problem-solving lessons was found. However, this increase in the application of heuristics did not result in students’ better performance on these non-routine word problems.     

High‐School Students' Approaches to Solving Algebra Problems that are Posed Symbolically: Results from an Interview Study

A cross‐curricular structured‐probe task‐based clinical interview study with 44 pairs of third year high‐school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students' use of graphing calculators to solve the problems is discussed.     

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.     

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.     

Coping strategies applied to comprehend multistep arithmetic word problems by students with above-average numeracy skills and below-average reading skills

This article discusses how 13-year-old students with above-average numeracy skills and below-average reading skills cope with comprehending word problems. Compared to other students who are proficient in numeracy and are skilled readers, these students are more disadvantaged when solving single-step and multistep arithmetic word problems. The difference is smaller for single-step word problems. Analysis of large-scale data as well as a case study suggested that students used knowledge of stereotype item formats and keywords to cope with comprehending word problems. Instances where students used prior experiences to form predispositions to word problems were observed in the case study. In addition, analyses in both studies revealed that errors caused by overuse of keywords were more frequent among the students with below-average reading skills.     

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem

Previous research has documented that preservice teachers (PSTs) struggle with understanding fraction concepts and operations, and misconceptions often stem from their understanding of the referent whole. This study expands research on PSTs’ understanding of wholes by investigating pictorial strategies that 85 PSTs constructed for a multistep fraction task in a multiplicative context. The results show that many PSTs were able to construct valid pictorial strategies, and the strategies were widely diverse with respect to how they made sense of an unknown referent whole of a fraction in multiple steps, how they represented the wholes in their drawings, in which order they did multiple steps, and which type of model they used (area or set). Based on their wide range of pictorial strategies, we discuss potential benefits of PSTs’ construction of their own representations for a word problem in developing problem solving skills.     

Comparing eye movements during mathematical word problem solving in Chinese and German

Language plays an important role in word problem solving. Accordingly, the language in which a word problem is presented could affect its solution process. In particular, East-Asian, non-alphabetic languages are assumed to provide specific benefits for mathematics compared to Indo-European, alphabetic languages. By analyzing students’ eye movements in a cross-linguistic comparative study, we analyzed word problem solving processes in Chinese and German. 72 German and 67 Taiwanese undergraduate students solved PISA word problems in their own language. Results showed differences in eye movements of students, between the two languages. Moreover, independent cluster analyses revealed three clusters of reading patterns based on eye movements in both languages. Corresponding reading patterns emerged in both languages that were similarly and significantly associated with performance and motivational-affective variables. They explained more variance among students in these variables than between the languages alone. Our analyses show that eye movements of students during reading differ between the two languages, but very similar reading patterns exist in both languages. This result supports the assumption that the language alone is not a sufficient explanation for differences in students’ mathematical achievement, but that reading patterns are more strongly related to performance.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.