Systems thinking and mathematical problem solving

Problem solving lies at the core of engineering and remains central in school mathematics. Word problems are a traditional instructional mechanism for learning how to apply mathematics to solving problems. Word problems are formulated so that a student can identify data relevant to the question asked and choose a set of mathematical operations that leads to the answer. However, the complexity and interconnectedness of contemporary problems demands that problem‐solving methods be shaped by systems thinking. This article presents results from three clinical interviews that aimed at understanding the effects that traditional word problems have on a student’s ability to use systems thinking. In particular, the interviews examined how children parse word problems and how they update their answers when contextual information is provided. Results show that traditional word problems create unintended dispositions that limit systems thinking.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

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.     

Coding strategic behaviour in mathematical problem solving

There has been some debate about the extent to which contextual'stories' and graphical images can help pupils to think their way through numerical problems. The degree to which contexts stimulate useful'models to think with' may vary considerably. Some contexts seem to encourage children to reason more effectively, but others have little impact either on pupils'performance, or on the way in which they tackle the computations required. In this study three sets of four questions involving subtraction and division were trialled with a total sample of 1795 Year 6 to 9 pupils. In each comparison, about a quarter of the pupils could answer the question only when it was presented in context, or only out of context, but not both. Pupils' methods of working were analysed to reveal differences in their approach to the different questions.     

Fostering creativity through instruction rich in mathematical problem solving and problem posing

Although creativity is often viewed as being associated with the notions of “genius” or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.     

Interactions between subgoals and understanding of problem situations in mathematical problem solving

The purpose of this paper is to investigate solver's use of subgoals in mathematical problem solving processes. For this purpose, the process in which a solver tackled a rather difficult problem will be analyzed, focusing on how he established subgoals and how these subgoals affected his solving activity. This analysis will imply an interactive relation between subgoals established by the solver and his understanding of the problem situation. That is, his understanding of the situation supported his generation of subgoals, and those subgoals influenced his understanding positively or negatively. His use of subgoals will be also examined from the viewpoint of metacognition, and this examination will suggest the difficulty of escaping from the influence of a subgoal.     

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.     

How young students communicate their mathematical problem solving in writing

This study investigates young students’ writing in connection to mathematical problem solving. Students’ written communication has traditionally been used by mathematics teachers in the assessment of students’ mathematical knowledge. This study rests on the notion that this writing represents a particular activity which requires a complex set of resources. In order to help students develop their writing, teachers need to have a thorough knowledge of mathematical writing and its distinctive features. The study aims to add to the body of knowledge about writing in school mathematics by investigating young students’ mathematical writing from a communicational, rather than mathematical, perspective. A basic inventory of the communicational choices, that are identifiable across a sample of 519 mathematical texts, produced by 9–12 year old students, is created. The texts have been analysed with multimodal discourse analysis, and the findings suggest diversity in students’ use of images, words, numerals, symbols and layout to organize their texts and to represent their problem-solving process along with an answer to the problem. The inventory and the indication that students have different ideas on how, what, for whom and why they should be writing, can be used by teachers to initiate discussions of what may constitute good communication.     

The methods of fostering creativity through mathematical problem solving

Which methods could be used to foster mathematical creativity in school situations? The following topics are treated with the respect to this question: 1. “Open-ended approach” and “From problem to problem”, 2. Relation to mathematical creativity, 3. Teacher’s belief and the mathematics textbook.     

A mathematical programming viewpoint for solving the ultimate pit problem

In 1965 Helmut Lerchs and Ingo Grossmann presented to the mining community an algorithm to find the optimum design for an open pit mine. In their words, “the objective is to design the contour of a pit so as to maximize the difference between total mine value of the ore extracted and the total extraction cost of ore and waste”. They modeled the problem in graph theoretic terms and showed that an optimal solution of the ultimate pit problem is equivalent to finding the maximum closure of their graph based model. In this paper, we develop a network flow algorithm based on the dual to solve the same problem. We show how this algorithm is closely related to Lerchs and Grossmann's and how the steps in their algorithm can be viewed in mathematical programming terms. This analysis adds insight to the algorithm of Lerchs and Grossmann and shows where it can be made more efficient. As in the case Lerchs and Grossmann, our algorithm allows us to use very efficient data structures based on graphs that represent the data and constraints.. 1782 1528 V 3     

Toward the problem-centered classroom: trends in mathematical problem solving in Japan

In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences.     

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.     

Learning to be an opportunistic word problem solver: going beyond informal solving strategies

Informal strategies reflecting the representation of a situation described in an arithmetic word problem mediate students’ solving processes. When the informal strategies are inefficient, teaching students to make way for more efficient ways to find the solution is an important educational issue in mathematics. The current paper presents a pedagogical design for arithmetic word problem solving, which is part of a first-grade arithmetic intervention (ACE). The curriculum was designed to promote adaptive expertise among students through semantic analysis and recoding, which would lead students to favor the more adequate solving strategy when several options are available. We describe the ways in which students were taught to engage in a semantic analysis of the problem, and the representational tools used to favor this conceptual change. Within the word problem solving curriculum, informal and formal solving strategies corresponding to the different formats of the same arithmetic operation, were comparatively studied. The performance and strategies used by students were assessed, revealing a greater use of formal arithmetic strategies among ACE classes. Our findings illustrate a promising path for going past informal strategies on arithmetic word problem solving.

     

Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: a critical review

Based on Verschaffel et al.’s conceptualisation of flexible strategy choice, this article provides a critical review on the literature concerning flexible representational choice in mathematical learning. We argue that, while flexibility in the selection of a representation to complete a mathematical task has traditionally been understood as choosing the representation(s) that match(es) the characteristics of the to-be-solved task, research evidence suggests that it also includes the ability to take into account the characteristics of the subjects interacting with the representations, as well as the context in which such interaction takes place. The instructional and research implications of acknowledging the subjectivity and contextuality of flexible representational choice are examined.     

Algebra word problem solving approaches in a chemistry context: Equation worked examples versus text editing

Text editing directs students’ attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three equation steps each of which comprises essential units of relevant information. In an experiment, students were randomly assigned to either text editing or equation worked examples condition in a regular classroom setting to learn how to solve algebra word problems in a chemistry context. The equation worked examples group outperformed the text editing group for molarity problems, which were more difficult than dilution problems. Empirical evidence supports the theoretical rationale in using equation worked examples to facilitate students’ construction of a coherent problem structure so as to develop problem skills and expertise to solve molarity problems.     

Heuristics for solving a capacitated multiple allocation hub location problem with application in German wagonload traffic

In this paper, we present a capacitated multiple allocation hub location problem, which arose from a network design problem in German wagonload traffic. We develop heuristic solution approaches based on local improvements. We solve the problem with the heuristics and CPLEX on test data sets provided by our partner Deutsche Bahn AG. The computational results are presented and compared.     

Revisiting mathematical problem solving and posing in the digital era: toward pedagogically sound uses of modern technology

The availability of sophisticated computer programs such as Wolfram Alpha has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is discussed. Using topics from algebra (equations) and elementary number theory (summation of powers of integers), the paper suggests ways of developing problems that are both technology-immune and technology-enabled in the sense that whereas software can facilitate problem solving, its direct application is not sufficient for finding an answer. Stemming from the author's work with secondary mathematics teacher candidates, this paper highlights the appropriate use of technology as support system for multiple ways of knowing and knowledge construction in the modern classroom.     

Free products and the word problem

An example is given to show that even in a variety whose free word problem (identity problem) is solvable, the free product of two recursive presentations with solvable word problems may have unsolvable word problem. Under some extra conditions on the syntax of the identities defining the variety however, the free product is shown to preserve solvability of the word problem for recursive presentations. The conditions can be checked mechanically, and common varieties such as semigroups and groups satisfy them, as well as many less familiar varieties. The results are obtained by using rewrite-completion techniques.Presented by Ralph Freese.     

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

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