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.     

Grade 3 students’ mathematization through modeling: Situation models and solution models with mutli-digit subtraction problem solving

In considering mathematics problem solving as a model-eliciting activity ( [Lesh and Doerr, 2003], [Lesh and Harel, 2003] and [Lesh and Zawojewski, 2008]), it is important to know what students are modeling for the problems: situations or solutions. This study investigated Grade 3 students’ mathematization process by examining how they modeled different types of multi-digit subtraction situation problems. Students’ modeling processes differed from one problem type to another due to their prior experiences and the complexity of the problems. This study showed that students make their own distinctions between solution and situation models in their mathematization process. Mathematics curricula and teaching should consider these distinctions to carefully facilitate different model development of and support student understanding of a content topic.     

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.     

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.     

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.     

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.     

Some mathematical models of the pulsating fountain problem

We consider the motion of the center of gravity of a variable-mass reservoir under the action of gravitation and the pressure of a water jet discharged from a vertical spout. It is assumed that the entire discharged amount of water enters the reservoir. A basic mathematical model is provided by the Cauchy problem for a Volterra integrodifferential equation. Numerical results are reported for particular parameter values.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 60–63, 1992;     

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     

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.     

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.     

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.     

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.     

Representational systems,learning, and problem solving in mathematics

This article explores aspects of a unified psychological model for mathematical learning and problem solving, based on several different types of representational systems and their stages of development. The goal is to arrive at a scientifically adequate theoretical framework, complex enough to account for diverse empirical results but sufficiently simple to be accessible and useful in mathematics education practice. Some perspectives on representational systems are discussed, and components of the model are described in relation to these ideas—including constructs related to imagistic thinking, heuristics and strategies, affect, and the fundamental role of ambiguity.     

Conceptual learning in a principled design problem solving environment

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five principles: (a) encouragement to produce multiple solutions; (b) creating collaborative situations; (c) socio-cognitive conflicts; (d) providing tools for checking hypotheses; and (e) inviting students to reflect on solutions. We describe how a problem solving task designed according to the above principles, promoted students' understanding of the area concept. We show that the design afforded the surfacing of multiple solutions and justifications in various modalities (including gestures) and initiated peer argumentation, leading to deep learning of the area concept.     

New mathematical models of the generalized vehicle routing problem and extensions

The generalized vehicle routing problem (GVRP) is an extension of the vehicle routing problem (VRP) and was introduced by Ghiani and Improta [1]. The GVRP is the problem of designing optimal delivery or collection routes from a given depot to a number of predefined, mutually exclusive and exhaustive node-sets (clusters) which includes exactly one node from each cluster, subject to capacity restrictions. The aim of this paper is to provide two new models of the GVRP based on integer programming. The first model, called the node formulation is similar to the Kara-Bekta? formulation [2], but produces a stronger lower bound. The second one, called the flow formulation, is completely new. We show as well that under specific circumstances the proposed models of the GVRP reduces to the well known routing problems. Finally, the GVRP is extended for the case in which the vertices of any cluster of each tour are contiguous. This case is defined as the clustered generalized vehicle routing problem and both of the proposed formulations of GVRP are adapted to clustered case.     

Historical evolution and mathematical models: A sociocultural algorithm

Though there are a lot of approaches to the problem of sociocultural evolution most of them are only one-sided, i.e., they deal only with either social or cultural processes. With few exceptions they are also only informal theories with no formal rigour. In this article we propose a theoretical model which considers both sides of the problem, that is the mutual interdependence of the evolution of social structures and of the culture of a society. A mathematical model, the sociocultural algorithm (SCA), based on these theoretical considerations maps several of the dynamic characteristics of sociocultural evolution, suggesting that universal principles underlie the dynamics of historical evolution.     

A hybrid tabu search/branch & bound approach to solving the generalized assignment problem

A new approach for solving the generalized assignment problem (GAP) is proposed that combines the exact branch & bound approach with the heuristic strategy of tabu search (TS) to produce a hybrid algorithm for solving GAP. The algorithm described uses commercial software to solve sub-problems generated by the TS guiding strategy. The TS approach makes use of the concept of referent domain optimisation and introduces novel add/drop strategies. In addition, the linear programming relaxation of GAP that forms part of the branch & bound approach is itself helpful in suggesting which variables might take binary values. Computational results on benchmark test instances are presented and compared with results obtained by the standard branch & bound approach and also several other heuristic approaches from the literature. The results show the new algorithm performs competitively against the alternatives and is able to find some new best solutions for several benchmark instances.     

The inverse problem for mathematical models of heart excitation

The inverse problem for mathematical models of heart excitation is stated; this problem is to determine the initial condition in the initial-boundary value problem for an evolutionary system of partial differential equations given the volume potential whose density is determined by the solution to the evolutionary system. It is proved that the solution of the inverse problem in the generic statement is not unique.