“I’m not very good at solving problems”: An exploration of students’ problem solving behaviours

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student's problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.     

Evaluation model of problem solving

Problem solving is a style of thinking, which transforms a given problem to the goal state through a so-called PS (problem solving) path. Different from the traditional GPS (General Problem Solver) approach, the focus in this paper is placed on how to judge the performance of PS paths, that is, the evaluation problem of problem solving.A series of PS paths point from the given source problem to the destination goal, then form a PSN (PS Network). This paper proposes an elaborated CPSN (Coordinate Problem Solving Network) as the evaluation model of problem solving. In CPSN, each problem is assigned a unique coordinate and then each PS path can have an evaluation vector. Several examples show such arrangement can give more insight to PS paths.Furthermore, an incremental learning algorithm is developed for the update of CPSN. When a new PS path is obtained, it is not necessary to recalculate the whole CPSN. Examples show such algorithm provides a more efficient way in finding new PS paths.     

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.     

Proving as problem solving: The role of cognitive decoupling

This paper discusses the process of proving from a novel theoretical perspective, imported from cognitive psychology research. This perspective highlights the role of hypothetical thinking, mental representations and working memory capacity in proving, in particular the effortful mechanism of cognitive decoupling: problem solvers need to form in their working memory two closely related models of the problem situation – the so-called primary and secondary representations – and to keep the two models decoupled, that is, keep the first fixed while performing various transformations on the second, while constantly struggling to protect the primary representation from being “contaminated” by the secondary one. We first illustrate the framework by analyzing a common scenario of introducing complex numbers to college-level students. The main part of the paper consists of re-analyzing, from the perspective of cognitive decoupling, previously published data of students searching for a non-trivial proof of a theorem in geometry. We suggest alternative (or additional) explanations for some well-documented phenomena, such as the appearance of cycles in repeated proving attempts, and the use of multiple drawings.     

Comparing representations and reasoning in children and adolescents with two-year college students

Problem solving and justification of a diversified group of two-year college students was compared with approaches of younger elementary and secondary school students working on the same tasks. The students in this study were engaged in thoughtful mathematics. Both groups found patterns, justified that their patterns were reasonable and, utilized similar strategies for their solutions and methods of justification. They were also able to make connections and build isomorphisms among the various problems.     

Shortest path problem with forbidden paths: The elementary version

This paper addresses the elementary shortest path problem with forbidden paths. The main aim is to find the shortest paths from a single origin node to every other node of a directed graph, such that the solution does not contain any path belonging to a given set (i.e., the forbidden set). It is imposed that no cycle can be included in the solution. The problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented. One is a Branch & Bound based algorithm, the other is a dynamic programming approach. Different versions of the proposed solution strategies are developed and tested on a large set of test problems.     

Students’ use of technological tools for verification purposes in geometry problem solving

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments - by providing more available tools compared to the traditional environment - might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.     

Levels of arithmetic reasoning in solving an open-ended problem

This paper presents the results of an experimental teaching carried out on 12-year-old students. An open-ended task was given to them and they had not been taught the algorithmic process leading to the solution. The formal solution to the problem refers to a system of two linear equations with two unknown quantities. In this mathematical activity, students worked cooperatively. They discussed their discoveries in groups of four and then presented their answers to the whole class developing a rich communication. This study describes the characteristic arguments that represent certain different forms of reasoning that emerged during the process of justifying the solutions of the problem. The findings of this research show that within an environment conducive to creativity, which encourages collaboration, exploration and sharing ideas, students can be engaged in developing multiple mathematical strategies, posing new questions, creating informal proofs, showing beauty and elegance and bringing out that problem solving is a powerful way of learning mathematics.     

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.     

An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.     

Students’ images of problem contexts when solving applied problems

This article reports findings from an investigation of precalculus students’ approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed that students who mentally constructed a robust structure of the related quantities were able to produce meaningful and correct solutions. In contrast, students who provided incorrect solutions consistently constructed an image of the problem's context that was misaligned with the intent of the problem. We also observed that students who caught errors in their solutions did so by refining their image of how the quantities in a problem's context are related. These findings suggest that it is critical that students first engage in mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs.     

Students’ use of technological features while solving a mathematics problem

The design of technology tools has the potential to dramatically influence how students interact with tools, and these interactions, in turn, may influence students’ mathematical problem solving. To better understand these interactions, we analyzed eighth grade students’ problem solving as they used a java applet designed to specifically accompany a well-structured problem. Within a problem solving session, students’ goal-directed activity was used to achieve different types of goals: analysis, planning, implementation, assessment, verification, and organization. As we examined students’ goals, we coded instances where their use of a technology feature was supportive or not supportive in helping them meet their goal. We categorized features of this applet into four subcategories: (1) features over which a user does not have any control and remain static, (2) dynamic features that allow users to directly manipulate objects, (3) dynamic features that update to provide feedback to users during problem solving, and (4) features that activate parts of the applet. Overall, most features were found to be supportive of students’ problem solving, and patterns in the type of features used to support various problem solving goals were identified.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

Children’s use of realistic considerations in problem solving: some English evidence

This is the second of two papers exploring children’s responses to an extended version of a division-with-remainder problem intended to elicit general rather than particular realistic considerations during mathematical problem solving. Responses to two problems are analyzed. The first is a ‘realistically’ contextualised item drawn from national tests in England whose ambiguities have been previously discussed (Cooper, 1992); the second is a version of this problem revised to encourage a wider range of realistic responses. In Cooper and Harries (2002), the responses of children at the end of their first year of secondary schooling were analyzed. Here the responses of children at the end of their primary schooling are analyzed and compared with the previous results. It is shown that many children, given suitable encouragement, are willing and able to enter into an extended form of realistic thinking during problem solving, although the original test item renders this invisible.     

A model of knowledge activation and insight in problem solving

This article presents a model of insight that offers predictions on how and when insights are likely to occur as an individual solves problems. This model is based on a fundamental trade‐off between the conscious cognition that underlies how people decide among alternatives and the unconscious cognition that underlies insight. I argue that the attention controls how much thought (i.e., knowledge activation) goes to conscious cognition, and whatever activation is left over will go to finding an insight. I validate this model by replicating the common pattern of insight in problem solving (preparation—impasse—incubation—verification). The model implies that 1) one should be able to increase the frequency of insight by lessening the demand for conscious cognition, 2) impasse is not necessary for insight, and 3) incubation time increases if a person engages in any activity with a high demand on attention. Understanding how insight occurs during problem solving provides practical suggestions to make people and groups more creative and innovative; it also provides avenues for future research on the cognitive dynamics of insight. © 2004 Wiley Periodicals, Inc. Complexity 9: 17–24, 2004     

Generalization and extension of a problem from an American mathematics competition

ABSTRACT

The purpose of these notes is to generalize and extend a challenging geometry problem from a mathematics competition. The notes also contain solution sketches pertaining to the problems discussed.     

The movement towards a more experimental approach to problem solving in mathematics using coding

Motivated by a problem proposed in a coding competition for secondary students, I will show on this paper how coding substantially changed the problem-solving process towards a more experimental approach.     

The role of problem representation and feature knowledge in algebraic equation-solving

Domain experts have two major advantages over novices with regard to problem solving: experts more accurately encode deep problem features (feature encoding) and demonstrate better conceptual understanding of critical problem features (feature knowledge). In the current study, we explore the relative contributions of encoding and knowledge of problem features (e.g., negative signs, the equals sign, variables) when beginning algebra students solve simple algebraic equations. Thirty-two students completed problems designed to measure feature encoding, feature knowledge and equation solving. Results indicate that though both feature encoding and feature knowledge were correlated with equation-solving success, only feature knowledge independently predicted success. These results have implications for the design of instruction in algebra, and suggest that helping students to develop feature knowledge within a meaningful conceptual context may improve both encoding and problem-solving performance.