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.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

Optimizing a dynamic order-picking process

This research studies the problem of batching orders in a dynamic, finite-horizon environment to minimize order tardiness and overtime costs of the pickers. The problem introduces the following trade-off: at every period, the picker has to decide whether to go on a tour and pick the accumulated orders, or to wait for more orders to arrive. By waiting, the picker risks higher tardiness of existing orders on the account of lower tardiness of future orders. We use a Markov decision process (MDP) based approach to set an optimal decision making policy. In order to evaluate the potential improvement of the proposed approach in practice, we compare the optimal policy with two naïve heuristics: (1) “Go on tour immediately after an order arrives”, and, (2) “Wait as long as the current orders can be picked and supplied on time”. The optimal policy shows a considerable improvement over the naïve heuristics, in the range of 7–99%, where the specific values depend on the picking process parameters. We have found that one measure, the slack percentage of the picking process, associated with the difference between the promised lead time and the single item picking time, predicts quite accurately the cost reduction generated by the optimal policy. Since relatively small-scale problems could be solved by the optimal algorithm, a heuristic was developed, based on the structure and properties of the optimal solutions. Numerical results show that the proposed heuristic, MDP-H, outperforms the naïve heuristics in all experiments. As compared to the optimal solution, MDP-H provides close to optimal results for a slack of up to 40%.     

Mathematical problem solving in small groups: Exploring the interplay of students' metacognitive behaviors,perceptions, and ability levels

This exploratory study extends our earlier work that identified the importance of metacognitive behaviors in mathematical problem solving in a small-group setting. In that study 27 seventh-grade students of varying ability were observed working in six small groups. The current investigation examines the perceptions of those students about themselves as problem solvers and about working in a small group. Data were obtained through videotapes of the students working in small groups and audiotapes of stimulated-recall interviews of the individual students. The results provided insight regarding the ways that beliefs, emotions and attitudes of students of varying ability influenced their own and their peers' metacognitive behaviors within their respective groups. The findings suggest a number of implications for teachers regarding the modality, level and frequency of assessment of group problem solving.     

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.     

Using Bayesian networks for bankruptcy prediction: Some methodological issues

This study provides operational guidance for building naïve Bayes Bayesian network (BN) models for bankruptcy prediction. First, we suggest a heuristic method that guides the selection of bankruptcy predictors. Based on the correlations and partial correlations among variables, the method aims at eliminating redundant and less relevant variables. A naïve Bayes model is developed using the proposed heuristic method and is found to perform well based on a 10-fold validation analysis. The developed naïve Bayes model consists of eight first-order variables, six of which are continuous. We also provide guidance on building a cascaded model by selecting second-order variables to compensate for missing values of first-order variables. Second, we analyze whether the number of states into which the six continuous variables are discretized has an impact on the model’s performance. Our results show that the model’s performance is the best when the number of states for discretization is either two or three. Starting from four states, the performance starts to deteriorate, probably due to over-fitting. Finally, we experiment whether modeling continuous variables with continuous distributions instead of discretizing them can improve the model’s performance. Our finding suggests that this is not true. One possible reason is that continuous distributions tested by the study do not represent well the underlying distributions of empirical data. Finally, the results of this study could also be applicable to business decision-making contexts other than bankruptcy prediction.     

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.     

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.     

On Waring’s problem for several algebraic forms

We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a grove, which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving

Along with neural dynamics (based on analog solvers) widely arising in scientific computation and optimization fields in recent decades which attracts extensive interest and investigation of researchers, a novel type of neural dynamics, called Zhang dynamics (ZD), has been formally proposed by Zhang et al. for the online solution of time-varying problems. By following Zhang et al.’s neural-dynamics design method, the ZD model, which is based on an indefinite Zhang function (ZF), can guarantee the exponential convergence performance for the online time-varying problems solving. In this paper, different indefinite Zhang functions, which can lead to different ZD models, are proposed and developed as the error-monitoring functions for the time-varying reciprocal problem solving. Additionally, for the goal of developing the floating-point processors or coprocessors for the future generation of computers, the MATLAB Simulink modeling and simulative verifications of such different ZD models are further presented for online time-varying reciprocal solving. The modeling results substantiate the efficacy of such different ZD models for time-varying reciprocal solving.     

When does data envelopment analysis outperform a naïve efficiency measurement model?

This paper compares the results from data envelopment analysis (DEA) to a naïve efficiency measurement model, which generates a scalar efficiency score by averaging all output–input ratios. Random data and real-life data are used to test the relative performance of the naïve model against various DEA models. The results suggest that the proposed the naïve model replicates DEA efficiency scores almost perfectly for constant return-to-scales and low heterogeneity in output–input data. It is therefore concluded that heterogeneity in output–input data is important to take advantage of the capability of DEA. It is also shown that heterogeneity is more relevant to efficiency measurement than the number of dimensions.     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.     

A case study of dynamic visualization and problem solving

This paper reports an example of a situation in which university students had to solve geometrical problems presented to them dynamically using the interactive computerized environment of the ‘MicroWorlds Project Builder’. In the process of the problem solving, the students used ten different solution strategies. The unsuccessful strategies were then classified into three main categories: distracting, reducing and confusing. One student group had to solve the same problem in its non-dynamic version. The results received from both groups were compared and analysed. Analysis of the solution strategies and the process of the categorization revealed that the percentage of success in both groups was similar and in the case of the given problem, the dynamic visual mode of the problem distracted the students’ attention away from proper handling of the solution of the problem.     

Ship-to-order supplies: Contract breachability and the impact of a manufacturer-owned direct channel

We address the issue of contract breachability in a supply chain involving a retailer and a manufacturer operating under ship-to-order contract terms and stochastic demands. The manufacturer is required to fulfill the retailer’s demands on a continuous basis with little or no advance notice. The issue in such an environment is whether the retailer can “naively” assume that she will get a very high fill rate from the manufacturer and therefore has no need for contract penalties in case the manufacturer’s inventory falls short. We suggest a stochastic calculus framework to study the problem and derive a condition when the retailer’s naïve assumption is justified since the probability of stock-outs of the manufacturer is negligible. That is, the ship-to-order contract will not be breached and the fill rate will be more than a predetermined threshold. Furthermore we find that although the manufacturer-owned direct channel generates more revenue and may reduce the volatility of both inventory and production orders, the ratio between expected direct channel and retail sales affects the benefits.     

Erratum to “A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling” [Appl. Math. Modell. 34 (2010) 655–670]

This study analyzes Mixed Integer Linear Program (MILP) proposed by G. Kannan, P. Sasikumar M. Devika, (2010) in their paper titled ‘A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling’, Applied Mathematical Modelling, (34, 655–670). The model in Kannan et al. (2010) is found to be inadequate for the problem described. It is erroneous/infeasible in terms of constraints, objective and variables. In this work, we list down the flaws in the published work and propose modifications to rectify the flaws. The revised model is presented and illustrated using hypothetical problems.     

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.     

Control decisions and personal beliefs: their effect on solving mathematical problems

The main purpose of this paper is to discuss how college students enrolled in a college level elementary algebra course exercised control decisions while working on routine and non-routine problems, and how their personal belief systems shaped those control decisions. In order to prepare students for success in mathematics we as educators need to understand the process steps they use to solve homework or examination questions, in other words, understand how they “do” mathematics. The findings in this study suggest that an individual’s belief system impacts how they approach a problem. Lack of confidence and previous lack of success combined to prompt swift decisions to stop working. Further findings indicate that students continue with unsuccessful strategies when working on unfamiliar problems due to a perceived dependence of solution strategies to specific problem types. In this situation, the students persisted in an inappropriate solution strategy, never reaching a correct solution. Control decisions concerning the pursuit of alternative strategies are not an issue if the students are unaware that they might need to make different choices during their solutions. More successful control decisions were made when working with familiar problems.     

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.