Location and Districting Problems in Services

Training in creative problem solving: Effects on ideation and problem finding and solving in an industrial research organization

Affective aspects are key mediators in the learning process. Whereas some of them can be associated with a certain discipline, others are situational and connected with specific activities that trigger positive emotions. This study analyzes these affective aspects based on two ways of approaching mathematical problems: problem posing and problem solving. In both cases, the starting point will be situations presented in multimodal representation, but with three different mediating elements: a real situation close to the students’ reality (text with data and image), a real situation far from the students’ reality (text with data and image), and a visual poem (hybrid text with implicit mathematical content that generates critical reading and provokes an aesthetic emotion). The aim is to explore the extent to which the mediating elements have affective and performance implications. To this end, an investigation was designed with future primary school teachers. As will be shown, the results, both in terms of performance and affective factors, are different for problem posing and problem solving. Problem posing based on a visual poem is a stimulating challenge for future teachers. However, in problem solving, as this study shows, the problem posed in a remote real situation is more successful in both performance and affective aspects.

Mathematical problem solving: What we know and where we are going

We examined the relations among problem-solving appraisal, self-reported study habits, and academic performance for 63 college students enrolled in a developmental course for academically unprepared students. The participants completed measures of self-appraise d problem-solving ability and study habits. Indexes of academic ability and performance were also collected. Regression analyses revealed that problem-solving appraisal was significantly predictive of study habits and semester grade-point average. The results indicate that problem-solving appraisal is related to study habits and academic performance among academically at-risk students, and they suggest that problem-solving appraisal encompasses more than social skill. The study of personal problem-solving ability and adjustment has accelerated considerably since Heppner's (1978) seminal review. Before that article the study of problemsolving skills was largely removed from real-life applications in general and counseling in particular. With the development of a valid and reliable measure of self-appraise d problemsolving ability, the Problem Solving Inventory (PSI; Heppner & Peterson, 1982; Heppner, 1988), an impressive literature has examined the relation between personal problem solving and adaptive behavior. This work has been guided by a theoretical conception of problem solving as a metacognitive variable (Butler & Meichenbaum, 1981) that serves a crucial function in the way a person perceives and experiences different aspects of the problem-solving process. Self-appraised problem solving, then, does not connote actual problemsolving skills but rather the individual perception of personal problem-solving ability, approach-avoidance style when one anticipates a problematic situation, and perceived control over personal abilities and subjective experience (Heppner, 1978, 1988). Research with the PSI has been predominately descriptive. Studies have found that research participants who appraise their problem-solving ability to be effective are less depressed and distressed than ineffective problem-solvers under general conditions (Heppner & Anderson, 1985; Heppner, Baumgardner, & Jackson, 1985; Heppner, Hibel, Neal, Weinstein, & Rabinowitz, 1982; Heppner, Kampa, & Brunning, 1987; Nezu, 1985, 1986a) or under stressful ones (Nezu, 1986b; Nezu, Nezu, Saraydarian, Kalmar, & Ronan, 1986). Other Appreciation is extended to Al Farrell for his consultation with regard to the statistical procedures used in this study. Correspondence concerning this article should be addressed to

This dissertation is to illustrate the current state of the art of DNA computing achievements, especially of new approaches to solve theoretical 3-SAT problems and the hitting-set problem. Beginning with Adleman’s breakthrough which is an molecular algorithm for the solution of a NP-complete, combinatorial problem, the directed Hamiltonian path problem (HPP). Today, many researchers all over the world concentrate on proposing new methods to solve engineering or application problems with a DNA computing approach. Satisfiability problem is given a Boolean formula, and decide if a satisfying truth assignment exists. ( ) ( ) … ( ) ( ) is an example of Boolean formula. k-SAT problem means that each clause has exactly k literals. Not-All-Equal (NAE) 3-SAT problem and One-In-Three (1IN3) 3-SAT problem are both NP-complete problems. In this dissertation, we present molecular solutions to find all true assignments (3-SAT problem) and furthermore find Not-All-Equal (NAE) solutions and One-In-Three (1IN3) solutions in DNA-based Supercomputing. Hitting-set problem assume that there exists a collection C of subsets of a finite set S, and a positive integer K  |S|, and we need to know if there is a subset  S with | |  K such that contains at least one element of each subset in C. In other words, is the subset that intersects every subset in C and is called the hitting-set. In this dissertation, a DNA-based algorithm is proposed to solve the small hitting-set problem. A small hitting-set is a hitting-set with the smallest K value, i.e., the hitting-set with the smallest number of elements. Furthermore, an algorithm is introduced to find the number of ones from 2n combinations and minimum numbers of ones represents the small hitting-set since K is expected to be as small as possible. The complexity of all the presented DNA-based algorithms is also discussed. We describe time complexity and volume complexity of three Algorithms, numbers of test tube used and the longest library strand in solution space of all three Algorithms. Finally, the simulated experiment is applied to verify correctness of the proposed DNA-based algorithm for solving the One-In-Three (1IN3) 3-SAT problem, and simulation of Not-All-Equal (NAE) 3-SAT problem is similar. Also, another simulated experiment is applied to our proposed DNA-based algorithm 6-2, in order to solve the well-known hitting-set problem. This research has been motivated by the benefit and the application of DNA computing and gives new methods to solve two 3-SAT problems and the hitting-set problem which are NP-complete.

This thesis documents the design, implementation and evaluation of a professional learning programme to support teachers in developing their ability to teach mathematics through problem solving. Teaching mathematics through problem solving is recognised as an effective way to teach mathematics. However, there are a number of different approaches with different descriptors such as problem posing and problem-based learning. In this study, teaching mathematics through problem solving is an approach where learners explore a problem or a task and their responses to the task are ‘orchestrated’ by the teacher to introduce and develop the mathematics to be learned. The development of this teaching approach involves the principles of task design and planning the sequencing of anticipated responses to be orchestrated. The professional development programme was informed by research on task design and the features of effective professional learning for teachers. The programme was built around the professional learning programme Lesson Study and was modified with a ‘pause’ in the research lesson and the use of remote visualisers to observe and analyse the work of the pupils. This latter feature was an adaption of the original programme design to respond to conditions arising in schools during the Covid-19 pandemic. Also a number of additional modifications were made to accommodate the ambitions of the programme. A qualitative approach was adopted using the methodology of design research and comprising two research cycles. The cohort in the first cycle were secondary school teachers and in the second cycle they were primary school teachers, all working in England. The aim of the study was to understand the effect of the programme on the participants’ knowledge, skills and confidence in teaching mathematics through problem solving. Existing frameworks in the literature on the known characteristics of effective professional learning were synthesised to evaluate the effectiveness of the programme. The main outcomes from this research confirm that the designed features of the PD programme were effective in supporting teachers in developing their ability to teach mathematics through problem solving. The use of task design principles in conjunction with the development of the teaching technique of orchestration resulted in gains in teacher confidence and subject and pedagogical knowledge. The pause in the research lesson and the use of visualisers to observe and analyse the pupils’ work provided the participants with a focused learning opportunity to develop the teaching approach. The evaluation of the programme using the designed analytical framework for CPD identified [a range of] known characteristics of effective CPD in the programme as designed and implemented. Three main implications emerged from the study, the first of which concerns the future development of teaching mathematics through problem solving. It was apparent that there were contextual barriers to the approach becoming an established part of teachers’ practice. To address these, the design of mathematics teaching resources would need to reflect the ambition to teach mathematics through problem solving. This is an issue for curriculum designers. There would also need to be recognition of the problem-solving skills that pupils require to access the problems, which would need to be taught either as part of teaching through problem solving or as a separate approach known as teaching for problem solving. The second implication identifies the challenges associated with introducing this type of programme into the current landscape of professional learning for teachers of mathematics in England, including the need for greater coherence between the professional learning of teachers of mathematics and their own workplace contexts. Finally, the third implication relates to the use of Lesson Study. This study described how the use of design research can develop Lesson Study variants that can make a positive contribution to professional learning environments for teachers without compromising the key principles of Lesson Study.

Despite problem solving being a core skill in chemistry, students often struggle to solve chemistry problems. This difficulty may arise from students trying to solve problems through memorising algorithms. Goldilocks Help serves as a problem-solving scaffold that supports students through structured problem solving and its elements, such as planning and evaluation. In this study, we investigated how first-year chemistry students solved problems, when taught with Goldilocks Help, and whether their problem-solving success and approaches changed over the course of one semester. The data comprised of student written problem-solving work, and was analysed using frequency analysis and grouped based on the problem-solving success and the extent of the demonstrated problem-solving elements. Throughout the course of semester, students exhibited increasingly consistent demonstration of structured problem solving. Nonetheless, they encountered difficulties in fully demonstrating such aspects of problem solving as understanding and evaluating concepts, which demand critical thinking and a firm grasp of chemistry principles. Overall, the study indicated progress in successful and structured problem solving, with a growing proportion of students demonstrating an exploratory approach as time progressed. These findings imply the need for incorporation of metacognitive problem-solving scaffolding, exposure to expert solutions, reflective assignments, and rubric-based feedback into wide teaching practice. Further research is required to extend the exploration of the effectiveness of metacognitive scaffolding, in particular via think-aloud interviews, which should help identify productive and unproductive uses of the problem-solving elements.

Recent studies suggest that worry is related to deficient problem-solving ability. This paper presents the relationship between worry and problem solving. Worry was measured with the Worry Domains Questionnaire (Tallis, Eysenck, & Mathews, 1992) and the Penn State Worry Questionnaire (Meyer, Miller, Metzger, & Borkovec, 1990); problem solving was assessed with the Social Problem-Solving Inventory (D'Zurilla & Nezu, 1990) and the Problem-Solving Inventory (Heppner & Petersen, 1982); and mood state was evaluated with the Beck Anxiety Inventory (Beck, Epstein, Brown, & Steer, 1988) and the Beck Depression Inventory (Beck, Rush, Shaw, & Emery, 1979). Subjects were 122 university students. The hypotheses were the following: (1) Worry would be negatively related to problem solving; and (2) cognitive and affective factors would play a more important role than behavioral deficits in limiting problem-solving ability. As both problem-solving scales are multidimensional, regression analyses predicted worry scores from the problem-solving scales. Results show that subscales describing theoretically sound problem-solving behaviors accounted for trivial or no variance in worry scores. On the other hand, even when mood state had been partialed out by hierarchical regression, subscales describing reactions to problematic situations predicted worry scores. Therefore, both hypotheses were confirmed. These results suggest that worry is related to problem orientation (i.e., immediate cognitive-behavioral-affective reactions to problematic situations) but not to problem-solving skills (i.e., articulating goals, generating solutions, making decisions, and implementing and verifying solutions). Clinical implications are discussed.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

The purpose of this study was to investigate whether a lengthened HyperCard treatment could promote problem-solving ability and reduce computer anxiety. It built upon a previous study by Reed and Liu (1994), which examined the comparative effects of the BASIC programming language and the HyperCard authoring language on students’ problem solving and computer anxiety. The Reed and Liu study found that after being exposed to the programming instruction for approximately 54 hrs, the problem-solving skills of the BASIC group improved significantly, whereas the problem-solving skills of the HyperCard group did not. This present research sought to answer the question “Can HyperCard promote problem solving if the treatment is lengthened and more of the programming components of HyperCard are included?” This study replicated a similar treatment, but lengthened the HyperCard instruction from 54 to 150 hrs of exposure to the language. The results showed that the lengthened treatment on HyperCard programming had an effect on problem solving. Students’ problem-solving scores increased significantly from pretreatment to the posttreatment. The most significant increase occurred between midtreatment and posttreatment, when students were involved in intense HyperCard programming. Students’ computer anxiety was reduced significantly, and a possible relationship between problem solving and computer anxiety was suggested. The implications of these findings with relation to teaching authoring languages such as HyperCard and traditional programming languages like BASIC were discussed.

In the information systems development (ISD) context, project teams are expected to find creative solutions to the problems encountered in the planning, development, and execution of projects. To efficiently leverage individual knowledge, team members must develop team cognition structures that facilitate their knowledge activities. This research examines the roles of team cognition and problem solving in successful ISD project teams. Data from 96 software development teams from four universities were obtained and analyzed. Relying on a multi-level conceptualization of team cognition, problem solving, and team creativity, our findings suggest that team cognition and problem solving on the relationship between task interdependence and team creativity are significant during software development, which means team cognition and problem solving can adequately account for how task interdependence influences team creativity as a mediator. Especially, process-based creativity is enhanced when team members have higher levels of team cognition or stronger problem solving abilities. It is concluded that in the context of ISD projects, team cognition (i.e., shared mental models, specialization, credibility, coordination) and problem solving (i.e., problem solving self-efficacy, problem solving skills) largely determine the extent of process-based creativity. The results are discussed in terms of the role of team cognition and problem solving in achieving such team creativity.

This thesis aims to contribute to a theoretical understanding of the determinants and outcomes of the use of explanations provided in knowledge-based systems. It is important to deepen our understanding of explanation use, because explanations from knowledge-based systems are thought to be valuable by researchers, systems developers, and users. Moreover, considerable effort can be expended in the design and development of explanation facilities. Nevertheless, no accepted and well-tested theory could be found to account for the use of explanations. Previous empirical studies have resulted in some contradictory findings. The theoretical model proposed in this study is taken from cognitive learning theory, a cognitive effort perspective applied to the use of knowledge-based systems, and Toulmin's model of argumentation. The determinants of the frequency of explanation use investigated were (i) the level of expertise of the user, (ii) the goal of the user (learning or problem solving), and (iii) the nature of the problem solving situation (collaborative or non-collaborative). Collaborative situations are those in which both user and system can contribute knowledge to the problem solving process. The outcomes of explanation use investigated were (i) learning by the user, (ii) problem solving performance, and (iii) users' confidence in the system. Two experiments were performed in laboratory settings. In the first experiment, advanced-level students in a taxation subject used an operational system for capital gains tax calculations. This experiment focussed on the effect of the user's goal, whether learning or problem solving. In the second experiment, members of the general public used a knowledge-based system designed to assist with personal financial planning. This experiment focussed on the effect of the nature of the problem solving situation, whether collaborative or non-collaborative. Two versions of the knowledge-based system were used: one with explanations and one without. In both experiments, measures of frequency of explanation use were obtained from traces of program use. The results of the study are congruent with a cognitive effort perspective. This perspective implies that explanations will be used more when users perceive that the advantage to be gained from the explanations outweighs the cognitive effort in accessing the explanations. As predicted from this perspective, a goal of learning rather than problem solving led to higher frequency of use of explanations. Also as predicted, a requirement for collaborative problem solving led to higher frequency of use of explanations. This latter effect, however, was observed only with users classed as practiced users of explanations on the basis of their performance in familiarisation activities undertaken in preparation for the experiments. Results offered some support for cognitive learning theory. Increased frequency of use of explanations led to increased problem solving performance under some conditions. In the first study, with non-collaborative problem solving, the use of an Answer Help explanation was significantly associated with improved performance. In the second study, the frequency of use of explanations and Answer Help in total was positively related to problem solving performance. This situation was observed in situations where explanations were available. There was no significant difference in performance between groups with and without explanations. There was some evidence that the positive relationship between explanations and improved performance was more noticeable when problems of a collaborative nature were undertaken. The expected relationship between increased frequency of use of explanations and an increase in the amount of learning taking place was not observed. The experimental conditions were such that learning could have occurred in ways other than through use of explanations. In this light, results are not inconsistent with cognitive learning theory. The level of expertise of the user was found to be related to higher problem solving performance. The level of expertise of the user was found also to be related to greater use of explanations in familiarisation activities. Both these relationships were observed only in the first experiment, where the measure used for level of expertise related specifically to the problem domain of the knowledge-based system. Results did not support the propositions that were based on Touhnin's model of argumentation. Possibly, confidence in machine advice is associated with explanations that can provide the backing for arguments. In this research, however, this relationship was not observed. It appeared, rather, that the degree of confidence expressed in a system was associated with the nature of the system. Confidence ratings were significantly related to the level of expertise of individuals, but the direction of this relationship was different for the two different systems. With the system presented as fully operational, the more expert individuals gave the system a higher confidence rating. With the system presented with disclaimers because it was still under research, the more expert individuals gave the system a lower confidence rating.

In increasingly complex environments, TRIZ offers a versatile set of tools and processes for problem solving. However, when it comes to defining which TRIZ tool(s) should be used in a given problem situation, many TRIZ users appear to be uncertain and seem to follow personal preferences rather than a structured process. The Adaptive Problem Sensing and Solving (APSS) Model provides a general process that is applied for TRIZ tool selection here. Based on the Cynefin framework, which allows sensing the characteristics of a problem situation, the authors suggest identifying which of the domains are actually available and provide strategies on how to extend the number of available domains. The user can set up the problem-solving process such that the solutions will more likely satisfy the respective expectations on the solution.

Innovative studies in the field of mathematics education, especially in recent years, have focused on problem solving, emphasizing that it is a process skill that should be addressed at every grade level and in every area of mathematics. The fact that many national and international exams try to evaluate students' problem-solving skills is an indication of this importance. Heavy emphasis is placed on problem solving with the aim of improving students’ skills in transferring the knowledge they learn at school to daily life and producing alternative solutions to the problems they may encounter in different situations. One of the methods used to achieve this aim is the drama method. Drama method can enable students to encounter problem situations in classroom environments. Students who encounter problem situations in the classroom environment become active in the process, internalize the problem and can produce appropriate solutions from different perspectives. In this context, the present study was conducted to examine the effect of teaching problems with natural numbers using the drama method on students’ problem-solving achievement and problem-solving attitudes. A quantitative research method was adopted and a pre-test/post-test quasi-experimental design with a control group was used. The study was conducted with two sixth-grade classes of a public school in a province in the north of Turkey. Data were obtained using the Problem-Solving Achievement Test and Mathematics Problem-Solving Attitude Scale. Predictive statistics were used in the analysis of the obtained data. It was concluded that the drama method was effective on students’ problem-solving skills and problem-solving attitudes.

Abstract: In this article, the author, who has written extensively about mathematical problem over the past 40 years, discusses some of his current thinking about the nature of problem-solving and its relation to other forms of mathematical activity. He also suggests several proficiencies teachers should acquire in order for them to be successful in helping students become better problem solvers and presents a framework for research on problem-solving instruction. He closes the article with a list of principles about problem-solving instruction that have emerged since the early 1970s.Keywords: mathematical activity, problem solving, problem-solving instruction, proficiencies for teaching, craft knowledge, research design, as a craft, teacher planning, metacognition..IntroductionMy interest in problem as an area of study within mathematics education began more than 40 years ago as I was beginning to think seriously about a topic for my doctoral dissertation. Since that time, my interest in and enthusiasm for problem solving, in particular problem-solving instruction, has not waned but some of my thinking about it has changed considerably. In this article I share some of my current thinking about a variety of ideas associated with this complex and elusive area of study, giving special attention to problem-solving instruction. To be sure, in this article I will not provide much elaboration on these ideas and careful readers may be put off by such a cursory discussion. My hope is that some readers will be stimulated by my ideas to think a bit differently about how mathematical problem solving, and in particular problem-solving instruction, might be studied.Setting the stageMost mathematics educators agree that the development of students' problemsolving abilities is a primary objective of instruction and how this goal is to be reached involves consideration by the teacher of a wide range of factors and decisions. For example, teachers must decide on the problems and problem-solving experiences to use, when to give problem particular attention, how much guidance to give students, and how to assess students' progress. Furthermore, there is the issue of whether problem is intended as the end result of instruction or the means through which mathematical concepts, processes, and procedures are learned. Or, to put it another way, should teachers adopt teaching for problem solving, -an ends approach-or teaching via problem solving -a means approach?1 (I say more about means and ends later in this article.) In my view, the answer to this question is that both approaches have merit; problem should be both an end result of learning mathematics and the means through which mathematics is learned (DiMatteo & Lester, 2010; Stein, Boaler, & Silver, 2003). Whichever approach is adopted, or if some combination of approaches is used, research is needed that focuses on the factors that influence student learning. Unfortunately, as far as I know, no prolonged, in-depth, programmatic research of this sort has been undertaken and, as a result, the accumulation of knowledge has been very slow. Moreover, the present intense interest in research on teachers' knowledge and proficiencies demands that future problem-solving research pay close attention to the mathematical and pedagogical knowledge and proficiencies a teacher should possess (cf., Ball, Thames, & Phelps, 2008; Hill, Sleep, Lewis, & Ball, 2007; Moreira & David, 2008; Zazkis & Leikin, 2010).But before discussing problem-solving instruction, let me first say a few things about mathematical problem solving. This short discussion will highlight how my thinking has changed about the nature of problem and other forms of mathematical activity.Some claims about Problem SolvingAmong the many issues and questions associated with problem-solving instruction I have worried about during my career, several have endured over time. …