När elever står inför uppgifter inom problemlösning är det ett flertal steg som eleven ska ta sig igenom: Förstå uppgiften, göra någon form av plan, sedan genomföra planen och i bästa fall även gå tillbaka till uppgiften och reflektera över lösning och svar. Syftet med studien var att undersöka och belysa lösningsstrategier inom problemlösning ur ett elevperspektiv. För att besvara syftet har en kvalitativ ansats varit till grund för studien, som bestod av två datainsamlingar: ett elevtest och efterföljande gruppintervjuer. Resultatet för den första datainsamlingen analyserades med ett begreppsligt ramverk som definierar och begreppsliggör lösningsstrategier. Det visade att eleverna till stor del valde mellan algebraisk lösning och blockmodellen. Elevperspektivet har belysts genom att elever efter ett genomfört test ombetts reflektera gällande deras val av lösningsstrategi. Intervjuerna analyserades utifrån en induktiv ansats där data tematiserades och påvisade fem centrala motiveringar till att elever väljer de strategier de väljer. Dessa var: enkelhet att tolka, tid spenderad på varje uppgift, erfarenhet och självförtroende, uppgiftens karaktär och lärarens undervisning. / When students face tasks in problem-solving, there are several steps they need to go through: understanding the task, devising some form of plan, executing the plan, and ideally, revisiting the task to reflect on the solution and answer. The purpose of the study was to investigate and illuminate problem-solving strategies from a student perspective. To address this purpose, a qualitative approach was employed, consisting of two data collection phases: a student test followed by group interviews. The results of the initial data collection were analyzed using a conceptual framework that defines and elucidates problem-solving strategies. It was found that students largely chose between algebraic solutions and block modeling. The student perspective was illuminated by having students reflect on their choice of solution strategy after completing the test. The interviews were analyzed using an inductive approach where data were themed, revealing six central motivations for why students choose the strategies they do. These were: simplicity of interpretation, time spent on each task, experience and self-confidence, the nature of the task, and teacher instruction.

In 2 experiments, I studied the effects of an Equivalence Based Instruction (EBI) math intervention on the emergence of untaught selection responses and abstraction to production responses. In Experiment I, using a multiple baseline design, I implemented the EBI intervention among a group of 17 first grade participants with varying levels of math prerequisites and verbal behavior development. The intervention sought to develop a comprehensive relational network for the part-whole relations involved in addition and subtraction operations. This intervention, informed by Verbal Behavior Development Theory, Relational Frame Theory, and research on math proficiency, utilized visual and verbal stimulus presentations of fact families to establish the concepts underlying addition and subtraction. The key concept was that of a fact-family, in which two parts are equivalent to the whole and the whole is equivalent to the sum of its parts. The goal of the EBI intervention was to establish a relational network involving pictures, number bonds, sentences, and equations such that the part-whole relations involved in fact-families could be related to both addition and subtraction. The EBI intervention consisted of 3 phases to build this relational network. In Phase I, participants learned to match sentences describing complete fact-families with pictures and number bonds. In Phase II, participants learned to match sentences describing incomplete fact-families with number bonds. In Phase III, participants learned to match incomplete number bonds with addition and subtraction equations presented in various topographies. Before and after each phase of the intervention, I assessed the degree to which participants acquired untaught responses as well as their performance on production, or problem-solving, probes. Results revealed that the combinatorially entailed response (i.e., matching pictures with number bonds) emerged for all participants, while the mutually entailed response (i.e., selecting sentences) emerged for only some participants. Participants generally improved their problem-solving following the intervention; however, further examination was needed to supplement initial visual analyses of the graphs. Accordingly, I conducted a series of statistical analyses to evaluate individual and group-level differences in responding during the EBI intervention. These analyses also sought to reveal whether math prerequisites or level of verbal behavior development were associated with performance during Phases I, II, and III. Results showed that the EBI intervention was associated with standardized math performance and problem-solving accuracy, and results suggested that verbal behavior development has a meaningful relation with rate of learning. In Experiment II, I aimed to evaluate the educational significance of the repertoires involved in the EBI intervention by conducting a correlational study with 32 additional first grade participants. This experiment revealed that the response-types targeted in Phase III of the intervention were significantly associated with standardized math performance.

The purpose of this study was to determine the relationship between High School learners‟ mathematics-related belief systems and their approaches to mathematics non-routine problem-solving. A mixed methods approach was employed in the study. Survey questionnaires, mathematics problem solving test and interview schedules were the basic instruments used for data collection. The data was presented in form of tables, diagrams, figures, direct and indirect quotes of participants‟ responses and descriptions of learners‟ mathematics related belief systems and their approaches to mathematics problem solving. The basic methods used to analyze the data were thematic analysis (coding, organizing data into descriptive themes, and noting relations between variables), cluster analysis, factor analysis, regression analysis and methodological triangulation. Learners‟ mathematics-related beliefs were grouped into three Learners‟ mathematics-related beliefs were grouped into three categories, according to Daskalogianni and Simpson (2001a)‟s macro-belief systems: utilitarian, systematic and exploratory. A number of learners‟ problem solving strategies were identified, that include unsystematic guess, check and revise; systematic guess, check and revise; trial-and-error; logical reasoning; non-logical reasoning; systematic listing; looking for a pattern; making a model; considering a simple case; using a formula; numeric approach; piece-wise and holistic approaches. A weak positive linear relationship between learners‟ mathematics-related belief systems and their approaches to non-routine problem solving was discovered. It was, also, discovered that learners‟ mathematics-related belief systems could explain their approach to non-routine mathematics problem solving (and vice versa). / Mathematics Education / D.Phil. (Mathematics Education)

The purpose of this study was to determine the relationship between High School learners‟ mathematics-related belief systems and their approaches to mathematics non-routine problem-solving. A mixed methods approach was employed in the study. Survey questionnaires, mathematics problem solving test and interview schedules were the basic instruments used for data collection. The data was presented in form of tables, diagrams, figures, direct and indirect quotes of participants‟ responses and descriptions of learners‟ mathematics related belief systems and their approaches to mathematics problem solving. The basic methods used to analyze the data were thematic analysis (coding, organizing data into descriptive themes, and noting relations between variables), cluster analysis, factor analysis, regression analysis and methodological triangulation. Learners‟ mathematics-related beliefs were grouped into three Learners‟ mathematics-related beliefs were grouped into three categories, according to Daskalogianni and Simpson (2001a)‟s macro-belief systems: utilitarian, systematic and exploratory. A number of learners‟ problem solving strategies were identified, that include unsystematic guess, check and revise; systematic guess, check and revise; trial-and-error; logical reasoning; non-logical reasoning; systematic listing; looking for a pattern; making a model; considering a simple case; using a formula; numeric approach; piece-wise and holistic approaches. A weak positive linear relationship between learners‟ mathematics-related belief systems and their approaches to non-routine problem solving was discovered. It was, also, discovered that learners‟ mathematics-related belief systems could explain their approach to non-routine mathematics problem solving (and vice versa). / Mathematics Education / D.Phil. (Mathematics Education)

This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

Studenten und Konstrukteure mit verschiedener konstruktionsmethodischer Ausbildung wurden bezüglich ihres Vorgehens beim Konstruieren und ihrer Konstruktionsleistungen untersucht. Dabei zeigte sich, dass die Wirkung der Ausbildung für drei Teilleistungen (Konzepte erstellen, Konzepte bewerten, Entwürfe erstellen) verschieden ist. Für die konstruktionsmethodischen Empfehlungen für die frühen Phasen wird eine neue Konzeption vorgeschlagen. Der Einfluss kognitiver Variablen (Arbeitsgedächtnis, heuristische Kompetenz, Interferenz) wurde untersucht.

This thesis Pupils' problem solving strategies at lower and upper secondary level brings a focus on the issue outlined by the name. At the same time, it focuses on pupils' strategies to solve problems that are closely related to information literacy. At first I define basic terms for the area of word problems. Consecutively I focus on the theoretical knowledge from the area of problem solving strategies themselves. The content of the experimental part of the thesis is the survey of pupils' solutions of eight selected tasks, by which I was looking for an answer of the three basic questions of this thesis. My experiment is divided into two branches. The first branch of the experiment took place at the lower level of the multi-year grammar schools. The second branch of the experiment took place at the higher grade of the multi-year grammar schools and, to a small extent, at the secondary school. The theoretical part contains views of various authors on issue of problems and word problems. I present and compare these individual approaches. The result is the demarcation of the terms needed for the experimental part of the work. The main aim of the experimental part of the thesis is to find the answers of three basic questions of this thesis, where I was using data from lower and higher grades of...

This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

Problem-solving ability is now the most sought-after trait in up-and-coming executives, according to a survey of 1 000 executives conducted by Caliper Associates, reported in the Wall Street Journal by Hal Lancaster (Hoenig, 2002:338). This trait would include the ability to solve human performance problems, something many people tend to steer clear of. According to Piskurich (2002:57-58) and Rothwell, Hohne and King (2000:67-71), the most common problem-solving tools that are used when solving human performance problems are brainstorming, cause-and-effect analysis, and the five why’s technique. Although techniques such as these have proven to be robust and useful, what is required to solve human performance problems is a logical and verifiable process that can establish a data point about which relevant information can be recognized and gathered, and against which the conclusion can be evaluated, to have confirmed knowledge of the root cause of the problems. Unfortunately, existing root cause analysis processes tend to focus on processes and systems, rather than on individual performance (Bowling, 2003). The main objective of this study was to develop a root cause analysis process that would uncover the root cause(s) of uncontrolled variation(s) in human performance and prevent the recurrence of events causing the variation. In addition to addressing individual human performance incidents, it is also necessary continually to manage people’s performance to detect and address any occurrences (or recurrences) of performance variations. Therefore, in addition to the main objective, the study also aimed to develop a Human Performance Management Model that incorporated the root cause analysis process as a problem-solving tool. Action research was used in this study, because of the cyclical iterative nature of this type of research, and because it is a rigorous, responsive and flexible process. The study consisted of three cycles. The end result was a structured root cause analysis process – the Human Performance Variation Analysis (HPVA) process – that enables the systematic collection of valid and reliable information, as is required to solve variation in human performance. The HPVA process is a three-part process that consists of 11 steps. The process is in turn a tool that forms part of a ten-step Human Performance Management Model. The study contributes to the body of knowledge on human performance management by presenting the following: • a systematic root cause analysis process that uncovers the root causes of human performance problems effectively and consistently and that controls these causes of problems in a way that prevents the problems from recurring; and • a Human Performance Management Model that will help to sustain the new, improved performance; prevent the same or similar performance problem(s) in other areas of the organisation; and ultimately, create an environment and culture of continuous human performance improvement. / Thesis (PhD)--University of Pretoria, 2009. / Human Resource Management / unrestricted

The main aim of this thesis was to investigate decision making from a broad developmental perspective to clarify the role of the underlying mechanisms influencing it. Problem solving and cognitive inhibitory capacity were chartered initially through the use of hypothetical vignettes depicting socially relevant situations and through the use of the Stroop task, to tap into automatic inhibitory capacity. Initial assumptions that prefrontal cortical refinement would denote enhanced social problem ability were not confirmed. Experience emerged as distinct factor in problem solving/decision-making, with the youngest participants equally as effective in producing solutions to situations that they had the most experience in. A shift in development is observed with maturation denoting greater experience and this being applied directly to problem solving and decision-making situations. Education was identified as a possible contributory factor in decision-making and this was explored in a cross-cultural study that tapped into a non-schooled population. The results reinforced the centrality of experience in shaping decision-making. Decision-making in regards to the use of experience was then looked at through real life decision-making situations, where adolescents were asked to provide their knowledge or experience of situations where risk was involved. Adolescents possessed the necessary knowledge to distinguish between optimal and sub-optimal decisions in terms of the consequences that risk behaviours carried with them. However, many still chose to engage in risky behaviours. This paradox could also be explained by actual experience, with the suggestion that positive experience in a peer group was serving as a pool from which adolescents drew to make future decision-making. If risk behaviours were not experienced adversely, the likelihood of their repetition was high. Taken together the findings suggest that adolescents are well equipped with the cognitive skills to make decisions. Compared to younger children, they have more experience of a greater range of situations from which to extrapolate responses from. They also have a great deal of knowledge and information about the negative consequences associated with a range of challenging situations and risk-taking behaviours. However, when faced with decisions in the social domain, the behaviour of friends and perceptions of what other people are doing are powerful influences on adolescent decisions.