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)

Este trabalho consiste em uma pesquisa acerca da análise do potencial heurístico resultado da não solução dos três problemas clássicos da matemática grega, via regra do uso exclusivo do compasso e da régua não graduada. Para uma melhor compreensão deste potencial, apresentaremos o histórico de tais problemas, fazendo posteriormente uma síntese geral sobre as principais concepções de filósofos e matemáticos sobre Heurística. Em seguida, demonstraremos algumas soluções alternativas para estes problemas, identificando nelas processos heurísticos. Finalmente introduziremos tais processos na resolução de problemas matemáticos, acompanhadas de possíveis implicações pedagógicas para o ensino dessa ciência. / This work consists of research about the potential of heuristic analysis result of no solution of the three classical problems of Greek mathematics, via rule of exclusive use of the compass and no graduated scale. For a better understanding of this potential, it presents the history of such problems, then making a general overview about the main ideas of philosophers and mathematicians on Heuristics. Then we demonstrate some alternative solutions to these problems, identifying them heuristic processes. Finally we introduce such processes in mathematical problem solving, accompanied by possible pedagogical implications for the teaching of science.

By manipulating constraints and goals, this study had generated some ill-posed problems in "Fractions" which were packed into 2 mathematical tasks for teacher uses in an intended exploration of their perceived effectiveness of teaching mathematical problem-solving against their student responses through the lens of the theory of formative assessment. Each ill-posed problem was characterized by certain descriptive "instability" that users would have to define own sets of mathematical assumptions for problem-solving inquiries. 3 highly qualified, experienced, and trained mathematics teachers were purposefully recruited, and instructed to acquire and mark student responses without any prior teaching and intervention. Each of these teachers' perceptions of ill-posed problems was acquired through a semi-structured clinical case-interview. All teachers in common demonstrated only individual singular mathematical problem-solving inquiries as major instructional adjustments during evaluation, even though individuals had ample opportunities in manipulating the described intention of each problem. Although some could realize inquiries from students being alternative to own used, not all would intend to change initial instructional plans of each problem and could design dedicated tasks in extending given problem-solving contexts for subsequent teaching and maintaining the described problem-solving intentions merely because of evaluation purposes. The resulting thick teacher perceptions were then analyzed by the Mayring's (2015) Qualitative Content Analysis (QCA) method for exploring particularly those who could intend to influence and get influenced by students' used mathematical assumptions in interviews. Certain unanticipated uses of assumptions of student individuals and groups were evidently found to have influenced cognitively some teachers' further problem-solving inquiries at some interview instants and stimulated their perception changes. In the lack of subject implementation in mathematics education for the theory of "formative" assessment (Black & Wiliam, 2009), based on its definition, these instants should be put as their potential creations of and/or capitalizations upon certain asynchronous moments of contingency according to their planning of instructional adjustments for more comprehensive learning and definite growths of mathematical inquiries of students according to individuals' needs of problem-solving. Due to QCA, these perception changes might be characterized by four certain inductively formed categories of scenarios of perceptions, which were summarized as 1) Evaluation Perception, 2) Assumption Expansion Perception, 3) Assumption Collection Perception, and 4) Intention Indecision Perception. These scenarios of perceptions might be used to explore teachers' intentions, actions, and coherency in accounting for students' used assumptions in mathematical inquiries for given problem-solving contexts and extensions of given intentions of mathematical inquiries, particularly in their designs of mathematical tasks. Teacher uses of ill-posed problems were shown to have provided certain evidences in implementing formative assessments which should substantiate a subject implementation of its theory in the discipline of mathematics education. Methodologically, the current study also substantiate how theory-guided designs of ill-posed problems as well as generic plain text analysis through QCA have facilitate effectiveness comparisons of instructional adjustments within a teacher, across different teachers, decided prior knowledge, students of prior mathematical learning experiences, and students in different levels of schooling and class size.

Este trabalho consiste em uma pesquisa acerca da análise do potencial heurístico resultado da não solução dos três problemas clássicos da matemática grega, via regra do uso exclusivo do compasso e da régua não graduada. Para uma melhor compreensão deste potencial, apresentaremos o histórico de tais problemas, fazendo posteriormente uma síntese geral sobre as principais concepções de filósofos e matemáticos sobre Heurística. Em seguida, demonstraremos algumas soluções alternativas para estes problemas, identificando nelas processos heurísticos. Finalmente introduziremos tais processos na resolução de problemas matemáticos, acompanhadas de possíveis implicações pedagógicas para o ensino dessa ciência. / This work consists of research about the potential of heuristic analysis result of no solution of the three classical problems of Greek mathematics, via rule of exclusive use of the compass and no graduated scale. For a better understanding of this potential, it presents the history of such problems, then making a general overview about the main ideas of philosophers and mathematicians on Heuristics. Then we demonstrate some alternative solutions to these problems, identifying them heuristic processes. Finally we introduce such processes in mathematical problem solving, accompanied by possible pedagogical implications for the teaching of science.

Made available in DSpace on 2014-06-11T19:31:40Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-06-23Bitstream added on 2014-06-13T20:22:40Z : No. of bitstreams: 1 bergamo_ga_dr_bauru.pdf: 825872 bytes, checksum: d3ab2d8147131b1965b7c40817cb9e35 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho trata da fundamentação teórica do Método de Resolução de Problemas Ampliados (MRPA), que foi elaborado para a disciplina Matemática no Ensino Básico e trabalhado na Faculdade de Ciências/UNESP, no período 2000-2002, em cursos para professores de matemática. A partir de um aporte teórico inicial, foi reformulado conforme os professores relatavam aplicações em suas salas de aula. O MRPA visa uma educação emancipadora e propõe um tratamento não-internalista para a disciplina. Concebe os enunciados de matemática constituídos num campo semântico e amplia o significado dos conceitos, através de relações de similitude ou contraste, para os campos social, político e econômico. Na prática, parte de enunciados típicos de problemas escolares de matemática e amplia esses enunciados com questões dos outros campos. A fundamentação teórica principia com uma exposição sintética do materialismo histórico e dialético. Procede a uma caracterização do movimento dialético das categorias e apresenta uma maneira do método marxista, bem como das categorias do materialismo histórico e dialético, ser aplicado em Educação, trabalhando com as categorias produção de conhecimento e produção de conhecimento escolar enquanto formas particulares da produçao em geral. A escola existente é caracterizada como instituição social de potenciação... / This study is about the theoretical foundations of the Método de Resolução de Problemas Ampliados - MRPA (Broadened Problem Solving Method), developed and applied in the course Mathematics in Elementary Education, proffered to mathematics teachers at the State University of São Paulo (UNESP) in the period from 2000 to 2002. The theoretical base that formed the starting point for the study was reformulated over the course of the study to reflect the teacher's reports of their own experiences when applying the method in the classroom. The MRPA involves a view of education as emancipatory, and proposes a non-internalistic approach for the course. Mathematical enunciations are conceived of as being constituted within a semantic field, and the meaning of the concepts is broadened, through their relations of similarity and contrast, to the social, political, and economic spheres. In practice, one begins with mathematical statements used in the school and broadens them by introducing questions... (Complete abstract, click electronic access below)

The dissertation thesis deals with mathematical problem-solving at lower secon- dary level, as viewed from the perspective of heuristic strategies. The aim of the thesis is to comprehensibly summarize the results of research which began in 2012 and runs until now. The results concern both with theoretical and empirical parts of our research. This research study was conducted in fifteen lower secondary and upper secondary classes. Three dimensional classification of use of heuristic strategies and the structure of heuristic strategies' characteristics were developed by the author, and these constructs are presented in this work. The theory of mathematical problem and mathematical problem solving method is an integral part of this thesis too. Furthermore, the author presents a summary of all strategies used in the experiments; each strategy is fully described and illustrated by an appropriate example. The results of several short-term research studies (three months) and a longitudinal research study (sixteen months) are analysed in the empirical part of the thesis. This part also strives to find answers to several research questions, e.g.: " Could certain strategies be taught in a short-term period (three months)?", " Which strategies are suitable for an average pupil?" or " Are the pupils able to...

Fehlende Kompetenzen in Mathematik und Naturwissenschaften werden von Studierenden als ein Grund für den Studienabbruch in Ingenieurwissenschaften angegeben (Heublein et al., 2017). Welche Kompetenzen für Studierende zu Beginn des Ingenieurstudiums relevant sind, ist jedoch bisher wenig empirisch untersucht. Das Ziel der vorliegenden Studie ist, relevante mathematische Kompetenzen von Ingenieurstudierenden zu analysieren und dabei sowohl Wissensbestände als auch die Anwendung von Wissen und die Zusammenhänge zwischen beiden Bereichen zu berücksichtigen. Dazu wurde eine Studie im Mixed-Methods Design entwickelt. In dieser werden die Studierenden hinsichtlich ihrer Dispositionen in Mathematik und Physik zu Beginn des Studiums und am Ende des ersten Studienjahres mit quantitativen Methoden getestet. Zu diesen beiden und einem weiteren Zeitpunkt am Ende des ersten Semesters wurden zudem die situationsspezifischen Fähigkeiten bei der Bearbeitung von Mathematik- und Physikaufgaben mit Hilfe eines theoretischen Rahmens zum mathematischen Problemlösen mit qualitativen Methoden untersucht. Dieser Theorierahmen umfasste für die Mathematikaufgaben die Aspekte Heurismen (Bruder & Collet, 2011; Schoenfeld, 1980) und Problemlösephasen (Polya, 1957) sowie das Modell der Epistemic Games (Tuminaro, 2004) zur Analyse der Bearbeitung von Physikaufgaben. Die Ergebnisse zeigen Zusammenhänge zwischen mathematischen und physikali-schen Dispositionen. Zusätzlich wird die Bedeutung von Aspekten des Problemlösens deutlich, um die Prozesse bei den Bearbeitungen von Mathematik und Physikaufgaben im ersten Studienjahr zu analysieren. Auf Grundlage der qualitativen Beschreibungen konnten Cluster von Fällen von Studierenden gebildet werden. Mit Hilfe dieser Cluster zeigen sich Zusammenhänge zwischen den Dispositionen und situationsspezifischen Fähigkeiten bei den besonders leistungsstarken und leistungsschwachen Studierenden. / Missing competences in mathematics and sciences are cited by students as a reason for the drop-out in engineering sciences (Heublein et al., 2017). However, the competences that are relevant for students at the beginning of their engineering studies have so far not been investigated in an empirical way. The aim of this study is to analyse relevant mathematical competences of engineering students, taking into account both knowledge and the application of knowledge and the interrelationships between the two. A study in mixed method design was developed for this purpose. In this study, students are tested with regard to their dispositions in mathematics and physics at the beginning of their studies and at the end of the first year of their studies using quantitative methods. At these two points in time and a further time at the end of the first semester, the situation-specific skills in processing math and physics tasks were examined with the help of a theoretical framework for solving mathematical problems, using qualitative methods. This theoretical framework included for the mathematical tasks the aspects heuristics (Bruder & Collet, 2011; Schoenfeld, 1980) and problem solving phases (Polya, 1957) as well as the model of Epistemic Games (Tuminaro, 2004) for the analysis of the processing of physical tasks. The results show interrelationships between mathematical and physical dispositions. In addition, it became clear that there is a need of problem solving aspects in order to analyse the processes involved in the working on maths and physics tasks in the first year of studies. Based on the qualitative descriptions, clusters of student cases could be formed. These clusters show the interrelationships between dispositions and situation-specific skills of particularly high-performing and underperforming students.

Magister Educationis - MEd / This study argues that not all children in the Junior Primary phase benefit from the Problem Centred Approach in mathematics that was adapted by the Research, Unit for Mathematics at the University of Stellenbosch (RUMEUS). \One of the reasons could be that not all pupils can construct their own knowledge and methods. There are the highly capable pupils who cope well with this approach. These pupils are able to solve mathematical problems with little or no teacher interaction. Then there are the average and weaker pupils who cannot solve a mathematical problem on their own. These pupils need strategies and skills to solve problems and they need the teacher to mediate these strategies and skills to them, which will help these pupils to become autonomous problem solvers. ,Working in groups can, to some extent, supplement mediation or teacher interaction. Peer group teaching can be effective, whereby pupils are placed in groups so that the more capable pupils can teach concepts or make concepts clearer to the average or weaker pupils). There is, however, the possibility that when pupils of mixed abilities are placed in groups of four there might be one pupil who might refuse to work with the group. This pupil will work on her own and will not share ideas with the other members of the group. If this happens, mediation is necessary for those pupils who cannot solve a mathematical problem on their own. The purpose of this study is to investigate how exposure to mediation can improve pupils' problem solving abilities. As directions for my research I've chosen the first six criteria of Feuerstein's Mediated Learning Experiences (MLE). The first three parameters: intentionality and reciprocity, mediation of transcendence and mediation of meaning _are conditions for an interaction to qualify as MLE. Mediation of competence and regulation of behaviour are functions of specific experiences that combine with the first three to make an adult-child interaction one of mediated learning. Mediation of sharing behaviour . can be added. Here the child and the mediator are engaged in a shared quest for structural change in the child. In addition to this, the five mechanisms of mediational teaching, i.e. process questioning; challenging or asking reasons; bridging; teaching about rules; and emphasising order, predictability, system, sequence and strategy are also used in the implementation of mediation as described by Haywood. Two methods of investigation were chosen. The pupils' problem solving abilities were studied by means of eight word sums, of which the first four word sums were done in the pre-test and the other four word sums in the post-test. After the pre-test and before the post-test there was a period of mediational teaching for the experimental group. During this period and during the post-test the control group was denied mediation. After this research, mediation was also available for the control group. Two pupils from the experimental group were then chosen for further in-depth, think-aloud, person-to-person interviews. The aim of the interviews was to determine why these pupils could not solve the problem in the pre-test, but could successfully solve the post-test question. The results of the word sums in the pre-test and the post-test were compared. The role of strategies and thinking skills is concentrated on in the results. Mediation was not equally successful in all of the four different types of problem sums. Questions one and five contained two or more numbers and here pupils tended to either plus or minus these numbers. Questions two and six also contained numbers, but this is a problem situated in a real life situation. Questions three and seven contained no numbers and questions four and eight compelled pupils to first work out a plan. Mediation was most successful in problem sums situated in a real life situation, followed by problem sums which compelled pupils to first work out a plan, and then by problem sums where there were no numbers. Mediation was least; successful in problem sums that contained two or more numbers. Analysis of these results shows that with mediation there is an improvement in the pupils' problem solving abilities; Mediation can be viewed as S-H-O-H-R, in which the human mediator (H) is interposed between the stimulus (S) and the organism (0), and between the organism and the response (R). We can argue that the Problem Centred Approach without mediation can produce individuals who are little, if at all, affected by their encounter and interaction with new situations. Due to the lack of support in the Problem Centred Approach to Mathematics, it is the aim of this mini-thesis to propose mediation as an essential component in the Problem Centred Approach to Mathematics in the Junior Primary phase.

Vi lever i ett samhälle där datorer och annan digitalteknik blir allt mer central i vår vardag. Sveriges regering har därför ålagt Skolverket att stärka elevernas digitala kompetens. Som en del av detta införs programmering som ett digitalt verktyg i matematikundervisningen både i grundskolan och på gymnasiet. Det krävs dock i nuläget inga kurser i programmering för att bli en legitimerad matematiklärare. Syftet med undersökningen som presenteras i denna rapport är att undersöka matematiklärares uppfattningar som uppkommit på grund av att Skolverkets revidering av läroplanerna i matematik. Denna revidering innebär att vissa matematikkurser på gymnasiet innefattar att programmering skall användas som problemlösningsverktyg. Underlaget till denna undersökning är en transkribering och tematisering av kvalitativa intervjuer med tio matematiklärare, samt tidigare forskning. Undersökningen fann en viss oro bland lärarna som till stor del handlade om bristande kunskap i programmering samt problematiken med att hinna med att få in ytterligare ett moment i undervisningen. Under intervjuerna framgick det att lärarna var allmänt fundersamma om vilka digitala verktyg de skulle använda för att lösa detta nya krav. En majoritet av lärarna förordade dock Excel och Geogebra. Det framkom ett visst missnöje med att detta nya krav infördes med mycket kort varsel. Många lärare förväntade sig och litade på att läroboksförfattarna skulle komma med en uppdatering av läroböckerna i matematik. En uppdatering som förväntades innefatta programmering och som därmed skulle lösa den nya pedagogiska utmaningen. / Computers and other digital technology are becoming increasingly important in our society. Due to that, the Swedish Government has instructed their National Agency for Education to strengthen the students' digital competence. One outcome of this was that programming become a part of teaching mathematics both in primary and upper secondary school. Programming is not a part of the mandatory studies needed to become a certified mathematics teacher. The purpose of this study is to investigate the ideas, attitudes and ideas of mathematics teachers that have arisen because of the National Agency for Education's revision of the curricula in mathematics. According to this revision of the curricula, students should use programming as a problem-solving tool. The basis for this study is a transcription of qualitative interviews with ten mathematics teachers and an examination of previous research. This study found that there was some concern among the teachers. Most of the concern was about lack of knowledge in programming. The majority of teachers preferred to use Excel and Geogebra as a digital tool to teach programming. Many teachers expressed spontaneously a general dissatisfaction with the impact that calculators already have in mathematics education. There was some dissatisfaction with the introduction of this new requirement at very short notice. Many teachers expected and trusted that the textbook authors would come up with an update of the textbooks in mathematics. An update that would thus solve their new educational challenge.