THESIS Pupils' problem solving strategies at lower and upper secondary level ABSTRACT Thesis Pupils' problem solving strategies at lower and upper secondary level deals with pupils' strategies of solving mathematical problems that we can observe at primary and secondary school. The theoretical part summarizes basic concepts in the field of mathematical problems and pupils' strategies. The main aim of the experimental part of the thesis is finding new knowledge from this field at the second level of Czech education. I focused primarily on tasks involving data, addiction and statistics, including the concept of a mathematical function that is deeply linked to these educational contents. In the experimental part, there are selected tasks, expected or discovered strategies, statistics of chosen strategies and the success of the solutions and strategies found, in some cases even the transcription of pupils' errors. Keywords Problem, word problem, assignment, strategy, error, analysis, success rate, solving problems.

Magister Educationis - MEd / The research explores the effects of Community of Enquiry practices on the teaching and learning of Mathematics and Literacy in two local primary schools. After the 1994 elections, both the government and education system changed in South Africa. With the introduction of Outcomes Based Education (OBE), critical outcomes that emphasized thinking and collaboration became a vital part of the curriculum. Soon after, the Education system adopted the National Curriculum Statement (NCS) and thereafter the Revised National Curriculum Statement (RNCS), which maintained these outcomes. The Curriculum and Assessment Policy Statement (CAPS) was introduced to the Foundation Phase in 2012 and to the Intermediate Phase in 2013 with the Critical Outcomes, (which emphasizes thinking) now stated as the aims of CAPS. However, no guidelines are provided regarding classroom practice. The approach to teaching these aims is not made clear. Lipman’s Philosophy for Children (P4C) is one way of working towards these aims, and promoting thinking and is consistent with many of Vygotsky’s ideas. He initiated ideas about cognitive development in which he refers to the importance of dialogue in which one is able to talk and communicate with others. Vygotsky also emphasised scaffolding where the teacher provides the learner with clues and suggestions in order to develop better problem- solving techniques and thinking habits. His concept of the zone of proximal development (ZPD) refers to the individual’s ability to accomplish more or to perform a challenging task with the proper assistance. The development of language is considered important within his theory as Vygotsky believes that individuals are born only with lower mental processes and develop their thinking ability (higher mental processes) by acquiring the thinking tools developed in a particular culture, the most important of which is language. The research followed a qualitative research methodology. The study explored the perceptions of both educators and learners after an intervention based on Philosophy for Children. Qualitative data involved two group interviews with teachers, one with the Cognitive Education Co-ordinator and interviews with four focus groups of selected Grade 5 and 7 learners (12 per group) whose teachers implemented Lipman’s Community of Enquiry pedagogy in the classroom the previous year. Quantitative data included a learner self-rating scale. All the educators of the two schools, who were involved in the classroom Community of Enquiry training, were invited to participate in the study, as were selected learners from the two Grade 5 and 7 classes at each school. I made use of thematic analysis of the interview data from both learners and teachers. Themes within the interviews were identified. Themes pertaining to teacher perception of self-change, teacher perception of learner change, and learner perceptions of self change were identified. During thematic analysis, the three research sub-questions were underlined. These were: (1) What are the teachers’ perceptions of self-change? (2) What are the teachers’ perceptions of learner change? (3) What are the learners’ perceptions of self-change? The conclusion of the study was that P4C has the potential to affect the teachers professionally and to influence the learners positively in Mathematics and Literacy classrooms. Ongoing support in cognitive education is vital in order to reach the aims required for the new CAPS curriculum.

I conducted a study among the middle school students to examinetheir attitudes to school mathematics both inside and outside of the classroom. Also, I have examined teachers’ perceptions of how everyday math examples can be integrated into the classroom experience. The purpose of the study is to investigate the effect of using everyday mathematics in teaching and how this will contribute to a further development of students' ability to think mathematically in different contexts. Parallels between prior research efforts and my study “The art of thinking mathematically - School mathematics in everyday life”are drawn by using qualitative analysis techniques such as the organized empirical data based on student survey questionnaire and teacher interviews. The theoretical perspectives utilized in the study are about the pragmatic perspective and its centralized concepts emphasizing continuity, experience, and interaction. The socio-cultural perspective sheds light on interaction between people and perceptions of themselves and others. Within a broader context I will address the proximal developmental zone (ZPD) by Vygotsky, scaffolding and mediating as those concepts apply to the aforementioned issues. In conclusion I will provide what advantages and disadvantages do teachers and students experience in linking everyday mathematics to the school mathematics.  I will give also approaches for teachers and suggestions for further research and investigation.

Magister Educationis - MEd / The study is located within the project: Relevance of School Mathematics Education (ROSME) of the Department of Didactics at the University of the Western Cape. The research is undertaken in the belief that Mathematics enables creative and logical reasoning about contextualised problems in the realm of the physical and social world as well as in the discipline mathematics itself. Relevance of school Mathematics has the implied notion of contextual issues. This research attempts to investigate the contextual issues that teachers have to deal with in Mathematics education. Given the results of the TIMMS report, this research aims to investigate, through comparison, the context preferences between a selected group of practicing teachers in South Africa and those of their counterparts in South Korea.

The purpose of this study was to develop a model to predict the college mathematics courses a freshman could expect to pass by considering their high school mathematics preparation. The high school information that was used consisted of the student's sex, the student's grade point average in mathematics, the highest level of high school mathematics courses taken, and the number of mathematics courses taken in high school. The high school sample was drawn from graduated Seniors in the State of Utah for 1979. The college sample was drawn from the fall semester 1980 at Utah State University, Weber State College, University of Utah, Westminster College, and Brigham Young University. The model was developed using ACT Scores as the dependent variable with the high school data in one equation and the college data in another equation and then predicting from high school to college using the ACT Scores as the bridge. The results showed that those students that had courses in the higher levels of mathematics in high school, were significantly more successful! in college mathematics. The level of mathematics was more significant than the grades received in mathematics. Females who had had higher levels of mathematics in high school were as successful! as males on that level.

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This study retrofitted a Diagnostic Classification Model (DCM) known as the Fusion model onto non-diagnostic test data from of the University of Chicago School Mathematics Project (UCSMP) Algebra and Geometry Readiness test post-test used with Transition Mathematics (Third Edition, Field-Trial Version). The test contained 24 multiple-choice middle school math items, and was originally given to 95 advanced 6th grade and 293 7th grade students. The use of these test answers for this study was an attempt to show that by using cognitive diagnostic analysis techniques on test items not constructed for that purpose, highly predictable multidimensional cognitive attribute profiles for each test taker could be obtained. These profiles delineated whether a given test taker was a master or non-master for each attribute measured by the test, thus allowing detailed diagnostic feedback to be disseminated to both the test takers and their teachers. The full version of the non-compensatory Fusion model, specifically, along with the Arpeggio software package, was used to estimate test taker profiles on each of the four cognitive attributes found to be intrinsic to the items on this test, because it handled both slips and guesses by test takers and accounted for residual skills not defined by the four attributes and twenty-four items in the Q-matrix. The attributes, one or more of which was needed to correctly answer an item, were defined as: Skills— those procedures that students should master with fluency; e.g., multiplying positive and negative numbers; Properties—which deal with the principles underlying the mathematics concepts being studied, such as being able to recognize and use the Repeated-Addition Property of Multiplication; Uses—which deal with applications of mathematics in real situations ranging from routine "word problems" to the development and use of mathematical models, like finding unknowns in real situations involving multiplication; and, Representations—which deal with pictures, graphs, or objects that illustrate concepts. Ultimately, a Q-matrix was developed from the rating of four content experts, with the attributes needed to answer each item clearly delineated. A validation of this Q-matrix was obtained from the Fusion model Arpeggio application to the data as test taker profiles showed which attributes were mastered by each test taker and which weren’t. Masters of the attributes needed to be acquired to successfully answer a test item had a proportion-correct difference from non-masters of .44, on average. Regression analysis produced an R-squared of .89 for the prediction of total scores on the test items by the attribute mastery probabilities obtained from the Fusion model with the final Q-matrix. Limitations of the study are discussed, along with reasons for the significance of the study.

Although the Internet is widely used in many areas, its use in school mathematics is at best in its infancy. Studies show that Mathematics teachers have fewer uses for the Internet than teachers of other disciplines. Hence, this research adopted a mixed method approach to investigate what mathematics materials are on the Internet, how teachers are teaching mathematics with the Web and mathematic students' perceptions and engagement with the Internet. This research reviewed the World Wide Web for mathematics materials and found three major groupings of online resources namely interactive resources, non-interactive resources, and communications possibilities. A typology of Web objects was constructed and a database based on a Task-Web object approach was proposed for teacher use. A broad survey was used to elicit information about Internet usage among mathematic teachers. A total of 103 mathematics teachers responded and 15 were interviewed to gain further insight into their usage. Observations of Internet use were also conducted in the classrooms of 4 teachers. The results show that most teachers would like to use the Internet more in their teaching of mathematics but many do not know where and how to do so in an effective way. Statistics, Business Mathematics and Number operations appear to be the more popular topics. Using statistics data from the Web seem to be the Web feature that is most common followed by using the Internet as a resource centre for word problems. Web communications are seldom used. Common constraints teachers face include lack of time, difficulty in planning, lack of knowledge of good Web sites that map to curricula, slow download times, and limited booking times. Students perceive doing activities on the Internet as better than from the textbook because of the amount and variety of information, the better explanations and the change in mode of presentation. Students who have a low comfort level with mathematics wish their teachers would use the Internet. The power of interactive activities on the Internet to engage and motivate these students is due to a variety of reasons such as the element of game play, a change from the routine, its ability to present different conceptual visuals, the independent self paced learning, and quick feedback that came with the use of the Internet. The Internet also enabled students to access difficult to find information and saved them time. The findings also suggest that teachers' persistence in using the Internet could bring about a routine that helps students settle down to the task and stay on task. Teachers' choice and discernment of Web-based activities that are engaging and motivating are paramount to the success of this learning tool. Four Web-based strategies for teaching mathematics were documented and a model of underlying knowledge for teacher practice with the Web was suggested.

Skolmatematiken och matematikdidaktiken har under de senaste årtiondena genomgått en förändring från ett historiskt fokus på ren räkning och utantillkunskaper mot alltmer processorientering. Det pågår en aktiv debatt om hur framgångsrik dagens matematikundervisning egentligen är då de svenska elevernas resultat i internationella jämförelser så som PISA är inte lysande. Historiskt har den svenska matematikundervisningen hämtat influenser från amerikansk matematikdidaktikutveckling. I detta examenarbete görs jämförelse av förespråkad matematikdidaktik vid två olika lärarkurser i Sverige och en lärarkurs på Stanford, USA. Syftet är att undersöka likheter och skillnader i synen på ”god” matematikundervisning på dessa kurser. Som huvudsaklig analysmetod valdes en diskursanalys. De tre olika lärarkurserna ses som tre diskurser. Fyra frågor ställts till respektive diskurs: Vad lyfts fram om matematikdidaktik? Hur talas det omdetta? Vad utesluts eller tonas ner? Vad framställs som god matematik-undervisning? Huvudinriktningen mot en processorienterad matematik är tydlig i alla tre diskurserna. Samtidigt så nämns i diskurserna att ”kunna vissa saker utantill är också viktigt” så det är inte helt entydigt men ändå en tydlig riktning. Alla diskurser tar också upp uppgifternas betydelse för lärandet. Val av uppgifter är en viktig del av matematikdidaktiken. Några skillnader som framkommer är att den amerikanska diskursen lyfter fram betydelsen av mjuka faktorer som attityd, självförtroende, motivation, tilltro, uppmuntran betydligt mer än de två svenska. Sammanfattningsvis visar min analys av de tre diskurserna att den amerikanska diskursen tydligare lyfter fram värderingar och undervisar lärarstudenterna i vad som är god matematikundervisning. God matematikundervisning innefattar många mjuka aspekter som motivation, självförtroende och jämlikhet. Budskapet i de två svenska diskurserna är sakligare och med mer bredd – god matematikundervisning omfattar ett spektrum av förmågor, kunskaper, ämnesområden. Lärarstudenten får ett ”smörgåsbord” och får sedan, på gott och ont, plocka ihop sin egen tallrik av hur matematikundervisningen ska bedrivas. / In recent decades, school mathematics and mathematics education have undergone a change from a historical focus on pure arithmetic and facts knowledge towards an increasingly process orientation. There is an active debate about how successful today's mathematics education really is and the Swedish students' results in international comparisons such as PISA are not brilliant. Historically, Swedish mathematics teaching has taken influences from American mathematics didactic trends. In this thesis, a comparison is made of advocated mathematic education at two different teacher courses in Sweden and a teacher course at Stanford, USA. The purpose is to investigate similarities and differences in the view of “good mathematics education” in these courses. A discourse analysis was chosen as the main analysis method. The three different teacher courses are seen as three discourses. Four questions are asked for each discourse: What mathematics didactics is highlighted? How is this talked about? What is excluded or toned down? What is presented as good mathematics teaching? In all three discourses a clear focus on a process-oriented mathematics is seen. At the same time, it is mentioned in the discourses that "knowing certain things by heart is also important" so it is not completely unambiguous but still a clear direction. All discourses also address the importance of the math problems. Choice of problems and exercises is an important part of mathematics didactics. One difference that emerge is that the American discourse highlights the importance of soft factors such as attitude, self-confidence, motivation, confidence, encouragement significantly more than the other two. In summary, my analysis of the three discourses shows that the American discourse more clearly highlights values and educates student teachers what is good mathematics teaching. Good mathematics education includes many soft aspects such as motivation, self-confidence and equality. The message in the two Swedish discourses is more objective and with more breadth - good mathematics education encompasses a spectrum of abilities, knowledge, subject areas. The teacher student gets a "smorgasbord" and then has to fill his own plate with theories and methods how the mathematics teaching should be conducted.

The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks associated with each manipulative. Qualitative research methods such as observation, teacher interviews, and student focus group interviews were employed. Quantitative data analysis techniques were used to analyze pretest and posttest data of middle school students (n=76). ANCOVA, analysis of covariance, uncovered statistically significant differences in favor of the control group. Differences in posttest scores, triangulated with qualitative data, suggested that concrete and virtual manipulatives require more classroom time because of administrative issues and because of time needed to learn how to operate the manipulative in addition to necessary time to learn mathematics content. Teachers must allow students enough time to develop conceptual understanding linking the manipulatives to the mathematics represented. Additionally, a discussion of unique benefits and drawbacks of each manipulative sheds light on the use of manipulatives in middle school mathematics.