This study examined the effects of a learning environment (embodying many of De Corte et al.’s, (2004) CLIA-model components) on secondary students’ mathematical beliefs. Such mathematical beliefs have been of interest to the research community due to their expected impact on students’ willingness to engage in mathematical problem-solving. This research adopted an action research methodology using a quasi-experimental sequential explanatory mixed methods design. Data was collected using the Mathematics Related Beliefs Questionnaire (MRBQ) and a number of focus groups and individual interviews were undertaken. The sample selected (age 13-14) was from a population of convenience. There was one treatment class (N=22) and three control classes (N=45). The classroom intervention was of six months duration and was carried out by the researcher teacher in a secondary community school. Findings revealed no significant positive effects on students’ beliefs from the new learning environment about the teacher’s role in the classroom, their personal competence and the relevance to their lives and mathematics as an inaccessible subject. A more negative outcome for the fourth factor of the MRBQ scale, ‘mathematics as an inaccessible subject’, resulted for all participants (experimental and control combined) with a moderate effect of eta2=0.09. Findings from the qualitative data indicated the experimental participants found mathematics to be a difficult but useful subject. Findings, overall, revealed no significant differences between the experimental and control classes, indicating the new learning environment had not had a positive impact on the beliefs examined. Possible factors identified were the length of the intervention, the ages of participants and the socio-economic status of the majority taking part in this study. Qualitative data also indicated participants in the treatment class had found some of the activities used in the intervention to be interesting and enjoyable. Responses to the use of group work indicated participants were both willing and able to enter into communities of learners. Other results showed that participants with the highest achievement scores appeared to be the most confident learners of mathematics. Participants appeared to accept the need to have patience and perseverance when solving difficult problems but this was not translated into action in the classroom. The importance of understanding mathematics appeared to be accepted by participants. Implications for methodology, research and practice are discussed in light of these findings.

Students learn many different concepts throughout their mathematical careers. In order to be successful in mathematics, students should be able to transfer these mathematical concepts learned in one situation and apply them to a new situation. There are many factors that might affect how students transfer knowledge; however research has focused mostly upon cognitive factors, even though affective factors might also exhibit a strong influence. This study examined how students' mathematical beliefs, specifically beliefs about mathematics education and self-efficacy, affect the transfer process. Data were collected from three middle school students. These data were collected through in-class observations, students' written work, and student interviews. The analysis of this data has shown that students' beliefs about mathematics education can affect specific steps in the transfer process as well as the transfer process in general. The data also showed that self-efficacy can influence the transfer process, but that this influence is mediated by the students' other mathematical beliefs.

數學能力向來被視為解決問題能力的重要基礎。隨著知識經濟時代的來臨，數學愈加受到重視，數學教師的教學效能亦面臨新的挑戰。為了協助師培生發展更具創意與多元的教學方式，本研究在中學數學科教材教法之課堂中，提供學生以知識翻新（knowledge-building）(Scardamalia, 2002) 教育理念為核心的課程設計，並使用知識論壇(Knowledge Forum™)數位學習平台作為本課程的線上輔助學習環境。研究主要目的在利用教室中與論壇上的知識翻新活動以提昇師培生對數學相關信念之反思與瞭解。研究對象為9位師培生。本研究透過混合研究設計以蒐集資料，資料來源包括：（1）整學期師培生在「知識論壇」平台上之討論與貼文；（2）期初與期末的數學信念開放式問卷；與（3）師培生的試教。資料分析方式如下：(1)平台上的貼文主要使用知識論壇分析工具(Analytic Toolkit for Forum)探討師培生在平台上的互動與知識翻新歷程；（2）數學信念開放式問卷主要以兩種觀點進行內容分析並交叉檢證；與（3）試教過程(錄影檔)以開放式問卷所產生之編碼進行影片內容的分析。 研究結果發現：（1）以知識翻新理論為原則之課程設計有助於師培生發展更建構取向之數學信念；（2）使用知識論壇平台有助於師培生分享知識與自我反思；（3）數學信念與教學實踐呈現相互影響的關係；以及（4）透過知識翻新的課程設計，師培生逐漸理解多元與彈性的教學對學生學習的重要性。本研究根據研究結果提出以下幾點建議以供未來師培教育之參考：（1）數學師資培育必須將專業學術知識的學習與成熟信念的養成作更密切的銜接；（2）師培機構進行課程設計時，應提供更多元開放的教學方式來幫助師培生學習；（3）師培教育應鼓勵師培生發展更能適應未來學生學習之教學方法。 / The purpose of this study was to investigate the impact of knowledge building pedagogy and technology on preservice teachers’ views on the nature of, and teaching practices in, mathematics. Participants were nine preservice teachers who took a university course titled “High-School Mathematics Teaching.” A software program called Knowledge Forum was employed to enable an online knowledge building environment for preservice teachers to explore, reflect, and discuss about the nature of, and teaching practices in, mathematics. Data mainly came from three sources: (1) preservice teachers’ online posting and discussion recorded in a KF database, (2) a survey about mathematical beliefs with eight open-ended questions, and (3) preservice teachers’ teaching practice videos. To analyze, (1) online discussion records automatically recorded in Knowledge Forum were analyzed using descriptive statistics; (2) open-ended survey questions were content-analyzed following an open-coding procedure; and (3) two cases of preservice teachers’ teaching practice were further explored by employing video analysis techniques. The main findings were as follows: (1) engaging in knowledge building was found to help preservice teachers develop more constructivist-oriented mathematics beliefs; (2) use of Knowledge Forum was helpful for preservice teachers in sharing their knowledge and reflecting on their teaching; (3) mathematics beliefs and teaching practice could mutually influence each other; (4) after engaging in knowledge building for a whole semester, preservice teachers were able to realize the importance of capitalizing on more diversified ways of teaching to enhance student learning. Building on the results, this study made the following three suggestions: (1) teacher education program should pay more attention to integrating the learning of pedagogical content knowledge into the development of more mature mathematical beliefs among preservice teachers; (2) teacher education program should encourage more diversified and flexible ways of teaching practice when designing its courses; and (3) teacher education program should encourage preservice teachers to develop more adaptive teaching practices to help students learn in the future.

I det svenska undervisningslandskapet har extraundervisning, från både privata och offentliga aktörer, under senaste decenniet fått en mer och mer betydelsefull och naturlig plats. Där både ideella och vinstdrivande organisationer under de senaste årtionden erhållit statliga subventioner eller bidrag. Det blir därför viktigt att skapa en förståelse för vilken typ av matematikundervisning som ges av dessa organisationer. Av särskilt intresse för denna studie är de ideella organisationerna, och dess volontärer, som erbjuder kostnadsfri undervisning i form av räknestugor i matematik för elever från gymnasieskolan. Tidigare forskning om de svenska ideella räknestugornas undervisning och dess volontärer lyser med sin frånvaro. Syftet med detta arbete är att skapa förståelse för verksamheten inom de ideella räknestugorna, dess volontärer och vilken matematikundervisning som där ges. För att undersöka volontärernas undervisning så genomfördes dels intervjuer med volontärerna, om deras föreställning om matematik som ämne och om matematikundervisning, dels observationer av volontärernas undervisning inom räknestugans verksamhet. Analys och tolkning av observationer och intervjuer tar utgångspunkt i tidigare forskning om lärares föreställning om matematik, lärande och dess undervisning. De intervjuade och observerade volontärerna gav en samstämmig bild, där föreställningen av matematik som kom till uttryck i intervjuer stämde väl överens med vad som observerades i undervisning. Av den tidigare forskningens tre föreställningar av matematiken gav de intervjuade och observerade volontärerna uttryck för två av de tre föreställningarna. Matematik som verktyg var ett förhållningssätt som uttrycktes av samtliga volontärer i både intervju och undervisning. Perspektivet matematik som problemlösning uttrycktes inte av någon av volontärerna i undervisning, och hade en tämligen tillbakaskjuten roll i den föreställning om matematik som uttrycktes i intervjuer. Slutsatsen är att utifrån vilka föreställningar av matematik som synliggörs inom undervisning kan räknestugor medföra vissa begränsningar sedda som undervisningsform, då den observerade undervisningen primärt gav uttryck för en föreställning om matematik som verktyg. Medan de platonska synsätten förekom till viss del och föreställningen av matematik som problemlösning ej kom till utryck i undervisningen. / In the Swedish educational landscape, supplementary tutoring, from both private and public organizations, has in the last decade gained a more and more significant and natural place. Where both non-profit and for-profit organizations have received government subsidies or grants in recent decades. It is therefore of important to gather an understanding of the type of mathematics education provided by these organizations. Of particular interest for this study are the non-profit organizations, and its volunteers, offering free tutoring in mathematics for student in the upper secondary school.Previous research on the non-profit supplementary tutoring in Sweden, and its volunteers, is lacking. The purpose of this work is to create an understanding of the activities in the non-profit supplementary tutoring, its volunteers and the mathematics teaching that is there provided. In order to investigate the volunteer’s educational practice, both interviews with the volunteers regarding their beliefs about mathematics as a school subject and their beliefs about mathematical education where conducted, and also on observations of the volunteer’s educational practice where done. Analysis and interpretation of observations and interviews are based on previous research about teachers’ beliefs of mathematics, learning and its teaching. The volunteers gave a coherent picture, where the beliefs of mathematics expressed in interviews was in agreement with what was observed in teaching practice. Of the previous research’s three beliefs of mathematics, the interviewed and observed volunteers, expressed two of the three beliefs. The Instrumentalists beliefs, mathematics as a tool, was expressed by all volunteers in both interview and teaching. The perspective of mathematics as problem solving was not expressed by any of the volunteers in their tutoring practice, and was almost invisible in the beliefs of mathematics expressed in interviews. The conclusion is that, from the beliefs of mathematics expressed in tutoring practice, the supplementary tutoring can be seen to have certain limitations as a form of education, since the observed practice primarily expressed the belief of mathematics as a tool. The platonic perspective appeared to some extent and the notion of mathematics as problem solving was not expressed in teaching.