Programmering är ett relativt nytt innehåll i matematikämnet för årskurs 1–3 då det infördes i läroplanen 2018. Därmed finns ett intresse att undersöka hur lärare bedriver programmeringsundervisning. Denna studie genomfördes i syfte att bidra med kunskap om hur lärare i grundskolans tidiga år planerar och genomför undervisning omprogrammering. För att uppnå syftet användes frågeställningar kring hur lärare planerar undervisning om programmering, hur lärare arbetar med programmering i matematikundervisningen, samt vilka kunskaper lärare behöver för att undervisa om programmering. Fem semistrukturerade intervjuer har genomförts med verksamma lärare för att samla in data som analyserats utifrån det teoretiska ramverket Mathematical Knowledge for Teaching (MKT). Studiens resultat sammanställer vilka lärarkunskaper som framkommit inom programmering i relation till kunskapskategorierna knowledge of content and curriculum (KCC), knowledge of content and students (KCS), common content knowledge (CCK) samt knowledge of content and teaching (KCT) inom MKT. Resultatet visar att stegvisa instruktioner är ett centralt programmeringsinnehåll vid lärarnas planering, samt att både analoga och digitala arbetssätt används i matematikundervisningen. Vid analoga arbetssätt kan eleverna programmera varandra som robotar, och vid digitala arbetssätt kan eleverna arbeta i programmet ScratchJr.

I examined the development of three Prospective Secondary Mathematics Teachers' (PSMTs) understandings of connections between concepts in Abstract Algebra and high school Algebra, as well as their use of this understanding while engaging in the teaching practice of noticing students' mathematical thinking. I drew on the theory, Knowledge of Nonlocal Mathematics for Teaching, which suggests that teachers' knowledge of advanced mathematics can become useful for teaching when it first helps reshape their understanding of the content they teach. I examined this reshaping process by investigating how PSMTs extended, deepened, unified, and strengthened their understanding of inverses, identities, and binary operations over time. I investigated how the PSMTs' engagement in a Mathematics for Secondary Teachers course, which covered connections between inverse functions and equation solving and the abstract algebraic structures of groups and rings, supported the reshaping of their understandings. I then explored how the PSMTs used their mathematical knowledge as they engaged in the teaching practice of noticing hypothetical students' mathematical thinking. I investigated the extent to which the PSMTs' noticing skills of attending, interpreting, and deciding how to respond to student thinking developed as their mathematical understandings were reshaped. There were key similarities in how the PSMTs reshaped their knowledge of inverse, identity, and binary operation. The PSMTs all unified the additive identity, multiplicative identity, and identity function as instantiations of the same overarching identity concept. They each deepened their understanding of inverse functions. They all unified additive, multiplicative, and function inverses under the overarching inverse concept. They also strengthened connections between inverse functions, the identity function, and function composition. They all extended the contexts in which their understandings of inverses were situated to include trigonometric functions. These changes were observed across all the cases, but one change in understanding was not observed in each case: one PSMT deepened his understanding of the identity function, whereas the other two had not yet conceptualized the identity function as a function in its own right; rather, they perceived it as x, the output of the composition of inverse functions. The PSMTs had opportunities to develop these understandings in their Mathematics for Secondary Teachers course, in which the instructor led the students to reason about the inverse and identity group axioms and reflect on the structure of additive, multiplicative, and compositional inverses and identities. The course also covered the use of inverses, identities, and binary operations used while performing cancellation in the context of equation solving. The PSMTs' noticing skills improved as their mathematical knowledge was reshaped. The PSMTs' reshaped understandings supported them paying more attention to the properties and strategies evident in a hypothetical student's work and know which details were relevant to attend to. The PSMTs' reshaped understandings helped them more accurately interpret a hypothetical student's understanding of the properties, structures, and operations used in equation solving and problems about inverse functions. Their reshaped understandings also helped them give more accurate and appropriate suggestions for responding to a hypothetical student in ways that would build on and improve the student's understanding. / Doctor of Philosophy / Once future mathematics teachers learn about how advanced mathematics content is related to high school algebra content, they can better understand the algebra content they may teach. The future teachers in this study took a Mathematics for Secondary Teachers course during their senior year of college. This course gave them opportunities to make connections between advanced mathematics and high school mathematics. After this course, they better understood the mathematical properties that people use while equation solving, and they improved their teaching practice of making sense of high school students' mathematical thinking about inverses and equation solving. Overall, making connections between the advanced mathematics content they learned during college and the algebra content related to inverses and equation solving that they teach in high school helped them improve their teaching practice.

In the fall of 2018, programming was implemented in the swedish curriculum and then became a new element in mathematics education for grades 1-3. Teachers who took their degree before the implementation, lacks education in programming and there is interest in finding out how teaching about programming is conducted since it became part of the curriculum. The purpose of this study was to contribute with knowledge about how teachers have implemented programming in their teaching even though they lack education in it. Four semi-structured interviews have been conducted where the data from the interviews has been analyzed from Mathematical Knowledge for Teaching. The result shows that teachers without education in programming find it difficult to plan instruction in programming by themselves. In the teaching of programming the teachers focus on the central concepts in programming and that the programming should interest the students. It was difficult for teachers to assess the students in programming and the only assessment that teachers make is the formative assessment. / Hösten 2018 implementerades programmering i den svenska läroplanen och blev då ett nytt moment inom matematikundervisningen för årskurs 1-3. Lärare som innan detta tog sin lärarexamen saknar utbildning inom programmering och det finns intresse att ta reda på hur undervisningen kring programmering bedrivs sedan det blev en del av läroplanen. Syftet med denna studie var att bidra med kunskap om hur lärare har implementerat programmering i sin undervisning trots att de saknar utbildning inom det. Fyra stycken semistrukturerade intervjuer har gjorts där datan från intervjuerna har analyserats utifrån Mathematical Knowledge for Teaching. Resultatet visar på att lärare utan utbildning inom programmering har svårigheter att på egen hand planera undervisning i programmering. Under genomförandet av undervisningen fokuserar lärarna på att befästa centrala begrepp inom programmering och att väcka ett intresse hos eleverna. Det upplevdes svårt för lärarna att bedöma eleverna inom programmering och den enda bedömning som lärarna gör är den formativa bedömningen.

In the fall of 2018, programming was implemented in the swedish curriculum and then became a new element in mathematics education for grades 1-3. Teachers who took their degree before the implementation, lacks education in programming and there is interest in finding out how teaching about programming is conducted since it became part of the curriculum. The purpose of this study was to contribute with knowledge about how teachers have implemented programming in their teaching even though they lack education in it. Four semi-structured interviews have been conducted where the data from the interviews has been analyzed from Mathematical Knowledge for Teaching. The result shows that teachers without education in programming find it difficult to plan instruction in programming by themselves. In the teaching of programming the teachers focus on the central concepts in programming and that the programming should interest the students. It was difficult for teachers to assess the students in programming and the only assessment that teachers make is the formative assessment. / Sammanfattning Hösten 2018 implementerades programmering i den svenska läroplanen och blev då ett nytt moment inom matematikundervisningen för årskurs 1-3. Lärare som innan detta tog sin lärarexamen saknar utbildning inom programmering och det finns intresse att ta reda på hur undervisningen kring programmering bedrivs sedan det blev en del av läroplanen. Syftet med denna studie var att bidra med kunskap om hur lärare har implementerat programmering i sin undervisning trots att de saknar utbildning inom det. Fyra stycken semistrukturerade intervjuer har gjorts där datan från intervjuerna har analyserats utifrån Mathematical Knowledge for Teaching. Resultatet visar på att lärare utan utbildning inom programmering har svårigheter att på egen hand planera undervisning i programmering. Under genomförandet av undervisningen fokuserar lärarna på att befästa centrala begrepp inom programmering och att väcka ett intresse hos eleverna. Det upplevdes svårt för lärarna att bedöma eleverna inom programmering och den enda bedömning som lärarna gör är den formativa bedömningen.

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Doctor of Philosophy / Curriculum and Instruction Programs / David S. Allen / This study examined pedagogical content knowledge (PCK) and Mathematics knowledge for teaching (MKT) from a perspective that blends existing definitions, questions, and methodologies into a unique method of collecting and analyzing data. Many studies on MKT have been done using qualitative methodology, usually with the researcher interviewing or testing individual teachers and analyzing that data using qualitative methods. A smaller number of studies have attempted to measure MKT using a quantitative approach, often times involving paper pencil tests with multiple choice and some open ended questions. Current research is also heavily weighted towards pre-service elementary teachers in the area of MKT for division by fractions (Depaepe et al, 2013). Although it may be true many pre-service elementary teachers have difficulties with division by fractions (Li & Kulm, 2008; Ma, 1999; Tirosh, 2000), we do not know if these problems persist for in-service elementary teachers or with secondary teachers at any stage. This study used a survey created by the researcher as a synthesis of existing questions from research on MKT with regards to division by fractions. This survey was delivered through an online format. The qualitative data in this research was then coded into quantitative data using a rubric developed by the researcher producing MKT scores that could be analyzed using statistical methods and generalized to a larger population. This study examined the Mathematical Knowledge for Teaching held by elementary, middle, and secondary in-service teachers in the domain of division by fractions. In particular this research asked if there was a relationship between the educational background, training, and experience of teachers and their MKT. Multiple analysis including independent t-tests, independent one-way ANOVAs, and Multiple regression analyses revealed that Middle school teachers and teaching at a middle school were significant predictors of increased MKT scores when compared with other groups of teachers. The type of teaching license, the type of degree held, total experience, and grade level experience were all found to have no significant relationship to MKT scores.

nÃo hà / Este trabalho pretendeu registrar prÃticas educativas da capoeira cearense a partir do inÃcio do sÃculo XXI, perÃodo em que a capoeira obteve maior prestÃgio polÃtico no Brasil e no mundo em virtude das polÃticas patrimonialistas vigentes. A imersÃo nos saberes e modos de ensinar a capoeira foi realizada atravÃs de uma anÃlise sobre as prÃticas educativas do Centro Cultural Ãgua de Beber - CECAB, ou seja, um Estudo de Caso. Portanto, trata-se de uma pesquisa qualitativa intensiva em documentos e relatos orais, cuja relevÃncia consistiu em responder ao seguinte questionamento: Do que à constituÃdo o Campo de Saberes da Capoeira na atualidade? O processo de coleta de dados se desenvolveu a partir do registro histÃrico de aspectos dos principais projetos empreendidos pelo grupo que, alÃm da prÃtica ritualÃstica da Capoeira, tambÃm promove cursos de qualificaÃÃo, eventos e espetÃculos, em uma perspectiva de pesquisar a diversidade cultural de um modo geral e da Capoeira. Entre os resultados obtidos, destacamos que o Campo de Saberes da Capoeira assume relaÃÃes com campos cientÃficos tradicionais da educaÃÃo formal e com os saberes populares tradicionais. Contudo, nÃo se refere a um campo eminentemente disciplinar com estruturas rÃgidas e imutÃveis, mas sim um campo livre de atuaÃÃo de educadores carismÃticos e com perfil de lideranÃa. à um campo em que alguns saberes sÃo postos em evidÃncia de acordo com os interesses coletivos de determinado perÃodo e pela motivaÃÃo oriunda das prÃticas corporais, oferecendo subsÃdios para propostas de ensino ativo e crÃtico. Portanto, concluÃmos que os projetos investigados sÃo aÃÃes educativas coletivas que, atravÃs de recursos pÃblicos e privados, perpetuam conhecimentos sobre valores e elementos culturais africanos e indÃgenas, numa perspectiva multicultural. / This work sought to know pedagogical practices of capoeira from the beginning of the XXI century, capoeira period gained greater political prestige in Brazil and the world due to patrimonialist policies. The methodology used was the Case Study, that is, an intensive qualitative research in documents and oral reports of a certain group of capoeira of the city of Fortaleza, the Cultural Center Capoeira Water of Drink- CECAB. The relevance of this study is to contribute with knowledge about the Field of Knowledge that constitutes this socioeducational space in constant expansion. The data collection process developed from the historical record of aspects of the main projects undertaken by the group which, in addition to the ritualistic practice of Capoeira, also promotes qualification courses, events and shows with a view to researching cultural diversity in a general way and Capoeira. Among the results obtained, we emphasize that the Capoeira Field of Knowledge assumes relations with traditional scientific fields of formal education and with traditional popular knowledge. However, it does not refer to an eminently disciplinary field with rigid and cast structures, but rather a free field of action of charismatic educators with a leadership profile. It is a field in which some knowledges are highlighted according to the collective interests of a given period and the motivation derived from corporal practices, offering subsidies for active and critical teaching proposals. Therefore, we conclude that the projects investigated are collective pedagogical actions that, through public and private resources, perpetuate knowledge about African and indigenous cultural values and elements, from a multicultural perspective.

This study investigated how student teachers used their mathematical knowledge for teaching and pedagogical knowledge to design and modify mathematical tasks. It also examined the relationship between teacher knowledge and the cognitive demands of a task. The study relied heavily on the framework in Hill, Ball, and Shilling (2008), which describes the different domains of knowledge in mathematical knowledge for teaching, and the framework on the cognitive demands of mathematical tasks in Stein, Smith, Henningsen, and Silver (2000). Results of the study indicated that the student teachers used their common content knowledge when they lacked sufficient knowledge in other domains, especially specialized content knowledge, to perform a particular job of teaching. There was often a decrease in the cognitive demands of a task when it was modified by the student teachers. These drops were often associated with a lack of specialized content knowledge.

This thesis concerns teaching in mathematics teacher education and is based on the implementation of a learning study at teacher training. The overall purpose was to investigate in what way teacher training could facilitate and improve student teachers’ Mathematical Knowledge for Teaching (MKT). In the learning study design, MKT was conceptualized as an object of learning with a meta-character, which meant that it was applicable to and transferable between different content areas of mathematics. This made it possible to vary the mathematical content between lessons but to keep the object of learning constant. Four critical features of the object of learning were found, giving insight in some of the problems related to teacher education. Student teachers had to be able to formulate proper aims for a lesson and to give detailed descriptions of elements of MKT for coherence in their MKT to occur. A focus on student teachers’ role as mathematics teachers had to be established and finally, sufficient mathematical knowledge was found to be a prerequisite for their MKT to develop. The study shows that enactment of these critical features improved the teaching by the teacher educators, which in its turn improved the student teachers’ learning with regard to MKT. The study also indicates that the prescribed design is worth considering for future collaborative efforts of improving teaching where other objects of learning with a similar meta-character are involved.

Knowledge of curriculum is a significant component of mathematical knowledge for teaching. However, clearly understanding knowledge of curriculum requires further refinement and substantial research. This study consists of three papers that aim to explore prospective and practicing middle school teachers’ Knowledge of Curriculum for Teaching Simple Algebraic Equations (KCTE). The first paper reviews trends in and the evolution of standards and policies and synthesizes significant findings of research on mathematics curriculum and Knowledge of Curriculum for Mathematics Teaching (KCMT). Through this synthesis, the paper examines policy changes and research relevant to mathematics curriculum and KCMT and anticipates future research approaches and topics that show promise. Building on the context provided by the first paper, the following two papers investigate KCTE from the perspectives of prospective and practicing middle school mathematics teachers. For the second paper, data was collected from a convenience sample of 58 prospective middle school mathematics teachers and a subsample of six participants. The findings of this study identify patterns of key mathematical topics in the teaching sequence of simple algebraic equations, compare the participants’ sequences with experts’, reveal participants’ orientations toward KCTE, draw connections between participants’ KCTE and their knowledge of content and teaching, and establish relationships between participants’ KCTE and their knowledge of content and students. Four middle school mathematics teachers participated in the third study. The results indicate that state-level intended curriculum is the most prevailing component of participants’ KCTE. Furthermore, from a vertical view of curriculum, participants’ awareness of their students’ lack of basic mathematical knowledge impacted their KCTE. The paper also identifies the role of the state-level intended curriculum in participants’ KCTE, alternative approaches to curriculum implementation that participants used to respond to the multiple intelligences of their students, and the participants’ lack of lateral curriculum knowledge in KCTE. Together, these three papers offer a closer look at KCMT with a focus on simple algebraic equations. This research broadens our understanding of prospective and practicing middle school teachers’ KCMT and discusses implications for professional development.