The overall purpose of this concurrent mixed methods study was to develop an online performance assessment using content questions taken from a reputable seventh and eighth grade standardized assessment that effectively evaluated and allowed preservice middle grades mathematics teachers to demonstrate their mathematics knowledge for teaching in the four main content strands of algebra, probability and statistics, geometry, and number and operations. In addition, this study examined differences in mathematics knowledge for teaching in enrollment characteristics, in courses taken and currently taking, and in three different cohorts, each at different stages in the program, of 122 preservice middle grades mathematics teachers at a large public university in central Texas. Constant comparative analysis and descriptive statistics revealed average scores on the seventh and eighth grade content questions. The middle grades preservice teachers’ content understanding and pedagogical understanding responses indicated several misunderstandings and misinterpretations in the middle grades mathematics they were tested on. Content knowledge, content understanding, and pedagogical understanding together made up a preservice teacher’s mathematics knowledge for teaching. The study revealed that although preservice middle grades teachers could answer a content question correctly; they did not necessarily understand the process they used to arrive at their answer. In addition, their lack of explanation and knowledge of how to complete the problem correctly was transferred to their pedagogical understanding of the same problem. There was a general indication of increasing mathematics knowledge for teaching for each content strand across enrollment characteristics (freshmen, sophomore, etc.) and cohorts. However, there was a noticeable decrease in average mathematics knowledge for teaching scores during middle grades preservice teachers’ junior year. Special integrated mathematics and pedagogy courses (MASC) and the middle grades methods course had the greatest affect on preservice teachers’ mathematics knowledge for teaching each content strand scores. Recommendations are also included in the study which may be used to help shape reform initiatives in teacher education programs throughout the United States.

Lärare planerar undervisning och väljer vad eleverna ska arbeta med. Denna studie syftar till att undersöka lärares tankar om vilket matematiskt innehåll elever möter samt vilka förmågor elever tränar i arbetet med en uppgift i matematik. Studien genomfördes genom intervjuer med fem lärare som fick svara på frågor kring en uppgift i matematik. De övergripande frågorna handlade om matematiskt innehåll och förmågor, men även uppgiftens koppling till Lgr11 efterfrågades. Lärarnas svar kopplades till de specifika kunskaper som behövs för att undervisa i matematik. Studien visade att lärarna i hög utsträckning sorterade in uppgiften inom området kombinatorik och att de framförallt såg uppgiften som en problemlösningsuppgift. När lärarna diskuterade det matematiska innehållet var det ingen som framhöll kombinatorikens olika generella formler och heller ingen som angav kombinatoriken som en grund för att förstå begreppet sannolikhet. Lärarna motiverade främst användandet av uppgiften med de i kursplanen beskrivna långsiktiga målen, förmågorna, men frågetecken kring hur lärarna tolkar dessa förmågor dök upp under studiens gång. I analysen av uppgiften visade lärarna prov på flera aspekter av Mathematical knowledge for teaching,  även om studien också visade att de specifika kunskaper som krävs för undervisning i matematik kan utvecklas

Although studies of teacher preparation programs have documented positive changes in mathematical knowledge for teaching with preservice teachers in mathematics content courses, this study focused on the impact of a mathematics methods course and follow-up student teaching assignment. The presumption was that preservice teachers would show growth in their mathematical knowledge during methods since the course was structured around active participation in mathematics, research-based pedagogy, and was concurrent with a two-day-per-week field experience in a local elementary school. Survey instruments utilized the computer adaptive test version of the Mathematical Knowledge for Teaching (MKT) measures from the Learning Mathematics for Teaching Project, and the Attitudes and Beliefs (towards mathematics) survey from the Mathematical Education of Elementary Teachers Project. A piecewise growth model analysis was conducted on data collected from 176 participants at 5 time-points (methods, 3 time-points; student teaching, 2 time-points) over a 9 month period. Although the participants' demographics were typical of U.S. undergraduate preservice teachers, findings suggest that initial low-level of mathematical knowledge, and a deep-rooted belief that there is only one way to solve mathematics problems, limited the impact of the methods and student teaching courses. The results from this study indicate that in (a) number sense, there was no significant change during methods (p = .392), but a significant decrease during student teaching (p < .001), and in (b) algebraic thinking, there was a significant decrease during methods (p < .001), but no significant change during student teaching (p = .653). Recommendations include that the minimum teacher preparation program entry requirements for mathematical knowledge be raised and that new teachers participate in continued professional development emphasizing both mathematical content knowledge and reform-based pedagogy to continue to peel away deep-rooted beliefs towards mathematics.

Mathematical argumentation is fundamental to doing mathematics and developing new knowledge. Working from the view that mathematical argumentation is also integral to teaching and learning mathematics, this study investigated teachers’ use of mathematical knowledge for teaching (MKT) to support student participation in mathematical argumentation. Classroom observations were made of three case-study teachers’ implementation of a three-day curriculum unit on mathematical argumentation and supplemented with paper and pencil assessments of teachers’ MKT. Teaching moves, or teachers’ actions directed toward supporting argumentation, were identified as a unit of discourse in which MKT-in-action appeared. Teachers’ MKT showed up in three types of teaching moves including: Revoicing by Reformulation, Responding to Student Difficulties, and Pressing for Generalization in Defining. MKT that was evident in these moves included knowledge of core information in argument, heuristic methods, and vii formulation of mathematical definition through and in argumentation. Findings highlight that supporting mathematical argumentation requires teachers to have a sophisticated understanding of the subject matter as well as how concepts develop through argumentation. Findings have limitations in understanding complex teaching practices by considering MKT as a single factor. The study has implications on teacher learning and MKT assessments. / text

Syftet med föreliggande studie är att beskriva hur det matematiska innehållet beskrivs och förmedlas av lärare vid introduktion av matematiklektioner. Observationer av åtta grundskolelärare har genomförts. Balls, Thames och Phelps (2008) modell för MKT har använts för att sortera och analysera studiens empiri. Resultatet visar att lärarna kombinerar både ämnes- och pedagogiska kunskaper vid introduktion av matematiklektioner. Dock missar samtliga lärare att förklara och förmedla målet med matematiklektionen till eleverna. Resultatet diskuteras utifrån ett pragmatiskt perspektiv och en av slutsatserna som dras är att modellen för MKT kan användas av lärare som underlag vid planering och genomförande av matematiklektioner.

This thesis is an investigation into conceptions of ‘understanding mathematics in depth’, as articulated by two specific groups of novice secondary mathematics teachers in the UK. Most participants in the sample interviewed have completed one of two government funded mathematics subject knowledge enhancement courses, which were devised with an aim of strengthening students’ understanding of fundamental mathematics. Qualitative data was drawn from semi-structured interviews with 21 subjects and more in-depth case studies of two of the sample. The data reveals some key themes common to both groups, and also some clear differences. The data also brings to light some new emergent theory which is particularly relevant in novice teachers’ contexts. To provide background context to this study, quantitative data on pre-service mathematics Postgraduate Certificate in Education (PGCE) students is also presented, and it is shown that, at the university in the study, there is no relationship between degree classification on entry to PGCE, and effectiveness as a teacher as measured on exit from the course. The data also shows that there are no significant differences in subject knowledge and overall performance on exit from PGCE, between students who have previously followed a subject knowledge enhancement course, and those who have followed more traditional degree routes.

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.