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

A longstanding problem in mathematics education has been to determine the knowledge that teachers need in order to teach mathematics effectively. It is generally agreed that teachers need a more advanced knowledge of the mathematical content that they are teaching. That is, teachers must know more about the content that they are teaching than their students and also know more than simply how to "do the math" at a particular grade level. At the same time, research does not clearly indicate what advanced mathematical knowledge (AMK) is useful in teaching or how it can be developed and identified in teachers. In particular, the potential AMK that is useful for teaching is too vast to be enumerated and may involve a great deal of tacit knowledge, which might be difficult to detect through observations of practice alone. In the last decade, researchers have identified that teaching practice entails a specialized knowledge of mathematics but the role of advanced mathematical knowledge in teaching practice remains unclear. However, the construct of horizon content knowledge (HCK) has emerged in the literature as a promising tool for characterizing AMK as it relates specifically to teaching practice. I propose an operationalization of HCK and then use that as a lens for analyzing the knowledge resources that a fourth and fifth grade teacher draw on in their encounters with the mathematical horizon. The analysis identifies what factors contribute to teachers' encounters with the horizon, characterizes the knowledge resources, or HCK, that teachers draw on to make sense of mathematics they engage with during their horizon encounters, and explores how HCK affords and constrains teachers' ability to navigate mathematical territory. My findings suggest that experienced teachers' HCK includes a situated, professional teaching knowledge that, while sometimes non-mathematical in nature, informs their understanding of mathematical content and teaching decisions. This professional teaching knowledge guides how teachers use and generate mathematical structures that sometimes align with established mathematical structures and in other cases do not. These findings have implications regarding the way in which the development of AMK is approached relative to teacher education, ongoing professional development, and curriculum design.

Submitted by Isaac Francisco de Souza Dias (isaac.souzadias@ufpe.br) on 2016-02-29T18:35:41Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) TESE Giselda Magalhaes Moreno Nobrega.pdf: 801122 bytes, checksum: d01aa5909fdb53816240291d133f546c (MD5) / Made available in DSpace on 2016-02-29T18:35:41Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) TESE Giselda Magalhaes Moreno Nobrega.pdf: 801122 bytes, checksum: d01aa5909fdb53816240291d133f546c (MD5) Previous issue date: 2015-02-11 / CNPQ / O pensamento sobre o possível (e consequentemente sobre o impossível e a certeza) configura-se enquanto raciocínio abstrato, sendo esse um dos aspectos que caracteriza o desenvolvimento cognitivo do sujeito. Isso porque conjecturar acerca de possíveis exige habilidades hipotético-dedutivas, que permitem ao sujeito pensar sobre situações que não constituem uma realidade imediata. Apesar da importância do tema, há na literatura uma escassez de estudos acerca da concepção de possível. O estudo encontrado data de 1985, e consiste em um conjunto de experimentos realizados por Piaget e colaboradores. Na visão piagetiana, as noções sobre o possível começam a emergir por volta dos sete, oito anos. Porém, estudos recentes no campo da Psicologia da Educação Matemática têm mostrado que crianças de cinco anos já são capazes de conjecturar sobre o possível em situações que envolvem noções iniciais de probabilidade e análise combinatória. Diante disso, este estudo teve por objetivo investigar a concepção do possível em crianças no âmbito do conhecimento matemático e não-matemático. Participaram deste estudo 180 crianças de ambos os sexos, alunas de duas escolas particulares da cidade de Recife, igualmente divididas em seis grupos de participantes em função da escolaridade, que variava do Infantil III ao 5º ano. Cada participante respondeu a um conjunto de 36 perguntas acerca de ocorrência de três tipos de situações: possíveis, impossíveis e certas de acontecer – tanto no domínio de conhecimento matemático como não-matemático. Especificamente no domínio da matemática, as questões eram referentes a probabilidade e análise combinatória. Os dados foram analisados em função do desempenho (acertos) e das justificativas dadas pelas crianças. Os resultados mostraram que aos cinco anos as crianças já são capazes de pensar sobre a possibilidade (ou não) de ocorrência de situações hipotéticas a elas apresentadas. Essa concepção de possível se desenvolve ao longo do tempo, de modo que as crianças mais velhas não só apresentam um melhor desempenho como também se tornam mais capazes de justificar suas respostas de maneira fundamentada. Os dados mostraram também que com exceção das crianças do Infantil III (que evidenciam um desempenho superior nas perguntas de conhecimento não-matemático), não houve diferença significativa de desempenho entre as perguntas de conhecimento matemático e não-matemático. Tal fato sugere que desde muito cedo as crianças já se mostram aptas a conjecturar sobre o possível em diferentes contextos. No que se refere aos tipos de questões (possibilidade, impossibilidade e certeza), constatou-se que as perguntas do tipo possibilidade foram mais facilmente respondidas no âmbito do conhecimento matemático do que no âmbito do conhecimento não-matemático. Já as perguntas do tipo impossibilidade mostraram-se mais fáceis no conhecimento não-matemático. Nas questões do tipo certeza o desempenho das crianças foi semelhante nesses dois domínios de conhecimento. O fato das crianças apresentarem facilidade em pensar sobre as possibilidades no âmbito do conhecimento matemático abre caminhos para aprofundar a abordagem dos conteúdos de probabilidade e análise combinatória no contexto escolar. Se crianças a partir do 3º ano do Ensino Fundamental já demonstram ter o entendimento de situações probabilistas, talvez seja o momento de aprofundar mais as noções de probabilidade trabalhadas em sala de aula. A partir do 3º ano as crianças apresentaram um desempenho significativamente melhor em probabilidade do que em combinatória, sugerindo a existência de um “freio” no desenvolvimento deste conceito, que é essencialmente escolar. Ao que parece, os conteúdos de análise combinatória ou não estão sendo trabalhados nas series iniciais, ou a maneira como se conduz esse trabalho não permite que as crianças se apropriem e desenvolvam o seu raciocínio acerca dos princípios da combinatória. / The thought of the possible (and consequently of the impossible, as well as “ertainty) is configured in the realm of abstract reasoning, being one of the aspects to characterize the cognitive development of a subject. That can be justified by the fact that conjectures about possibilities and outcomes, requires hypothetical-deductive skills that allow the subject to think about situations that are not an immediate reality. Despite the importance of the issue, there is a shortage, in the literature, of studies on the conception of possible. The study found dates back to 1985 and consists of a set of experiments conducted by Piaget and colleagues. According to their view, the notions about the possible begin to emerge when the individual is about seven or eight years old. However, recent studies in the field of Psychology of Mathematics Education have shown that children under five years old are already able to conjecture about the possible in situations involving basics of probability and combinatory analysis. This study aimed to investigate the designation of possible in children under the mathematical as well as non-mathematical knowledge. The study included 180 children of both sexes, two private school students in the city of Recife, equally divided into six groups of participants according to level of education, ranging from Childhood III to 5th Grade. Each participant answered a set of 36 questions about occurrence of three types of situations: possible, impossible and certain to happen - both in the mathematical and non-mathematical knowledge domains. Specifically in the mathematical domain, the questions concerned probability and combinatory analysis. Data were analyzed in terms of performance (right/wrong) and the justifications given by subjects. The results showed that at the age of five, children are capable of thinking about the possibility (or not) of the occurrence of hypothetical situations presented to them. This design can develop over time, in the sense that older children not only incur in a better performance including well fundamented justifications to their responses. The data also showed that with the exception of children from Childhood III (that show superior performance in non-mathematical knowledge questions), there was no significant difference in performance between the mathematical and non-mathematical knowledge questions. This suggests that at early stages in life, children already show the ability to conjecture about the possible in different contexts. With regard to the types of questions (possibility and impossibility and certainty), it was found that the questions of the type possibility were more easily answered in the context of the mathematical knowledge that under the non-mathematical knowledge. Nonetheless the questions of impossibility type proved to be better handled in the non-mathematical domain. In the certainty questions the children's performance was similar regarding both domains of knowledge. The fact that children presented their responses regarding possibility with ease within the mathematical knowledge paves the way to deepen the approach of combinatorial probability of content and analysis in the school context. If children from the 3rd year of elementary school have already demonstrated understanding of probabilistic situations, it may be time to deepen more the notions of probability explored in the classroom. Still in this context, children from the 3rd year on, the subjects had significantly better performance in probability than in combinatorial analysis, suggesting the existence of a brake on the development of this concept, which is essentially established by school. Apparently, the combinatorial contents are either not being developed in the early school years, or the way that this work is conducted does not allow children to take ownership and develop their own thinking about such principles.

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.

I It was unclear how the teacher education curriculum at a regional university in the south central region of the United States developed mathematical knowledge for teaching (MKT) in prospective elementary teachers. Understanding how MKT develops during teacher training is important because MKT has been linked to student achievement. The purpose of this sequential explanatory mixed methods study was to examine how prospective elementary teachers' MKT developed while enrolled in a math and science strategies course. Guided by Ball et al.'s MKT framework and Silverman and Thompson's development of this framework, this study investigated changes in prospective teachers' MKT levels and teacher candidates' perceptions of instructional tasks that assisted in the development of MKT during the course. During the quantitative phase, teacher candidates (N = 30) completed the Number Concepts and Operations assessment as a pre- and posttest. Paired t test results showed no significant changes in candidates' MKT levels. During the qualitative phase, volunteers were interviewed about their perceptions of how the course influenced their development of MKT. Thematic analyses revealed that teacher candidates recognized instruction that developed MKT, perceived the strategies course to have little to no influence on MKT, and felt unprepared to teach math. Findings were used to develop a revised curriculum plan for developing prospective teachers' MKT. The findings may lead to positive social change in the form of curriculum revisions aimed at developing teacher candidates' MKT to improve future instruction. The project may be shared with other colleges to improve curriculum with the goal of improving the quality of math instruction statewide.

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.