Orientador: Dr. Alessandro Jacques Ribeiro / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Neurociência e Cognição, 2017. / O presente estudo preocupou-se em investigar se e como o conceito anel de polinomios e (re)construido por professores de Matematica que lecionam na Educacao Basica. Neste estudo, discutiu-se a importancia da construcao de conceitos matematicos, tendo por um lado a contribuicao do campo da ciencia cognitiva, principalmente os trabalhos Anna Sfard, e as recentes pesquisas sobre conhecimento matematico para o ensino, na area de Educacao Matematica, utilizando, essencialmente os trabalhos de Debora Ball, Jose Carrillo e seus colaboradores. Adotou-se a metodologia de pesquisa de natureza quantiqualitativa e o processo de coleta de dados se desenvolveu por meio de encontros do Curso de Extensao \O Ensino de Algebra para a Educacao Basica., ministrado na Universidade Federal do ABC, no estado de Sao Paulo, e conduzido por professores universitarios integrantes do programa Observatorio da Educacao. Para a producao de dados da pesquisa, utilizou-se como instrumentos metodologicos os questionarios e os registros escritos, elaborados/produzidos pelos professores-participantes da pesquisa durante os encontros do referido curso. A partir da analise dos dados, os resultados da investigacao apontam para a necessidade de promover acoes que possam ampliar e aprofundar o conhecimento especifico matematico desses professores, dado que ninguem pode ensinar o que nao sabe. Tambem evidenciamos a necessidade de um (re)pensar sobre o ensino de polinomios, uma vez que tal a analise dos dados realizados revelou que os professores desconhecem procedimentos para se operar com polinomios. Espera-se com este estudo chamar a atencao das politicas publicas para a necessidade de investimento na formacao continuada dos professores de matematica e, por conseguinte, na valorizacao da carreira docente. / This study intended to investigate mathematical knowledge building for teaching the concept of polynomial with Math teachers in Basic Education. In this study, the importance of building mathematical concepts was discussed, considering contributions from the cognitive sciences field, particularly Anna Sfard`s works, and recent research on mathematical knowledge for Algebra teaching from the Math Education field, using essentially works by Ball, Carrillo and their collaborators. A qualitative-quantitative approach for our research design was adopted, and the data gathering process was developed from meetings of a Extracurricular Course on Algebra Teaching for Basic Education, offered at the Federal University of ABC, in São Paulo State, conducted by university professors from the Education Observatory program. In producing the research data, questionnaires and written registers were used, elaborated by research participants themselves during the Course`s meetings. From the analysis of the data, the research results point to the need to promote actions that can broaden and deepen the specific mathematical knowledge of these teachers, given that no one can teach what they do not know. We also show the need for a new thinking about the teaching of polynomials, once such analysis of the data revealed that teachers are not aware of procedures to operate with polynomials. It is hoped that this study will draw the attention of public policies to the need for investment in the continuing education of mathematics teachers and, consequently, in the valorization of the teaching career.

Made available in DSpace on 2014-12-17T15:04:46Z (GMT). No. of bitstreams: 1 AnildaPSG.pdf: 471746 bytes, checksum: ca6e9ed04b0deccfe6bb0e02877d63fe (MD5) Previous issue date: 2006-05-19 / The central question of the present study is to identify the epistemological knowledge that the teachers-trainees possess regarding the characteristics (properties) of the decimal numbering system; its purpose is to offer a contribution to the pedagogic practice of the teachers who work within the Basic Literacy Cycle, in terms of what concerns both the acquisition of contents and the development of the knowledge that helps them in the elaboration of adequate strategies to working with the Decimal Numbering System in the classroom. The study is based on the constructivist sociointeractionist approach to teaching Mathematics and it constitutes, in itself, a methodological intervention with the teachers-trainees engaged in the Professional Qualification Program in Basic Education of the Federal University of Rio Grande do Norte. The foundations of the study were found in investigations of researchers who had carried out studies on the construction of numerical writing, showing, for instance, that the construction process of ideas and procedures involved in groupings and changes to base 10 take a lot longer to be accomplished than one can imagine. A set of activities was then elaborated which could not only contribute to the acquisition of contents but that could also make the teachers-trainees reflect upon their teaching practices in the classroom so that in this way they will be able to elaborate more consistent didactic approaches, taking into consideration the previous knowledge of the students and also some obstacles that often appear along the way. Even when teachers have access to the most appropriate dicactic resources, the lack of knowledge of the content and of the real meaning of that content make the Decimal Numbering System, a subject of fundamental importance, be taught most times in a mechanical way. The analisys of the discussions and behaviours of the teachers-trainees during the activities reavealed that they made them reflect upon their current practices in the classroom and that, as a whole, the aims of each of the activities carried out with the teachers-trainers were reached / O presente estudo tem como quest?o central identificar o conhecimento epistemol?gico que os professores-alunos possuem a respeito das caracter?sticas (propriedades) do sistema de numera??o decimal, e tem como finalidade oferecer uma contribui??o para a pr?tica pedag?gica dos professores que atuam nas turmas de Ciclo B?sico de Alfabetiza??o, tanto no que se refere ? aquisi??o de conte?dos quanto ao aprimoramento de conhecimentos que os auxiliem na elabora??o de estrat?gias adequadas para o trabalho com o Sistema de Numera??o Decimal em sala de aula. O estudo est? baseado na proposta construtivista sociointeracionista para o ensino da Matem?tica e se constitui numa interven??o metodol?gica, com os professores-alunos ligados ao Programa de Qualifica??o Profissional para a Educa??o B?sica da Universidade Federal do Rio Grande do Norte. Buscaram-se fontes de sustenta??o em investiga??es de pesquisadores que realizaram estudos sobre a constru??o das escritas num?ricas, mostrando, por exemplo, que o processo de constru??o das id?ias e procedimentos envolvidos nos agrupamentos e trocas na base 10 leva muito mais tempo para ser realizado do que se pode imaginar. Foi elaborado um conjunto de atividades as quais pudessem n?o s? contribuir para a aquisi??o de conte?dos, mas fizessem os professores-alunos refletir sobre suas pr?ticas em sala de aula, para que, assim, eles pudessem elaborar propostas did?ticas mais consistentes, levando em conta os conhecimentos pr?vios dos alunos e alguns obst?culos que se interp?em nessa trajet?ria. Mesmo o professor dispondo dos mais apropriados recursos did?ticos, a falta de dom?nio do conte?do, do real significado desse conte?do, faz com que o Sistema de Numera??o Decimal, tema de fundamental import?ncia, seja ensinado, na maioria das vezes, de forma mec?nica. A an?lise das discuss?es e o comportamento dos professores-alunos durante a realiza??o das atividades revelaram que estas provocaram reflex?es sobre as pr?ticas de sala de aula como tamb?m que, de modo geral, foram atingidos os objetivos propostos em cada atividade realizada com os professores-alunos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this text are presented results of a research whose objective was analyze the appropriations of the intuitive method of Pestalozzi for the teaching of elementary mathematical knowledge in Brazilian periodicals of late of XIX and early of XX centuries. The theoretical contribution came from the use of Chartier (2003) to talk about appropriation, Valente (2015) about the elementary mathematical knowledge, Ragazzini (2001) about historical sources and Pestalozzi (1889, 2003) aiming an understanding about the intuitive method. In reading the works of Pestalozzi it was identified in relation the intuitive method started of elements of number, form and word, aiming the development of the child’s faculties or capacities that he had intuition as a foundation, and was based on perception, observation, use of the senses, in the graduality of teaching and sought to assist in the development of the child as a human person. The resources used Brazilian periodicals of late of XIX and early of XX centuries found in the digital repository of UFSC and that had some reference to Pestalozzi, method or some the principles. As results, can be emphasize a circulation of Pestalozzi in these periodicals, since it was referenced in publications of eight Brazilian states: Alagoas, Amazonas, Bahia, Minas Gerais, Paraná, Rio de Janeiro, Rio Grande do Norte and São Paulo. To indentify appropriation, understood as use or interpretation, were presented at a first moment those that in the articles had explicit references to Pestalozzi, and at a second moment, the articles that brought the intuitive method or some the principles. It is also worth noting that in the periodical Pestalozzi was pointed out as a forerunner of modern pedagogy, active school, natural education, and culture of the senses etc, all this refers to the intuitive method. He was also taken as an example of master and authority in Education. In relation to the elementary mathematical knowledge, were identified appropriation about the contents of fractions, count, sum and calculations called elementary arithmetical knowledge, geometric solids and drawing called elementary geometric knowledge. It is also emphasize that for the application of method intuitive, the elementary mathematical knowledge were indicated objects such as: Parker Letters, mechanical counters and geometric solids. / Este trabalho é resultado de uma pesquisa que teve por objetivo analisar apropriações dos princípios do método intuitivo de Pestalozzi, para o ensino de saberes elementares matemáticos em periódicos brasileiros do final do século XIX e início do século XX. Como fundamentação teórica para os principais conceitos foram utilizados Chartier (2003), para falar sobre apropriação, Valente (2015), para os saberes elementares matemáticos, Ragazzini (2001), a respeito de fontes históricas e Pestalozzi (1889, 2003), visando a um entendimento sobre o método intuitivo. Na leitura das obras de Pestalozzi foi identificado, em relação ao método intuitivo, que o mesmo partia dos elementos do número, forma e palavra e objetivava o desenvolvimento das faculdades ou capacidades da criança e que tinha a intuição como fundamento, e era pautado na percepção, observação, no uso dos sentidos, na gradação do ensino e buscava auxiliar no desenvolvimento da criança como pessoa humana. Como fontes forma utilizados periódicos brasileiros do final do século XIX e início do século XX encontrados no repositório digital da UFSC e que apresentavam alguma referência a Pestalozzi, ao método ou a pelo menos um de seus princípios. Como resultados, pode-se enfatizar uma circulação de Pestalozzi nesses periódicos, uma vez que ele foi referenciado em publicações de oito estados brasileiros: Alagoas, Amazonas, Bahia, Minas Gerais, Paraná, Rio de Janeiro, Rio Grande do Norte e São Paulo. Para identificar apropriação, entendida como uso ou interpretação, foram apresentadas em um primeiro momento aquelas que nos artigos tinham referências explícitas a Pestalozzi, e em um segundo, os artigos que traziam o método intuitivo ou um de seus princípios. Destaca-se também que, nos periódicos, Pestalozzi foi apontado como um precursor da pedagogia moderna, da escola ativa, da educação natural, da cultura dos sentidos etc., tudo isso remete ao método intuitivo. Além disso, ele também foi tido como exemplo de mestre e autoridade no que se refere à Educação. Em relação aos saberes elementares matemáticos foram identificadas apropriações acerca dos conteúdos fração, contagem, soma e cálculo denominados de saberes elementares aritméticos, e sólidos geométricos e desenho chamados de saberes elementares geométricos. Ressalta-se ainda, que para aplicação do método intuitivo aos saberes matemáticos foram indicados objetos como: cartas de Parker, contadores mecânicos e sólidos geométricos.

In this text are presented results of a research whose objective was to identify indications of appropriations of theories of Edward Lee Thorndike for the teaching of elementary mathematical knowledge in pedagogical journals that circulated in Brazil between 1920 and 1960. For this, pedagogical journal that circulated at the time were examined, for example, Revista do Ensino, Revista Brasileira de Estudos Pedagógicos and Revista do Ensino. The theoretical contribution came from the use of Thorndike works, such as: The Principles of Teaching Based on Psychology (1905), The Thorndike Arithmetics (1917), The new methods in Arithmetic (1921) and The Psychology of Arithmetic (1922). As a result, it was possible to verify that Thorndike's theories began to be appropriated through Brazilian pedagogical journals from references to works such as The Thorndike Arithmetics (1917) and The Psychology of Arithmetic (1922), cited respectively in the Revista do Ensino, 1930, state of Minas Gerais, and by Murgel (1929), whose dates precede the publication of the work translated A Nova Metodologia da Aritmética, 1936. The authors of the articles made interpretations and uses of aspects for the teaching of elementary mathematical knowledge, in relation, mainly, to aspects of problem solving and tests, to criticize the problems with fanciful statements that would be difficult for students to see in a real situation, and ways to arouse student interest, by working the reasoning and the habit formation through by controlling time for to learning and monitoring of school development. Such identifications were associated to the orientations for teachers at the time of the chronological limit of this research. Thus, it was possible to affirm that there was appropriation of theories of Edward Lee Thorndike in the pedagogical journals that circulated between 1920 and 1960 in Brazil. / Neste texto são apresentados resultados de uma pesquisa cujo objetivo foi identificar indícios de apropriações de teorias de Edward Lee Thorndike para o ensino dos saberes elementares matemáticos em revistas pedagógicas que circularam no Brasil entre 1920 e 1960. Para isso, foram examinadas revistas pedagógicas que circularam à época, por exemplo, Revista do Ensino, Revista Brasileira de Estudos Pedagógicos e Revista de Ensino. Como referencial teórico foram utilizadas obras de Thorndike, como: The Principles of Teaching Based on Psychology (1905), The Thorndike Arithmetics (1917), The new methods in Arithmetic (1921) e The Psychology of Arithmetic (1922). Como resultados, foi possível constatar que as teorias de Thorndike começaram a ser apropriadas por meio das revistas pedagógicas brasileiras a partir de referências às obras The Thorndike Arithmetics (1917) e The Psychology of Arithmetic (1922), citadas, respectivamente, na Revista do Ensino, de 1930 do estado de Minas Gerais, e por Murgel (1929), cujas datas são anteriores a publicação da obra traduzida A Nova Metodologia da Aritmética, de 1936. Os autores dos artigos efetuaram interpretações e usos de princípios para o ensino dos saberes elementares matemáticos, em relação, principalmente, à aspectos da resolução de problemas e aos testes, para criticar os problemas com enunciados fantasiosos, que dificilmente seriam vistos pelos alunos em uma situação real, e às maneiras de despertar o interesse do aluno, trabalhando o raciocínio e a formação de hábitos por meio do controle do tempo para a aprendizagem e do acompanhamento do desenvolvimento escolar. Tais identificações estavam associadas às orientações para professores à época. Assim, é possível afirmar que houve apropriação das teorias de Edward Lee Thorndike em revistas pedagógicas que circularam entre 1920 e 1960 no Brasil.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-06-06T12:26:44Z No. of bitstreams: 2 Dissertação - Greiton Toledo de Azevedo - 2017.pdf: 9210718 bytes, checksum: 06fc74404cdd4449bbb59c52527fd71a (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-06-06T12:27:00Z (GMT) No. of bitstreams: 2 Dissertação - Greiton Toledo de Azevedo - 2017.pdf: 9210718 bytes, checksum: 06fc74404cdd4449bbb59c52527fd71a (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-06-06T12:27:00Z (GMT). No. of bitstreams: 2 Dissertação - Greiton Toledo de Azevedo - 2017.pdf: 9210718 bytes, checksum: 06fc74404cdd4449bbb59c52527fd71a (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-04-10 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This work has as main objective to understand the process of the construction of mathematical knowledge from the preparation and development of digital games (games) by students of Elementary School, in their intrinsic relation with the didactic pedagogical practices of Basic Education. The possibility of this construction of knowledge is defended without leaving aside the challenges that are limited in the temporality of the events of the school scene. For this purpose, a mathematics project was developed within a public school, located in a city in the metropolitan region of Goiânia, which is a fertile field of research, entitled Mattics, in the school counterpart, with the proposal to produce digital games, while mobilizing the construction of mathematical knowledge of the 16 participants of the research. The actions developed, based on the qualitative assumption, were based on the use of the Scratch programming language, which was developed at the Massachusetts Institute of Technology, articulated with exploratory-investigative activities of mathematics. From the interrelationship of the empirical materials produced in the project, a path was coursed that sought theoretical support both in aspects of the production of digital games and in the construction of mathematical knowledge by the students in a constructionist environment. According to the data collected and analyzed, we found out that the results achieved, in this research, give us indications to understand the process of building knowledge from the production of games as a dynamic movement, which conjugates ideas / mathematical meanings and which is not necessarily part of formal concepts throughout the process of a non-linear production. The construction is based on the active participation of the student in the environment. A production that is not absent from external factors and influences how the students think/ argue when producing their game when interacting with their local environment. / Este trabalho tem como principal objetivo compreender o processo da construção de conhecimento matemático a partir da elaboração e desenvolvimento de jogos digitais (games) por estudantes do Ensino Fundamental, em sua intrínseca relação com as práticas didático- pedagógicas da Educação Básica. Defende-se com isso a possibilidade dessa construção de conhecimento sem deixar de lado os desafios que se circunscrevem na temporalidade dos acontecimentos do cenário escolar. Para isso, foi desenvolvido no âmbito de uma escola pública, localizada em uma cidade da região metropolitana de Goiânia, um projeto de matemática, que se constitui como campo fértil de investigação, intitulado Mattics, no contraturno escolar, com a proposta de se produzir jogos digitais, ao mesmo tempo que mobilizasse a construção de conhecimento matemático dos 16 participantes da pesquisa. As ações desenvolvidas, tendo como pano de fundo o pressuposto qualitativo, estiveram alicerçadas no uso da linguagem de programação Scratch, que foi desenvolvida no Massachusetts Institute of Technology, articulada com atividades exploratório-investigativas de matemática. A partir do intercruzamento dos materiais empíricos produzidos no projeto, percorreu-se um caminho que procurou sustentação teórica tanto em aspectos da produção de jogos digitais, quanto da construção de conhecimento matemático pelos estudantes em um ambiente construcionista de aprendizagem. De acordo com os dados coletados e analisados, percebemos que os resultados alcançados, nesta pesquisa, nos dão indícios para entender o processo de construção de conhecimento a partir da produção de jogos como um movimento dinâmico, que conjuga ideias/significados de matemática e que não parte necessariamente de conceitos formais ao longo do processo de uma produção não linear. A construção se fundamenta pela produção quando há participação ativa do estudante no ambiente. Uma produção que não se ausenta de fatores externos e influencia a forma como aluno passa a pensar/discutir/argumentar ao produzir o seu jogo quando interage com o seu meio local.

Lärare som undervisar i matematik förväntas kunna mer avancerad matematik än vad de undervisar om. Men formell matematikkunskap anses inte vara tillräckligt för att lärare ska kunna undervisa så att ämnesinnehållet blir begripligt för eleverna, de behöver även pedagogical content knowledge (PCK). Begreppet belyser en speciell form av ämneskunskap för undervisning och skiljer sig från den matematikkunskap som används av andra välutbildade vuxna. Det har föreslagits att olika arrangemang av kollegialt och praktikbaserat lärande kan utveckla lärares PCK. Ett exempel på ett sådant arrangemang är learning study. Den här avhandlingen handlar om den kunskap om lärande och undervisning i matematik som studiens lärare utvecklar då de deltar i learning studies och utforskar sin praktik utifrån ett variationsteoretiskt perspektiv. Det yttersta syftet med en learning study är att utveckla elevernas lärande om specifika lärandeobjekt, genom att undersöka vad som kan vara kritiskt för elevernas lärande. I ett samarbetsprojekt med fyra högstadielärare genomfördes två learning studies i matematik, under ett år. Lärargruppen undersökte vad eleverna behöver lära för att de ska förstå i) varför en kvot kan vara större än talet i täljaren och ii) olika representationer av konstanterna k och m i räta linjens ekvation. Under learning study-arrangemangets olika steg samlades studiens empiri in och denna består av filmade lektioner, inspelade möten där lärargruppen planerade och analyserade undervisning och elevers lärande, skriftliga elevtest samt elevintervjuer. Studien har en variationsteoretisk utgångspunkt, vilket innebär att lärande förklaras ske när en person ser något på ett nytt och mer kvalitativt sätt, genom att personen urskiljer aspekter som han/hon inte tidigare har urskilt. Studien visar de två lärandeobjektens kritiska aspekter samt hur de kritiska aspekterna gradvis förändrades och specificerades. Förändringen var ett resultat av att lärargruppen fick syn på avgörande detaljer om på vilket sätt eleverna förstod ämnesinnehållet samt hur skilda sätt att förstå kunde användas i undervisningen för att utveckla elevernas lärande. Där av titeln att få syn på avgörande skillnader. Denna form av utvecklad kunskap om lärandeobjektet kan ses som ett bidrag om PCK och vad det kan vara. / It is a common view that teachers need more than formal content knowledge to teach and to make the content comprehensible to others. They also need pedagogical content knowledge, or PCK (Shulman, 1986). It has been suggested that different teacher collaboration approaches may support teachers’ development of PCK (Chapman, 2013, Davis & Renert, 2014; Steele & Rogers, 2012). This thesis aims to provide insights into the kind of knowledge about teaching and learning mathematics that teachers develop through their participation in a specific collaboration approach called learning study. Four teachers of mathematics and their 74 students (aged 15−16 years) participated in two learning studies over the course of one year. The foremost aim of a learning study is to enhance student learning about specific objects of learning and to identify what is critical for the students’ learning (Marton & Tsui, 2004). The objects of learningin the two learning studies were to understand that dividing with a denominator between 0 and 1 gives a quotient larger than the numerator and to understand different representations of the constants b and m in the equation of the straight line. During the two learning studies data were collected from 8 video-recorded lessons, 2 written student tests, student interviews, and 14 audio-recorded sessions in which the teachers and I (PhD student) planned, analysed and revised teaching and student learning. The analysis was based on variation theory (Marton & Tsui, 2004) and focused on what participants considered to be critical aspects of the objects of learning and on the components embedded in that knowledge. The result shows the identified critical aspects of the two objects of learning and, furthermore, how the teachers’ knowledge about those critical aspects gradually changed and became more refined and specified in relation to their students’ understanding. The thesis provides an insight into the value of the teachers’ enhanced knowledge of the object of learning, in relation to how PCK can be understood.

The thesis is based on the constructivistic approach to mathematics at primary school and geometry topics situated in the environment of square grid. It researches the extent to which pupils are able to use the experience with the square grid in the project method. For this purpose, observed two groups of pupils aged 10-11 years, experimental and control group. Work with the experimental group was situated in two weeks.

El presente estudio tuvo como objetivo describir el conocimiento especializado de un profesor de matemáticas experto, que enseña el tema de la resolución de problemas de adición y sustracción a los estudiantes. Realizamos un análisis de contenido de los 22 documentos seleccionados en la matriz, empleando un sistema de categorías dentro del subdominio de Conocimiento de los Temas (KoT) del modelo de Conocimiento Especializado del Profesor de Matemáticas (MTSK). / The aim of this study was to describe the specialized knowledge of an expert math teacher while teaching the topic of solving addition and subtraction problems to students. A content analysis of 22 documents was performed by using a category system within the subdomain of Knowledge of Topics (KoT) of the model Mathematics Teacher's Specialized Knowledge model (MTSK). / Trabajo de investigación

This paper examines the mathematical knowledge for teaching (MKT) in Indonesia, specifically in school geometry content. A translated and adapted version of the MKT measures developed by the Learning Mathematics for Teaching (LMT) project was administered to 210 Indonesian primary and junior high teachers. Psychometric analyses revealed that items related to classification of quadrilaterals were difficult for these teachers. Further interactions with teachers in a professional development setting confirmed that teachers held a set of exclusive definitions of quadrilaterals.

<p>  </p> <p>Mathematics teacher educators have suggested that approximations of practice provide preservice mathematics teachers (PMTs) with opportunities to engage with, develop, and demonstrate subdomains of Mathematical Knowledge for Teaching ([MKT], Ball et al., 2008) because MKT provides a way for PMTs to understand how to contextualize their discipline-specific content knowledge for effective mathematics teaching and learning. However, the affordances and limitations of commonly used forms of approximations of practice (i.e., lesson planning and peer teaching) coupled with reflective practices to engage PMTs in subdomains of MKT are still being explored. In this study, I investigated how lesson planning, peer teaching, and associated reflections individually and collectively afforded opportunities for PMTs to demonstrate and develop the MKT subdomains. Eleven PMTs enrolled in a secondary mathematics methods course at a large Midwestern University participated in the study. My dissertation comprises three sub-studies (Sub-study “1”, “2”, and “3”), and I produced three manuscripts to individually report findings from those sub-studies. I investigated how lesson planning, peer teaching, and reflections afforded opportunities for PMTs to demonstrate and describe MKT subdomains in Sub-studies 1, 2, and 3, respectively. The findings across the sub-studies suggested that several MKT subdomains (e.g., Knowledge of Content and Teaching, Knowledge of Content and Students) were evidenced in the PMTs’ planned teacher and student actions (e.g., selecting mathematical tasks, formulating and sequencing questions), and in-the-moment actions and decisions (e.g., mathematically representing students’ responses, implementing mathematical tasks). Several aspects of MKT subdomains (e.g., evaluate the diagnostic potential of tasks) were strongly evidenced only in the PMTs’ lesson plans whereas other aspects (e.g., modifying tasks based on students’ responses) were evidenced only in peer teaching. These findings suggested that various forms of approximations of practice (planned and enacted actions) created unique opportunities for the PMTs to engage with and demonstrate MKT. I also found that the PMTs reflected on some subdomains of MKT that were not evidenced in their approximated practices, indicating that how PMTs describe the MKT subdomains is not entirely a result of what subdomains they engage in during approximations of practice. My findings also revealed limitations of using approximations of practice to engage PMTs with MKT subdomains. The MKT subdomains that required the PMTs to think about students’ alternative mathematical concepts, big mathematical ideas, and non-standard mathematics problem-solving strategies were least evidenced across the approximations of practice and reflections. These findings have two primary implications for mathematics teacher educators. First, I invite mathematics teacher educators to engage PMTs in multiple forms of approximations of practice to optimize their opportunities to engage with, demonstrate, and develop the MKT subdomains. Second, I suggest potential instructional activities (e.g., inviting PMTs to reflect on their roles as students and teachers during peer teaching) that could be incorporated into approximations of practice to address the existing limitations. Broadly, I invite mathematics teacher educators to design instructional activities at the intersection of mathematics content and pedagogy, collaborating with colleagues to enhance these opportunities across programs.</p> <p>  </p>