In order to investigate ways pre-service student teachers (PSTs) might learn to teach with high-level tasks and effectively incorporate student thinking into their lessons a teaching experiment was designed and carried out by the cooperating teacher/researcher (CT). The intervention was for the CT to interject into the lessons of the PSTs during moments of opportunity. By interjecting a small question or comment during the lesson the CT hoped to support the learning of both the students of mathematics in the class and the PSTs. This in-the-moment interjecting was meant to enhance and underscore the situated learning of the PSTs within the context of actual practice. Essentially the PSTs learned how to manage and improve the discourse of the classroom in the moment of the discourse. This study utilized both an ongoing analysis of the data during collection in order to inform the instruction provided by the CT and a retrospective analysis of the data in order to develop an understanding of the developmental sequence through which PSTs progressed. The results suggest the interjections provided to the PSTs served multiple roles within the domains of mathematical development for the students of mathematics and pedagogical development for the PSTs. A classification of the interjections that occurred and the stages of development through which PSTs passed will be discussed. Implications from this work include increased attention to the groundwork leading up to the student teaching experience as well as an adjustment to the role of cooperating teacher to be more that of a teacher educator.

Syftet med föreliggande licentiatavhandling är att, utifrån ett specialdidaktiskt perspektiv, bidra med kunskap om lågpresterande elevers grundläggande antals- och taluppfattning, och hur utvecklingen av denna kan stödjas. Den övergripande forskningsfråga som besvaras är vilka aspekter, som ur ett specialdidaktiskt perspektiv, är särskilt betydelsefulla att beakta vad gäller lågpresterande elevers kunskapsutveckling inom grundläggande antals- och taluppfattning. Avhandlingen består av två delstudier. I delstudie 1, som är en systematisk litteraturöversikt, studeras vad som är kännetecknande för lågpresterande årskurs F-3-elever och hur de definieras i forskningslitteraturen. I den andra studien, delstudie 2, undersöks vilket kunnande gällande tals del-helhetsrelationer som förskoleklasselever utvecklar i en undervisningsinsats där konceptuella subitiseringsaktiviteter fokuseras. Specialdidaktikens förebyggande och stödjande roll utgör studiens övergripande förståelseram där de lågpresterande elevernas kunskapsutveckling förstås i förhållande till vilket lärande som möjliggörs i undervisningen. Det matematiska innehållet är grundläggande antals- och taluppfattning med fokus på konceptuell subitisering. Teorier om barns antals- och taluppfattningsutveckling (Baroody m.fl., 2009; Nunes &Bryant, 2007; Sayer m.fl. 2016), inbegripet teorier om subitisering (Clements m.fl., 2019; Kaufman m.fl., 1949) och groupitizing (Starkey &McCandliss, 2014), har utgjort den innehållsliga utgångspunkten. För att tolka specialdidaktikens specifika bidrag och krafter i relation till allmän matematikdidaktisk kompetens har ramverket Mathematical Knowledgefor Teaching (MKT) (Ball m.fl., 2008) använts, mer specifikt de tre delarna specialized content knowledge, knowledge of content and students och knowledge of content and teaching. Resultatet av syntesen visar att den specifika kompetens som krävs i relation till innehållet (specialized content knowledge), är fördjupad kunskap om centrala aspekter och vanliga trösklar i elevers kunskapsutveckling inom grundläggande antals- och taluppfattning för att motverka framtida matematiksvårigheter. Även fördjupad kunskap om elevers individuella variationer vad gäller att förstå och hantera antal och tal (knowledge of content and students) för att tidigt kunna identifiera elever i svårigheter är centralt och slutligen fördjupad kunskap om hur lågpresterande elevers antals-och taluppfattning kan stödjas och svårigheter förebyggas och överbryggas i undervisningen (knowledge of content and teaching). Specialdidaktikens bidrag förstås som krafter som hjälper till att balansera relationen mellan den svagpresterande eleven, läraren och matematikinnehållet i undervisningen, så att lärande möjliggörs. Specialdidaktisk kompetens kan därmed sägas komplettera den allmänna ämnesdidaktiska kompetensen (MKT) genom sitt bidrag med fördjupad kunskap om hur elever som inte utvecklas som förväntat i matematik kan stödjas, i grundläggande antals- och taluppfattning, vilket bildar Specialdidactic Mathematical Knowledge for Teaching eller SMKT.

Orientador: Prof. Dr. Alessandro Jacques Ribeiro / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Ensino, História, Filosofia das Ciências e Matemática, 2014. / Essa é uma pesquisa a qual fora desenvolvida no programa de pós-graduação em Ensino, História e Filosofia das Ciências e Matemática na Universidade Federal do ABC, em Santo André, cujo título é: "Analisando a mobilização de conhecimentos algébricos de professores de educação básica: O momento de preparação de aulas sobre equações.". Os objetivos dessa pesquisa consistem em mapear, investigar e compreender quais os conhecimentos algébricos que são mobilizados por professores quando estão elaborando suas aulas sobre equações para a Educação Básica. Adotou-se uma abordagem qualitativa como metodologia de pesquisa e os dados foram obtidos através de questionários e da análise documental das aulas preparadas pelos professores dessa pesquisa; gravações em áudio dos encontros os quais os professores preparam suas aulas em duplas. Os seis sujeitos de pesquisa são pessoas que preparam aulas para a Educação Básica nos conteúdos matemáticos tanto para seu ofício como professor(a) efetivo ou contratado, quanto para o desenvolvimento de pesquisa associado aos projetos de formação inicial ou continuada. Com isso, para fundamentar essa pesquisa inclusive nas análises dos dados, foram utilizados os trabalhos de Shulman (1986 e 1987) e Ball e equipe (2008). Estes últimos autores sugerem o quadro teórico do "Conhecimento Matemático para o Ensino", que é o "conhecimento matemático necessário para realizar o trabalho de ensinar matemática", além da existência de dois subdomínios, a partir dos trabalhos de Shulman: (i) Conhecimento Comum do Conteúdo e Conhecimento Especializado do Conteúdo; e (ii) Conhecimento do Conteúdo e os Estudantes e Conhecimento do Conteúdo e o Ensino. Após analisarmos os dados, baseados na perspectiva do conhecimento matemático para o ensino, pudemos identificar, entre outros, os seguintes conhecimentos algébricos: Reconhecimento de que uma sentença matemática não é equação (Conhecimento Comum do Conteúdo); Compreensão dos multisignificados do símbolo "=" (Conhecimento Especializado do Conteúdo); Reconhecimento dos conteúdos prévios para que os alunos possam compreender e participar de uma aula sobre equações (Conhecimento do Conteúdo e os Estudantes); Utilização de uma abordagem etimológica das palavras "equação" e "igualdade", com o objetivo de promover uma discussão destes conteúdos em sala de aula (Conhecimento do Conteúdo e o Ensino) e, por fim, Reconhecer que o conteúdo de equação, em especial a equação polinomial de 1º grau, tem forte relação e importância para o conteúdo de inequações, funções e outros conteúdos mais avançados (Conhecimento Curricular). / This is a research which had been developed in Master¿s program in Teaching, History and Philosophy of Sciences and Mathematical at the Federal University of ABC, in Santo André, whose title is "Analyzing the mobilizations of algebraic knowledge from basic education teachers: The moment to prepare lessons about equations". The objectives of this research are to map, investigate and understand which algebraic knowledge that was mobilized by teachers when working out their classes about equations for Basic Education. We adopted a qualitative approach to research methodology and data were collected through questionnaires and documentary analysis of the lessons had been prepared by the teachers of this research; audio recorded in meetings when teachers had planned their lessons in pairs. Those six teachers are people who prepare lessons for Basic Education in mathematical content to their craft both as a teacher actual or engaged, and for the development of research projects associated with initial or continuing training. Thus, to support this research including data analysis, the work of Shulman (1986 and 1987) and Ball et al. (2008) were used. Ball et al. suggest the theoretical framework of "Mathematical Knowledge for Teaching" which is the "Mathematical Knowledge needed to perform the job of teaching math" beyond the existence of two subdomains, from the works of Shulman: (i) Common Content Knowledge and Specialized Content Knowledge; also (ii) Knowledge of Content and Students, and Knowledge of content and Teaching. After analyzing the data, based on the perspective of mathematical knowledge for teaching, we identified, among others, the following algebraic knowledge: Recognition of a mathematical sentence is not an equation (Common Content Knowledge); Multimeaning understanding of the "=" symbol (Specialized Content Knowledge); Recognition of prior knowledge, so that students can understand and participate in an equations class (Content Knowledge and Students); Use an etymological approach of the words "equation" and "equality" with the aim of promoting a discussion of such content in the classroom (Content Knowledge and Teaching) and, finally, recognize that the contents of the equation, especially linear polynomial equation, has a strong relationship and importance to the content of inequalities, functions and other more advanced contents (Curricular Knowledge).

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.

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.

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.

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>

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