South African learners underachieve in mathematics. The many different factors that influence this underachievement include mathematics teachers' role in teaching mathematics with understanding. The question arises as to how teachers' mathematical content knowledge states can be transformed to positively impact learners' achievement in mathematics. In this study, different kinds of teachers' knowledge needed for teaching mathematics were discussed against the background of research in this area, which included the work of Shulman, Ma and Ball. From this study an important kind of knowledge, namely mathematical content knowledge for teaching (MCKfT), was identified and a teacher's ability to unpack mathematical knowledge and understanding was highlighted as a vital characteristic of MCKfT. To determine further characteristics of MCKfT, the study focussed on the nature of mathematics, different kinds of mathematical content knowledge (procedural and conceptual), cognitive processes (problem solving, reasoning, communication, connections and representations) involved in doing mathematics and the development of mathematical understanding (instrumental vs. relational understanding). The influence of understanding different problem contexts and teachers' ability to develop reflective practices in teaching and learning mathematics were discussed and connected to a teacher's ability to unpack mathematical knowledge and understanding. In this regard, the role of teachers' prior knowledge or current mathematical content knowledge states was discussed extensively. These theoretical investigations led to identifying the characteristics of MCKfT, which in turn resulted in theoretical criteria for the development of MCKfT. The theoretical study provided criteria with which teachers' current mathematical content knowledge states could be analysed. This prompted the development of a diagnostic instrument consisting of questions on proportional reasoning and functions. A qualitative study was undertaken in the form of a diagnostic content analysis on teachers' current mathematical content knowledge states. A group of secondary school mathematics teachers (N=128) involved in the Sediba Project formed the study population. The Sediba Project is an in-service teacher training program for mathematics teachers over a period of two years. These teachers were divided into three sub-groups according to the number of years they had been involved in the Sediba Project at that stage. The teachers' current mathematical content knowledge states were analysed with respect to the theoretically determined characteristics of and criteria for the development of MCKfT. These criteria led to a theoretical framework for assessing teachers' current mathematical content knowledge states. The first four attributes consisted of the steps involved in mathematical problem solving skills, namely conceptual knowledge (which implies a deep understanding of the problem), procedural knowledge (which is reflected in the correct choice of a procedure), the ability to correctly execute the procedure and the insight to give a valid interpretation of the answer. Attribute five constituted the completion of these four attributes. The final six attributes were an understanding of different representations, communication of understanding in writing, reasoning skills, recognition of connections among different mathematical ideas, the ability to unpack mathematical understanding and understanding the context a problem is set in. Quantitative analyses were done on the obtained results for the diagnostic content analysis to determine the reliability of the constructed diagnostic instrument and to search for statistically significant differences among the responses of the different sub-groups. Results seemed to indicate that those teachers involved in the Sediba Project for one or two years had benefited from the in-service teacher training program. However, the impact of this teachers' training program was clearly influenced by the teachers' prior knowledge of mathematics. It became clear that conceptual understanding of foundation, intermediate and senior phase school mathematics that should form a sound mathematical knowledge base for more advanced topics in the school curriculum, is for the most part procedurally based with little or no conceptual understanding. The conclusion was that these teachers' current mathematical content knowledge states did not correspond to the characteristics of MCKfT and therefore displayed a need for the development of teachers' current mathematical content knowledge states according to the proposed criteria and model for the development of MCKfT. The recommendations were based on the fact that the training that these teachers had been receiving with respect to the development of MCKfT is inadequate to prepare them to teach mathematics with understanding. Teachers' prior knowledge should be exposed so that training can focus on the transformation of current mathematical content knowledge states according to the characteristics of MCKfT. A model for the development of MCKfT was proposed. The innermost idea behind this model is that a habit of reflective practices should be developed with respect to the characteristics of MCKfT to enable a mathematics teacher to communicate and unpack mathematical knowledge and understanding and consequently solve mathematical problems and teach mathematics with understanding. Key words for indexing: school mathematics, teacher knowledge, mathematical content knowledge, mathematical content knowledge for teaching, mathematical knowledge acquisition, mathematics teacher education / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2007.

abstract: This study contributes to the ongoing discussion of Mathematical Knowledge for Teaching (MKT). It investigates the case of Rico, a high school mathematics teacher who had become known to his colleagues and his students as a superbly effective mathematics teacher. His students not only developed excellent mathematical skills, they also developed deep understanding of the mathematics they learned. Moreover, Rico redesigned his curricula and instruction completely so that they provided a means of support for his students to learn mathematics the way he intended. The purpose of this study was to understand the sources of Rico's effectiveness. The data for this study was generated in three phases. Phase I included videos of Rico's lessons during one semester of an Algebra II course, post-lesson reflections, and Rico's self-constructed instructional materials. An analysis of Phase I data led to Phase II, which consisted of eight extensive stimulated-reflection interviews with Rico. Phase III consisted of a conceptual analysis of the prior phases with the aim of creating models of Rico's mathematical conceptions, his conceptions of his students' mathematical understandings, and his images of instruction and instructional design. Findings revealed that Rico had developed profound personal understandings, grounded in quantitative reasoning, of the mathematics that he taught, and profound pedagogical understandings that supported these very same ways of thinking in his students. Rico's redesign was driven by three factors: (1) the particular way in which Rico himself understood the mathematics he taught, (2) his reflective awareness of those ways of thinking, and (3) his ability to envision what students might learn from different instructional approaches. Rico always considered what someone might already need to understand in order to understand "this" in the way he was thinking of it, and how understanding "this" might help students understand related ideas or methods. Rico's continual reflection on the mathematics he knew so as to make it more coherent, and his continual orientation to imagining how these meanings might work for students' learning, made Rico's mathematics become a mathematics of students--impacting how he assessed his practice and engaging him in a continual process of developing MKT. / Dissertation/Thesis / Ph.D. Mathematics 2011

abstract: A teacher’s mathematical knowledge for teaching impacts the teacher’s pedagogical actions and goals (Marfai & Carlson, 2012; Moore, Teuscher, & Carlson, 2011), and a teacher’s instructional goals (Webb, 2011) influences the development of the teacher’s content knowledge for teaching. This study aimed to characterize the reciprocal relationship between a teacher’s mathematical knowledge for teaching and pedagogical goals. Two exploratory studies produced a framework to characterize a teacher’s mathematical goals for student learning. A case study was then conducted to investigate the effect of a professional developmental intervention designed to impact a teacher’s mathematical goals. The guiding research questions for this study were: (a) what is the effect of a professional development intervention, designed to perturb a teacher’s pedagogical goals for student learning to be more attentive to students’ thinking and learning, on a teacher’s views of teaching, stated goals for student learning, and overarching goals for students’ success in mathematics, and (b) what role does a teacher's mathematical teaching orientation and mathematical knowledge for teaching have on a teacher’s stated and overarching goals for student learning? Analysis of the data from this investigation revealed that a conceptual curriculum supported the advancement of a teacher’s thinking regarding the key ideas of mathematics of lessons, but without time to reflect and plan, the teacher made limited connections between the key mathematical ideas within and across lessons. The teacher’s overarching goals for supporting student learning and views of teaching mathematics also had a significant influence on her curricular choices and pedagogical moves when teaching. The findings further revealed that a teacher’s limited meanings for proportionality contributed to the teacher struggling during teaching to support students’ learning of concepts that relied on understanding proportionality. After experiencing this struggle the teacher reverted back to using skill-based lessons she had used before. The findings suggest a need for further research on the impact of professional development of teachers, both in building meanings of key mathematical ideas of a teacher’s lessons, and in professional support and time for teachers to build stronger mathematical meanings, reflect on student thinking and learning, and reconsider one’s instructional goals. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2016

abstract: This dissertation reports three studies of students’ and teachers’ meanings for quotient, fraction, measure, rate, and rate of change functions. Each study investigated individual’s schemes (or meanings) for foundational mathematical ideas. Conceptual analysis of what constitutes strong meanings for fraction, measure, and rate of change is critical for each study. In particular, each study distinguishes additive and multiplicative meanings for fraction and rate of change. The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development. The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items. The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions. Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2016

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Agradecemos a compreensão. on 2017-04-12T20:08:29Z (GMT) / Submitted by TAÍS ALVES MOREIRA BARBARIZ null (taisbarbariz@ggmail.com) on 2017-04-17T15:32:10Z No. of bitstreams: 1 A CONSTITUIÇÃO DO CONHECIMENTO MATEMÁTICO EM UM CURSO DE MATEMÁTICA À DISTÂNCIA.pdf: 7379684 bytes, checksum: a336c7f5cdd4d9463e623834e106ee3d (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-04-17T16:10:44Z (GMT) No. of bitstreams: 1 barbariz_tam_dr_rcla.pdf: 7379684 bytes, checksum: a336c7f5cdd4d9463e623834e106ee3d (MD5) / Made available in DSpace on 2017-04-17T16:10:44Z (GMT). No. of bitstreams: 1 barbariz_tam_dr_rcla.pdf: 7379684 bytes, checksum: a336c7f5cdd4d9463e623834e106ee3d (MD5) Previous issue date: 2017-03-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Esta pesquisa tem por meta compreender a constituição de conhecimento matemático, tomando como foco a experiência vivenciada no mundo-vida da Educação a Distância. O desdobramento dos estudos persegue a questão, objetivo da investigação: Como se constitui o conhecimento matemático quando se está junto à Matemática, ao computador e aos cossujeitos? A pesquisadora assume, para isso, a postura filosófica-fenomenológica, entendendo que a Fenomenologia busca a ir-às-coisas-mesmas, não deduzindo consequências de pressupostos teóricos. Assim, a pesquisadora foca sua análise nas vivências em que o sentido vai se fazendo para ela. Para a constituição dos dados foi projetado um curso na modalidade à distância sobre Geometria, tomando como inspiração o tratado em dois capítulos de duas obras de Hans Freudenthal que tratam dessa parte da Matemática. Os procedimentos que conduzem a investigação tomam como dados, constituídos para esse fim, dois momentos distintos. O primeiro momento se deu na temporalidade da preparação do curso, quando se constituíram os dados que tiveram como solo os registros da pesquisadora, sujeito da investigação, que buscou, de modo atentivo, dar-se conta do por ela percebido nesse movimento, descrevendo essa percepção tal como a ela aparece no fluxo de sua lembrança. O segundo momento selecionado para análise e interpretação se constituiu de um dos diálogos, destacado entre todos os que ocorreram durante a realização do curso. Este diálogo mostrou-se exemplar pelo fato de apresentar diferentes maneiras de participações nas atividades do curso, como a apresentação de outros autores, que não os indicados no curso, para dialogar e auxiliar nas compreensões dos assuntos tratados, e, também por trazer outros alunos no movimento do diálogo em que um comenta a fala do outro, ratificando-a ou trazendo-a em sua própria reflexão. Todos os registros, do primeiro e do segundo momento, foram interpretados como um único movimento, à luz da interrogação que conduz esta pesquisa. A esta interpretação seguiu-se o movimento de metainterpretação, onde a pesquisadora busca transcender às compreensões constituídas por meio da pesquisa. Nesse momento, a pesquisadora compreendeu, ainda, abranger a busca de sentido que isso que está em constituição faz para o sujeito que indaga pelo que diz para ele. Ao explicitar o como se constitui o conhecimento matemático, estando junto a cossujeitos, na realidade do ciberespaço, a pesquisadora deu-se conta de que sua pesquisa se dá em uma direção que aprofunda compreensões a respeito dos modos pelos quais se dá a produção de conhecimento pelos seres humanos com mídias. / This research aims to understand the constitution of mathematical knowledge, focusing on the experience lived in the Distance Education life-world. The studies unfolding pursues the question which is the aim of the investigation: How is mathematical knowledge constituted when one is close to Mathematics, the computer and co-subjects? The researcher assumes, for this, the philosophical-phenomenological position, understanding that Phenomenology aims to go-to-things-themselves, without deducing consequences of theoretical presuppositions. The researcher thus focuses her analysis on her living experiences in which it is making sense to her. For the constitution of data, a Geometry distance learning course was projected, inspired in two chapters of two Hans Freudenthal works that deals with this branch of Mathematics. The investigation procedures took two distinct moments as data, which was constituted specifically for this purpose. The first moment occurred in the temporality of the preparation of the course, when the constituted data had the researcher, the investigations subject, own records as its soil. She has attentively sought of her perceptions in the movement to describe this as it shows in her remembrance flow. The second moment selected for analysis and interpretation consisted of one of the dialogues, highlighted among all occurred during the realization of the course. This dialogue showed itself exemplary because it presented different ways of participating in the course activities, such as the presentation of authors other than those indicated in the course, dialoguing and helping in understanding some matters discussed, and also to bring other students into the dialogue movement in which one comments the speech of the other, ratifying it or bringing it in his/her own reflection. All records, from the first and second moments, were interpreted as a single movement, in the light of the interrogation which drives this research. This interpretation was followed by the meta-interpretation movement, where the researcher seeks to transcend the understandings constituted through the research. The researcher further understood that it encompasses the search for sense that what is in constitution makes for the subject who inquires what it tells her. In explaining how mathematical knowledge is constituted, being close to co-subjects, in the reality of cyberspace, the researcher realized that she understood that her research takes place in a direction that deepens understandings about the ways in which knowledge production takes place by humans with media.

abstract: The research indicated effective mathematics teaching to be more complex than assuming the best predictor of student achievement in mathematics is the mathematical content knowledge of a teacher. This dissertation took a novel approach to addressing the idea of what it means to examine how a teacher's knowledge of mathematics impacts student achievement in elementary schools. Using a multiple case study design, the researcher investigated teacher knowledge as a function of the Mathematics Teaching Cycle (NCTM, 2007). Three cases (of two teachers each) were selected using a compilation of Learning Mathematics for Teaching (LMT) measures (LMT, 2006) and Developing Mathematical Ideas (DMI) measures (Higgins, Bell, Wilson, McCoach, & Oh, 2007; Bell, Wilson, Higgins, & McCoach, 2010) and student scores on the Arizona Assessment Collaborative (AzAC). The cases included teachers with: a) high knowledge & low student achievement v low knowledge & high student achievement, b) high knowledge & average achievement v low knowledge & average achievement, c) average knowledge & high achievement v average knowledge & low achievement, d) two teachers with average achievement & very high student achievement. In the end, my data suggested that MKT was only partially utilized across the contrasting teacher cases during the planning process, the delivery of mathematics instruction, and subsequent reflection. Mathematical Knowledge for Teaching was utilized differently by teachers with high student gains than those with low student gains. Because of this insight, I also found that MKT was not uniformly predictive of student gains across my cases, nor was it predictive of the quality of instruction provided to students in these classrooms. / Dissertation/Thesis / Ph.D. Curriculum and Instruction 2013

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-10-18T10:41:45Z No. of bitstreams: 1 michelenazaretdealmeida.pdf: 746804 bytes, checksum: d71ce3caccd003fcc60473733b0adf77 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-10-25T12:05:34Z (GMT) No. of bitstreams: 1 michelenazaretdealmeida.pdf: 746804 bytes, checksum: d71ce3caccd003fcc60473733b0adf77 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-10-25T12:06:12Z (GMT) No. of bitstreams: 1 michelenazaretdealmeida.pdf: 746804 bytes, checksum: d71ce3caccd003fcc60473733b0adf77 (MD5) / Made available in DSpace on 2016-10-25T12:06:13Z (GMT). No. of bitstreams: 1 michelenazaretdealmeida.pdf: 746804 bytes, checksum: d71ce3caccd003fcc60473733b0adf77 (MD5) Previous issue date: 2008-03-24 / Este trabalho pretende compreender como o conhecimento matemático é construído por trabalhadores da construção civil, no exercício de suas funções num canteiro de obras. Para isso, busca questionar o modelo de conhecimento que vem se constituindo como pensamento hegemônico há tanto tempo, o modelo cartesiano/ocidental, com o intuito de constituir um olhar crítico sobre ele. Pautando-se na compreensão do conhecimento como construção que acontece nas práticas sócio-culturais, essa pesquisa procura compreender como um trabalhador da construção civil constitui seus conhecimentos matemáticos em situações de trabalho ao mesmo tempo em que se constitui na pessoa que é. A pesquisa se pretende afim com a perspectiva Etnomatemática, já que essa vertente da educação matemática possibilita novos olhares sobre a educação e sobre a matemática, na tentativa de compreender o conhecimento como um modo que as pessoas possuem de se constituir no mundo, diante da realidade a qual vivenciam. A pesquisa pretende se constituir em uma abordagem qualitativa, pois busca compreender como o fenômeno se constitui para aqueles que o experimentam, seus modos de significá-lo, as concepções que dele têm. / This work aims to understand how the mathematical knowledge is built by employees of the construction, in the exercise of its functions in a gantry works. To do this, search questioning the model of knowledge that has been constituted as hegemonic thought long, the model Cartesian/Occidental in order to be a critical eye on him. Guiding up in the understanding of knowledge as construction is happening in the socio-cultural practices, such research seeks to understand how an employee of the construction build their mathematical knowledge in situations of working at the same time that is the person who is. The research is to be related with the prospect Ethnomathematics, since this aspect of mathematics education allows new visions on education and on the math, in an attempt to understand the knowledge as a way that people have to be in the world, ahead of reality which experience. The research seeks to be an qualitative approach, because search understand how the phenomenon is constituted for those that experience, their modes of mean it, the concepts that have it.

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.

Made available in DSpace on 2017-07-21T20:31:49Z (GMT). No. of bitstreams: 1 Deise.pdf: 1359946 bytes, checksum: abee968ad9a1d6edb91d46ccb3651239 (MD5) Previous issue date: 2006-11-17 / This research aimed to study the evolution of possible and the construction of the logical-mathematical knowledge through rules games with elementary students, 5th and 6th degrees. The leading questions were: Have they difficulty at learning Mathematics? Do they manage the four basic operations? Do rules game helps the evolution of possible and the construction of the logical-mathematical knowledge? As theoretical reference were used works from Piaget (1972a, 1972b, 1973, 1977, 1985, 1996, 1998), Grando (1995, 2000), Macedo (1992, 1997, 2005a, 2005b), Brenelli (1996), Piantavini (1999) and Ciasca (1994, 2003). The general objective was to show that rule game contributes for the evolution of possible and the construction of the logical-mathematical knowledge. The specific objectives were to identify the operation level, of possible and mathematical knowledge of the research subjects; to realize the educative intervention based on rule games; to show the construction process of logical-mathematical knowledge and the evolution of possible. The methodology used was a case study focused on Piaget´s clinical method. The research was done with seven boys, 12 years old, studying 5th and 6th degree in a public school. Procedures of data collect were: registration on the field diary, movie making, observation and educative intervention with rule games as Mathematical Detective (REISDOEFER, 1999) and Contig 60® (REGATO, 1980, 1986). Analysis and discussion of the data were based on Piaget´s epistemological theory. The results showed that: students had difficulty with four basic mathematical operations, mainly division; they had different operation levels and of possible; the rule games used were efficient for the mathematical learning; it was showed great interest on learning by games, social interaction, evolution of the four basic operation learning and raising of new possible and rules. It was brought to a conclusion that using rule games helped the evolution process of possible and the construction of logical-mathematical knowledge; transcended the game representation as a resource to motivation and fixation of contents; propitiated action, integration, reflection and transposition of learning difficulties, contributing for social inclusion. Keywords: teaching-learning; games; Piaget; logical-mathematical knowledge; level / Esta pesquisa teve por objeto de estudo a evolução dos possíveis e a construção do conhecimento lógico-matemático via jogo de regras em alunos de 5ª e 6ª séries. As questões orientadoras foram: Encontram dificuldade de aprendizagem em Matemática? Dominam as quatro operações fundamentais? O jogo de regras favorece a evolução dos possíveis e a construção do conhecimento lógico-matemático? Teve como referencial teórico obras de Piaget (1972a, 1972b, 1973, 1977, 1985, 1996, 1998), Grando (1995, 2000), Macedo (1992, 1997, 2005a, 2005b), Brenelli (1996), Piantavini (1999) e Ciasca (1994, 2003). O objetivo geral foi demonstrar que o jogo de regras contribui para a evolução dos possíveis e construção do conhecimento lógico-matemático. Especificamente: identificar o nível operatório, de possíveis e de conhecimento matemático dos sujeitos da pesquisa; proceder a intervenção educativa mediada por jogos de regras; demonstrar o processo de construção do conhecimento lógico-matemático e a evolução dos possíveis. Utilizou-se como metodologia o estudo de caso com enfoque no método clínico piagetiano. Participaram da pesquisa sete meninos com idade de 12 anos, alunos de 5ª e 6ª séries de uma escola pública inserida em uma Instituição Abrigo. Os procedimentos de coletas de dados constituíram-se de: registro em diário de campo, filmagem, observação e intervenção educativa com os jogos de regras Detetive Matemático (REISDOEFER, 1999) e Contig 60® (REGATO, 1980, 1986). A análise e discussão dos indicadores fundamentaram-se na teoria epistemológica piagetiana. Os resultados evidenciaram: que os alunos tinham dificuldade com as quatro operações fundamentais, principalmente a divisão; caracterizaram-se por diferentes níveis operatórios e de possíveis; os jogos de regras detetive Matemático e Contig 60® se mostraram eficazes para a aprendizagem de Matemática; destacou-se o interesse pelo lúdico, as antecipações de jogadas, a interação social, a evolução da aprendizagem das quatro operações fundamentais e o surgimento de novos possíveis e novas regras. Conclui-se que o uso do jogo de regras: favoreceu o processo de evolução dos possíveis e a construção do conhecimento lógico-matemático; transcendeu a representação do jogo como recurso para motivação e fixação de conteúdos; propiciou a ação, integração, reflexão e superação de dificuldades de aprendizagem, contribuindo para a inclusão social. Palavras-chave: ensino-aprendizagem, jogos, Piaget, conhecimento lógico-matemático, nível de possíveis.