The aim of this study is to investigate how elementary school teachers use conceptual understanding in the teaching of the variable concept, namely, how these teachers connect the variable concept with other areas of mathematics, such as algebraic expressions.  The empirical data was obtained by qualitative methods comprising interviews with five mathematic elementary school teachers. In addition three observations in three classrooms were made; one observation is in grade 4 and the other two are in grade 6.The theoretical framework is based on Kilpatrick et al. (2001) theories: conceptual understanding, strategic competence and procedural fluency. Furthermore the theory of representations of Bergsten et. al (1997) and Persson (2010) were used in the theoretical framework. The results of this study show that none of the five participating teachers connects the variable concept to other areas of mathematics, such as algebraic expressions. However, two of them mention the variable concept in equations, problem-solving, algebraic expressions and shapes, but without explaining the meaning of the variable concept or how that concept is used and integrated into these mentioned areas of mathematics. Furthermore the study shows that the other three participating teachers mention the variable only to one to two areas of mathematics. These areas are problem- solving and shapes. These teachers do not explain how the variable concept is used in the mentioned areas of mathematics, but they focus on how the calculation will be performed to solve the data they used. The participating teachers do not use any representations to explain the variable concept and how it’s connected to the other areas of mathematics. In conclusion, the variable concept was not explained and its connection to other mathematical areas was not addressed, which due to the absence of usage of conceptual understanding in teaching the variable concept.

Made available in DSpace on 2014-06-11T19:31:42Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-13Bitstream added on 2014-06-13T18:42:42Z : No. of bitstreams: 1 olimpiojunior_a_dr_rcla.pdf: 1589834 bytes, checksum: 60dddfc290943f0766aa1097afc56731 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A partir da integração oralidade-escrita-CAS/MAPLE, eu investiguei compreensões emergentes sobre os conceitos de função, limite, continuidade e derivada, produzidas por ingressantes em um curso de Matemática oferecido por uma universidade pública do estado de São Paulo. A investigação, sob o balizamento do paradigma interpretativo e caracterizado pela metodologia qualitativa, desenvolveu-se com a realização de experimentos com oito voluntário(a)s. Os dados para a análise inicial constituíram-se de respostas individuais escritas em linguagem natural e de videotapes das interações entre duplas de participantes e o MAPLE. Esta análise produziu quatro episódios tematizando conflitos emergentes sobre o conceito de derivabilidade, a definição de derivada, o conceito de limite e a comparação entre os gráficos de uma função e de sua derivada. Cinco categorias de interação entre duplas de participantes e o MAPLE foram descritas. Três níveis de compatibilidade entre compreensões materializadas a priori pela escrita e as emergentes da interação participantes-MAPLE foram identificados. A análise inicial sugere que abordagem é apropriada à materialização de tais compreensões. A análise final sugere que os conflitos emergentes poderiam ter suas raízes numa limitada compreensão do conceito de função. A pesquisa também sugere uma maior e mais intensiva exploração da natureza dinâmica do Cálculo Diferencial. / From the integration of orality, writing and the CAS-MAPLE, I investigated understandings that emerge about the concepts of function, limit, continuity and derivative produced by full-time first-year students of mathematics from a public university in the state of São Paulo, Brazil. The research, implemented under the guidelines of the interpretive paradigm and of the qualitative methodology, was characterized by experiments, which were conducted with eight volunteer participants. The data consisted of individual written answers in natural language and videotapes of the interactions between pairs of participants and the MAPLE. The initial analysis is on four episodes focusing on emerging conflicts on the concept of differentiability, the definition of derivative, the concept of limit, and the comparison between the graph of a function f and the graph of its derivative. Five interaction categories between pairs of participants and the MAPLE were described. In addition, three levels of compatibilities between a priori participants' writings and the mentioned interactions were identified. The initial analysis suggests that the chosen approach is appropriate to the materialization of such understandings. The final analysis suggests that the conflicts that emerged from the experiments could have their roots in a limited understanding of the concept of function. The research also suggests a more intensive exploration of the dynamical nature of the differential calculus.

The aim of this study is to investigate how elementary school teachers use conceptual understanding in the teaching of the variable concept, namely, how these teachers connect the variable concept with other areas of mathematics, such as algebraic expressions.  The empirical data was obtained by qualitative methods comprising interviews with five mathematic elementary school teachers. In addition three observations in three classrooms were made; one observation is in grade 4 and the other two are in grade 6.The theoretical framework is based on Kilpatrick et al. (2001) theories: conceptual understanding, strategic competence and procedural fluency. Furthermore the theory of representations of Bergsten et. al (1997) and Persson (2010) were used in the theoretical framework. The results of this study show that none of the five participating teachers connects the variable concept to other areas of mathematics, such as algebraic expressions. However, two of them mention the variable concept in equations, problem-solving, algebraic expressions and shapes, but without explaining the meaning of the variable concept or how that concept is used and integrated into these mentioned areas of mathematics. Furthermore the study shows that the other three participating teachers mention the variable only to one to two areas of mathematics. These areas are problem- solving and shapes. These teachers do not explain how the variable concept is used in the mentioned areas of mathematics, but they focus on how the calculation will be performed to solve the data they used. The participating teachers do not use any representations to explain the variable concept and how it’s connected to the other areas of mathematics. In conclusion, the variable concept was not explained and its connection to other mathematical areas was not addressed, which due to the absence of usage of conceptual understanding in teaching the variable concept.

Orientador: Marcelo de Carvalho Borba / Banca: Arthur Belford Powell / Banca: Nilson José Machado / Banca: Edna Maura Zuffi / Banca: Renata Zotin Gomes de Oliveira / Resumo: A partir da integração oralidade-escrita-CAS/MAPLE, eu investiguei compreensões emergentes sobre os conceitos de função, limite, continuidade e derivada, produzidas por ingressantes em um curso de Matemática oferecido por uma universidade pública do estado de São Paulo. A investigação, sob o balizamento do paradigma interpretativo e caracterizado pela metodologia qualitativa, desenvolveu-se com a realização de experimentos com oito voluntário(a)s. Os dados para a análise inicial constituíram-se de respostas individuais escritas em linguagem natural e de videotapes das interações entre duplas de participantes e o MAPLE. Esta análise produziu quatro episódios tematizando conflitos emergentes sobre o conceito de derivabilidade, a definição de derivada, o conceito de limite e a comparação entre os gráficos de uma função e de sua derivada. Cinco categorias de interação entre duplas de participantes e o MAPLE foram descritas. Três níveis de compatibilidade entre compreensões materializadas a priori pela escrita e as emergentes da interação participantes-MAPLE foram identificados. A análise inicial sugere que abordagem é apropriada à materialização de tais compreensões. A análise final sugere que os conflitos emergentes poderiam ter suas raízes numa limitada compreensão do conceito de função. A pesquisa também sugere uma maior e mais intensiva exploração da natureza dinâmica do Cálculo Diferencial. / Abstract: From the integration of orality, writing and the CAS-MAPLE, I investigated understandings that emerge about the concepts of function, limit, continuity and derivative produced by full-time first-year students of mathematics from a public university in the state of São Paulo, Brazil. The research, implemented under the guidelines of the interpretive paradigm and of the qualitative methodology, was characterized by experiments, which were conducted with eight volunteer participants. The data consisted of individual written answers in natural language and videotapes of the interactions between pairs of participants and the MAPLE. The initial analysis is on four episodes focusing on emerging conflicts on the concept of differentiability, the definition of derivative, the concept of limit, and the comparison between the graph of a function f and the graph of its derivative. Five interaction categories between pairs of participants and the MAPLE were described. In addition, three levels of compatibilities between a priori participants' writings and the mentioned interactions were identified. The initial analysis suggests that the chosen approach is appropriate to the materialization of such understandings. The final analysis suggests that the conflicts that emerged from the experiments could have their roots in a limited understanding of the concept of function. The research also suggests a more intensive exploration of the dynamical nature of the differential calculus. / Doutor

Problemlösning är en central del av Lgr11 och dagens matematikundervisning. Detta gör att de läromedel som används i de svenska skolorna måste behandla problemlösning om de ska kunna ge eleverna möjlighet att nå de utsatta målen i Lgr11. I årskurs 6 ska eleverna få betyg i matematik vilket gör det viktigt att undersöka om läromedlen når upp till målen i lgr11 runt om problemlösning. Studien är gjord som en läromedelsanalys, av Matte Direkt Borgen 6a och 6b. Min studie är en kvantitativ studie i två delar. Först analyserade jag hur många av de problem som enligt författaren anser är problem är problemuppgifter eller rutinuppgifter. Av det resultatet kunde jag sedan analysera hur väl problemlösningsförmågan, begreppsförståelse och resonemangsförmågan gick att öva via uppgifterna i läromedlet. Resultatet visade att bara 59 av de 137 uppgifter som författaren anser är problemuppgifter gick att klassificera som problemuppgifter. Förmågorna som övades via uppgifterna spelade dock ingen roll om det var problem- eller rutinuppgifter. Här övade alla uppgifter någon av de tre undersökta förmågorna. / Problem solving is a central part of Lgr11 and today's mathematics education. This allows the teaching materials used in the Swedish schools must treat the problem if they are to provide students with the knowledge to reach the vulnerable requirements of Lgr11. In Year 6, pupils get grades in mathematics, making it important to conduct research about learning reaches Lgr11 targets around the problem solving. The study is designed as a teaching material analysis, of Matte Direkt Borgen 6a and 6b. My study is a quantitative study in two parts. First, I analyzed how many of the problems according to the author considers to be problems are problems tasks or routine tasks. Of the result, I could then analyze how well the problem-solving ability, conceptual understanding and reasoning ability went to practice using the information in the teaching material. The results showed that only 59 of the 137 tasks which the author believes is the problem tasks could be classified as problem tasks. The abilities who was exercised by the data, however, does not matter if it was a problem or routine tasks. Here practiced all the information any of the three investigated abilities.

Teaching mathematics towards mathematical proficiency cannot easily be described since mathematical proficiency is complex and no term alone capture all aspects. The purpose of the study was to examine how mathematical proficiency in conceptual understanding can be discriminated through the concept of subtraction in grade 1 and 3 primary school. Question raised by the study was 1. What different levels in quality can be discriminated in the ways students subtracts?   2. Can progression be discriminated within subtraction operations between grade 1 and 3? Concepts like mathematical artefacts, capability and ability have been useful tools when analyzing the material. Developing mathematical thinking in students is about developing conceptual understanding anchored in the process rather than focus on procedures and outcome. When subtracting with comprehension one knows that subtraction undoes addition since subtraction is the inverse operation of addition.      The methods used are qualitative studies using observations and interviews. The material consists of notes from observations, recordings of interviews and written solutions. The study developed an analysis tool in order to compare the different ways of thinking when doing subtraction activities. Developing the tool was critical in order to compare and discriminate different levels of mathematical thinking. The results indicate that the Swedish curricula in mathematics do not support mathematical teaching towards mathematical proficiencies because it’s lack of concrete measurable goals.

In our study, the problem area is that concept teaching in mathematics is too low to be able to achieve mathematical understanding. Mathematics also has two different parlance ​​that make teaching difficult and create a lack of clarity about which one of these parlances to use at which occasion and how these parlances ​​should be interpreted and how they are linked.  We have conducted an interview study with semi-structured interviews. The respondents consist of 16 teachers who were selected through our inclusion- and exclusion criteria. The material was transcribed, coded, and categorized, and we used the socio-cultural perspective as a theoretical framework to analyse the material.  One conclusion we can draw from the results of this interview study is that the teachers in this study express that they are aware that cooperative learning is beneficial for students' understanding of concepts, but that organizational problems such as lack of planning time, too few educators and too large classes make it is difficult to conduct a developing concept teaching. This can be a reason why the textbook takes up a lot of space in the classroom. Another conclusion we can draw is that the teachers in this study express that they have subject knowledge and that their pedagogical knowledge at certain times is not sufficient, especially when it comes to supporting students who need extra support and students with Swedish as a second language. Another conclusion we can draw from the results is that the students have difficulty transferring the concrete to the abstract in mathematics. This may be because the language used by the students is the everyday one, which does not correspond to what the students encounter in the textbooks and on tests. As the students get the smallest speaking space, they also rarely get the chance to use any of the language. The use of everyday language is not advocated in the socio-cultural perspective, but can be questioned when students do not encounter the correct language in everyday life, and can become an element in the teaching that can not relate to everyday life.

The learning difficulties prevalent among mathematics students are widely documented. This case study explores the difficulties experienced by National Certificate (Vocational) Level 2 mathematics students at a Technical and Vocational Education and Training (TVET) College in Gauteng in the learning of functions. The primary research question was: What conceptual and procedural knowledge difficulties do NC(V) L2 students experience when learning the concept of functions in mathematics? Qualitative data was generated from the students (n=17) through lesson observations, test responses and interviews. The convenient sample of students all belonged to one L2 mathematics class. The findings revealed that the students experience conceptual knowledge difficulties in terms of defining a function, identifying functions, translating between representations of functions, and interpreting the behaviour of a function. The findings also revealed procedural knowledge difficulties prevalent among the students. The errors that students committed in this regard included factorisation errors, structural errors, misapplication and modification of the rules. In general, the findings indicated that the students in this sample lack procedural knowledge and conceptual understanding of functions. The lesson observations revealed a prevalence of several misconceptions regarding functions among the students which were seemingly not recognised nor remedied by the teacher. The teacher’s instructional practices thus appeared to be among the possible sources of the difficulties that the students experience in the learning of functions. This finding was also confirmed by the students during the interviews. Shaky foundations from previous grades were also a factor. / Dissertation (MEd)--University of Pretoria, 2020. / Science, Mathematics and Technology Education / MEd / Unrestricted

Eftersom läroboken har en dominerande roll i den svenska matematikundervisning är det av vikt att läroböckernas innehåll är av god kvalitet och arbetar mot de mål som skolans styrdokument strävar mot. Begreppsförståelsen kan förstås som grunden i det matematiska tänkandet. Denna förmåga utvecklas i takt med att eleven får nya erfarenheter och kunskaper kring matematiska problem i olika sammanhang. Ett möte med uttrycksformer av stor variation skapar erfarenheter och hjälper eleven utveckla sin begreppsförmåga. Syftet med denna studie är att få en förståelse för vilka uttrycksformer eleven får möta i det marknadsdominerande läromedlet för grundskolans lägre åldrar samt hur eleven får möjlighet att använda uttrycksformerna som ett verktyg i utvecklandet av begreppsförmågan. Resultatet av studien visar att eleverna får få tillfällen att möta olika uttrycksformer och lära sig använda dem som ett stöd för att utveckla sin begreppsförståelse i matematik. Studien belyser även vikten av elevers kunskap att översätta uttrycksformer till en annan uttrycksform för att erhålla en bredare förståelse för begreppens betydelse. / Since the textbook has a dominant role in Swedish mathematics education, it is important that content is in good quality and works towards the goal of Swedish education. Conceptual understanding can be understood as the basis of mathematical thinking and develops as the student gains new experiences and knowledge of mathematical problems in different contexts. A meeting with a variety of representations can create new experiences and help the student develop a god conceptual understanding. The purpose of this study is to understand what forms of representations the pupils may encounter in the market-dominant textbook for the elementary school. As well the possibility the students have using representation as a tool to create conceptual understanding. The results of the study show that the students are given few opportunities to meet a variation of representations and learn how to use them as a support to gain the conceptual understanding. The study also highlights the importance of students’ ability to translate representations to gain a broader conceptual understanding.

For this action research project, I wanted to examine my practice of teaching mathematics. Specifically, I encouraged students to improve their communication skills during my math class through daily discussion and writing tasks. After establishing a class set of sociomathematical norms, the students solved problems provided by the Every Day Counts: Calendar Math program and used verbal and written formats to describe their problem solving methods and reasons. My study showed the effects of using discussion and writing to help students develop their conceptual understanding of mathematical ideas. Focus was placed on the quality of daily discussions and written tasks both at the beginning of the study and continually as the study progressed. Through daily discussions, monthly written assessments, and student interviews, the study helped to determine the importance of developing students' mathematical communication skills and building conceptual understanding of mathematical ideas.