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Student Use of Mathematical Content Knowledge During Proof ProductionVan de Merwe, Chelsey Lynn 11 June 2020 (has links)
Proof is an important component of advanced mathematical activity. Nevertheless, undergraduates struggle to write valid proofs. Research identifies many of the struggles students experience with the logical nature and structure of proofs. Little research examines the role mathematical content knowledge plays in proof production. This study begins to fill this gap in the research by analyzing what role mathematical content knowledge plays in the success of a proof and how undergraduates use mathematical content knowledge during proofs. Four undergraduates participated in a series of task-based interviews wherein they completed several proofs. The interviews were analyzed to determine how the students used mathematical content knowledge and how mathematical content knowledge affected a proof’s validity. The results show that using mathematical content knowledge during a proof is nontrivial for students. Several of the proofs attempted by the students were unsuccessful due to issues with mathematical content knowledge. The data also show that students use mathematical content knowledge in a variety of ways. Some student use of mathematical content is productive and efficient, while other student practices are less efficient in formal proofs.
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Criteria for effective mathematics teacher education with regard to mathematical content knowledge for teaching / Mariana PlotzPlotz, Mariana January 2007 (has links)
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.
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Criteria for effective mathematics teacher education with regard to mathematical content knowledge for teaching / Mariana PlotzPlotz, Mariana January 2007 (has links)
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.
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The role of the pedagogical content Knowledge in the learning of quadratic functionsIbeawuchi, Emmanuel Ositadinma 06 1900 (has links)
This study investigates to what extent educators’ pedagogical content knowledge affects learners’ achievement in quadratic functions. The components of pedagogical content knowledge (PCK) examined are: (i) mathematical content knowledge (MCK), (ii) knowledge of learners’ conceptions, and misconceptions, and (iii) knowledge of strategies. The participants were seventeen mathematics educators and ten learners from each educator’s class. The sample of educators was a convenient sample, while the sample of learners was selected by means of random sampling. A mixed method design was used to execute the study. Data about educators’ MCK, and knowledge of learners’ misconceptions were collected by means of a questionnaire. An interview was used to gather data about educators’ knowledge of strategies. Data on learners’ achievements and misconceptions was collected by means of a questionnaire. Descriptive statistics were used to describe the effect of each component of the educators’ PCK on learners’ achievements. The result indicates that the achievement of learners who are taught by educators who have strong PCK is higher than the achievement of learners who are taught by educators who have weak PCK. / Mathematical Sciences / M. Ed. (Mathematics Education)
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Yngre barns möte med matematikGustafsson, Liselotte, Runnqvist, Elisabeth, Nathansohn, Teresia January 2009 (has links)
Purpose: The purpose of the study is to find out what mathematical content primary school children encounter in their free options at school. Through observation, the study defines mathematical areas that primary school students encounter in their free options at school. We want the study to show the reader the mathematics that students continuously meet without associating it with regular mathematics as taught in school. A number of mathematical areas have been defined in the analysis of the observations. These areas have subsequently been discussed more thoroughly. Finally, the areas have been arranged in a grid system to clarify the results. In our study, we have discovered that mathematics exists in all the observed situations the students participated in. We believe that observation as a method can give teachers a tool for helping students associate practical actions during their free options with the more theoretical aspects of formal teaching of mathematics. We discuss this further in the study.
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Yngre barns möte med matematikGustafsson, Liselotte, Runnqvist, Elisabeth, Nathansohn, Teresia January 2009 (has links)
<p>Purpose: The purpose of the study is to find out what mathematical content primary school children encounter in their free options at school.</p><p>Through observation, the study defines mathematical areas that primary school students encounter in their free options at school. We want the study to show the reader the mathematics that students continuously meet without associating it with regular mathematics as taught in school.</p><p>A number of mathematical areas have been defined in the analysis of the observations. These areas have subsequently been discussed more thoroughly. Finally, the areas have been arranged in a grid system to clarify the results.</p><p>In our study, we have discovered that mathematics exists in all the observed situations the students participated in.</p><p>We believe that observation as a method can give teachers a tool for helping students associate practical actions during their free options with the more theoretical aspects of formal teaching of mathematics. We discuss this further in the study.</p>
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The role of the pedagogical content Knowledge in the learning of quadratic functionsIbeawuchi, Emmanuel Ositadinma 06 1900 (has links)
This study investigates to what extent educators’ pedagogical content knowledge affects learners’ achievement in quadratic functions. The components of pedagogical content knowledge (PCK) examined are: (i) mathematical content knowledge (MCK), (ii) knowledge of learners’ conceptions, and misconceptions, and (iii) knowledge of strategies. The participants were seventeen mathematics educators and ten learners from each educator’s class. The sample of educators was a convenient sample, while the sample of learners was selected by means of random sampling. A mixed method design was used to execute the study. Data about educators’ MCK, and knowledge of learners’ misconceptions were collected by means of a questionnaire. An interview was used to gather data about educators’ knowledge of strategies. Data on learners’ achievements and misconceptions was collected by means of a questionnaire. Descriptive statistics were used to describe the effect of each component of the educators’ PCK on learners’ achievements. The result indicates that the achievement of learners who are taught by educators who have strong PCK is higher than the achievement of learners who are taught by educators who have weak PCK. / Mathematical Sciences / M. Ed. (Mathematics Education)
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Opportunities for the development of understanding in Grade 8 mathematics classroomsde Jager, Gerdi January 2016 (has links)
Learner performance in South Africa is poor in comparison with other countries as
a result of poor teaching. At the core of the concern about learners' performance in
mathematics in South Africa lies a controversy regarding how mathematics should
be taught. The purpose of this study was to explore Grade 8 mathematics teachers'
creation and utilisation of opportunities for learners to develop mathematical
understanding in their classrooms. To accomplish this, an explorative case study
was conducted to explore three mathematics teachers' instructional practices by
using Schoenfeld et al.'s (2014) five dimensions of Teaching for the Robust
Understanding of Mathematics (TRU Math) scheme, namely, the mathematics,
cognitive demand, access to mathematical content, mathematical agency, authority
and identity and uses of assessment. The three participants were conveniently
selected from three private schools in Mpumalanga. The data collected consist of a
document analysis, two lessons observations and a post-observation interview per
teacher.
This study revealed that only one of the three teachers applied all Schoenfeld et
al.'s (2014) TRU Math dimensions. The dimension identified which the teachers
applied most in their classrooms was the mathematics. The dimensions identified
where teachers still lack skills were cognitive demand, access to mathematical
content, agency, authority and identity, and uses of assessment. This study
revealed that the content of most tasks and lessons was focused and coherent, and
built meaningful connections. However, the content did not engage learners in
important mathematical content or provided opportunities for learners to apply the
content to solve real-life problems. Due to the small sample used, the results from
this study cannot be generalised. However, I hope that the findings will contribute
to student-teacher training and in-service teacher training in both government and
private schools. Future research could possibly build on this study by examining
the learners and how they learn with understanding by using the TRU Math
dimensions. / Dissertation (MEd)--University of Pretoria, 2016. / Science, Mathematics and Technology Education / MEd / Unrestricted
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Exploring mathematical literacy : the relationship between teachers’ knowledge and beliefs and their instructional practicesBotha, Johanna Jacoba 15 February 2012 (has links)
South Africa is the first country in the world to offer Mathematical Literacy as a school subject. This subject was introduced in 2006 as an alternative to Mathematics in the Further Education and Training band. The purpose of this subject is to provide learners with an awareness and understanding of the role that mathematics plays in the modern world, but also with opportunities to engage in real-life problems in different contexts. A problem is the beliefs some people in and outside the classroom have regarding this subject such as teachers believing ML is the dumping ground for mathematics underperformers (Mbekwa, 2007). Another problem is the belief of some principals that any nonmathematics teacher can teach ML. In practice there is Mathematics teachers who teach ML in the same way that they teach Mathematics; non-Mathematics teachers who in many cases lack the necessary mathematical content knowledge and skills to teach ML competently; and Mathematics teachers who adapted their practices to teach ML using different approaches than those required for teaching Mathematics. Limited in-depth research has been done on the ML teachers, what they believe and what knowledge is required to teach this subject effectively and proficiently. The purpose of this study is to investigate the way in which ML is taught in a limited number of classrooms with the view to exploring the relationship between ML teachers’ knowledge and beliefs and their instructional practices. According to Artzt, Armour-Thomas and Curcio (2008) the instructional practice of the teacher plays out in the classroom where teachers’ goals, knowledge and beliefs serve as the driving force behind their instructional efforts to guide and mentor learners in their search for knowledge. To accomplish this aim, an in-depth case study was conducted to explore the nature of teachers’ knowledge and beliefs about ML as manifested in their instructional practices. A qualitative research approach was used in which observations and interviews served as data collection techniques enabling me to interpret the reality as I became part of the lives of the teachers. My study revealed that there is a dynamic but complex relationship between ML teachers’ knowledge and beliefs and their instructional practices. The teachers’ knowledge, but not their stated beliefs were reflected in their instructional practices. Conversely, in one case, the teacher’s instructional practice also had a positive influence on her knowledge and beliefs. It was further revealed that mathematics teacher training and teaching experience played a significant role in the productivity of the teachers’ practices. The findings suggest that although mathematical content knowledge is required to develop PCK, it is teaching experience that plays a crucial role in the development of teachers’ PCK. Although the study’s results cannot be generalised due to the small sample, I believe that the findings concerning the value of teachers’ knowledge and the contradictions between their stated beliefs and practices could possibly contribute to teacher training. Curriculum decision-makers should realise that the teaching of ML requires specially trained, competent, dedicated teachers who value the subject. This exploratory study concludes with recommendations for further research. / Thesis (PhD)--University of Pretoria, 2011. / Science, Mathematics and Technology Education / unrestricted
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Máximos e mínimos na Educação Básica: abordagens elementares sem derivadas / Maxima and minima in Basic Education: elementary approaches without derivativesRocha, Wilian Oliveira 28 May 2019 (has links)
Nosso objetivo com este trabalho é contribuir para o aperfeiçoamento da ação educativa do professor de matemática na Educação Básica, tanto em formação inicial quanto em formação continuada. Apresentamos algumas abordagens elementares para estudo de Máximos e Mínimos que utilizam conteúdos próprios dos anos finais do Ensino Fundamental e do Ensino Médio embasados principalmente na obra Maxima and Minima without Calculus (NIVEN, 1981). Discutimos os conceitos de Conhecimento Especializado e de Horizontes de Conteúdo Matemático como justificativa para a relevância do uso deste material, que foram cunhados por pesquisadores da Universidade de Michigan, liderados por Deborah Ball no artigo Content Knowledge for Teaching: What Makes it Special? (2008). Trazemos uma análise crítica da abordagem utilizada para o tema em alguns livros didáticos de Ensino Médio. Discorremos sobre os quatro conceitos de Médias aritmética, geométrica, harmônica e quadrática partindo de problemas que originaram tais conceitos. Mostramos ainda como podem ser naturalmente associados a medidas de segmentos definidos em quadrados, trapézios e semicírculos que evidenciam claramente certas desigualdades entre elas. A seguir, como aplicação de produtos notáveis e trinômios do segundo grau, apresentamos problemas algébricos e geométricos envolvendo máximos e mínimos e discutimos suas soluções. Estabelecemos e provamos algebricamente as desigualdades entre as quatro médias (de até quatro números positivos), que são aplicadas para a determinação de pontos de máximo ou mínimo de funções variadas em problemas contextualizados. Por fim generalizamos e provamos as desigualdades entre as médias para n números positivos e desenvolvemos várias outras aplicações. / This work intends to be a contribution to the improvement of the educational action of mathematics school teachers in both initial or continuous formation. We present some elementary approaches for the study of Maxima and Minima that use final years of Elementary and High School contents only, mainly based on Ivan Nivens book Maxima and Minima without Calculus (NIVEN, 1981). We discuss the concepts of Specialized and Horizons Knowledge of Mathematical Content as a justification for the relevance of the use of this material, which were been introducted by the University of Michigans researchers, led by Deborah Ball, in the article - Content Knowledge for Teaching: What Makes it Special? (2008). We bring a critical analysis of the approach employed for the topic (maxima and minima) in some high school textbooks. We discuss the four concepts of averages - arithmetic, geometric, harmonic and quadratic - starting from problems that originated them. We also show how they can be naturally associated with measures of segments defined in squares, trapezoids and semicircles so that we can clearly visualise certains inequalities between them. Next, as an application of notable products and of second degree trinomials, we present algebraic and geometric problems of maxima or minima and discuss their solutions. We establish and prove algebraically the inequalities between the four averages (up to four positive numbers), which are applied to determine maximum or minimum points of varied functions in contextualized problems. Finally we generalize and prove the averages inequalities for n positive numbers and we develop several applications.
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