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Preparing students to be life-long learners that are career and college ready is a goal of agricultural education. Changing expectations of education have pointed to agriculture educators as potential leaders in the STEM education movement. Literature related to STEM education in agricultural education is lacking in guidance for teachers, administrators, and curriculum developers in integrating academic content related to STEM content areas. A review of STEM education literature coupled with the framework of quantitative reasoning, lead to a conceptualization of a framework for integrative agricultural education. This framework was implemented through a case study to investigate collaborative efforts in curriculum development in agricultural education with a specific focus on integrating mathematics to develop students' quantitative reasoning skills. Teacher characteristics were identified that seemed to support the implementation of integrative agricultural education practices. Teaching and planning strategies were also identified in the case study. Recommendations suggest support of collaboration between agriculture and mathematics teachers would best support curriculum design and aid in the quality of instruction that follows. / Ph. D.

Esta pesquisa situa-se no campo dos processos cognitivos subjacentes à aprendizagem da matemática. Procurou-se avaliar a relação entre o raciocínio quantitativo e a memória de trabalho na aprendizagem da matemática em 30 alunos da 4ª série do Ensino Fundamental, com idades entre 9 e 10 anos em duas escolas públicas de Porto Alegre. Os participantes foram divididos igualmente em dois grupos, um com alto desempenho em matemática e outro com baixo desempenho em matemática. Avaliamos os processos cognitivos dos grupos através de tarefas que envolveram: desempenho matemático (DM), memória de trabalho (MT), memória de curto prazo (MCP), habilidades numéricas (HN) e raciocínio quantitativo (RQ). Os resultados foram analisados quanti-qualitativamente. Para verificar as correlações entre as funções avaliadas, foi utilizado o teste não paramétrico de Spearman. O grupo de alunos com alto desempenho em matemática apresentou melhores resultados em todas as funções avaliadas nessa pesquisa. Tal resultado sugere que todas essas habilidades desempenham um papel essencial na aprendizagem da matemática. Além disso, foram utilizadas medidas estatísticas para verificar a significância das diferenças entre os grupos. Os resultados apontam que houve diferença estatisticamente significativa entre os grupos quanto às funções avaliadas, exceto a MT. Destacamos o fato de que, o RQ foi a única função a correlacionar-se de forma estatisticamente significativa com todas as demais. A correlação entre RQ e MT foi estatisticamente significativa (p ≤ 0,05) ressaltando que os recursos da MT são importantes para o RQ, uma vez que evitam sobrecarga e garantem fluidez no raciocínio. Os resultados do estudo oferecem uma importante implicação educacional: a necessidade de incluir-se, ao longo do Ensino Fundamental, tarefas escolares que tenham em vista competências conceituais e procedurais, como tarefas desafiadoras em que os alunos devam manter a atenção, raciocinar quantitativamente demonstrando flexibilidade e adaptação na utilização de aprendizagens anteriores. / This research stands on the underlying cognitive processes and Math learning field. It intended to evaluate the relation between quantitative reasoning and working memory on the math learning with 30 elementary school fourth-grade students, with ages varying from 9 to 10 years in two Porto Alegre public schools. Participants were divided equally into two groups, one with high performance in mathematics, and other with low performance in mathematics. We evaluated the groups cognitive processes through tasks which involved: mathematical achievement (MA), working memory (WM), short-term memory (STM), numeracy skills (NS), and quantitative reasoning (RQ). Results were analyzed quantiqualitatively. The Spearman non-parametric test was used in order to verify the correlations among the evaluated functions. In this research, the students group with good performance in Mathematics presented better results in all evaluated functions. This result suggests that all these skills play an essential role on math learning. Statistical measures were used to verify the significance of the differences between the groups. Results indicate that there was statistically significant difference between the groups in relation to all evaluated function, except WM. We emphasize the fact that QR was the only function to be correlated in a statistically significant way with all the other functions. Correlation between QR and WM was statistically significant (p ≤ 0,05) highlighting the fact that the WM resources are important for QR, since they avoid overload and guarantee fluidity of reasoning. The study results offer an important educational implication: the need to include, along the elementary school, academic tasks that aim conceptual and procedural competencies, as challenging tasks in which the students must maintain attention, reason quantitatively, showing flexibility and adaptation in the use of early learnings.

Esta pesquisa situa-se no campo dos processos cognitivos subjacentes à aprendizagem da matemática. Procurou-se avaliar a relação entre o raciocínio quantitativo e a memória de trabalho na aprendizagem da matemática em 30 alunos da 4ª série do Ensino Fundamental, com idades entre 9 e 10 anos em duas escolas públicas de Porto Alegre. Os participantes foram divididos igualmente em dois grupos, um com alto desempenho em matemática e outro com baixo desempenho em matemática. Avaliamos os processos cognitivos dos grupos através de tarefas que envolveram: desempenho matemático (DM), memória de trabalho (MT), memória de curto prazo (MCP), habilidades numéricas (HN) e raciocínio quantitativo (RQ). Os resultados foram analisados quanti-qualitativamente. Para verificar as correlações entre as funções avaliadas, foi utilizado o teste não paramétrico de Spearman. O grupo de alunos com alto desempenho em matemática apresentou melhores resultados em todas as funções avaliadas nessa pesquisa. Tal resultado sugere que todas essas habilidades desempenham um papel essencial na aprendizagem da matemática. Além disso, foram utilizadas medidas estatísticas para verificar a significância das diferenças entre os grupos. Os resultados apontam que houve diferença estatisticamente significativa entre os grupos quanto às funções avaliadas, exceto a MT. Destacamos o fato de que, o RQ foi a única função a correlacionar-se de forma estatisticamente significativa com todas as demais. A correlação entre RQ e MT foi estatisticamente significativa (p ≤ 0,05) ressaltando que os recursos da MT são importantes para o RQ, uma vez que evitam sobrecarga e garantem fluidez no raciocínio. Os resultados do estudo oferecem uma importante implicação educacional: a necessidade de incluir-se, ao longo do Ensino Fundamental, tarefas escolares que tenham em vista competências conceituais e procedurais, como tarefas desafiadoras em que os alunos devam manter a atenção, raciocinar quantitativamente demonstrando flexibilidade e adaptação na utilização de aprendizagens anteriores. / This research stands on the underlying cognitive processes and Math learning field. It intended to evaluate the relation between quantitative reasoning and working memory on the math learning with 30 elementary school fourth-grade students, with ages varying from 9 to 10 years in two Porto Alegre public schools. Participants were divided equally into two groups, one with high performance in mathematics, and other with low performance in mathematics. We evaluated the groups cognitive processes through tasks which involved: mathematical achievement (MA), working memory (WM), short-term memory (STM), numeracy skills (NS), and quantitative reasoning (RQ). Results were analyzed quantiqualitatively. The Spearman non-parametric test was used in order to verify the correlations among the evaluated functions. In this research, the students group with good performance in Mathematics presented better results in all evaluated functions. This result suggests that all these skills play an essential role on math learning. Statistical measures were used to verify the significance of the differences between the groups. Results indicate that there was statistically significant difference between the groups in relation to all evaluated function, except WM. We emphasize the fact that QR was the only function to be correlated in a statistically significant way with all the other functions. Correlation between QR and WM was statistically significant (p ≤ 0,05) highlighting the fact that the WM resources are important for QR, since they avoid overload and guarantee fluidity of reasoning. The study results offer an important educational implication: the need to include, along the elementary school, academic tasks that aim conceptual and procedural competencies, as challenging tasks in which the students must maintain attention, reason quantitatively, showing flexibility and adaptation in the use of early learnings.

abstract: This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change. / Dissertation/Thesis / Ph.D. Mathematics 2012

Esta pesquisa situa-se no campo dos processos cognitivos subjacentes à aprendizagem da matemática. Procurou-se avaliar a relação entre o raciocínio quantitativo e a memória de trabalho na aprendizagem da matemática em 30 alunos da 4ª série do Ensino Fundamental, com idades entre 9 e 10 anos em duas escolas públicas de Porto Alegre. Os participantes foram divididos igualmente em dois grupos, um com alto desempenho em matemática e outro com baixo desempenho em matemática. Avaliamos os processos cognitivos dos grupos através de tarefas que envolveram: desempenho matemático (DM), memória de trabalho (MT), memória de curto prazo (MCP), habilidades numéricas (HN) e raciocínio quantitativo (RQ). Os resultados foram analisados quanti-qualitativamente. Para verificar as correlações entre as funções avaliadas, foi utilizado o teste não paramétrico de Spearman. O grupo de alunos com alto desempenho em matemática apresentou melhores resultados em todas as funções avaliadas nessa pesquisa. Tal resultado sugere que todas essas habilidades desempenham um papel essencial na aprendizagem da matemática. Além disso, foram utilizadas medidas estatísticas para verificar a significância das diferenças entre os grupos. Os resultados apontam que houve diferença estatisticamente significativa entre os grupos quanto às funções avaliadas, exceto a MT. Destacamos o fato de que, o RQ foi a única função a correlacionar-se de forma estatisticamente significativa com todas as demais. A correlação entre RQ e MT foi estatisticamente significativa (p ≤ 0,05) ressaltando que os recursos da MT são importantes para o RQ, uma vez que evitam sobrecarga e garantem fluidez no raciocínio. Os resultados do estudo oferecem uma importante implicação educacional: a necessidade de incluir-se, ao longo do Ensino Fundamental, tarefas escolares que tenham em vista competências conceituais e procedurais, como tarefas desafiadoras em que os alunos devam manter a atenção, raciocinar quantitativamente demonstrando flexibilidade e adaptação na utilização de aprendizagens anteriores. / This research stands on the underlying cognitive processes and Math learning field. It intended to evaluate the relation between quantitative reasoning and working memory on the math learning with 30 elementary school fourth-grade students, with ages varying from 9 to 10 years in two Porto Alegre public schools. Participants were divided equally into two groups, one with high performance in mathematics, and other with low performance in mathematics. We evaluated the groups cognitive processes through tasks which involved: mathematical achievement (MA), working memory (WM), short-term memory (STM), numeracy skills (NS), and quantitative reasoning (RQ). Results were analyzed quantiqualitatively. The Spearman non-parametric test was used in order to verify the correlations among the evaluated functions. In this research, the students group with good performance in Mathematics presented better results in all evaluated functions. This result suggests that all these skills play an essential role on math learning. Statistical measures were used to verify the significance of the differences between the groups. Results indicate that there was statistically significant difference between the groups in relation to all evaluated function, except WM. We emphasize the fact that QR was the only function to be correlated in a statistically significant way with all the other functions. Correlation between QR and WM was statistically significant (p ≤ 0,05) highlighting the fact that the WM resources are important for QR, since they avoid overload and guarantee fluidity of reasoning. The study results offer an important educational implication: the need to include, along the elementary school, academic tasks that aim conceptual and procedural competencies, as challenging tasks in which the students must maintain attention, reason quantitatively, showing flexibility and adaptation in the use of early learnings.

abstract: This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

This study examined the use of ranking task exercises in physics as a means to elicit student's quantitative and/or qualitative understanding of four different physics concepts. Each ranking task exercise in physics asked students to examine several different scenarios that contain a number of quantitative features and then arrange the scenarios in an ordered sequence according to some other quantitative feature. In this study, students completed four different ranking task exercises as part of their coursework in their high school physics class. The responses of students to these ranking task exercises were scored, analyzed, and categorized according to the extent to which a student's response was primarily quantitative or primarily qualitative in nature. The results show that while students relied on a combination of both qualitative and quantitative representations as they completed the exercises, the majority of students used qualitative representations in their solutions to the ranking task exercises in physics. While the students' qualitative and quantitative representations supported the students' rankings of the scenarios in each ranking task exercise, the qualitative representations used by the students provided insight into the student's current understanding of the physics concepts being investigated. The findings suggest that regardless of the representation used by the student to complete the ranking task exercise, students had difficulty in correctly ranking the scenarios in all of the ranking task exercises used in this study. While the students used both quantitative and qualitative representations in their solutions to ranking task exercises in physics that contained two quantitative variables, the study found that students relied exclusively on qualitative representations in their solutions to the ranking task exercise in physics that contained four quantitative variables. / CITE/Mathematics and Science Education

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