On Aspects of Mathematical Reasoning : Affect and Gender

This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning. 

Teachers' conceptions about students' mathematical reasoning : Gendered or not?

This study looks at how upper secondary school teachers gender stereotype aspects of students' mathematical reasoning. Girls were attributed gender symbols including insecurity, use of standard methods and imitative reasoning. Boys were assigned the symbols such as multiple strategies especially on the calculator, guessing and chance-taking. 

Writing in the Mathematics Classroom: Does It Have an Effect on Students’ Mathematical Reasoning?

In this action research study of my classroom of 5th grade mathematics, I investigate how to improve students’ written explanations to and reasoning of math problems. For this, I look at journal writing, dialogue, and collaborative grouping and its effects on students’ conceptual understanding of the mathematics. In particular, I look at its effects on students’ written explanations to various math problems throughout the semester. Throughout the study students worked on math problems in cooperative groups and then shared their solutions with classmates. Along with this I focus on the dialogue that occurred during these interactions and whether and how it moved students to a deeper level of conceptual understanding. Students also wrote responses about their learning in a weekly math journal. The purpose of this journal is two-fold. One is to have students write out their ideas. Second, is for me to provide the students with feedback on their responses. My research reveals that the integration of collaborative grouping, journaling, and active dialogue between students and teacher helps students develop a deeper understanding of mathematics concepts as well as an increase in their confidence as problem solvers. The use of journaling, dialogue, and collaborative grouping reveals themselves as promising learning tasks that can be integrated in a mathematics curriculum that seeks to cultivate students’ thinking and reasoning.

Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students' problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.

Differential effects of single -sex versus coed education on the mathematical reasoning ability, verbal reasoning ability, and self-concept of high school girls

This study explored the differential effects of single-sex versus coed education on the cognitive and affective development of young women in senior year of high school. The basic research question was: What are the differential effects of single-sex versus coed education on the development of mathematical reasoning ability, verbal reasoning ability, or self-concept of high school girls?^ This study was composed of two parts. In the first part, the SAT verbal and mathematical ability scores were recorded for those subjects in the two schools from which the sample populations were drawn. The second part of the study required the application of the Piers-Harris Children's Self-Concept Scale to subjects in each of the two sample populations. The sample schools were deliberately selected to minimize between group differences in the populations. One was an all girls school, the other coeducational.^ The research design employed in this study was the causal-comparative method, used to explore causal relationships between variables that already exist. Based on a comprehensive analysis of the data produced by this research, no significant difference was found to exist between the mean scores of the senior girls in the single-sex school and the coed school on the SAT 1 verbal reasoning section. Nor was any significant difference found to exist between the mean scores of the senior girls in the single-sex school and the coed school on the SAT 1 mathematical reasoning section. Finally, no significant difference between the mean total scores of the senior girls in the single-sex school and the coed school on the Piers-Harris Children's Self-Concept Scale was found to exist.^ Contrary to what many other studies have found in the past about single-sex schools and their advantages for girls, this study found no support for such advantages in the cognitive areas of verbal and mathematical reasoning as measured by the SAT or in the affective area of self-concept as measured by the Piers-Harris Children's Self-Concept Scale. ^

Brief report: preliminary proposal of a conceptual model of a digital environment for developing mathematical reasoning in students with Autism Spectrum Disorders

There is clear evidence that in typically developing children reasoning and sense-making are essential in all mathematical learning and understanding processes. In children with autism spectrum disorders (ASD), however, these become much more significant, considering their importance to successful independent living. This paper presents a preliminary proposal of a digital environment, specifically targeted to promote the development of mathematical reasoning in students with ASD. Given the diversity of ASD, the prototyping of this environment requires the study of dynamic adaptation processes and the development of activities adjusted to each user’s profile. We present the results obtained during the first phase of this ongoing research, describing a conceptual model of the proposed digital environment. Guidelines for future research are also discussed.

Using Non-Traditional Activities to Enhance Mathematical Connections

In this action research study of my classroom of seventh grade mathematics, I investigated the use of non-traditional activities to enhance mathematical connections. The types of nontraditional activities used were hands-on activities, written explanations, and oral communication that required students to apply a new mathematical concept to either prior knowledge or a realworld application. I discovered that the use of non-traditional activities helped me reach a variety of learners in my classroom. These activities also increased my students’ abilities to apply their mathematical knowledge to different applications. Having students explain their reasoning during non-traditional activities improved their communications skills, both orally and in writing. As a result of this research, I plan to incorporate more non-traditional activities into my curriculum. In doing so, I hope to continue to increase my students’ abilities to solve problems. I also plan to incorporate the use of written explanations of my students’ mathematical reasoning in order to continue to improve their communication of mathematics.

A resolução de problemas e o cálculo mental na educação pré-escolar e no ensino do 1.º ciclo do ensino básico: uma reflexão em contexto de estágio

Mestrado (PES II), Educação Pré-Escolar e Ensino do 1º Ciclo do Ensino Básico, 1 de Julho de 2014, Universidade dos Açores.

O jogo e o desenvolvimento do raciocínio matemático na educação pré-escolar e no ensino do 1.º ciclo do ensino básico

Mestrado (PES II), Educação Pré-Escolar e Ensino do 1º Ciclo do Ensino Básico, 13 de Fevereiro de 2015, Universidade dos Açores.

Explorar e aprender : os materiais didáticos no contexto da aprendizagem da Matemática na Educação Pré-escolar e no 1.º Ciclo do Ensino Básico

Mestrado (PES II), Educação Pré-Escolar e Ensino do 1º Ciclo do Ensino Básico, 17 de Junho de 2015, Universidade dos Açores.

A survey about experiments on notions of time in children k-1 and k-2

In this paper we explore the importance of analyzing the exercises that the manuals have in Mathematics study, because the difficulty of identifying some errors on them can interfere with the capabilities of children. We work with some exercises related to the theme of temporal notions, based on a survey of textbooks from the 1st and 2nd grade (K-1 and K-2). Our concern is to alert about the importance of reflecting on the content of the books, in order to promote a teaching-learning process tailored to the needs of children. The activities present in the manuals should allow children to develop their logical- mathematical reasoning, for later be able to understand and apply Mathematics. To this end, we present some reflection about the exercises of manuals, and we give our opinion about what is the correct and incorrect. Also, some activities are suggested, among which were implemented with children of the 2nd grade, K- 2, along the experiments that support our work.

Da resolução de problemas à explicitação do raciocínio matemático : uma experiência em contexto de estágio

A resolução de problemas é um processo fundamental na aprendizagem da matemática. Neste artigo, apresenta-se uma reflexão sobre a importância deste processo matemático e de como ele pode ser conduzido de forma a estimular o raciocínio matemático através da promoção da comunicação, em contexto de sala de aula. O trabalho foi realizado na etapa final de formação de educadores e professores no contexto do pré-escolar e do primeiro ciclo do ensino básico. Em resultado das atividades realizadas, discute-se o papel da utilização de uma heurística ao longo da resolução de problemas, a importância na escolha de estratégia para a interação com os alunos, bem como o desenho intencional de materiais didáticos. A experiência enquadra-se numa abordagem qualitativa de design de experiência de ensino.

O jogo no ensino da matemática : explorações no contexto da educação Pré-escolar e do 1.º ciclo do Ensino Básico

Mestrado (PES II), Educação Pré-Escolar e Ensino do 1.º Ciclo do Ensino Básico, 17 de Junho de 2015, Universidade dos Açores.