This thesis reports on a qualitative study in which three fifth grade children were presented with six nonroutine mathematical problems involving six different 3-D pyramids, constructed out of multi-link cubes1. The children were videotaped while they worked without any adult help as a cooperative group to solve the pyramid problems. During these sessions, the students produced various 3-D cube models, 2-D drawings, and written records of arithmetic calculations as their solutions to the six problems. Through the lens of enactivism, this study describes and interprets the coevolutionary processes of the group's path of mathematical understandings as it unfolded during the six videotaped sessions. The results revealed building, drawing, and numbering as modes of representation of this group's problem solving work. An analysis of these three modes of representation explored the co-emergence of the children's individual and collective understandings, as well as the interrelationships which existed between their spatial structuring and their use of numerical operations in solving the pyramid problems. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

Understanding the processes that allow all students to successfully learn mathematics has been an important objective for most education systems including those in Canada. Educational systems however, have not achieved this goal as many students with low socioeconomic status, females, and minority students fail to achieve an adequate knowledge of mathematics. Much of the discussion regarding this lack of achievement concerns classroom resources and practices, school policies within educational systems, and the specific domain of mathematics achievement considered. This study conceptualizes a successful mathematics classroom in terms of its level of mathematics achievement and how equitably achievement is distributed. The study employs multilevel models and the Canadian data from the Third International Mathematics and Science Study to address three main research issues: 1) the extent to which differences in mathematics achievement is attributable to gender, family background, classrooms, and the province where a student attends school; 2) whether the variation in achievement is specific to a mathematics domain; and 3) whether the variation among six provinces (Newfoundland, New Brunswick, Ontario, Alberta, British Columbia, and Quebec) in the levels of their mathematics achievement is associated with various aspects of school policy and practices. The analyses indicate a slight male advantage in mathematics achievement, and a large, significant gap in achievement associated with the socioeconomic status (SES) of the students' families. Students from low SES backgrounds are disadvantaged as they tend to have relatively low achievement in mathematics within classrooms, especially in Proportionality, Measurement, and Fractions. The most successful classrooms are those in which students from disadvantaged backgrounds excel in mathematics. Disadvantaged students excel in mathematics classrooms in which there are fewer groupings, the mathematics teachers are specialized, and in schools with lower pupil-teacher ratio. Mathematics achievement is equitably distributed in provinces with high mathematics achievement levels. Provincial achievement levels are stable across mathematics domains; that is, provinces with high achievement levels in one domain also tend to have high achievement levels in other domains. On average, Quebec's mathematics achievement is higher than the other provinces in all mathematics domains, and at all levels of SES. This high achievement level in Quebec is partially attributed to higher teacher specialization, lower pupil-teacher ratio, and lower withinschool remedial tracking. The study recommends a comprehensive longitudinal study employing multilevel models with a focus on what other provinces can learn from Quebec's advantage in mathematics. Such a study should conceptualize successful mathematics classrooms as those in which an average student excels in mathematics and where mathematics achievement is equitably distributed. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

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This project will provide support for teachers who have solid understanding of math as the goal for the students in their classrooms. Using the district adopted course of study as the foundation, this project will provide a curriculum supplement for the first grade.

Mathematical connection has received increasing attention and become one major goal in mathematics education. Two types of connections are distinguished: (a) between-concept connection, which cuts across two concepts; and (b) within-concept connection, which links two representations of one concept. For example, from the theoretical probability to experimental probability is a between-concept connection; generate a graph of a circle from its equation is a within-concept connection. Based on the directionality, unidirectional and bidirectional connections are discerned. Bidirectional connection portrays a pair of a typical and a reverse connection. The benefits of connections, especially bidirectional connections, are widely endorsed. However, researchers indicated that students and even teachers usually make unidirectional connections, and underlying reasons may be the curriculum and cognitive aspects. Previous studies have reported differences in learning opportunities for bidirectional connections in U.S. and Chinese textbook problems, but few have explored the high school level. This study addressed this issue by comparing the directionality of mathematical connections and textbook-problem features in popular U.S. (the UCSMP series) and Chinese (the PEP-A series) high school mathematics textbook problems. The results indicated that the between-concept condition and unidirectional connections dominated textbook problems. Mathematical topic, contextual feature, and visual feature were most likely to contribute to different conditions of connections. Overall, problems dealing with quadratic relations from Chinese textbooks presented a vigorous network of more unique and total between-concept connections with balanced typical and reverse directions than the U.S. counterparts. Problems from U.S. textbooks showed a denser network of (a) within-concept connections in two topics and (b) between-concept connections in probability and combinatorics than the Chinese counterparts, but still exhibited an emphasis on specific concepts, representations, and directionality. The study reached a generalized statement that the new-to-prior knowledge direction was largely overlooked in textbook problems. The results have implications for adopting graph theory and Social Network Analysis to visualize and evaluate mathematical connections and informing mathematics teachers and textbook authors to pay attention to the new-to-prior knowledge connection.

Mathematics of the Islamic medieval world is still not sufficiently studied. As a result, a goldmine of Islamic medieval books and materials lie unexplored. One manuscript that certainly deserves attention is al-Bāhir fi’l-Jabr (The Shining Treatise on Algebra) of al-Samaw’al ibn Yahya al-Maghribi, a twelfth century mathematician. Al-Bāhir fi’l-Jabr is a manuscript written in Arabic and has never been translated except for a few excerpts in French. The purpose of this study was to explore the mathematical and pedagogical contribution of al-Samaw’al through an analysis of al-Samaw’al’s mathematical techniques and methods in al-Bāhir fi’l-Jabr. Moreover, the treatise provides a precise description of the “arithmetization of algebra”, and gives an accounting of the original ideas of another mathematician, al-Karaji, whose original documents have been lost. To develop a comprehensive picture of al-Samaw’al’s mathematical techniques and methods in al-Bāhir fi’l-Jabr (and his contribution to algebra in particular) this research has been based mainly on a careful analysis of al-Samaw’al’s manuscript in MSS Aya Sofia numbered 2718 (116ff), stored in the Suleymaniye library (Istanbul, Turkey). This study of al-Bāhir fi’l-Jabr focuses on an overview of how al-Samaw’al dealt with signed numbers, exponents and polynomial operations. Furthermore, this study describes the al-Samaw’al’s “method of the tables,” certain algorithms he employed, as well as his work on the binomial theorem, binomial coefficients, and the tabular arrangement known today as Pascal’s triangle. Most importantly, the study attempts to show the pedagogical approaches of al-Samaw’al.

The study investigated the effectiveness of Problem-based learning (PBL) in the Further Mathematics classrooms in Nigeria within the blueprint of pre-test-post-test non-equivalent control group quasi-experimental design. The target population consisted of all Further Mathematics students in the Senior Secondary School year one in Ijebu division of Ogun State, Nigeria. Using purposive and simple random sampling techniques, two schools were selected from eight schools that were taking Further Mathematics. One school was randomly assigned as the experimental while the other as the control school. Intact classes were used and in all, 96 students participated in the study (42 in the experimental group taught by the researcher with the PBL and 54 in the control group taught by the regular Further Mathematics teacher using the Traditional Method (TM)). Four research questions and four research hypotheses were raised, answered, and tested in the study. Four research instruments namely pre-test manipulated at two levels: Researcher-Designed Test (RDT) (r = 0.87) and Teacher- Made Test (TMT) (r = 0.88); post-test manipulated at two levels: RDT and TMT; pre-treatment survey of Students Beliefs about Further Mathematics Questionnaire (SBFMQ) (r = 0.86); and post-treatment survey of SBFMQ were developed for the study. The study lasted thirteen weeks (three weeks for pilot study and ten weeks for main study) and data collected were analysed using Mean, Standard deviation, Independent Samples t-test statistic, and Analysis of Variance. Results showed that there were statistically significant differences in the mean post-test achievement scores on TMT (t=-3.58, p<0.05), mean post-test achievement scores on RDT (t=-5.92, p<0.05) and mean post-treatment scores on SBFMQ (t=-6.22, p<0.05) between students exposed to the PBL and those exposed to the TM, all in favour of the PBL group. Results also revealed that there was statistically significant difference in the post-test achievement scores on TMT at knowledge (t= -23.97, p<0.05) and application (t= -11.41, p<0.05) but not at comprehension (t= -0.50, p>0.05, ns) levels of cognition between students exposed to the PBL and the TM. Based on the results, the study recommended that the PBL should be adopted as alternative instructional strategy to the TM in enhancing meaningful learning in Further Mathematics classrooms and efforts should be made to integrate the philosophy of PBL into the pre-service teachers’ curriculum at the teacher-preparation institutions in Nigeria. / Science and Technology Education / D. Phil. (Mathematics, Science and Technology Education)

This thesis is the partial product of a project begun in an effort to build a unified course in advanced high school mathematics beyond the first two years of elementary study. It is a course intended for the academic student who is preparing for the fields of engineering, the sciences, or pure mathematics. It is a course aimed at a unification and greater correlation of the various courses and topics, and at an association on the various mathematical processes involved in the various fields. It is also a course aimed at mastery of mathematics and of its operations as a whole, as a unity not as separate compartments. Furthermore, it is a course which aims to give individual instruction in a classroom situation and to meet as nearly as possible the individual needs and abilities of the students in the class. For the present, attention has been focused upon the problem of unification of the units and topics as presented in the traditional courses in advanced high school mathematics - intermediate algebra, trigonometry, and solid geometry.

The need to improve early mathematics education in the United States is very clear, given the importance of early mathematics learning and the consistently poor math performance of children from low-SES backgrounds. It is crucial to engage these disadvantaged children in meaningful math learning as early as possible. With the continuous infiltration of technology into our lives, the powerful affordances of tablet computers may enable the development of powerful math applications. Given the limited evidence of using well-designed math applications to enhance young children’s math learning, the primary goal of this dissertation was to examine the efficacy of a research-based math software application on low-income preschoolers’ number sense performance. Twenty-eight 4- to 5-year-old intervention participants completed MathemAntics (MA) training 15 minutes a day, 3 days a week for 4 weeks. The other 28 control participants stayed in their classroom and received general class instruction. All participants were given a pretest and a posttest to evaluate their number sense performance. Results indicated that the intervention group outperformed the control group on number sense assessment, and the intervention participants’ mean standardized Addition & Subtraction gain was the highest among the seven subtests, with the mean Standardized Compare Quantities gain being the lowest. Results also indicated that prior knowledge of identifying numerals predicted the overall post-assessment performance and the lack of knowledge on Addition & Subtraction predicted the standardized overall gain. The MA training analyses showed that the participants improved adequately on most of the MA activities during intervention. The use of MA tools was also discussed. The results provided direct evidence for demonstrating the efficacy of MA and added valuable information to the field of math software design. The results of this study also suggested that future studies can examine how the MA activities can be effectively integrated into the math curriculum and whether the MA activities are appropriate for home numeracy development.

This dissertation is dedicated to the historical review of female mathematics education during the Progressive Era, from 1890 through 1920. This time period is known in the United States for multiple social reforms. Secondary schools experienced rapid expansion in enrollment, and the purpose and direction of education underwent development and change. During this era, a secondary education, which had been available to few, came to be accepted as a necessity for a majority of children in the United States. During this period of development and change, the educational system encountered several challenges. One such challenge was to tailor curricula according to the needs of different students. In parallel with increased enrollment, labor market demands experienced changes as well, and these changes were especially prominent in the urban areas. Historical evidence documents that the purpose of girls’ and boys’ education was often considered to be distinct. This belief stemmed from the idea that girls and boys had different career paths upon high school graduation. Therefore, differentiation of curricula received needed support and allowed schools to provide an elective system of subjects within high schools. This dissertation provides historical analysis of the mathematics education available for girls in the Progressive Era, focusing on the purpose of mathematics education, on curricula differentiation facilitated by an elective system, and on social factors that affected girls’ enrollment into the mathematics classes when the election of mathematics classes was allowed.