Thesis (Ph. D.)--Illinois State University, 1997. / Title from title page screen, viewed June 8, 2006. Dissertation Committee: Graham A. Jones, John A. Dossey (co-chairs), Robert L. Fisher, Cynthia W. Langrall, Jane O. Swafford. Includes bibliographical references (leaves 251-257) and abstract. Also available in print.

This dissertation analyzes differences found in Spanish-speaking middle school and high school students in algebra-based problem solving. It identifies the accuracy differences between word problems presented in English, Spanish and numerically based problems. The study also explores accuracy differences between each subgroup of Spanish-speaking students in each category. It identifies specific strategies used by successful students when solving algebra problems. The study also sought to identify factors that could serve to predict Spanish-speaking students' ability to accurately solve algebra word problems presented in English and Spanish. A heterogeneous urban sample composed of one hundred and fifty two middle school and high school students were given an assessment composed of pre-approved algebra-based problems and a biographical information sheet. Specific students were then chosen for individual interviews in which researcher sought to gain more in depth information about student's reaction to assessment. The study found that the average accuracy rate for Hispanics non-ELL and non-Hispanic students was significantly higher for numerically based problems than Spanish word problems. Similarly, the average accuracy rate for Hispanics non-ELL and non-Hispanic students was significantly higher in English word problems that in Spanish word problems. Results showed that there was a significant difference in the overall performance of the assessment between Hispanic ELL and Hispanic non-ELL students. On one particular set, set C (Spanish word problems), findings showed that Hispanic ELL students performed better than Hispanic non-ELL students and non-Hispanic students. All other subgroup comparisons did not show a significant difference. The study found that students who were most successful in the assessment: (a) used previous linguistics knowledge and memory of previously seen mathematical problems properly; (b) highlighted the question being asked; (c) used key words to identify mathematical principles and to aid in the translation process; (d) used diagrams, tables and graphs to organize data; (e) showed work and had all processes laid out clearly; and (f) displayed a clear verification process for their answer as strategies for successfully answering the problems. As it was evident through the study, the diversity in the Spanish speaking population and their needs exposes the need for teaching methods, which are inclusive of all populations. Schools must be sensitive to the diversity in which students learn and aim to individualize the teaching for every student. As Hispanics become the largest minority in the United States, understanding the diverse needs of Spanish speaking students in the classroom will be necessary for the development of a better educated society.

The main objective of this study was to explore how Deaf learners orientate to a problem solving mathematics curriculum. The study took the form of an autoethnography situated in critical pedagogy. Purposeful sampling was used to select Grade 9 learners from a local school for the Deaf as participants. Data was collected from the learners using a structured questionnaire, viz. Students Orientation to Mathematics (SOM), as well as through focus group sessions and personal interviews. In addition, teachers’ and parents were interviewed to ascertain the general orientation of Deaf learners to mathematics and to identify barriers that may prevent these learners from progressing optimally in their studies of mathematics. Although the learners had difficulties in accessing particular pedagogical aspects of problem solving mathematics, the findings showed a slight increase in the learners’ study attitude and study habits towards mathematics and in their problem solving skills. Moreover, the learners particularly enjoyed the activity element of the problem solving curriculum. At the end of the study the learners indicated that they preferred a modeling problem solving approach to a more traditional way of teaching mathematics. Although the study points out that implementing a problem solving curriculum into a Deaf classroom is not necessarily straightforward, it does suggest that with exposure Deaf learners can develop a propensity for working within a cognitively rich problem solving environment.

Although the importance of problem solving as an essential component of mathematics learning and doing has consistently been recognized, recent research has only just begun to identify and describe the complex set of variables influencing the endeavor. Therefore, the aim of this study was to empirically investigate the relationships between several of these variables: mathematical expertise (as measured by the advanced nature of the mathematics courses students have taken, and are enrolled in), the cognitive ability known as inhibitory control (the ability to inhibit or suppress an immediate response to a stimulus, and engage in deeper, more reflective thought), and facets of college students’ psychoeducational profile (e.g., academic habits of mind, future orientation, self-limiting beliefs), which provide information about the nature of college students’ learning and development. In this study, one hundred and thirty college students, enrolled in different levels of mathematics courses (from introductory courses to major courses in mathematics) were administered a modified version of the Cognitive Reflection Test (an instrument designed to measure the ability to activate one’s inhibitory control capacities) and a survey instrument designed to measure domain-general and mathematics-specific psychoeducational facets of their academic profile. Information about membership to other subgroups (e.g., gender, academic major, mathematics courses taken in high school) helped to further contextualize the findings. The majority of all participants did not correctly solve any of the problems of the modified version of the Cognitive Reflection Test which required inhibitory control. However, those with a greater level of mathematical expertise (i.e., those taking more advanced mathematical courses) performed significantly better than their peers on these problems and exhibited more desirable responses on the psychoeducational survey instrument. Responses to items of the survey instrument that measured behaviors, habits, and experiences that limit students in their conception of, approach to, and engagement with mathematics indicate the presence of a psychoeducational facet specific to mathematics that cannot be sufficiently explained by domain-general facets also under measure. These limiting characteristics related to mathematics were also significantly related to students’ performance on the modified version of the Cognitive Reflection Test, indicating a potential relationship between such characteristics and problem solving success on inhibitory control tasks. Considering the measures of mathematical expertise utilized in the current study, the social nature of mathematics learning may help explain the development of both inhibitory control ability and limiting beliefs in mathematics. The current study extended the methods utilized in previous research to examine the relationships between inhibitory control and mathematical expertise in college students while also investigating these in relation to particular psychoeducational variables known to influence learning and development of college students. The findings of this small-scale empirical study provide a modest step forward in these areas of research by providing another lens through which to view several phenomena already being extensively investigated by other researchers.

In this study, the role of the problem-based approach on the performance of Grade 9 learners in solving word problems is investigated. Traditional approaches have produced learners whose performance in mathematics is not satisfactory and who are not sufficiently equipped with critical and problem skills that are necessary in this dynamic world. Problem-based approach is among the current reform efforts recommended for teaching and learning mathematics. For this approach to be successful, learners need vital tools such as problem solving strategies, which many learners in this country lack. The emphasis in this study was therefore on exposing learners to a variety of problem solving strategies through the problem-based approach. Problems solved throughout the investigation were non-routine, word problems. The results show that awareness of these strategies improves learners' problem solving performance and attitudes towards mathematics. Based on this investigation, recommendations are made concerning effective implementation of this approach to the teaching and learning of mathematics. / Educational Studies / Thesis (M. Ed. (Specialisation in Mathematical Education))

In this study, the role of the problem-based approach on the performance of Grade 9 learners in solving word problems is investigated. Traditional approaches have produced learners whose performance in mathematics is not satisfactory and who are not sufficiently equipped with critical and problem skills that are necessary in this dynamic world. Problem-based approach is among the current reform efforts recommended for teaching and learning mathematics. For this approach to be successful, learners need vital tools such as problem solving strategies, which many learners in this country lack. The emphasis in this study was therefore on exposing learners to a variety of problem solving strategies through the problem-based approach. Problems solved throughout the investigation were non-routine, word problems. The results show that awareness of these strategies improves learners' problem solving performance and attitudes towards mathematics. Based on this investigation, recommendations are made concerning effective implementation of this approach to the teaching and learning of mathematics. / Educational Studies / Thesis (M. Ed. (Specialisation in Mathematical Education))

Visual modeling concept of integrated constructs (essence) of mathematical objects in teacher training of humanistic area is presented as technology of education in problem solving. The main goal of innovative approach is student’s activity in mathematics on generating of concrete essence manifestations on concepts, methods, theorems, algorithms, procedures and so on. Such student’s activity should be: · Success in an area of actual interests and person’s experience and reached by perception; · Have high level of variability in visual modeling; · Success in domain of reflection process stimulation. Similar creative behavior of persons is typical for actors, dancing, and figure skating and so on. Now we show that such technology will be fruitful for teacher training in mathematics for humanistic specialties.

The purpose of this study was to explore solution strategies that can enhance the achievement of low-performing Grade 12 learners in the following mathematical aspects: finding the general term of a quadratic sequence, factorising third degree polynomials, determining the centre and radius of a circle, and calculating the angle between two lines. A convenience sample of twenty-five low-performing Grade 12 learners from a secondary school in Capricorn District of Limpopo Province participated in the study which adopted a repeated-measures research design. Learners were exposed to multiple solution strategies and data were collected using achievement tests. Findings indicated significant differences in learners‟ average scores due to the solution strategies used. In determining the general term of a quadratic sequence, learners‟ scores were significantly higher when they used formula and the table method than with the method of residues and solving simultaneous equations. Synthetic division made learners to achieve better scores than long division and equating coefficients in factorising third degree polynomials. The use of formulae to find the centre and radius of a circle made learners to have better achievement scores than completing the square. In calculating the angle between two lines learners‟ scores were better using formula and the cosine rule than using theorems. It was concluded that exposing low-performing Grade 12 learners to multiple solution strategies would enhance their achievement in the mathematical aspects explored in the study. Some of the solution strategies that made learners to achieve better results were not in the prescribed mathematics textbooks. The study therefore recommends that mathematics teaching should not be textbook-driven and that low-performing Grade 12 learners should not be regarded as beyond redemption. / Mathematics Education / M.Sc. (Mathematics, Science and Technology Education)

Although proof has long been viewed as a cornerstone of mathematical activity, incorporating the mathematical practice of proving into classrooms is not a simple matter. In this dissertation I aim to advance research on proof by addressing this issue. In particular, I explore the role proof might play in promoting the learning of mathematics in the classroom. I do this in a series of three articles (organized as three chapters), which are preceded by an introductory chapter. The introductory chapter situates the remaining chapters in the context of mathematics education research. In the second chapter I explore what the literature on proof tells us about what role proof might play in the promotion of learning in the mathematics classroom. In this chapter I also compare the ways in which proof is purported to promote learning in the mathematics classroom with the roles it is purported to play in the field of research mathematics. In the third chapter I look at empirical data to explore ways engaging in proof and proving might create opportunities for student learning. In particular, my analysis led me to focus on how identifying and reflecting on the key idea of a proof can create opportunities for learning mathematics. The final chapter is an article for a practitioner journal and discusses implications for practice based on the two preceding articles.

The purpose of this study was to explore solution strategies that can enhance the achievement of low-performing Grade 12 learners in the following mathematical aspects: finding the general term of a quadratic sequence, factorising third degree polynomials, determining the centre and radius of a circle, and calculating the angle between two lines. A convenience sample of twenty-five low-performing Grade 12 learners from a secondary school in Capricorn District of Limpopo Province participated in the study which adopted a repeated-measures research design. Learners were exposed to multiple solution strategies and data were collected using achievement tests. Findings indicated significant differences in learners‟ average scores due to the solution strategies used. In determining the general term of a quadratic sequence, learners‟ scores were significantly higher when they used formula and the table method than with the method of residues and solving simultaneous equations. Synthetic division made learners to achieve better scores than long division and equating coefficients in factorising third degree polynomials. The use of formulae to find the centre and radius of a circle made learners to have better achievement scores than completing the square. In calculating the angle between two lines learners‟ scores were better using formula and the cosine rule than using theorems. It was concluded that exposing low-performing Grade 12 learners to multiple solution strategies would enhance their achievement in the mathematical aspects explored in the study. Some of the solution strategies that made learners to achieve better results were not in the prescribed mathematics textbooks. The study therefore recommends that mathematics teaching should not be textbook-driven and that low-performing Grade 12 learners should not be regarded as beyond redemption. / Mathematics Education / M.Sc. (Mathematics, Science and Technology Education)