This dissertation reports the results of a three-phase study on the learning of algebra. The first phase involved interviews with ten seventh graders (12 and 13 years of age) to uncover some of their pre-algebraic notions on equations and equation solving. Six of these novice subjects were kept on for the second phase: a two-month teaching experiment on equation solving which emphasized the symmetric procedure of performing the same operation on both sides. The pretest and two posttest interviews of the second phase included both equation-solving and equivalent-equations tasks. The third phase involved interviews with nine more-experienced algebra students from grades eight to eleven, to investigate their equation-solving procedures, errors, and methods of determining the equivalence of equation-pairs. Their approaches were compared with those of the novices on the same tasks. / The study uncovered two distinct paths followed in the learning of equation solving: one by those already predisposed toward inversing; the other by those with a predisposition toward using surface operations. The latter group was more receptive to the procedure taught during the teaching experiment. A relationship was found to exist between subjects' view of the literal term in equations and their preferred equation-solving method. Novices with an inversing preference applied learned principles to equivalence tasks, but not to equation solving. More-expert subjects relied on inversing for both. / Theoretical implications of these findings concern: the processes used in the early stages of learning a new domain, the modeling of the procedures used to determine equivalence, the relationship between errors and structural knowledge, and the representation of word problems by equations. Finally, the characterization of an arithmetic approach and an algebraic approach to the learning of equation solving is used to suggest a basis for a theory of algebra learning.

This study set out to investigate how the use of a Smart Board impacts on the teaching and learning of algebraic functions. The research took place in a school equipped with Smart Boards in each Mathematics classroom. Data collection involved lesson observations in three classes over three lessons each. The teachers and learners were interviewed post observation and the data obtained were analysed according to Sfard‘s three-phase model framework to determine if the learners had a procedural or object view of a function after having been taught on a Smart Board. The findings show that by using a Smart Board learners had both procedural and object view of functions however, much of the teaching occurred in a way which would have been possible without the use of a Smart Board, indicating that teachers did not fully utilise the potential of such a technological tool. However, it emerged that visualisation played an important role in allowing learners to operate on functions as objects. So while the visualization that technology enables encouraged reification or allowed teachers and learners to operate on functions as a whole or even on families of functions, this appeared simply to 'speed up‘ the normal teaching-learning process rather than promote the explorative and investigative aspect of learning. Still, it must be acknowledged that this kind of practice is bound to strengthen these learners‘ function concepts as was evident in the ways they appeared to operate confidently on the objects as shown in the study. It must be acknowledged that teachers were extremely enthusiastic about the possibilities of the technology and were inspired to use technology more in their lessons to allow learners‘ visualisation of concepts. Positive comments made by learners showed that they too, were also motivated by the use of the Smart Board. / Thesis (M.Ed.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

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The purposes of the study were: (1) to determine the effects of four testing strategies upon performance in college algebra, attitude towards mathematics, and attrition rate; (2) to determine the effects of two types of frequent testing upon performance, attitude, and attrition rate, (3) to determine the effects of different frequencies of in-class testing upon performance, attitude, and attrition rate; and (4) to draw conclusions which might help in selecting testing methods for college algebra classes.

Set-based reasoning and conditional language are two critical components of deductive argumentation and facility with proof. The purpose of this qualitative study was to describe the role of truth value and the solution set in supporting the development of the ability to reason about classes of objects and use conditional language. This study first examined proof schemes – how students convince themselves and persuade others – of Algebra I students when justifying solutions to routine and non-routine equations. After identifying how participants learned to use set-based reasoning and conditional language in the context of solving equations, the study then determined if participants would employ similar reasoning in a geometrical context. As a whole, the study endeavored to describe a possible trajectory for students to transition from non-deductive justifications in an algebraic context to argumentation that supports proof writing. First, task-based interviews elicited how participants became absolutely certain about solutions to equations. Next, a teaching experiment was completed to identify how participants who previously accepted empirical arguments as proof shifted to making deductive arguments. Last, additional task-based interviews in which participants reasoned about the relationship between Varignon Parallelograms and Varignon Rectangles were conducted. The first set of task-based interviews found that a majority of participants displayed ritualistic proof schemes – they viewed equations as prompts to execute processes and solutions as results, or “answers.” Approximately half of participants employed empirical proof schemes; they described convincing themselves or others using a range of arguments that do not constitute valid proof. One particularly noteworthy finding was that no participants initially used deductive justifications to reach absolute certainty. Participants successfully adopted set-based reasoning and learned to use conditional language by progressively accommodating a series of understandings. They later utilized their new ways of reasoning in the geometrical context. Participants employed the implication structure, discriminated between necessary and sufficient conditions, and maintained a disposition of doubt toward empirical evidence. Finally, implications of these findings for pedagogues and researchers, as well as future directions for research, are discussed.

It was with an idea of establishing a tool to aid in vocabulary growth that E. L. Bellhorn, Ft. Lauderdale, Florida, and the writer, during the summer quarter of 1948 at Florida State University, built a test for ability to recognize and to apply algebraic language. Realizing their inexperience in such an important matter as developing a testing device which would meet the requirements of highly specialized experts in the field of testing, they took no credit in attempting to devise new techniques or devices, but followed rather slavishly the steps in procedure in good test construction. This study grew out of the desire to improve the test items to answer such questions as: 1. how much is student performance affected by the inability to recall an exact word at a specific instant? 2. Can the student choose the right word when it is coupled with a wrong one? 3. What would be the effect if the range of choice of response-words is increased? 4. What would happen if non-verbal items are introduced? / Typescript. / "August, 1950." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaves 62-64).

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Effort was made to identify teaching competencies of value to the pre-algebra mathematics teacher, to construct a teaching competencies diagnostic inventory based on pre-algebra mathematics teaching competencies selected from those identified, to determine if such a diagnostic inventory could be validated using student observations on an instrument similar to the diagnostic inventory, and to investigate the responses on the two inventories using correlational analysis and factor analysis. Nine educators involved in the in-service and pre-service education of pre-algebra mathematics teachers identified 28 competencies of particular value to the pre-algebra mathematics teacher. From this list of competencies, 10 competencies were selected for use in the study. The teacher diagnostic inventory is composed of teaching situation descriptors and personal data items. The teacher is asked to indicate how closely his classroom behavior is described by the situation description. The student inventory consists of situations similar to those described in the teacher inventory and personal data items regarding the students' sex and selected attitudes. The student is asked to indicate his perception of his teacher1s behavior in each situation described. The teacher diagnostic inventory was distributed to a sample of pre-algebra mathematics teachers employed in 28 schools located in southeastern Oklahoma. Copies of the student inventory were distributed by each participating teacher to the students in one of his pre-algebra mathematics classes. Based on the analyses of the responses of 73 teachers and 1602 students, it was concluded that the teacher's responses could not be validated using the students' observations. Factor analytic studies of the teacher responses and the student responses revealed the presence of eight factors among the teacher inventory items and the presence of six factors among the student inventory items. These factors were identified according to the items with loadings of .35 and more on each factor. A discussion of the relationships between the item scores and responses and the factors contained in the two inventories is included. / Ed. D.

This study characterizes the mathematical knowledge upon which secondary school algebra teachers draw when pondering problem situations that could arise in the teaching and learning of solving algebraic equations, as well as examines the potential connections between teachers' knowledge and their academic backgrounds and teaching experiences. Seventy-two middle school and high school algebra teachers in Texas participated in the study by completing an academic background questionnaire and a written-response assessment instrument. Eight participants were then invited for followup semi-structured interviews. The results revealed three topic areas in equation solving in which teachers' mathematical subject matter understanding should be strengthened: (a) the balancing method, (b) the concept of equivalent equations, and (c) the properties of linear equations in their general forms. The participants provided a wide range of instances of student misconceptions and difficulties in learning how to solve linear and quadratic equations, as well as a variety of strategies for helping students to improve their understanding. Teachers' subject matter knowledge played a central or prerequisite role in their reasoning and decision-making in specific contexts. When the problem contexts became broader or more general, teachers drew from across the three basic domains of mathematical knowledge for teaching (knowledge of the mathematical subject matter, knowledge of learners' conceptions, and knowledge of didactic representations) and showed individual preferences. Overall, teachers tended to rely more heavily upon their knowledge of students' specific or general learning characteristics. Statistical analyses suggest that teachers who majored in mathematics and who had the most experience in teaching first-year or more advanced algebra courses performed significantly higher on the assessment than their counterparts, and there is a linear relationship between teachers' performance and the number of advanced mathematics course they have taken. Neither course-taking in mathematics education nor number of years of algebra teaching made a significant difference in their performance. Results are either unclear or inconsistent about the role of teachers' (a) use of algebra textbooks, (b) prior experience with a method or a manipulative, and (c) participation in professional development activities. Teachers also rated (a) collaborating with and learning from colleagues and (b) dealing with student conceptions and questions as highly influential on their professional knowledge growth.

The pass rate of first-year university mathematics students at the University of KwaZulu-Natal (Pietermaritzburg Campus) has been low for many years. One cause may be weak algebra skills. At the time of this study, revision of high school algebra was not part of the major first year mathematics course. This study set out to investigate if it would be worthwhile to spend tutorial time on basic algebra when there is already an overcrowded calculus syllabus, or if students refresh their algebra skills sufficiently as they study first year mathematics. Since it was expected that remediation of algebra skills would be found to be worthwhile, two other questions were also investigated: Which remediation strategy is best? Which errors are the hardest to remediate? Five tutorial groups for Math 130 were randomly assigned one of four remediation strategies, or no remediation. Three variations of using cognitive conflict to change students’ misconceptions were used, as well as the strategy of practice. Pre- and post-tests in the form of multiple choice questionnaires with spaces for free responses were analysed. Comparisons between the remediated and non-remediated groups were made based on pre- and post-test results and Math 130 results. The most persistent errors were determined using an 8-category error classification developed for this purpose. The best improvement from pre- to post-test was 12.1% for the group remediated with cognitive conflict over 5 weeks with explanations from the tutor. Drill and practice gave the next-best improvement of 8.1%, followed by self-guided cognitive conflict over 5 weeks (7.8% improvement). A once-off intervention using cognitive conflict gave a 5.9% improvement. The group with no remediation improved by 2.3%. The results showed that the use of tutorintensive interventions more than doubled the improvement between pre-and post-tests but even after remediation, the highest group average was 80%, an unsatisfactory level for basic skills. The three most persistent errors were those involving technical or careless errors, errors from over-generalising and errors from applying a distorted algorithm, definition or theorem. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.