abstract / Education / Doctoral / Doctor of Philosophy

Typescript (Photocopy) / Bibliography: leaves 370-397 / iii, 397 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--Dept. of Education, University of Adelaide, 1990

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

published_or_final_version / Curriculum Studies / Doctoral / Doctor of Philosophy

This study is an investigation into the performance of learners in algebra using the levels of understanding as measured by the ICCAMS diagnostic instrument. The study was conducted in two phases. The first phase of the study consisted of an analysis of the scripts of a sample of 29 learners in Grade 9 who had written the test administered by the Wits Maths Connect-Secondary unit at Wits University. The scripts of the same 29 learners in Grade 10 were analysed to determine the progression within the levels of these learners from Grades 9 to 10. Eighteen learners progressed from a lower to a higher level. During the analysis of the tests it was found that the conjoining error was the main obstacle to some learners in progressing from moving from level 1 to level 3. During phase 2 of the study, a sample of 6 learners was selected from the original 29 learners. These learners completed a written task to investigate errors made in algebra in Grade 11. Interviews were conducted with these learners based on a written task. The analysis of the interviews and written task illustrated the problems learners experienced with level 2 questions, particularly with respect to the conjoining error.

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This research investigated the impact of "embedded teaching" and "learner-controlled" instruction on student learning of algebra in a controlled computer-tutoring environment. Three versions of a computer tutor were developed to establish three experimental conditions. Condition 1 corresponds to a conventional "lecture-demonstration-practice" in which conceptual knowledge is presented by the computer tutor as a coherent entity prior to engagement in problem-solving activities (Lecture-Demonstration-Practice). Condition 2 reflects "embedded teaching" in which before students begin practice, the computer tutor uses examples to demonstrate problem-solving processes, introducing concepts and principles, as they become relevant (Embedded-Teaching Condition). Condition 3 is a "learner-controlled" instruction in which students engage directly in problem-solving activities without receiving any prior formal instruction, but in which they are provided with instructional assistance and demonstrations upon request (Learner-Controlled Instruction). / Twenty-seven high-school students participated in the experiment over a 1-month period. Students were divided into three groups based on their pre-test scores, each group was then assigned randomly to one of the three experimental conditions. The computer tutor was used as the sole source of instruction. Pre- and posttests were administered to measure the changes in students' algebraic abilities. A multivariate analysis of the pre- and posttest results indicates that overall student performance in all three conditions improved significantly over time, as measured by the ability to construct algebraic representations and the ability to made estimates using the various representations ( F (2, 23) = 46.6, p < 0.01). In particular, students in Lecture-Demonstration-Practice Condition demonstrated a higher level of accuracy (89.51%) than students in the Embedded-Teaching and Learner-Controlled Instruction did (61.1% and 63.3% respectively). Moreover, all students in Lecture-Demonstration-Practice Condition completed the posttest successfully, whereas only 56% of students in the other two conditions passed the posttest. / This research demonstrates that students learn more effectively from instruction that emphasizes the coherent representations of the symbol system of algebra. It is postulated that such coherent representations enable students to make sense of the subsequent examples to be studied and the problems to be solved thus leading to better problem-solving performance. This research has implications for the development of instructional theories and educational computer applications.

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).

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.