It was observed that the students in Mathematics 100 made many errors which appeared to be due directly to a lack of knowledge of the fundamental facts of elementary algebra. This thesis studied the effect of an organized remedial program given during the regular class-room periods on the mean class marks and the numbers of failures. This involved a determination of the facts and abilities necessary for success in Mathematics 100, and the measurement of the students' knowledge of these fundamentals. Moreover, an analysis of the results might provide a method for selecting those students who would probably have difficulty passing the course. The program of the experiment had three parts, namely: the construction of the test of fundamentals; the selection of control and experimental groups on the basis of the scores made on this test, an intelligence test, and the Mathematics 100 Christmas examination; and the administration of the remedial program in the experimental group. The differences between the mean scores made by these groups on the April examination were evaluated by four procedures. In the first of these, the method of gains, the significance of the difference between the gains was determined. In the other three methods, the effect of the remedial program was studied by determining the significance of the difference between the mean April scores of: (a) a control and an experimental group equated on the basis of their Christmas scores (method 2); (b) a control and an experimental group composed of pairs of students, chosen from the two original groups, and matched on the basis of their Christmas marks (method 3); (c) the actual marks obtained by the experimental group and its scores predicted from its Christmas marks using the regression equation of the control group (method 4). The following table gives the critical ratios and the chances of a significant difference being obtained by using the above four methods. (Tables omitted) It may be concluded, therefore, that the remedial program raised the mean class mark of the experimental group, since there are at most four chances in one hundred that the difference could be due to chance. Furthermore, the program reduced the number of failures, although the reduction was not statistically significant. / Education, Faculty of / Graduate

No abstract included. / Arts, Faculty of / Philosophy, Department of / Graduate

The problem under investigation in this study was to find out what relationship a unit in modular arithmetic might have to Grade 6 pupils' skill in computing the division of fractions and to their understanding of the mathematical basis of the algorithm. It was hypothesized that a unit in modular arithmetic would aid in developing skill in computing and understanding of the algorithm. The study was conducted with a sample of 58 Grade 6 students from the same school. The subjects were assigned to two treatment groups. Both groups received a review of fraction concepts at the beginning of the study. Following this, one group was taught modular arithmetic while the other group reviewed adding and subtracting of fractions. Then both groups were taught multiplication and division of fractions. Following the instruction period, both groups were tested for ability to compute division of fractions. To test understanding of the division of fractions algorithm, an interview inventory test was administered to all subjects in both groups. A statistical analysis of the data from these tests revealed no support for the hypotheses. The conclusion was that teaching modular arithmetic to the Grade 6 pupils participating in the study did not appear to improve their ability to compute division of fractions nor their understanding of the mathematical basis of the division of fractions. / Education, Faculty of / Graduate

An exploratory investigation into the effect of co-operative group learning, involving tutoring, on the achievement and attitudes of 174 grade eight pupils in new mathematics is described. Three volunteer teachers and six volunteered mathematics classes were involved. Five hypotheses concerning test performance and one concerning attitudes were advanced. Using the scores obtained in the mathematics sections of the Stanford Achievement Test (Advanced) and an entering behaviour test of prior mathematics learning, to establish similarity of the groups, instruction was carried out over a period of twelve weeks. A retention test was given two months later. Attitude scores from data collected by Semantic Differential before and after the experiment were analysed using a model for multidimensional analysis of Semantic Differential attitude data (McKie and Foster, 1972). Achievement in algebra learning and retention reached the .05 level of statistical significance, favouring the experimental group. No differences for treatment occurred for instructor effects, instructor by group interaction or attitudes at the .05 level of statistical significance. Conclusions for further research and practice are drawn. / Education, Faculty of / Graduate

This was a study to determine the effects of two instructional procedures for a multiplication algorithm on the ability of elementary school children to extend this algorithm to the solving of computational tasks involving the use of a higher-order algorithm. Each of two groups was given preliminary instruction in solving multiplication problems via the application of the distributive law. After this readiness phase was completed, students were randomly assigned to either a Tl or T2 treatment group. The Tl subjects were taught a rote-type standard multiplication algorithm for determining the solution of 2 x 1 and 3 x 1 products. No explicit instruction was given to indicate the relationships between the two learning tasks, viz. the acquisition of the distributive law and the standard multiplication algorithm. Unlike the Tl instructional sequence, the T2 instructional sequence was designed to promote the learning of the relationships between the series of learning tasks. That is, the T2 subjects were taught a standard multiplication algorithm that required the explicit use of the distributive law and other acquired algebraic skills. It was hypothesised that this continual integration of learning tasks would enable the T2 subjects to exhibit superiority over the Tl subjects in extending their standard multiplication algorithm to computational tasks requiring the use of an untaught higher-order algorithm. A total of 238 subjects and 8 teachers were used in all phases of the experiment. A mixed model of analysis of variance was used to validate the performance hypothesis. It was found that the Tl subjects were significantly better than the T2 subjects in the performance of the standard multiplication algorithm. An analysis of covariance was performed to determine the validity of the transfer hypothesis. A subject's score on the performance test was used as a covariate in order to equate the disparate computational abilities of the Tl and T2 subjects. Although the mean score of the T2 subjects was higher than that of the Tl subjects on the transfer test, this difference was not statistically significant. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

A study was conducted to determine the interaction effect, if any, between the field-independence construct and two instructional strategies, a pattern strategy which used diagrams extensively and an algebraic strategy which used algebraic field properties familiar to the child and was devoid of diagrams. Two algorithms classified as simple and two algorithms classified as complex formed the content of the instructional materials. One half the children in each of twelve grade five classes, which were participating in a study conducted by a doctoral student, were randomly selected to form the sample of the study. The Children's Embedded Figures Test was individually administered to the sample. Three null hypotheses were tested each at < = .05. These were: (1) There is no significant difference in mean post-test scores between students taught by a pattern instructional strategy and students taught by an algebraic instructional strategy; (2) There is no significant difference in mean post-test scores between groups of students differing in degree of field independence; (3) There is no significant interaction between students' degree of field independence and instructional strategy. Multiple linear regression techniques were used to analyse the data. The results of the study indicated that extreme field independent children did respond differently to the two instructional strategies, although for the sample as a whole the two strategies did not produce significantly different results. For extreme field independent students, the algebraic strategy was superior to the pattern strategy. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This was a study to determine the effects of inquiry training in elementary mathematics, in both group and individualized situations. Three experimental conditions were used. One group of grade six students received problems on area calculation and arithmetic rate problems in which not enough information was given, requiring them to acquire more data by asking questions on blank cards given to them individually for that purpose. They were then given prepared response cards in reply. A second group received the same problems with not enough information, but all questions and all responses were made orally in a classroom so that all class members could simultaneously receive the information which one student requested. A third group received the same problem sheets, but with all necessary data present, and no new information was given them. Two short training sessions were held dealing with area calculation and solution of arithmetic rate problems to familiarize the subjects with the nature of the learning materials. A total of 64 subjects participated in all phases of the experiment. All subjects were given an IQ test and a pretest, parallel to the final cirterion test, consisting of problems of both an arithmetic and geometric nature, as well as area calculation. The criterion test, at the conclusion of the instructional period, consisted of the same three subdivisions, including items both similar and dissimilar to the training materials. Each item on both the pretest and the criterion test was answerable by "not enough information" as well as by three numerical possibilities. Both pretest and criterion tests were marked twice, once to give an actual score and once to determine the correct use of the "not enough information" response. Each of the three subtests - arithmetic, geometric, and area calculation - was marked separately from the total test, as well, and they were then scored again to consider them as tests measuring the correct use of "not enough information". Thus a total of eight separate criterion scores with corresponding pretest scores were involved. An analysis of covariance was performed using the IQ and pretest scores as covariates. The results indicated that the group inquiry training approach was the most successful of the three approaches in that these students scored significantly higher marks on five of the eight scores. No significant differences were found among the other three scores. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

The purpose of this study was to investigate if a relationship exists between the understandings of students and those of their teachers for a specific concept in mathematics. A review of literature revealed that no study-had attempted to examine the relation between student and teacher understanding of a specific concept in mathematics although several had investigated the relation between teacher and student understanding of general mathematical concepts, usually in arithmetic. The single concept chosen for the present study was intuitive limit concepts as prescribed for Mathematics 8 students in British Columbia schools. The following null hypothesis was established and tested: For Mathematics 8 classes of better students there is no significant correlation between teacher understanding of intuitive limit concepts and student understanding of intuitive limit concepts. Measures of understanding were obtained by the use of two testing instruments constructed by the investigator, one for students and one for teachers. The preliminary student test constructed was checked for content validity and given a trial use. The reliability of the test was calculated and an item analysis made to determine which items to use in the final form of the test. The teacher test constructed used hypothetical answers to student test items. Teacher test items were taxonomized according to Bloom. Fourteen classes of Mathematics 8 students of better ability and their teachers were tested using the final form of each test. Class means for student tests were adjusted by analysis of covariance to allow for Initial differences in intelligence and mathematics achievement. Calculation of the coefficient of correlation between these adjusted means and teacher scores gave a result of 0.09. This correlation was not significant. Thus the null hypothesis tested was accepted. / Education, Faculty of / Graduate

This disquisition presents a qualitative study that investigated the complicit nature of theory and practice in mathematics teaching. Situated within an ecological perspective, this research interrogates the role that theory plays as a cognizing domain in which one's pedagogy of teaching mathematics co-exists and co-evolves. A systemic exploration of mathematics and the teaching and learning of it is conducted and assessed against tenets of complexity, sustainability, languaging, co-emergence, integration, and recursion. This study reveals the impact that theoretical discourses have on the kind of place and the forms of mathematics that are enabled and disabled through the metaphors, perceptions of mathematical understanding, and conceptions of time that are embodied and enacted by the teacher and her students. This research involved the explication of the teacher's assumed theoretical and practical patterns of teaching mathematics. The expressive forms in which this disquisition is written provide interpretive snapshots that document the teacher's conceptual journey from that of a heavily mechanistic, linear, and hierarchical mindset towards the development of an ecologically coherent theoretical domain for teaching. The classroom vignettes of the teacher, another teacher with whom she collaborated, and the second and third grade students span a course of two and half school years. These vignettes focus on the teacher's work in occasioning ecological forms of teaching, learning, and mathematics in the classroom. The analysis of these episodes revealed stark differences from that of her previous teaching practice not only in the nature of the students' understandings, their ways of acting and being mathematical but also, in the kinds of mathematics that arose during the lessons. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

One of the most striking facts disclosed in national reports is the large number of students who avoid mathematics courses, especially electives. The problem has become a serious public concern because it bears social and individual consequences: (a) a technologically advanced society demands a mathematically literate workforce, yet a large number of students drop out of mathematics; (b) inadequate preparation in mathematics seriously limits future educational and occupational opportunities of individuals. Although research on school and teacher effects has revealed the effects of school structure and policies and teaching practices on mathematics achievement, researchers have paid little attention to the course of students' academic careers. Even the few existing studies are compromised by serious methodological flaws. Researchers, thus, have not been able to provide policymakers with reliable answers to their basic concerns about mathematics participation. This study tackles these problems, employing the six-wave data from the Longitudinal Study of American Youth (LSAY). The primary purposes of this study are (a) to estimate the probability of students' dropping out of mathematics, conditional on psychological and sociological variables, including sex, socioeconomic status (SES), prior mathematics achievement, prior attitude toward mathematics, prior mathematics anxiety, and prior self-esteem, over a five-year period from grade 8 to 12, (b) to identify conditions that affect the probability, and (c) to determine whether there are critical transition points, and if so, whether certain factors have stronger effects at these points. Survival analysis is used to overcome the difficulties conventional statistical techniques have in modeling probability Analyses of mathematics participation indicate that (a) students are most likely to drop out of mathematics in grade 12; (b) males are more likely than females to participate in mathematics in grade 12; (c) the effect of SES decreases over grades; (d) prior attitude toward mathematics is as important as prior mathematics achievement, and their effects are almost constant over grades; (e) the longitudinal effect of prior mathematics achievement or prior attitude toward mathematics depends on students' sex and SES. Analyses of participation in advanced mathematics show that (a) students are most likely to drop out of advanced mathematics in grade 12; (b) males are more likely than females to participate in advanced mathematics in grade 12, and sex differences are similar across different levels of SES; (c) there is a male advantage in participation in advanced mathematics even when there is a male disadvantage in SES; (d) SES plays a critical role in the early grades, and socioeconomic differences are similar across different levels of mathematics achievement or attitude toward mathematics; (e) prior attitude toward mathematics has the strongest effect in the later grades, whereas the effect of prior mathematics achievement decreases over grades; (f) the effect of prior mathematics achievement varies across different levels of attitude toward mathematics, and vice versa; (g) the longitudinal effect of prior mathematics achievement or prior attitude toward mathematics depends on students' sex and their initial mathematics achievement and attitude toward mathematics. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate