"Since arithmetic is a skill that presents almost insurmountable difficulty for some children there is a need for investigation of the methods and materials of teaching. Assuming that the meaning theory is a desirable way of teaching numbers the writer has endeavored to organize some ideas and objectives for developing quantitative understanding in children at the first and second grade levels. The writer has not attempted an exhaustive study of the problem but has focused her attention on principles that would be of immediate aid during the coming school year"--Introduction. / "August, 1952." / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: Elizabeth Hamlin, Professor Directing Paper. / Includes bibliographical references (leaf 42).

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There is no abstract available for this thesis.

There is no abstract available for this thesis.

Bidirectional Translation, devised by the author, is a structured approach to the teaching of addition and subtraction which aims to give children greater understanding of arithmetical operations. The approach systematically involves both: the translation of numerical representations into hypothetical, real world contexts; and the extraction of the appropriate numerical operations from hypothetical, real world contexts. It is this emphasis on translation from and to both the numerical representation and realistic contexts which gives rise to the name, Bidirectional Translation. An experimental group of 90 primary one children were taught to add and subtract (within 10) by the method of Bidirectional Translation. Post-test comparison of the experimental subjects' performance with that of a control group showed significantly superior performance on the part of the experimental subjects in terms of the utilizability of addition, the evocability of addition, the utilizability of subtraction and the evocability of subtraction for five different classes of verbal context, namely: Part-Part Whole, Separating, Joining, Equalizing and Comparison contexts. In all instances the probability of the results being chance ones were less than 5% and in most, were less than 1%. In both the experimental and control groups, most children performed better when they were required to utilize concepts than when they were required to evoke concepts. Similarly they performed better when they were required to add than when they were required to subtract. The differences, however, were not always significant. It is suggested that the effectiveness of the methodology of Bidirectional Translation is rooted in a structure which allows the child to make his/her thinking explicit and which allows the teacher to monitor this.

Modern theory of arithmetic instruction supports the idea that the development of understandings of basic mathematical principles produces a desirable type of learning. This is a reaction against the traditional method of instruction which places emphasis upon mechanical drill procedures, devoid of meanings. This study is an attempt to deter-mine what relationship, if any, exists between computational ability and understanding of fundamental processes. The investigation has been limited to the area of decimal fractions. Two tests -were developed for the purpose of the investigation. The test in computation was constructed and validated using pupils of the junior high school level as testees. Student-teachers constituted the personnel for the construction and validation of the test in understandings. The investigation, of relationship was performed using 236 Normal School students as testees. The tests, which had been constructed for use in the study, were administered at the beginning of the school term. The data obtained from the investigation were analyzed and the following conclusions were formulated: 1. There is a positive correlation of considerable magnitude between the scores on the test in computation and the scores on the test in understandings. ( r = .640 ). This is an indication that there is a tendency for the scores to vary in the same direction. 2. When the factor of intelligence is held constant, there is a net correlation of marked magnitude -which is somewhat less than, the apparent coefficient. This indicates that the common factor of intelligence has an influence upon the relationship between the two variables. 3. The magnitude of the relationship between scores in understanding and intelligence test scores is an indication of common elements in both these tests. 4. The relationship between, the scores in computation and the intelligence test scores is not high. A high intelligence does not appear to be a prerequisite for high achievement in computation. 5. There is evidence that ability in computation is not essential for high achievement in understandings and vice versa, nor do high scores in one of these factors guarantee high scores in the other. 6. Although a study of the scatter diagram suggests that success in computation is more probable if it is accompanied by a high degree of understanding, it cannot be inferred from the data that one variable is the cause or the effect of the other. / Education, Faculty of / Graduate

The meaning theory of teaching arithmetic requires that those who do the teaching understand the mathematical bases of arithmetic. This study was concerned with determining the extent to which one teacher-training institution was successful in raising the level of understanding of arithmetic on the part of its students during its usual year's programme. The 280 testees were students of the Vancouver Normal School. The data were obtained through the administration of Glennon's Test of Basic Mathematical Understandings at the beginning and at the end of the school year. The analysis of the data led to the following conclusions: 1. The teacher-training programme was effective in bringing about small but significant gains in the students' achievement of the basic mathematical understandings contained in Glennon's Test. The testees knew an average of 61.5 per cent of the understandings at the beginning of the study and an average of 66 per cent of them at the end of the study. 2. The understandings related to the decimal system of notation and to the integers and processes were well known to the students on both tests whereas those understandings related to decimals and processes, fractions and processes, and the rationale of computation were considerably less well known on both tests. 3. Significant gains were made in the areas of the decimal system of notation, integers and processes, fractions and processes, and the rationale of computation. The gains were fairly equal for the four areas of arithmetic. The section on decimals and processes showed no significant gains. The order of difficulty for the five areas of arithmetic included in Glennon's Test remained essentially the same for the test and retest. 4. Superior gains in achievement of the basic mathematical understandings were made by the students who were in the lowest quarter of the cases in the initial test as compared with those attained by the students who were in the highest quarter of the cases in the initial test. 5. The test items tended to be 3.5-choice rather than 5-choice items for this group of testees. As determined by the number of actual choices per item the level of understanding for this group was higher at the end of the year than it was at the beginning of the year. 6. 83 per cent of the rejection of the misleads which took place was made in favour of the correct answers to the items whereas 17 per cent of the rejection was made in favour of the incorrect alternatives. The latter finding indicates that the change (perhaps gain) which takes place in relation to certain items is incompletely assessed by the usual statistical procedures. 7. The mean of the gross changes involved in the shift of responses to and from the correct answers in the test-retest situation averaged six times the mean of the differences between the net changes in the responses. It is evident that minor changes are considered in present methods of estimating reliability, whereas the major changes are obscured and so ignored. In spite of the apparent inconsistency of the responses of the group to the test items in this study, reliability was .94, indicating that further research is necessary in the area of test reliability. / Education, Faculty of / Graduate

M.Sc., Faculty of Science, University of the Witwatersrand, 2011 / This study was concerned with the constitution of the ‘legitimate text’ - a key construct in Basil Bernstein’s (2000) theory of the pedagogic device. The question the study sought to understand was: what is constituted as the legitimate text across the mathematics education literature, the official curriculum document, in the official assessment texts, and in the textbook used in the classroom observed for the topic of number pattern. These sources were compared with what was constituted as the legitimate text in a sequence of five lessons based on number patterns in Grade 11 in an inner city school. This was a qualitative case study, the methodology of which was framed by Bernstein’s theory which explains the sociological nature of knowledge, the implicitness and explicitness of the communication for the acquisition of the legitimate text and hence opportunities to learn. One teacher was observed while teaching number pattern to a G11 class in an inner-city high school in Johannesburg in South Africa. A sequence of five lessons was videotaped and transcribed. The documents were analysed. One broad evaluative event with numerous sub-events called input objects were used to chunk the data into more manageable units of analysis. A framework emanating from the literature and from the analysis of the curriculum was used to present and categorise the legitimate text from the documents and the classroom. Kieran’s (2007) model of school algebra was used to do the analysis as well as Dowling’s (1998) model of domains of practice. The results of the study showed that the documents did not align with each other in terms of what they constituted as the legitimate text. It was found that the teacher aligned with the curriculum document. The results revealed that the teacher preferred working with numeric contexts. The consequence of this misalignment was that the documents created an additional work load for the teacher to understand and interpret them (documents).

It was the purpose of this investigation to examine the effects of sex, race, intelligence, and socioeconomic status upon the reading and arithmetic achievement test scores of sixth grade pupils in one school district. This was accomplished by combining these factors to establish 32 pupil performance categories. Norms, based upon these 32 categories, were established for both the level of performance at the end of sixth grade and gain in performance based upon the increment in reading and arithmetic between fifth and sixth grades.

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