This study was conducted in one of the education districts in the Eastern Cape Province of South Africa. The purpose was to analyse the effectiveness of the concrete/semi-concrete/abstract (CSA) approach and drill-practice instructional strategies on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. The following two objectives were set. First, to identify the learners’ challenges in studying addition and subtraction of algebraic fractions in grade 10; and second to analyse the effectiveness of the CSA approach and drill-practice instructional strategies on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. Both threshold concepts and troublesome knowledge, Polya’s problem-solving techniques, CSA Approach theory and Drill-practice theory were all pertinent as a theoretical framework for the study. Positivism research paradigm was adopted for the study and it afforded the researcher opportunity to employ quantitative research approach. Based on the research question of this study, an experimental design was chosen as a suitable descriptive design. Purposive sampling method was used to select three schools which involved 135 grade 10 mathematics learners. Stratified random sampling method was thereafter employed to select 45 learners from each school for the study. The learners were grouped in each school as top, average and weak based on their performance in Algebra in term one. Pre-questionnaire and post-questionnaire were used to obtain data regarding challenges learners experience in simplifying addition and subtraction of algebraic fractions. Ethical clearance from the relevant school and university authorities were obtained. On the first two days, the researcher briefed the school authorities and learners and explained to them the purpose and details of the study. Day three was used to administer the pre-questionnaire test, thereafter, the next ten days were used to teach addition and subtraction of both numeric and algebraic fractions with same and different numerators and denominators. The next two days were used for revision and the last day was used to administer the postquestionnaire test out 25 marks. The respondent rate was 98.5%. The data collected were analysed by using SPSS version 16.10. Both descriptive and inferential statistics were used to analyse the data. The pre-questionnaire scores revealed that majority of the learners’ perceived fractions as two separate entities and as a result add or subtract numerator to numerator and denominator to denominator. It was also discovered that learners had a challenge in finding LCM of algebraic fractions. A t-Test for independent means was used to test the following hypotheses at 𝛼 = 0.05: 𝐇𝟎: The CSA approach and drill-practice intervention has no significant effect on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions; 𝐇𝟏: The CSA approach and drill-practice will significantly enhance Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. The t-Test revealed a p-value of 0.139 which was statistically significant at 𝛼 = 0.05. Therefore, the researcher rejected the null hypothesis and concluded that the CSA approach and drill-practice have significantly enhanced the Grade 10 learners’ ability to simplify algebraic fractions.

I. Statement of the problem. The subject of abstraction is of very great importance not only in science, in logic and mathematics, or in the theory of knowledge, but also in art. It might be said that abstraction is a process in which consideration is given to some aspect or feature of a complex whole to the neglect of the remainder; but this statement is both too vague and too restrictive to cover all the cases in which abstraction is commonly said to occur. In order to avoid here the mistake involved in the "paradox of analysis", exemplified by saying that "centaur" and "medusa" mean the same thing (since there never was either the one nor the other), it is necessary to recognize that the abstractness of a work of art is something quite different from that of science, mathematics, logic or epistemology. This difference does not lie in the meaning of "abstraction", but in the purpose for which abstraction is used. In both art and science abstraction is the recognition of a relational structure or "form" apart from the specific thing in which it is exemplified. But the word "form" has different meanings in various fields. A logician or mathematician may question what sense it makes to call anything "form" except the logical form of discourse, the structure of propositions expressed either in ordinary language or in the refined symbolism of the pure sciences; an artist, in his turn, may ask how one can speak of the "form" as something invisible and intangible, as for example, the series of natural numbers, or elaborate mathematical equations, when for him "form" must be sensible. The problem of abstraction in art is complicated further by the fact that, although, (speaking of painting and sculpture) abstraction is achieved by the use of the technical device of schematized shapes (usually called "abstractions"), contemporary art critics and artists are divided as to how abstraction "works". Roughly speaking, they are divided into two camps, and the defence of each position rests on a different view of the world. One theory of art experience presupposes a world consisting of mysterious entities which, by interaction with another equally mysterious entity, the ego, produce sensations. Configurations of these sensations result in artistic form in two basic varieties: one, the naturalistic, the product of an "affirmative" attitude toward the world, is bound to concrete experiences, in fact, to physical things; the other, the non-naturalistic or "abstract", the consequence of "negative" attitude toward the world, essentially artificial and difficult, results from modern man's "dread of space" or "fear of nothingness", etc. I shall call this theory in its two aspects the "existentialist theory" of art experience. Another theory of art experience takes a rational view of the universe (recognizing, however, certain limits to man's reason) and proposes that that experience is closely bound up with sense perception and cognition analyzable in logical terms. In this tradition abstraction is viewed as an intellectual process, capable of being discussed in terms other than "positive" and "negative". Accordingly, the work of art is so constructed that its categorical elements are in common with reality, and combined to represent a coherent structure. Abstraction in art is defined as the constant experimenting with syntactical combinations of the language elements of that special type of language which is the actual work of art. I shall call this the "rational theory" of art experience. Discussion of this view of abstraction will involve the discussion of what is meant by "work of art", "internal" and "external" logic of the structure of works of art, what I mean by such banalities as "reality" and "truth". In a word, the discussion will turn upon the meaning and purpose of all artistic activity. II. Method of investigation. In discussing the problem outlined above I am not going to declare which of the two views of the world is right, which is wrong, but I will support the theory which appears to me the more exciting and which seems to afford the more satisfactory explanation of what is meant by abstraction in art. In doing it I shall use as supporting arguments my own conclusions about works of recent and contemporary masters along with quotations from their manifestos, autobiographies and other writings, as well as the ideas of critics and historians about them. Also I shall not shrink from using classical sources, because, to my mind, the phenomenon of abstraction cannot be reserved for contemporary art alone. It cannot be denied, however, that the use of the adjective "abstract", synonymous with "modern", "non-objective", "non-representational", etc., has become quite frequent and has acquired special importance since the turn of the century. The term "abstract art" is generally understood as denoting cubism, futurism and expressionism, and demonstrating a common effort on the part of contemporary artists, musicians and writers on the one hand, and scientists, philosophers and mathematicians on the other, to solve similar or identical problems. Therefore some attention will be directed to contemporary developments in these disciplines along with art. In my treatment of the subject I shall stress the philosophical implications rather than the historical significance. Above all, my approach will be determined by the influence upon me of the Cambridge philosophers and the adherents of philosophical movements related to them, but especially the writings of Ludwig Wittgenstein will be seen to have inspired the concepts developed in this thesis. III. General conclusions. The general conclusions arrived at in this thesis might be stated in the form of the following principles: (a) All art is abstract. (b) In art the aim is to depict reality. (c) A work of art is constructed so that its categorical elements are in common with reality and combined to represent a coherent structure (correspondence theory). (d) A work of art is an interpreted fact, and thus represents a prototype truly or falsely. Truth is arrived at when inquiry is stopped. (e) Since the elements of a work of art are combined in a definite way, its truth value will be determined upon the examination of the "internal" logic of its structure, together with the "key of interpretation." (f) It makes sense to say that there are many realities and one way of interpreting them, just as it makes sense to say that there is one reality and many ways of interpreting it. (g) Abstraction in art is the constant experimenting with syntactical combinations of the "language elements" of that special type of language which is the actual work of art. / Arts, Faculty of / Art History, Visual Art and Theory, Department of / Graduate

Thesis (M.F.A.)--Rochester Institute of Technology, 1993. / Typescript. Includes bibliographical references (leaf [32]).

Thesis (M.F.A.)--Rochester Institute of Technology, 1992. / Typescript. Includes bibliographical references (leaf 42).

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In my paintings and prints, I create to understand my experiences. I make layered, repetitive marks and gestures that consist of spherical masses, orbits, and cellular forms. The shapes represent aspects of reproduction and symbolize my thoughts and ideas about procreation. In making pieces, I employ a mathematical system that describes growth patterns found in nature. I use this system as a way of echoing natural structures, as well as a way of focusing my attention. I create multiple small pieces and present them in large, gridded formats. These pieces are abstracted narratives of my thoughts and feelings.

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