Over the past several decades there has been an emphasis on educational research pertaining to learners’ performance in Mathematics and on finding methods to improve learner performance in this subject. In South Africa, Grade 12 learners’ results in Mathematics from 2010 to 2013 were unsatisfactory as shown in DBE, 2013a. The teachers are challenged to find new teaching methods that will make the subject more interesting and appealing to the learners (Oliver & Makar, 2010 in Goos, 2010). The purpose of this study was to investigate the effect of using computers in the teaching and learning of Mathematics with special reference to the topic of linear functions in order to improve learner performance. The literature reviewed shows that the use of computers not only improves the learners’ performance but also changes their attitude towards Mathematics (Bester & Brand, 2013). The quantitative research approach was used to gather the data, namely the quasi- experimental, non-equivalent control group pre-test-post-test design. Two intact classes formed part of the research study, that is an experimental group (n=50) and control group (n=50). The experimental group learnt the concept of linear function using GeoGebra software. The control group learnt the same concept through the traditional pen and paper method. The data were analysed using the SPSS on ANOVA. The results indicated that there was a significant difference between the mean scores of the experimental group (μ=70.5) and the control group (μ=47.5). From the results it was evident that the use of computers had a positive effect on learners understanding of linear functions as reflected in their performance and on their attitude towards Mathematics, as seen in the questionnaire responses. / Mathematics Education / M. Ed. (Mathematics Education)

As estruturas de dados concretas (cds) são quaternas (C, V, E, l-) que contêm um conjunto C de células, um conjunto V de valores, um conjunto E de eventos e uma relação de habilitação l-. O conjunto de estados de uma cds é um domínio concreto que pode ser considerada a parte "abstrata" das cds. Da mesma maneira tem-se que os domínios de eventos (que são generalizações dos domínios concretos) são a parte abstrata das estruturas de eventos. Mostra-se a relação dos domínios concretos e domínios de eventos com os espaços coerentes, assim como também das teias de espaços coerentes com as cds e estruturas de eventos. Intuitivamente, uma cds é uma teia de um espaço coerente se toda célula c de C não é habilitada por nenhum evento (ou equivalentemente, é habilitada pelo conjunto vazio), isto é, V C E C, 0 F c. Outra forma de expressar isto é dizer que uma cds e uma teia de um espaço coerente se o conjunto de estados da cds é um espaço coerente. Definem-se os algoritmos lineares como sendo estados de uma cds no estilo dos algoritmos seqüenciais do Curien ([CUR 86]). Em particular as cds consideradas são teias de espaços coerentes. Mostra-se como obter a cds !A—>B, a partir de uma função estável f. A —> B. O algoritmo linear desta cds possui todas as estratégias de computação (seqüenciais e paralelas) que computam a função subjacente f, o que implica que os algoritmos lineares podem ser considerados meta-algoritmos. Mostra-se que para toda estratégia de computação seqüencial de um algoritmo linear, existe um algoritmo seqüencial de Curien que computa a mesma função, e vice-versa. A definição de estratégia de computação é dada de maneira tal que permite se dar semântica a segmentos de programas. Define-se uma operação de composição de estratégias, de forma tal que se pode obter uma estratégia de computação de um programa, a partir da composição das estratégias dos segmentos. / The concrete data structures, or cds, (C, V, E, l-) consists of a set C of cells, a set V of values. a set E of events and an enabling relation l-. The set of states of a cds is a concrete domain, that can be considered the "abstract" counterpart of the cds. In the same way we have that the events domains (that are more general that the concretes domains) are the abstract counterpart of the events structures. We show the relation between the concretes domains and events domains with the coherence spaces, as just as the relation between the cds and events structures with webs of coherence spaces. Intuitivelly, a cds is a web of a coherence space if any cell c is not enabled for any event, i.e. Vce C, 0 F c. We can say that a cds is a web of a coherence space if the set of states of the cds is a coherence space. We define the linear algorithms as states of a cds following the Curien's sequential algorithms ([CUR 86]). In particular the cds considered are webs of coherence spaces. We show how to obtain a cds !A—>B from a stable function f. A —> B. The linear algorithm of this cds contain all the computational strategies (sequentials and parallels) that compute the subjacent function f; this implies that the linear algorithms can be considered a kind of meta-algorithms. We show that for all sequential computational strategy of a linear al gorithm exists a Curien's sequential algorithm that compute the same function and conversely. We define the computational strategies in such a way that we can give semantic of segments of programs. We define a composition operation for strategies. This operation has the advantage that we can obtain the computational strategy of a program as the composition of the segments of it.

One major challenge facing mathematics education in South Africa in general and Limpopo in particular, is learners’ underachievement and lack of motivation to learn the subject. Some studies have shown that one of the topics that learners dread is linear functions. Many teachers also find it difficult to teach the topic effectively. Studies in other parts of the world have advocated the integration of graphing software with the teaching and learning of functions to enhance learners’ learning of mathematics. This study therefore investigated the effect of integrating GeoGebra graphing software into the teaching of linear functions on the achievement of Grade 9 learners. The study was guided by APOS theory which, in accordance with constructivist theories, posits that an individual needs to construct the necessary cognitive structures in order to make sense of mathematical concepts. A total of 127 Grade 9 learners from four schools in a circuit in Mopani district of Limpopo Province participated in the study which followed a pretestpost- test quasi-experimental study design. Two schools, namely B (35 learners) and D (33 learners) formed the experimental groups while school A (31 learners) and school C (28 learners) were the control groups. Data were collected using an achievement test and analyzed using descriptive and inferential statistics. The pretest results showed that the groups were of comparable cognitive abilities. The post-test results showed that there was a significant difference between the mean scores of the experimental groups and control groups. There were also statistically significant differences between group treatment means (p < .05). Bonferroni post-hoc test results showed that there were no statistically significant differences between treatments A and C. The results showed that the learners in the two control groups were of comparable cognitive abilities. The implications of the findings are discussed and recommendations made. / Mathematics Education / M. Sc. (Mathematics Education)

Many U.S. students do not perform well on mathematics assessments with respect to algebra topics such as linear functions, a building-block for other functions. Poor achievement of U.S. middle school students in this topic is a problem. U.S. eighth graders have had average mathematics scores on international comparison tests such as Third International Mathematics Science Study, later known as Trends in Mathematics and Science Study, (TIMSS)-1995, -99, -03, while Singapore students have had highest average scores. U.S. eighth grade average mathematics scores improved on TIMMS-2007 and held steady onTIMMS-2011. Results from national assessments, PISA 2009 and 2012 and National Assessment of Educational Progress of 2007, 2009, and 2013, showed a lack of proficiency in algebra. Results of curriculum studies involving nations in TIMSS suggest that elementary textbooks in high-scoring countries were different than elementary textbooks and middle grades texts were different with respect to general features in the U.S. The purpose of this study was to compare treatments of linear functions in Singapore and U.S. middle grades mathematics textbooks. Results revealed features currently in textbooks. Findings should be valuable to constituencies who wish to improve U.S. mathematics achievement. Portions of eight Singapore and nine U.S. middle school student texts pertaining to linear functions were compared with respect to 22 features in three categories: (a) background features, (b) general features of problems, and (c) specific characterizations of problem practices, problem-solving competency types, and transfer of representation. Features were coded using a codebook developed by the researcher. Tallies and percentages were reported. Welch's t-tests and chi-square tests were used, respectively, to determine whether texts differed significantly for the features and if codes were independent of country. U.S. and Singapore textbooks differed in page appearance and number of pages, problems, and images. Texts were similar in problem appearance. Differences in problems related to assessment of conceptual learning. U.S. texts contained more problems requiring (a) use of definitions, (b) single computation, (c) interpreting, and (d) multiple responses. These differences may stem from cultural differences seen in attitudes toward education. Future studies should focus on density of page, spiral approach, and multiple response problems.