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Thesis (Ph. D.)--University of Hawaii at Manoa, 1994. / Includes bibliographical references (leaves 104-107). / Microfiche. / viii, 107 leaves, bound ill. 29 cm

The early years of schooling are a crucial part of a student’s education. Recent years have seen the implementation of new literacy and numeracy programs in primary school classrooms. The key area of mathematics (numeracy) has been closely monitored and funded by political and educational bodies (Clarke, Cheeseman, Gervasoni, Gronn, Horne, McDonough, Montgomery, Roche, Sullivan, Clarke, & Rowley, 2002; Association of Independent Schools of South Australia, 2004). The new numeracy programs have been introduced into the school curriculum to ensure that all students’ needs are catered for in the classroom program. However, standardised testing using pencil and paper is still being used as the accepted form of assessment. The Victorian State Government uses the Achievement Improvement Monitor (AIM) to assess students’ mathematical achievement levels. This pencil and paper test is conducted for students in years three, five, seven and nine and is used to sort the students into a percentile group. Other than the ‘Early Numeracy in the Classroom’ program (2002) used by Victorian schools as a prep. to three program, where a one-on-one interview is used as a form of assessment, there is currently no program that offers students the opportunity to choose how best to represent their own mathematical understanding. Although, the learning needs of students are being better catered for within the classroom, students are being disadvantaged by the way in which they are assessed.

Recent international reports, for example TIMSS (2011 & 2015), point to serious challenges in South African learner performance in Mathematics and Science. Of greatest concern is that research findings (e.g. Graven, Venkat, Westaway and Tshesane 2013) suggest that many South African learners show signs of mathematical knowledge gaps in the lower grades. Hence, there is a need to address challenges of this nature very early in Foundation Phase. This study was undertaken with a view to contribute towards addressing mathematical challenges encountered by learners in Foundation Phase This empirical enquiry was undertaken under the auspices of the South African Numeracy Chair Project (SANCP) at Rhodes University whose mission is to develop sustainable ways of improving quality teaching and learning of Mathematics in South Africa. A relatively new SANCP programme called Pushing for Progression (PfP) run as part of the after-school Maths Clubs to develop the number sense and four Operations in learners was used to achieve the research aims of this study. Research participants were drawn from the Maths Clubs established by the researcher in a small rural town of Ottosdal in the North West Province of South Africa. This Study is grounded on the Vygotskian perspective and uses the interpretivist qualitative research method for data collection and analysis. Sampling was done opportunistically by enlisting participants (12 teachers and 117 learners) on the basis of their availability and willingness to participate. Pre- and post-assessment of learners’ proficiency on the four Basic Operations was conducted at the beginning and at the end of the research project, respectively. This was done to determine the impact of the project on learner performance. Data analysis was done thematically and through the comparison of learner results of the pre- and post-assessment. The findings point to the effectiveness of the PfP Programme in learner performance. This can be deduced from improved scores between pre- and post-assessment and the observations made by participant-teachers on their respective club learners’ mathematical proficiencies. Accordingly, based on the findings, this study recommends, inter alia, that since the PfP programme is still in its early stages, similar research be conducted elsewhere. Additionally, the Department of Basic Education could consider exploring the PfP programme as one of several other strategies to help improve learner proficiency in Mathematics.

M.Ed. (Educational Psychology) / Please refer to full text to view abstract.

Thesis (M. Ed. (Mathematics)) -- University of Limpopo, 2022 / The purpose of this study was to document my enactment of the five key strategies of assessment for learning in my mathematics classroom to account for learners' mathematical understanding. I used a constructivism teaching experiment methodology to explore learners' mathematical activities as they interacted in the classroom. Twenty-five learners from my Grade 10 mathematics class took part in the study. Data were gathered through classroom observations, written work samples from learners, and the teacher's reflective journal. My enactment of the five key strategies enabled learners to participate in classroom discussions, collaborate with their peers, and use self-assessment tools while engaging in classroom interactions. The major findings revealed that, through my enactment of the five key strategies, learners developed conceptual understanding, procedural fluency and strategic competence of the concepts taught. In addition, practices such as the development of lesson plans detailing how the five key strategies will be enacted in the classroom, use of comment – only feedback for grading learners’ work, creating a conducive learning environment to allow the use of peer and self-assessment allowed for a meaningful enactment of assessment for learning in my classroom. Strategies four and five, whose primary goal is to encourage learners' participation in the lesson, were critical in promoting learners' mathematical understanding.

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Perspective and Purpose Recent investigations by Steffe, Richards, and von Glasersfeld (1979) have indicated that addition and subtraction problem-solving competencies are developmental in nature and that these competencies build upon counting abilities. They postulate that, in beginning addition and subtraction, a type of problem-solving strategy termed counting-all develops prior to another kind of strategy termed counting-on (for addition) and counting-back (for subtraction). If these tasks are developmental, one may assume that students approach the tasks in qualitatively different ways based upon their developmental levels. Neo-Piagetian researchers have postulated that a quantitative measure of development explains the qualitatively different ways in which children react to the same cognitive task at different stages of development. The measure, termed mental space or M-Space, describes the number of schemes which may be coordinated at one time. First graders, the majority of whom have an M-Space of a+2 or a+3, are capable of solving addition and subtraction word problems by utilizing the counting-all and/or the counting-on (back) strategies. Given this information, the purpose of this study was to determine what effect M-Space level has on the strategy a subject uses to solve problems when he is trained on a strategy which either matches or mismatches his M-Space level. Design To determine whether a match between M-Space and strategy demand is necessary or whether instruction will facilitate the chunking of schemes which allows the developmental task to be solved by a strategy which would otherwise be above the subject's M-Space level, the following steps occurred: one hundred thirty-nine first graders were pretested to identify those who could count to sixteen, perform numeral/number correspondence to sixteen, but could not solve addition and subtraction number fact problems to sixteen. One hundred fifteen subjects meeting these criteria were given the Cucumber Test and Backward Digit Span Test to assess their M-Space levels. After eliminating subjects before and during training, 50 subjects remained. Twenty-six subjects with an a+2 M-Space were divided into two training groups. Approximately half of the group was trained to use an a+2 strategy (the count-all strategy) to solve addition and subtraction word problems and the other half of the group was trained to use an a+3 strategy (the count-on (back) strategy). The same training procedure was used for the twenty-four subjects with an M-Space of a+3. Four to five weeks later, a delayed posttest consisting of four addition and four subtraction problems and one each of three types of transfer problems was presented. Results Mann-Whitney test results indicated that there were significantly fewer a+3 responses by the subjects with an a+2 M-Space who were trained to use an a+3 strategy than there were for subjects with an a+3 M-Space trained to use an a+3 strategy. However, there was no significant difference between those with an a+2 M-Space trained on an a+2 strategy and those with an a+2 M-Space trained on an a+3 strategy. Results of other research questions indicated that subjects gave similar responses to transfer problems which varied by material or additional variable; for subjects with an a+3 M-Space trained on an a+3 strategy, there were significantly more a+3 addition responses than subtraction responses; the implied comparison subtraction problem was answered incorrectly more often than straight take-away subtraction problems, and students tended to devise simple addition and subtraction problems and solve them by using memorized number facts. Discussion The findings indicate that more study is warranted for the application of the M-Space construct to a theory of how mathematical knowledge develops sequentially, the different ways in which addition and subtraction tasks can be conceptualized, and the instructional implications of applying a developmentally based theory of instruction to mathematics problem-solving. / Ph. D.

The primary purpose of this study was to examine the extent to which age-related bias exists when ACT Mathematics test scores are used as the sole predictor of future academic performance in entry-level college mathematics courses. A secondary purpose was to investigate the extent to which academic and attitudinal variables, in conjunction with ACT Math scores, a) lessen or eliminate the age-related bias, and b) enhance the prediction of course grades and posttest scores in freshman level mathematics courses. ACT Mathematics test scores were used to predict course grades and posttest scores of students enrolled in Developmental Math and General Math classes at Bluefield State College, WV, or one of its community college components. Course grades of Developmental Math students and posttest scores of General Math students were found to be under-predicted for nontraditional-age students and over-predicted for traditional-age students. No differences were found in predictions of posttest scores for Developmental Math students or in predictions of course grades for General Math students. When attitudinal and other academic variables were introduced to the regression equation, there was less evidence of prediction bias and a significant increase in the amount of variance explained in the criterion measures. / Ed. D.

The study reported here examined the move towards online assessments and addressed the question of whether or not different assessment tools affect student scores and student learning. The research activities covered a three-week period, from June 5, 2006 to June 23, 2006. During this time, seventeen third grade students served as their own control group by taking several math tests online and several math tests with paper and pencil. Results were compared to see if performance on computer-based tests would be more successful than pencil-and-paper tests. A follow-up survey to evaluate and interpret the quantitative results was also used. Findings revealed that 59% of the students did better on the paper tests. Further analysis revealed that the overall average of the computer tests was 71.9%, while paper tests revealed an average of 69.9%. In summary, these findings report that there is no significant difference in scores when taking a test on the computer or a test on paper. Suggestions for further research and recommendations are included.