Low levels of learner performance in Mathematics in the Senior Phase (Grades 7-9) in South Africa is often attributed to insufficient mathematics content knowledge among teachers. Although this view might be justifiable, it is often incorrect to assume that content knowledge alone will solve the problem of low performance in mathematics. This study, therefore, argues that understanding learner misconceptions and/or errors and their underlying intricacies could provide the basis for instructional decision making, subsequently improved performance in mathematics. The purpose of the study was to explore the implications of teachers’ understanding of learner errors for mathematics learning. The study was guided by qualitative methods using a case study design which involved data collection from two schools, followed by in-depth data analysis. Two theoretical lenses, namely, Cognitively Guided Instruction (CGI) and Constructivist theory were used to explore the main research question: What are the implications of the teachers’ understanding of learner errors on the learning of school mathematics in the Senior Phase (specifically Grade 9)? Data was collected through lesson observations, analysis of learners’ test responses and interviews. The findings revealed that teachers’ understanding of learner errors from written responses differed notably from intricacies of same errors emanating from interviewing the learners as well as the same errors analysed by the researcher. The implications of these findings suggest the likelihood of a mismatch between teachers’ instructional decision making and learner misconception/errors and this may hamper effective learning of mathematics. / Dissertation (MEd)--University of Pretoria, 2020. / Science, Mathematics and Technology Education / MEd / Unrestricted

The purpose of this study was to investigate the effect of teachers attending Cognitively Guided Instruction (CGI) professional development on students' problem solving strategies and the effect of students' use of strategies on their mathematics achievement as measured by a standardized test. First, the study analyzed the differences in students' use of strategies between treatment and control groups. The treatment was CGI professional development, and the teachers in the treatment group attended CGI workshops whereas the teachers in the control group did not. Next, the study analyzed the differences in the mathematics achievement of students between different strategy groups. A student posttest, which was ITBS (Math Problems and Math Computation), was used to compare students' mathematics achievement. A student pretest was used as a covariate. The results of this study showed that there were statistically significant differences in the students' use of strategies between the treatment and control groups at the second grade level. A greater percentage of treatment students used derived facts / recall strategies (the most advanced strategy for single-digit addition and subtraction) than control students did. The results related to the effect of students' use of strategies on their mathematics achievement showed that the students who used derived facst/recall strategies for single-digit problems had significantly higher mathematics achievement than students who used counting or concrete modeling strategies. Furthermore, the students who used invented algorithms for multi-digit problems had significantly higher mathematics achievement than the students who used standard algorithms.

abstract: The need for improved mathematics education in many of America's schools that serve students from low income households has been extensively documented. This practical action research study, set in a suburban Title I school with a primarily Hispanic, non-native English speaking population, is designed to explore the effects of the progression through a set of problem solving solution strategies on the mathematics problem solving abilities of 2nd grade students. Students worked in class with partners to complete a Cognitively Guided Instruction-style (CGI) mathematics word problem using a dictated solution strategy five days a week for twelve weeks, three or four weeks for each of four solution strategies. The phases included acting out the problem using realia, representing the problem using standard mathematics manipulatives, modeling the problem using a schematic representation, and solving the problem using a number sentence. Data were collected using a five question problem solving pre- and post-assessment, video recorded observations, and Daily Answer Recording Slips or Mathematics Problem Solving Journals. Findings showed that this problem solving innovation was effective in increasing the problem solving abilities of all participants in this study, with an average increase of 63% in the number of pre-assessment to post-assessment questions answered correctly. Additionally, students increased the complexity of solutions used to solve problems and decreased the rate of guessing at answers to word problems. Further rounds of research looking into the direct effects of the MKO are suggested as next steps of research. / Dissertation/Thesis / Ed.D. Educational Leadership and Policy Studies 2013

abstract: The purpose of this study is to impact the teaching and learning of math of 2nd through 4th grade math students at Porfirio H. Gonzales Elementary School. The Cognitively Guided Instruction (CGI) model serves as the independent variable for this study. Its intent is to promote math instruction that emphasizes problem-solving to a greater degree and facilitates higher level questioning of teachers during their instructional dialogue with students. A mixed methods approach is being employed to see how the use of the CGI model of instruction impacts the math achievement of 2nd through 4th grade students on quarterly benchmark assessments administered at this school, to see how students problem-solving abilities progress over the duration of the study, and to see how teacher practices in questioning progress. Quantitative methods are used to answer the first of these research questions using archival time series (Amrein & Berliner, 2002) to view trends in achievement before and after the implementation of the CGI model. Qualitative methods are being used to answer questions around students' progression in their problem-solving abilities and teacher questioning to get richer descriptions of how these constructs evolve over the course of the study. / Dissertation/Thesis / Ed.D. Educational Leadership and Policy Studies 2012

This study reports the development of an instrument intended to measure mathematics teachers' knowledge of Cognitively Guided Instruction (CGI). CGI is a mathematics professional development framework based on how students think about and solve problems and how that knowledge guides instruction for developing mathematical understanding. The purpose of this study was to (a) analyze and revise the original CGI Teacher Knowledge Assessment (CGI TKA), (b) administer the revised CGI TKA, and (c) analyze the results from the revised CGI TKA. As part of the revision of the original CGI TKA, distractor analysis identified distractors that could be improved. Experts in CGI content were interviewed to identify ways in which the content of the CGI TKA could be improved, and some new items were created based on their feedback. Formatting changes were also made to administer the assessment electronically.After the original CGI TKA was revised, the revised CGI TKA was administered to teachers who had been trained in CGI. Two hundred thirteen examinees completed the revised CGI TKA and the results were analyzed. Exploratory and confirmatory factor analyses showed 21 of the items loaded adequately onto one factor, considered to be overall knowledge of CGI. The Rasch model was used to estimate item difficulty and person abilities as well as to compare models using dichotomous and partial credit scoring. Advantages and disadvantages of using partial credit scoring as compared to dichotomous scoring are discussed. Except under special circumstances, the dichotomous scoring produced better fitting models and more reliable scores than the partial credit scoring. The reliability of the scores was estimated using Raykov's rho coefficient. Overall, the revised CGI TKA appears to validly and reliably measure teachers' CGI knowledge.