It has been assumed that mathematics testing indicates the development of mathematics concepts, but the linguistic demands of assessment have not been evaluated, especially for children with mild intellectual disabilities. 244 children (grades 2 – 5) were recruited from a larger reading intervention study. Using a multilevel longitudinal SEM model, baseline and post-intervention time points were examined for the contribution of item linguistic complexity, child language skills, and their potential interaction in predicting item level mathematics assessment performance. Item linguistic complexity was an important, stable, and negative predictor of mathematics achievement with children’s language skills significantly and positively predicting mathematics achievement. The interaction between item linguistic complexity and language skills was significant though not stable across time. Following intervention, children with higher language skills performed better on linguistically complex mathematics items. Mathematics achievement may be related to an interaction between children’s language skills and the linguistic demands of the tests themselves.

This doctoral research was concerned with the role of language and its implications for the learning, teaching and assessment of mathematics for children aged 4-9 years. Earlier research by the author had established language and assessment as bridges enabling learning although they had the potential to increase the divide between teacher and learner. Reflection raised the question on how children achieved in mathematics despite potential difficulties with language and assessment. Review of the literature concluded that resources and sociocultural norms were also bridges between learner and teacher. A model was established of the relationships and processes between all perceived variables that provided an external, theoretical structure to be evaluated against structuralist, pragmatic and integrational linguistic approaches and empirical outcomes. The overarching approaches adopted were institutional ethnography and case study. An appropriate methodology was devised whereby sophisticated ICT equipment captured all visual and speech events during classroom interactions. Frequency analysis at word level, content analysis at utterance level and discourse analysis at total speech level triangulated with content analysis of interviews and evaluation of documentation completed the empirical research. Data analysis revealed five registers of children’s talk. Evidence suggested that the peer-peer ‘conditioned talk’ used in focused group work was the most effective for learning as it enabled them to discern the small steps in the inferential leaps in discourse made by their teachers, work out problems together, inform their peers, share findings and reinforce each others’ learning. Learners’ language showed aspects of structural, pragmatic and integrational linguistics, confirming a conclusion of the literature review that the various linguistic approaches discussed should be used to support and not exclude each other. The contribution made to knowledge is the ethnomethodology provided by the model, ICT resource and the five registers of talk revealed by the linguistic approach to discourse analysis. Teachers would be able to understand nuances of language used by their pupils and acquire essential skills and tools to put into effect the personalised learning agenda. Peer-peer observation of teachers would be an appropriate platform for the observation of the different registers used by learners, the resources that generate those registers, and their most effective use to close the gap between natural language and the subject specific language of mathematics.

Educators need clear, actionable data to help them understand students' current levels of performance and students' probable trajectory toward college- and career-readiness in math if they are to make informed programmatic decisions to shape that trajectory. This study explored the relation between CBM-math in Grade 7 as a one-point, teacher accessible measure of student math skill and the students' performance on the Grade 8 EXPLORE-math test, a large-scale achievement test linked to one set of college- and career-readiness benchmarks. Results indicated that a moderate positive correlation and predictive relation exist between CBM-math and EXPLORE-math. Information was disaggregated by gender and for subgroups, including students eligible for special education, free or reduced meals, and English language development services. No difference in means for male and female students on either measure was identified, but eligibility for special education or for free or reduced lunch was associated with lower performance on both measures. Insufficient numbers of ELD students hindered detailed analysis, but none of the ELD students included in the study achieved the EXPLORE benchmark or the CBM normalized cut score based on the 40th percentile. An ROC analysis showed that easyCBM consistently predicted students who did not meet the EXPLORE benchmark, although results indicated that a higher cut score on easyCBM may be a more consistent predictor. The study adds to validity research on CBM and may be useful for educators seeking to identify students at risk of missing achievement benchmarks and make programmatic decisions to ensure students are on track to be college- and career-ready in math.

Technology may play a critical role in impacting student engagement, specifically within an assessment context. Using a mixed methods approach, I examined the relationship between varying degrees of technology-enhancements applied in a mathematics performance task on the outcome of student cognitive engagement. Using a counterbalanced quasi-experimental design, I evaluated the impact of three performance task platforms on student self-reported cognitive engagement in from a sample of students in grades 6-8 in Oregon, Washington, and North Carolina (N = 450). The three performance task platforms (a) included technology-enhanced (technology-based including animation and interactivity), (b) technology-enabled (computer-based without including animation and interactivity), (c) and paper-and-pencil. The measure used for cognitive engagement (CE-S-DSP & SOS) was a hybrid of previously used self-reporting tools and showed preferable reliability for the overall score of cognitive engagement. The data were not able to be explored using a 5-factor confirmatory factory analysis, due to model fit limitations. Results from the between subjects analysis of variance and did not suggest a relationship between performance task platform (modality type) and student cognitive engagement. Qualitative interview data indicated that students preferred using technology to take tests and overall showed favorability for the technology-enhanced performance task, specifically the interactivity and animations to help visualize and work through the problem. Yet, despite the positive links to technology-enhancements, there were features of paper-and-pencil tasks that students appreciated such as the ability to navigate between the items and the ability to take notes. Results indicated that just putting tests on computers may not be enough and technological affordance should be purposefully implemented. Findings from this study can help inform future use of platform type, technological enhancements employed, and strategies for technology use within an assessment context.

English language learner (ELL) is a term to describe students who are still acquiring English proficiency. In recent decades, ELLs are a very rapidly growing student group in United States. In school classrooms, ELLs are learning English and their academic subjects simultaneously. It is challenging for them to hear lectures, read textbooks, and complete tests in English despite of their inadequate English language proficiency (Ilich, 2013). As a result, the increasing number of ELLs in public schools has paralleled the increase in ELLs’ low mathematics performance (NCES, 2016). Due to the popularization of international large-scale assessments in the recent decade, it is necessary to analyze their psychometric properties (e.g., reliability, validity) so that those results can provide with evidence-based implications for policymakers. Educational researchers need to assess the validity for subgroups within each country. The Programme for International Student Assessment (PISA), as one of the influential large-scale assessments, allows researchers to investigate academic achievement and group membership from a variety of different viewpoints. The current study was to understand the nature and potential sources of the gaps in mathematics achievement between ELLs and non-ELLs. The nature of achievement gap was examined using three DIF methodologies including Mantel-Haenszel procedure, Rasch analysis, and Hierarchical Generalized Linear Model (HGLM) at the item level instead of total test level. Amon the three methods, HGLM was utilized to examine the potential sources of DIF. This method can take into account of the nested structure of data where items are nested within students, and students nested within schools. At the student level, sources of DIF were investigated through students’ variations in mathematics self-efficacy, language proficiency, and student socioeconomic status. At the school level, school type and school educational resource were investigated as potential sources of DIF after controlling for the student variables. The U.S. sample from PISA 2012 was used, and 76 dichotomously coded items from PISA 2012 mathematics assessment were included to detect DIF effects. Results revealed that ten common items are identified with DIF effects using MH procedure, Rasch analysis, and HGLM. These ten items are all in favor of non-ELLs.The decreasing number of items showing DIF effects in HGLM after controlling for student-level variables revealed mathematics self-efficacy, language proficiency, and SES are potential sources of DIF between ELLs and non-ELLs. In addition, the number of DIF items continued to decrease after controlling for both student and school-level variables. This finding proved that school type and school educational resources were also potential sources of DIF between ELLs and non-ELLs. Findings from this study can help educational researchers, administrators, and policymakers understand the nature of the gap at item level instead of the total test level so that United States can be competitive in middle school mathematics education. This study can also help guide item writers and test developers in the construction of more linguistically accessible assessments for students who are still learning English. The significance of this study lies in the empirical investigation of the gap between ELLs and non-ELLs in mathematics achievement at an item level and from perspectives of both students and schools.

This longitudinal study tracks how 5- to 7-year-olds perform with early number skills. The aim of this study is to diagnose at-risk mathematics students by distinguishing the skills that, if not mastered by the end of Kindergarten, lead to greater difficulty in mathematics in 1st grade. This study’s methodology is mixed as it follows an exploratory and inductive path in light of its use of a hypothesis, an interpretive path in light of its interest in the individual student, and a positivist path in light of its focus on developing rules from analyzed data. An oral diagnostic test based on a comprehensive collection of early number skills was used to test students as Kindergarteners and again as 1st graders. The test results created benchmarks, revealing how the majority of the students performed with early number skills. The test results also revealed that each early number skill is highly, moderately, or minimally predictive in terms of student placement by the end of 1st grade. When comparing the individual skill scores of each Kindergarten student to his/her total test results of 1st grade, the predictive power of each skill emerged. Performing poorly with skills that are minimally predictive did not seem to have an impact on how the Kindergarten student finished in 1st grade; performing poorly with moderately predictive skills had a greater impact on 1st grade placement; performing poorly with highly predictive skills in Kindergarten increased the likelihood that the student would finish in the lower attaining group in 1st grade. A third result of the test showed that certain skills serve as preconditions for other skills; success with certain skills usually meant success with other skills. These connections between skills point to a learning model called in this study “simultaneous pathways,” indicating that there are connections between certain skills, and that students can be learning on several pathways simultaneously. The impact of the predictive power of early number skills is that diagnosis becomes more effective. Early diagnosis means early remediation which may prevent at-risk students from falling further behind their peers. The benchmarks developed by this research will help teachers assess their students because they will know the general skill level of Kindergarteners and 1st graders. This oral diagnostic test informs curriculum development. If test results show that students are missing the skills that are highly predictive, teachers can address those gaps in order to insure mastery.

In this study, I investigate how successful provided response questions, such as multiple choice questions, are as an assessment format compared to the conventional constructed response questions. Based on the literature on mathematics assessment, I firstly identify an assessment taxonomy, consisting of seven mathematics assessment components, ordered by cognitive levels of difficulty and cognitive skills. I then develop a theoretical framework, for determining the quality of a question, with respect to three measuring criteria: discrimination index, confidence index and expert opinion. The theoretical framework forms the foundation against which I construct the Quality Index (QI) model for measuring how good a mathematics question is. The QI model gives a quantitative value to the quality of a question. I also give a visual representation of the quality of a question in terms of a radar plot. I illustrate the use of the QI model for quantifying the quality of mathematics questions in a particular undergraduate mathematics course, in both of the two assessment formats – provided response questions (PRQs) and constructed response questions (CRQs). I then determine which of the seven assessment components can best be assessed in the PRQ format and which can best be assessed in the CRQ format. In addition I also investigate student preferences between the two assessment formats. / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

The transition from school to tertiary study of mathematics is rightly coming under increasing scrutiny in research. This paper employs Tall’s model of the three worlds of mathematical thinking to examine key variables in teaching and learning as they relate to this transition. One key variable in the transition is clearly the teacher/lecturer and we consider the perspectives of both teachers and lecturers on teaching related matters relevant to upper secondary and first year tertiary calculus students. While this paper deals with a small part of the data from the project, which aims to model the transition, the results provide evidence of similarities and differences in the thinking of teachers and lecturers about the transition process. They also show that each group lacks a clear understanding of the issues involved in the transition from the other’s perspective, and there is a great need for improved communication between the two sectors.

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Mathematics, as a subject, is used in various scientific careers as a selection tool. It is regarded as the cornerstone of scientific literacy. However, since learners in South Africa do not perform optimally in mathematics they do not enjoy international recognition. Education renewal is ongoing, and South Africa currently follows an outcomes-based (OBE) approach. The teaching of mathematics cannot be renewed successfully if assessment methods are not regularly adapted to meet new developments in the field. The incorporation of an OBE approach at school level made it necessary to facilitate assessment renewal in tertiary mathematics at the Tshwane University of Technology (TUT). TUT is engaged in a merger of three institutions, which has made the development of new curricula and teaching material essential. Hence this a perfect time to introduce assessment renewal. The primary purpose of this thesis is to report on the research study and its results, and to make recommendations for improving the practice. The overarching research hypothesis in this study is that a suitable assessment would probably enhance the effectiveness of a student’s learning. The research focused on the following questions: -- To what extent are outcomes-based strategies effectively and regularly introduced in the teaching of mathematics at TUT? -- Will tertiary mathematics facilitators be prepared to implement outcomes-based strategies at TUT? -- To what extent are outcomes-based strategies strategies in subjects supported by mathematics implemented at TUT? -- How does the ecology of TUT affect the implementation of outcomes-based strategies? -- What other factors could influence the level of implementation of OBS at TUT? -- Have any of the mathematics facilitators at TUT received suitable and adequate training in the implementation of outcomes-based strategies? -- What are the possible implications of the study for TUT’s assessment policy? Action research was chosen as the research design because it is ideally suited to improving practice. Quantitative and qualitative data were collected through questionnaires, personal interviews, interviews with focus groups, observations, documentation and a reflective diary. The main findings are as follows:– -- OBE strategies are not being introduced throughout TUT in the teaching of mathematics. -- Group work and peer assessments are rare occurences. -- Some lecturers are convinced that new assessment methods would lower the standard of their teaching. -- Uncertainty about the merger and the varying teaching conditions at the different campuses tend to inhibit lecturers, making them less willing to undertake assessment renewals. -- TUT should review its admission criteria. -- The lecturers cited large class groups, a lack of marking assistance and ignorance about OBE as reasons for failing to undertake assessment renewal. The study prepared respondents for assessment renewal. In the interim, however, TUT has introduced a Policy on Teaching, Learning and Technology, whereby OBE has been selected as the teaching model for TUT. In future, respondents will receive training and guidance in the implementation of OBE. This study has hopefully made a significant contribution to this positive development. / Thesis (PhD (Curriculum Studies))--University of Pretoria, 2007. / Curriculum Studies / unrestricted