Web-based diagnosis of misconceptions in rational numbers

A thesis submitted to the Wits School of Education, Faculty of Humanities, University of the Witwatersrand in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016.This study explores the potential for Web-based diagnostic assessments in the classroom, with specific focus on certain common challenges experienced by learners in the development of their rational number knowledge. Two schools were used in this study, both having adequate facilities for this study, comprising a well-equipped computer room with one-computer-per-learner and a fast, reliable broadband connection. Prior research on misconceptions in the rational numbers has been surveyed to identify a small set of problem types with proven effectiveness in eliciting evidence of misconceptions in learners. In addition to the problem types found from prior studies, other problem types have been included to examine how the approach can be extended. For each problem type a small item bank was created and these items were presented to the learners in test batteries of between four and ten questions. A multiple-choice format was used, with distractor choices included to elicit misconceptions, including those previously reported in prior research. The test batteries were presented in dedicated lessons to learners over four consecutive weeks to Grade 7 (school one) and Grade 8 (school two) classes from the participating schools. A number of test batteries were presented in each weekly session and, following the learners’ completion of each battery, feedback was provided to the learner with notes to help them reflect on their performance. The focus of this study has been on diagnosis alone, rather than remediation, with the intention of building a base for producing valid evidence of the fine-grained thinking of learners. This evidence can serve a variety of purposes, most significantly to inform the teacher on each learners’ stage of development in the specific micro-domains. Each micro-domain is a fine-grained area of knowledge that is the basis for lesson-sized teaching and learning, and which is highly suited to diagnostic assessment. A fine-grained theory of constructivist learning is introduced for positioning learners at a development stage in each micro-domain. This theory of development stages is the foundation I have used to explore the role of diagnostic assessment as it may be used in future classroom activity. To achieve successful implementation into time-constrained mathematics classrooms requires that diagnostic assessments are conducted as effectively and efficiently as possible. To meet this requirement, the following elements of diagnostic assessments were investigated: (1) Why are some questions better than others for diagnostic purposes? (2) How many questions need to be asked to produce valid conclusions? (3) To what extent is learner self-knowledge of item difficulty useful to identify learner thinking? A Rasch modeling approach was used for analyzing the data, and this was applied in a novel way by measuring the construct of the learners’ propensity to select a distractor for a misconception, as distinct from the common application of Rasch to measure learner ability. To accommodate multiple possible misconceptions used by a learner, parallel Rasch analyses were performed to determine the likely causes of learner mistakes. These analyses were used to then identify which questions appeared to be better for diagnosis. The results produced clear evidence that some questions are far better diagnostic discriminators than others for specific misconceptions, but failed to identify the detailed rules which govern this behavior, with the conclusion that to determine these would require a far larger research population. The results also determined that the number of such good diagnostic questions needed is often surprisingly low, and in some cases a single question and response is sufficient to infer learner thinking. The results show promise for a future in which Web-based diagnostic assessments are a daily part of classroom practice. However, there appears to be no additional benefit in gathering subjective self-knowledge from the learners, over using the objective test item results alone. Keywords: diagnostic assessment; rational numbers; common fractions; decimal numbers; decimal fractions; misconceptions; Rasch models; World-Wide Web; Web-based assessment; computer-based assessments; formative assessment; development stages; learning trajectories

Widening the Knowledge Acquisition Bottleneck for Intelligent Tutoring Systems

Empirical studies have shown that Intelligent Tutoring Systems (ITS) are effective tools for education. However, developing an ITS is a labour-intensive and time-consuming process. A major share of the development effort is devoted to acquiring the domain knowledge that accounts for the intelligence of the system. The goal of this research is to reduce the knowledge acquisition bottleneck and enable domain experts to build the domain model required for an ITS. In pursuit of this goal an authoring system capable of producing a domain model with the assistance of a domain expert was developed. Unlike previous authoring systems, this system (named CAS) has the ability to acquire knowledge for non-procedural as well as procedural tasks. CAS was developed to generate the knowledge required for constraint-based tutoring systems, reducing the effort as well as the amount of expertise in knowledge engineering and programming required. Constraint-based modelling is a student modelling technique that assists in somewhat easing the knowledge acquisition bottleneck due to the abstract representation. CAS expects the domain expert to provide an ontology of the domain, example problems and their solutions. It uses machine learning techniques to reason with the information provided by the domain expert for generating a domain model. A series of evaluation studies of this research produced promising results. The initial evaluation revealed that the task of composing an ontology of the domain assisted with the manual composition of a domain model. The second study showed that CAS was effective in generating constraints for the three vastly different domains of database modelling, data normalisation and fraction addition. The final study demonstrated that CAS was also effective in generating constraints when assisted by novice ITS authors, producing constraint sets that were over 90% complete

Virtual Reality in Mathematics Education (VRiME):An exploration of the integration and design of virtual reality for mathematics education

This thesis explores the use of Virtual Reality (VR) in mathematics education. Four VR prototypes were designed and developed during the PhD project to teach equations, geometry, and vectors and facilitate collaboration.Paper A investigates asymmetric VR for classroom integration and collaborative learning and presents a new taxonomy of asymmetric interfaces. Paper B proposes how VR could assist students with Autism Spectrum Disorder (ASD) in learning daily living skills involving basic mathematical concepts. Paper C investigates how VR could enhance social inclusion and mathematics learning for neurodiverse students. Paper D presents a VR prototype for teaching algebra and equation-solving strategies, noting positive student responses and the potential for knowledge transfer. Paper E investigates gesture-based interaction with dynamic geometry in VR for geometry education and presents a new taxonomy of learning environments. Finally, paper F explores the use of VR to visualise and contextualise mathematical concepts to teach software engineering students.The thesis concludes that VR offers promising avenues for transforming mathematics education. It aims to broaden our understanding of VR's educational potential, paving the way for more immersive learning experiences in mathematics education

Instructional strategies in explicating the discovery function of proof for lower secondary school students

In this paper, we report on the analysis of teaching episodes selected from our pedagogical and cognitive research on geometry teaching that illustrate how carefully-chosen instructional strategies can guide Grade 8 students to see and appreciate the discovery function of proof in geometr