Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1974. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 72-77).

The “graph-as-picture misconception” (GAPM) occurs when an abstract representation (e.g., a line graph) is interpreted as a picture of an object (e.g., a mountain). Previous research on students' line graph interpretations has focused on secondary school level and above, thus this research extends the investigation of the GAPM to primary school level. Particularly, it investigates: which type of environment is more effective for improving young students' line graph interpretations; and how can be assessed their interpretations. A pilot study involved an improved version of Janvier's (1978) paper-and-pencil tasks (to create an interactive learning environment) and it investigated how to incorporate a card-sort task (to assess students' interpretations). Different touch-screen technologies were considered too. Two experiments were conducted. In experiment one, 37 participants (third to sixth year) were assessed in their graphical knowledge through a picture/diagram card-sort task and a “pictorial group” was formed using participants' interpretations. During the intervention, students performed an active or passive mode of a Racing Car activity in which they moved or watched a car along a track while its speed/distance graph was plotted concurrently alongside. The results suggested that a wide variety of pictorial interpretations exist and students seemed to benefit from the active modality. In experiment two, 38 fifth-year students performed different assessment tests. Extending experiment one, a “drawing the graph” mode and its passive modality were included. In that mode, students modified a plotted line of a speed/distance graph, which was used by the system to race a car along a track. Previous results were not confirmed: only students under the “drawing the graph” modality (including the “pictorial group”) significantly improved their interpretations; and different assessment tests seemed better to observe students' various interpretations. In conclusion, a learning environment that allows interaction with the representation could potentially improve students' interpretations, which might be better assessed through a rich set of tests.

Considering that geometry is taught according to certain principles that do not encourage creativity, I have decided to employ the mixed methods philosophical framework applying the concurrent transformative design in the form of an exploratory case study. The case study to (i) explore and design a model that influences learning using polygon pieces and mathematics dictionary in the teaching and learning of geometry to grade 8 learners; (ii) investigate if the measurement of angles and sides of polygons using polygon pieces assisted by mathematics dictionary promote learners’ comprehension of geometry and (iii) investigate how mathematics teachers should use polygon pieces along with mathematics dictionary to teach properties of triangles in order to promote learners’ conceptual understanding. Drawing from my research findings a model has been developed from the use of polygon pieces and mathematics dictionary. The model use of mathematics dictionary in teaching and learning geometry is to develop learners’ mathematics vocabulary and terminology proficiency. Polygon pieces are to enhance the comprehension of geometric concepts. The quantitative data emerged from marked scripts of the diagnostic and post-intervention tests, the daily reflective tests and intervention activities were analysed as percentages and presented in line and bar graphs. Qualitative data obtained from observation notes and transcribed interviews were analysed in three forms: thematically, constant comparison and keywords in context. These findings support other research regarding the importance of using physical manipulatives with mathematics dictionary in teaching and learning geometry. They align with other findings that stress that manipulatives are critical facilitating tools for the development of mathematics concepts. The investigations led into the designing of a teaching model for the topic under study for the benefit of the mathematics community in the teaching and learning of geometry, focusing on properties of triangles. The model developed during this study adds to the relatively sparse teaching models but growing theoretical foundation of the field of mathematics. / Mathematics Education / Ph. D. (Mathematics Education)