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Developing knowledge about how students acquire mathematical understanding is a focus of mathematics curricula and research, including the ability of students to move from manipulating concrete materials to abstract number properties when solving problems. This study, informed by the Numeracy Development Projects (Ministry of Education 2007a, 2008b) and the work of Pirie and Kieren (1989, 1992, 1994a, 1994b), examines the role of ‘imaging’ in supporting the development of students’ mathematical thinking and understanding. Imaging is an important phase of the teaching model advocated by the Numeracy Development Project. The context of this study is a primary school mathematics programme, which involved the teaching of two mathematics units that focused on the addition and subtraction of decimal fractions and whole numbers. There is considerable research about what is effective in mathematics education for diverse learners, and how students learn. There is, however, very limited research about the role of imaging in mathematical learning. This qualitative study adopted a case study approach and focused on a group of Year 6 students. Data collection methods included observation, interviews, field notes and document analysis. A thematic approach was used to analyse data and to develop and inform an emerging theoretical framework. During this study I developed a model, entitled A Model for the Development of Students’ Mathematical Understanding, which illustrates six mathematical resources students use as they solve problems. These resources are: materials, mental picture images, drawn picture images, transformed mental images, transformed drawn images and number properties. Students’ engagement with these six resources illustrates how they develop understanding of mathematical concepts. The students identified a preference for using drawn rather than mental images when solving problems. This study also emphasizes the complexity of the imaging process, and the fluid and multifaceted nature of learning in mathematics. This study serves to highlight the complexities of the teaching and learning process in mathematics for both teachers and students.

Students at university encounter quantitative information in tables and graphs or through prose in textbooks, journals, electronic sources and in lectures. The degree to which students are able to engage with this kind of information and draw their own conclusions, influences the extent to which they need to rely on the interpretation of others. In particular, students who are studying in non-mathematical disciplines often fail to engage seriously with such material for a number of reasons. These may include a lack of confidence in their ability to do mathematics, a lack of mathematical skills required to understand the data, or a lack of an awareness of the importance of being able to read and interpret the data for themselves. In this thesis, the successful choice and use of skills to interpret quantitative information is referred to as numeracy. The level of numeracy exhibited by a student can vary depending on the social or cultural context, his/her confidence to engage with the quantitative information, the sophistication of the mathematics required, and his/her ability to evaluate the findings. The first part of the thesis is devoted to the conceptualisation of numeracy and its relationship to mathematics. The empirical study that follows this is focused on an aspect of numeracy of importance to university students: the reading and interpreting of tables of data in a range of non-mathematical contexts. The students who participated in this study were enrolled in degree programs in the social sciences. The study was designed to measure the effectiveness of a one-hour intervention workshop aimed at improving the levels of the students’ numeracy. The short length of the intervention was dictated by practical and organisational constraints. This workshop involved reading and interpreting a table of data using strategies based on the SOLO taxonomy (Biggs & Collis, 1982). The SOLO taxonomy was developed mainly as a means of classifying the quality of responses across both arts and science disciplines. The categorisation uses five levels: prestructural, unistructural, multistructural, relational and extended abstract. It can be used as a diagnostic tool at all levels of education as it can be seen as a spiral learning structure repeating itself with increasing levels of abstraction. It can also be used as a teaching tool in feedback to students. A measuring instrument, also based on the SOLO taxonomy, was designed to gauge the levels of the students’ responses to these tasks. Each response was allocated a level that was subsequently coded as a number from zero to seven. Because the responses were in distinct ordered categories, it was possible to analyse the scores using the Rasch Model (Rasch 1960/80) for polytomous responses, placing both the difficulty of the tasks and the ability of the students on an equal interval scale. The Rasch Model was also used to evaluate the measuring instrument itself. Some adjustments were made to the instrument in the light of this analysis. It was found that it is possible to construct an instrument to distinguish between levels of students’ written responses for each of the chosen table interpretation tasks. The workshop was evaluated through a comparison of the levels achieved by individual students before and after the workshop. T-tests for dependent samples indicated a significant improvement (p < 0.01) in student performance.

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Thesis (Ed.D.)--Murdoch University, 2006. / Thesis submitted to the Division of Arts. Bibliography: leaves 147-150.

Thesis (Ed.D.)--Murdoch University, 2005. / Thesis submitted to the Division of Arts. Bibliography: leaves 217-243.

Thesis (M.Ed.)--Victoria University of Wellington, 2010. / Includes bibliographical references.

The relative magnitude of an event (number magnitude) can have direct implications on timing judgments. Previous studies have found that large magnitude numbers are perceived to have longer durations than those of smaller numbers. This bias can be accounted for in several ways; first, the internal clock model theorizes that stimulus magnitude directly interacts with the components of a dedicated cognitive timer by increasing pacemaker speed. Another explanation posits that different quantitative dimensions (space, time, size, intensity and number) are all represented within a common cortical metric thus facilitating interactions within and across dimensions. I have expanded on this framework by proposing that perceived duration is inferred using flexibly applied rules of thumbs (heuristics) in which information from a more accessible dimension (e.g., number magnitude) is substituted for duration. Three paradigms were used to test this theory. First, commonalities in how the intervals separating discrete stimuli of different magnitudes were judged was examined across a variety of quantitative dimensions (number, size, and colour saturation). Perceived duration judgments increased systematically as the magnitude difference between the stimuli increased. This finding was robust against manipulations to sequence direction, and order, suggesting that interval duration was estimated by substituting information regarding the absolute magnitude difference. Second, the impact of number magnitude on sound intensity judgments was examined. When target sounds were presented simultaneously with large digits, they were categorized as loud more frequently, suggesting that participants substituted number magnitude when performing difficult sound intensity judgments in a manner similar to when judging duration. Third, the repetition of magnitude information presented in either symbolic (Arabic digits) or non symbolic (numerosities) formats was manipulated prior to the presentation of a target number, whose duration was judged. The results demonstrated that large numbers were judged to last for longer durations relative to small numbers. Furthermore, context had an effect in which a greater discrepancy in the target’s numerical magnitude from the initial context sequence resulted in a longer perceived duration. The results across all three paradigms suggest that people generally employ information regarding one magnitude dimension (number) when making difficult perceptual decisions in a related dimension (time, sound intensity). / February 2016

A Research Report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements of the degree of Master of Science. October 2012, Johannesburg. / Several commentators describe the low performance of South African students in mathematics as ‗a crisis‘. In the Foundation Phase specifically, there is evidence of a lack of shift from concrete counting-based strategies to more abstract calculation-based strategies (Ensor et al., 2009; Schollar, 2009). Concrete counting-based strategies refer to actions where the learner cannot find the answer to a mathematical problem without using concrete objects. In contrast, abstract calculation-based strategies involve strategies where the child does not need concrete objects to find the answer, but can instead use mental calculations in which numbers have been transformed into abstract objects upon which operations can then be carried out. Ensor et al argue that the poor mathematical results in South Africa are the result of inefficient moves made by learners from counting to calculating. In their study, many students failed to move their thinking sufficiently forward from concrete counting actions to abstract thinking. The focus of this study is to investigate a sample of Grade 2 learners‘ strategies on tasks drawn from the Learning Framework in Number (LFIN) test and responses on number related questions in the Annual National Assessment tests (ANA). I use the Learning Framework in Number to describe the stage of learners in their shift from concrete to a more abstract way of thinking about number. The theory of reification refers to the turning of processes into objects, and in this research, the origin of an abstract object in reification is explored. I also aim to understand the kinds of information I can get from children‘s grasp of early number strategies, by looking at the responses of learners on the ANA and LFIN tests. My research question is: What do the two tests (ANA and LFIN) tell us about the strategies on early number used by a sample of Grade 2 learners in a township school in Gauteng? The two critical questions that follow from this are:  How does learner performance on number problems compare across the two tests?  What evidence in relation to concrete/abstract strategies is evident in the responses of learners in the two tests? My findings showed that the learners in the school that I investigated still relied a great deal on concrete counting methods to answer questions. In spite of this, the mean ANA mark were much higher than the LFIN mean. The low number range of the ANA test, (1-34 for most of the number related questions), made it possible for the learners to use concrete counting (fingers or tallies) to answer the questions. The relatively low LFIN mark range indicated that children had difficulties in moving to more abstract ways of working with number. The implications of the reliance on concrete counting is potential difficulties when the learners move into higher grades where the number range is much higher, making the use of concrete methods time consuming and error prone.

Quantitative literacy, or numeracy, has been discussed as an essential component of mathematics instruction. In recent years community colleges around the nation introduced a quantitative literacy alternative to the developmental algebra curriculum for students placed into remedial mathematics. The QL curriculum consists of problem situations that are meant to improve numeracy through a combination of collaborative work and a student-centered pedagogy. There is little research that investigates the enactment of that curriculum. Research in K-12 indicates that teachers’ beliefs influence the enactment of curriculum, but studies that connect instructional practices and faculty beliefs are scarce. This study employs a multiple qualitative case study approach to investigate the alignment between four community college instructors’ beliefs about teaching, learning, the nature of mathematics, and curriculum on their enacted practices in two different developmental mathematics courses at a large urban community college (UCC). One is a standard developmental algebra curriculum and the other curriculum is based on quantitative literacy. Data were collected through semi-structured interviews, classroom observations and field notes. The results indicate an alignment between the professed beliefs and enacted practices for all but one instructor in this study. The findings imply that curriculum plays a significant role when its intended design correlates with instructors’ belief systems. The study also discusses the differences in instructional practices across the quantitative literacy and elementary algebra curricula taught by the instructors in this study.