Equality statements as rules for transforming arithmetic notation

This thesis explores children’s conceptions of the equals sign from the vantage point of notating task design. The existing literature reports that young children tend to view the equals sign as meaning “write the result here”. Previous studies have demonstrated that teaching an “is the same as” meaning leads to more flexible thinking about mathematical notation. However, these studies are limited because they do not acknowledge or teach children that the equals sign also means “can be exchanged for”. The thesis explores the “sameness” and “exchanging” meanings for the equals sign by addressing four research questions. The first two questions establish the distinction, in terms of task design, between the two meanings. Does the “can be exchanged for” meaning for the equals sign promote attention to statement form? Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct? The final two research questions seek to establish how children might coordinate the two meanings, and connect them with their existing implicit knowledge of arithmetic principles. Can children coordinate “can be exchanged for” and “is the same as” meanings for the equals sign? Can children connect their implicit arithmetical knowledge with explicit transformations of notation? The instrument used is a specially designed notational computer-microworld called Sum Puzzles. Qualitative data are generated from trials with pairs of Year 5 (9 and 10 years), and in one case Year 8 (12 and 13 years), pupils working collaboratively with the microworld toward specified task goals. It is discovered that the “sameness” meaning is useful for distinguishing equality statements by truthfulness, whereas the “exchanging” meaning is useful for distinguishing statements by form. Moreover, a duality of both meanings can help children connect their own mental calculation strategies with transformations of properly formed notation

GEOSCIENCE EDUCATION RESEARCH: TRENDS AND APPLICATIONS IN UNDERGRADUATE COURSES

Water resources are progressively under pressure from anthropogenic uses. Students need to learn about water systems as they are the future decision-makers and problem solvers who will be faced with unknown challenges in the future. The overarching goals of this dissertation were: 1) to identify ways in which geoscience instructors are incorporating systems thinking and science modeling in their teaching along with the accompanying methods for improving systems thinking and modeling implementation and 2) explore how the implementation of science modeling and systems thinking increase student evaluation of models and the understanding of hydrologic content. Data for these studies came from the Geoscience Educators Research (GER) 2016 survey data, student assignments and interviews surrounding the Water Balance Model, and student responses from a sociohydrologic systems thinking assignment. First, GER survey data was analyzed with significant variation observed in reported frequency of science modeling and systems thinking (SMST) practices with the highest levels of SMST reported in the atmospheric and environmental sciences, those who emphasize research-based, student centered pedagogical methods, those who recently made course revisions, and those who reported high levels of participation in educational professional development. Therefore, to test if this was replicable in subsequent work, we examined a course at UNL, SCIL 109: Water in Society, a novel course. Courses in SCIL (Science Literacy) are housed in the College of Agricultural Sciences and Natural Resources, are interdisciplinary, and include both human and scientific dimensions. A case study emerged from this data presenting the use of a computer-based water model over three iterations of SCIL 109. Results indicate that students regardless of year in college, gender, or major can effectively reason about the Water Balance Model. Specific investigation into student performance and reasoning surrounding the Water Balance Model indicate that model evaluation and understanding of core hydrologic content increased from 2017 to 2018 in part due to a flipped classroom format. Finally, the systems thinking assignment from SCIL 109 was studied using mixed-methods to investigate student operationalization of a sociohydrologic system. Results show that students scored highest on problem identification from their written work and mechanism inclusion form their drawn models. Each of these studies contributes to the overall body of knowledge surrounding undergraduate geoscience education. Adviser: Cory T. Forbe

Sociohydrologic Systems Thinking: An Analysis of Undergraduate Students’ Operationalization and Modeling of Coupled Human-Water Systems

One of the keys to science and environmental literacy is systems thinking. Learning how to think about the interactions between systems, the far-reaching effects of a system, and the dynamic nature of systems are all critical outcomes of science learning. However, students need support to develop systems thinking skills in undergraduate geoscience classrooms. While systems thinking-focused instruction has the potential to benefit student learning, gaps exist in our understanding of students’ use of systems thinking to operationalize and model SHS, as well as their metacognitive evaluation of systems thinking. To address this need, we have designed, implemented, refined, and studied an introductory-level, interdisciplinary course focused on coupled human-water, or sociohydrologic, systems. Data for this study comes from three consecutive iterations of the course and involves student models and explanations for a socio-hydrologic issue (n = 163). To analyze this data, we counted themed features of the drawn models and applied an operationalization rubric to the written responses. Analyses of the written explanations reveal statistically-significant differences between underlying categories of systems thinking (F(5, 768) = 401.6, p \u3c 0.05). Students were best able to operationalize their systems thinking about problem identification (M = 2.22, SD = 0.73) as compared to unintended consequences (M = 1.43, SD = 1.11). Student-generated systems thinking models revealed statistically significant differences between system components, patterns, and mechanisms, F(2, 132) = 3.06, p \u3c 0.05. Students focused most strongly on system components (M = 13.54, SD = 7.15) as compared to related processes or mechanisms. Qualitative data demonstrated three types of model limitation including scope/scale, temporal, and specific components/mechanisms/patterns excluded. These findings have implications for supporting systems thinking in undergraduate geoscience classrooms, as well as insight into links between these two skills

Exploring multiplicative reasoning with grade four learners through structured problem solving

Research Report submitted to the Wits School of Education, Faculty of Science, University of the Witwatersrand, Johannesburg In partial fulfilment of the requirements For the degree of Master of Science (Mathematics Education) Johannesburg, 2017South Africa’s performance in mathematics education is ranked amongst the world’s worst. This performance is not only alarming at an international level, but also nationally. Annual National Assessments (ANA) conducted by the Department of Education have showed that the level of mathematics across the foundation and intermediate phase is poor with a pronounced dip in performance at a Grade 4 level (Department of Basic Education, 2014). Multiplication and division are common challenging areas that contribute to this poor performance. This is concerning as mathematics is globally recognised as a key competence for providing access to higher education and developing a country’s society and economy. My study, aimed at exploring multiplicative reasoning with Grade 4 learners through structured problem solving, is focused on the learning of multiplication and division within the context of an intervention concentrated on developing learners’ ability to model multiplicative situations. Shifts in the use of models were investigated following a smallscale intervention in which different modelling approaches (particularly ratio modelling) were introduced and developed. A control group was used to determine the usefulness of the intervention. Questions which I sought to answer were: (a) what kinds of multiplicative reasoning (models) are Grade 4 learners using prior to intervention, (b) what changes, if any, are seen in overall performance, across the intervention and control group, in the post-test, and, (c) what kinds of differences in model use were associated with the shifts in performance? The main dataset comprised of 61 pre- and post-test scripts across three Grade 4 classes in a former Model C school in a Johannesburg district. A sample of 15 interviews were also conducted across the classes. Document analysis and transcription notes were used to analyse data with a Realistic Mathematics Education (RME) framework informing my analysis. Findings from my study reveal that prior to intervention, Grade 4 learners presented limited multiplicative models which were predominantly confined to traditional algorithms. After the small-scale intervention, learners used a broader range of models with an emerging take up of ratio models. The success rate associated with the models presented by learners also improved. Limited and/or no changes in model use and their respective success rates were seen in the control group suggesting that the intervention program was useful. These findings suggest that, as a future recommendation, it would be worthwhile to investigate the outcomes of running a similar intervention in less privileged settings.MT 201

Theorizing teaching: current status and open issues

Presents practical implications for teaching and educating teachers. Examines systematically the issue of theorizing teaching. Enables collective thinking about issues that are of paramount importance in the field. This book is open access, which means that you have free and unlimited access

The mathematics teacher exchange and 'mastery' in England: The evidence for the efficacy of component practices

'Mastery’ is central to current policy in mathematics education in England, influenced by East Asian success in transnational assessments. We scrutinise the prospects for mastery pedagogies to improve pupil attainment in English primary schools. The Mathematics Teacher Exchange (MTE)—an element of the mastery innovation—involves teachers visiting Shanghai and then hosting Shanghai teachers in their schools. Informed by programme evaluation, core component practices are analysed, which were implemented by schools belonging to the first cohort of MTE schools. These consist of: varied and interactive teaching; meaningful and coherent mathematical activity; and full curriculum access for all. These elements are supported, optimally, by collaborative, embedded, and mathematically focused professional development. Details of the implemented pedagogy and forms of professional development are reported. Differences from prevailing practice in primary mathematics in England are highlighted. Evidence is reviewed from quasi-experimental trials, reviews and meta-analyses, and rigorous observational studies of the efficacy of practices similar to the MTE mastery pedagogy components in order to assess the prospects for increases in pupil attainment. The analysis suggests that many of the specific practices, if considered individually, have the potential to improve attainment, though overall policy ambitions may not be realised. Based on the review, component practices are identified for which existing evidence justifies immediate implementation by schools and teachers. In addition, practices that would benefit from further testing and evaluation are highlighted

AN EXAMINATION OF THE IMPACT OF COMPUTER-BASED ANIMATIONS AND VISUALIZATION SEQUENCE ON LEARNERS' UNDERSTANDING OF HADLEY CELLS IN ATMOSPHERIC CIRCULATION

Research examining animation use for student learning has been conducted in the last two decades across a multitude of instructional environments and content areas. The extensive construction and implementation of animations in learning resulted from the availability of powerful computing systems and the perceived advantages the novel medium offered to deliver dynamic representations of complex systems beyond the human perceptual scale. Animations replaced or supplemented text and static diagrams of system functioning and were predicted to significantly improve learners' conceptual understanding of target systems. However, subsequent research has not consistently discovered affordances to understanding, and in some cases, has actually shown that animation use is detrimental to system understanding especially for content area novices (Lowe 2004; Mayer et al. 2005). This study sought to determine whether animation inclusion in an authentic learning context improved student understanding for an introductory earth science concept, Hadley Cell circulation. In addition, the study sought to determine whether the timing of animation examination improved conceptual understanding. A quasi-experimental pretest posttest design administered in an undergraduate science lecture and laboratory course compared four different learning conditions: text and static diagrams with no animation use, animation use prior to the examination of text and static diagrams, animation use following the examination of text and static diagrams, and animation use during the examination of text and static diagrams. Additionally, procedural data for a sample of three students in each condition were recorded and analyzed through the lens of self regulated learning (SRL) behaviors. The aim was to determine whether qualitative differences existed between cognitive processes employed. Results indicated that animation use did not improve understanding across all conditions. However learners able to employ animations while reading and examining the static diagrams and to a lesser extent, after reading the system description, showed evidence of higher levels of system understanding on posttest assessments. Procedural data found few differences between groups with one exception---learners given access to animations during the learning episode chose to examine and coordinate the representations more frequently. These results indicated a new finding from the use of animation, a sequence effect to improve understanding of Hadley Cells in atmospheric circulation

Teaching and learning secondary school biology with diagrams

This thesis comprises a series of inter-related studies that examined: (1) diagrams presented in commonly used biology textbooks in Western Australian schools; (2) teachers’ use of diagrams as part of their normal teaching routines; (3) students’ perceptions of how they learn about diagrams in their lessons; and (4) students’ use of text and diagrams in explaining two phenomena in biology that had not been presented in class.Phase one of the research reports the results of an analysis diagrams presented in biology textbooks used by Western Australian students to examine their distribution pattern. Three types of diagrams (iconic, schematic, and charts & graphs) were investigated in science education based on the work of Novick (2006). Therefore, content analysis in this research entailed a systematic reading and categorizing of these diagrams from a number of secondary school textbooks. The textbook types include lower secondary general science textbooks, upper secondary biology textbooks, and biology workbooks. Descriptive statistics were conducted in order to provide first-hand data on exploring how diagrams are used in biology books. Findings of the study suggest that the three types of diagrams are distributed with unique patterns in the secondary biology textbooks.Phase two reports the investigation of biology teachers’ use of diagrams in their classroom teaching. Biology teachers’ teaching was observed in order to determine instructional methods related to diagrammatic teaching and learning in the natural environment. This study described and analysed how teachers of biology use the three different types of diagrams to introduce, explain and evaluate abstract biology concepts.The third phase of the research reports an analysis of how students think about their teachers’ instructional strategies when teaching with diagrams. An instrument was developed from a previously existing instrument to help students reflect upon their use of diagrams during their teachers’ instruction. The questionnaire data indicated that most participant students recognised teachers’ instructional methods in teaching diagrammatic representations as being explanatory tools, in representing biological concepts, and in help assessing their learning. The three dimensions identified by the questionnaire (Instruction with diagram, Assessment with diagrams and Student diagrammatic competence), demonstrated that students’ perceived experienced biology teachers as being more skillful in having diagrams to engage their learning.Phase four investigated students’ conceptual learning of diagrams alongside other modes of representations. The purpose of this phase was to determine how the students interpreted diagrams together with their counterpart – text – when learning three different biology concepts using an interview protocol. In each interview, students were presented with a biological concept with diagrammatic representation (iconic, schematic diagrams, and charts & graphs) together with textual representation (such as written text and chemical equations). The chapter concludes by showing that diagram and text serve different functional roles in students’ conceptual learning when one or both representations are presented. The results showed that diagram and text may constrain, construct or complementary each other so as to help students understand the complex concept.The final chapter draws together and discusses the findings generated in all of the previous studies in which diagrams were used in various aspects of secondary biological education, such as textbooks, classroom instruction, students’ perceptions, and representational learning with text. The limitations of the research are discussed and suggestions made for future research on the instructional usage of diagrams in biological teaching and learning

Transformative practice in teaching: How experienced teachers explain the profound transformative influences on their teaching practice.

The social transformation represented by a shift from the industrial economy to the knowledge economy presents a challenge to the education sector. That challenge is to provide high-quality teaching that results in improved outcomes for students who are developing the habit of continuous learning. A challenge such as this may be met by teachers transforming their teaching practice. This small-scale qualitative research project seeks an understanding of and insight into those factors which influence transformative practice for experienced teachers. It uses semi-structured interviews to gather the perspectives of seven experienced teachers and explores the themes derived from their stories of transformative practice in relation to themes derived from the literature. The literature reveals four significant dimensions of influence on transformative practice: professional development, individual factors, school factors, and an emerging theme of communities of practice. The research findings confirm that experienced teachers' perceptions of the profound transformative influences on their teaching practice are consistent with some of the literature. These congruencies include teachers working individually or collaboratively on problems of practice using a process of trial and error experimentation, and where workplace conditions support risk-taking and promote ownership of learning. The findings confirm that transformative practice is driven by powerful emotions that connect teachers to the learning needs of their students, and is sustained by intrinsic rewards. The research findings reveal two significant areas of divergence. The literature identifies the need for depth and breadth of content knowledge and assessment knowledge, and for critical reflection on the effectiveness of transformative teaching practices on student outcomes, neither of which were identified by participants as factors which influenced transformative practice. This indicated that teachers were unlikely to be developing local knowledge-of-practice, a necessary prerequisite for linking the purpose of transformative practice with its goal - to improve outcomes for students. Drawing on the understandings of, and insights into, transformative practice, this research presents a diagrammatic representation of a framework to illustrate the transformative influences on teaching practice. It also presents a knowledge-building learning cycle as a diagrammatic representation of the process required to generate knowledge-of-practice. This research project includes recommendations suggesting that teachers develop a rich understanding of the concept of knowledge-of-practice and embed this practice in their daily work. It recommends that the knowledge-building learning cycle is facilitated by leaders of learning who have the skills to activate teacher learning, and that during the knowledge-building learning cycle, teachers develop depth and breadth of content knowledge and assessment knowledge. It recommends that leaders of learning guard against transformative practice becoming an end in itself, and suggests that utilizing the knowledge-building learning cycle could lead to a new form of professionalism that is continuous and sustainable. This study proposes that by acting on these recommendations, leaders of learning may enhance their ability to influence transformative teaching practice where students receive high-quality teaching which simultaneously achieves improved outcomes for students