Multiple perspectives on the concept of conditional probability

Conditional probability is a key to the subjectivist theory of probability; however, it plays a subsidiary role in the usual conception of probability where its counterpart, namely independence is of basic importance. The paper investigates these concepts from various perspectives in order to shed light on their multi-faceted character. We will include the mathematical, philosophical, and educational perspectives. Furthermore, we will inspect conditional probability from the corners of competing ideas and solving strategies. For the comprehension of conditional probability, a wider approach is urgently needed to overcome the well-known problems in learning the concepts, which seem nearly unaffected by teaching

Enactivism and ethnomethodological conversation analysis as tools for expanding Universal Design for Learning: the case of visually impaired mathematics students

Blind and visually impaired mathematics students must rely on accessible materials such as tactile diagrams to learn mathematics. However, these compensatory materials are frequently found to offer students inferior opportunities for engaging in mathematical practice and do not allow sensorily heterogenous students to collaborate. Such prevailing problems of access and interaction are central concerns of Universal Design for Learning (UDL), an engineering paradigm for inclusive participation in cultural praxis like mathematics. Rather than directly adapt existing artifacts for broader usage, UDL process begins by interrogating the praxis these artifacts serve and then radically re-imagining tools and ecologies to optimize usability for all learners. We argue for the utility of two additional frameworks to enhance UDL efforts: (a) enactivism, a cognitive-sciences view of learning, knowing, and reasoning as modal activity; and (b) ethnomethodological conversation analysis (EMCA), which investigates participants’ multimodal methods for coordinating action and meaning. Combined, these approaches help frame the design and evaluation of opportunities for heterogeneous students to learn mathematics collaboratively in inclusive classrooms by coordinating perceptuo-motor solutions to joint manipulation problems. We contextualize the thesis with a proposal for a pluralist design for proportions, in which a pair of students jointly operate an interactive technological device

Drawing to learn in STEM

Scientists, mathematicians and engineers draw and model to create knowledge. This presentation will describe a guided inquiry approach to teaching and learning science that involves students actively creating visual and other representations to reason and explain as they explore the material world. The approach has been successfully used in a number of major professional learning initiatives in Victoria and NSW. Evidence will be presented of increased student engagement and quality learning flowing from the approach, which aligns classroom processes more authentically with processes of imaginative scientific discovery. Examples of activities and student drawings and model construction will be used to unpack the relationship between representation, reasoning and learning. Video evidence including that generated in the Science of Learning Research Centre (SLRC) classroom at the University of Melbourne, equipped with sophisticated video capture facilities, will be drawn on to explore ways in which drawing, gesture and talk are coordinated to imaginatively respond to material challenges. The presentation will explore the alignment of these sociocultural analyses to recent findings from neuroscience. Evidence will be presented that the creation of representations is central to quality learning across the STEM disciplines and for interdisciplinary STEM challenges

A Comparative Analysis of the Supernova Legacy Survey Sample with {\Lambda}CDM and the Rh=ctR_{\rm h}=ct Universe

The use of Type~Ia SNe has thus far produced the most reliable measurement of the expansion history of the Universe, suggesting that $\Lambda$CDM offers the best explanation for the redshift--luminosity distribution observed in these events. But the analysis of other kinds of source, such as cosmic chronometers, gamma ray bursts, and high-$z$ quasars, conflicts with this conclusion, indicating instead that the constant expansion rate implied by the $R_{\rm h}=ct$ Universe is a better fit to the data. The central difficulty with the use of Type~Ia SNe as standard candles is that one must optimize three or four nuisance parameters characterizing supernova luminosities simultaneously with the parameters of an expansion model. Hence in comparing competing models, one must reduce the data independently for each. We carry~out such a comparison of $\Lambda$CDM and the $R_{\rm h}=ct$ Universe, using the Supernova Legacy Survey (SNLS) sample of 252 SN~events, and show that each model fits its individually reduced data very well. But since $R_{\rm h}=ct$ has only one free parameter (the Hubble constant), it follows from a standard model selection technique that it is to be preferred over $\Lambda$CDM, the minimalist version of which has three (the Hubble constant, the scaled matter density and either the spatial curvature constant or the dark-energy equation-of-state parameter). We estimate by the Bayes Information Criterion that in a pairwise comparison, the likelihood of $R_{\rm h}=ct$ is $\sim 90\%$, compared with only $\sim 10\%$ for a minimalist form of $\Lambda$CDM, in which dark energy is simply a cosmological constant. Compared to $R_{\rm h}=ct$, versions of the standard model with more elaborate parametrizations of dark energy are judged to be even less likely.Comment: 31 Pages, 5 Figures, 1 Table. Accepted for publication in A

Exploring the episodic structure of algebra story problem solving

This paper analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent subjects. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attampts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complimentary activities, giving rise to a succession of problem solving episodes. While direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of "model-based reasoning" to identify, pursue, and verify quantitative constraints required for solution. The paper concludes by discussing the implications of these findings for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction

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

Making mathematics on paper : constructing representations of stories about related linear functions

This dissertation takes up the problem of applied quantitative inference as a central question for cognitive science, asking what must happen during problem solving for people to obtain a meaningful and effective representation of the problem. The core of the dissertation reports exploratory empirical studies that seek to answer the descriptive question of how quantitative inferences are generated, pursued, and evaluated by problem solvers with different mathematical backgrounds. These are framed against a controversy, described in Chapter 2, over the theoretical and empirical validity of current cognitive science accounts of problems, solutions, knowledge, and competent human activity outside of laboratory or school settings.Chapter 3 describes a written protocol study of algebra story problem solving among advanced undergraduates in computer science. A relatively open-ended interpretive framework for "problem-solving episodes" is developed and applied to their written solution attempts. The resulting description of problem-solving activities gives a surprising image of competence among an important occupational target for standard mathematics instruction.Chapter 4 follows these results into detailed verbal problem-solving interviews with algebra students and teachers. These provide a comparison across settings and levels of competence for the same set of problems. The results corroborate similar generative activities outside the standard formalism of algebra across levels of competence. Notable among these nonalgebraic problem-solving activities are "model-based reasoning tactics," in which people reason about quantitative relations in terms of the dimensional structure of functional relations described in the problem. These tactics support different activities within surrounding solution attempts and usually describe "states" in the problem's situational structure.Chapter 5 holds these activities accountable to local combinations of notation and quantity, reinterpreting results for model-based reasoning in an ecological analysis of material designs for constructing and evaluating quantitative inferences. This analysis brings forward important relations between what material designs afford problem solvers and the complexity of episodic structure observed in their solution attempts. The dissertation closes with a reappraisal of the relationship between knowledge, person, and setting and, I will argue, puts us on a more promising track for a descriptively adequate theoretical account of constructing mathematical representations that support applied quantitative inference