Mathematics and outdoor photography experience - exploration of an approach to mathematical education, based on the theory of Dewey’s aesthetics

Lehden nimi on vaihtunut. Oikea nimi julkaisuvuonna on "LUMAT: International Journal on Math, Science and Technology Education"Based on Dewey’s theory of art, aesthetics, and experiences and photographer Barnbaums’ writing about expanded perception through photography, we conducted a one-day experimental mathematics education unit. Using photography in outdoor conditions had a positive impact on teacher students’ perception of the use of photography for teaching mathematics. To study the changes in students’ visual attention deeper, we used gaze-tracking to analyse one student’s visual attention when walking outdoors after the activity. The gaze data indicated that more visual attention was given to objects he had photographed or discussed during the group activity in comparison to other objects.Based on Dewey’s theory of art, aesthetics, and experiences and photographer Barnbaums’ writing about expanded perception through photography, we conducted a one-day experimental mathematics education unit. Using photography in outdoor conditions had a positive impact on teacher students’ perception of the use of photography for teaching mathematics. To study the changes in students’ visual attention deeper, we used gaze-tracking to analyse one student’s visual attention when walking outdoors after the activity. The gaze data indicated that more visual attention was given to objects he had photographed or discussed during the group activity in comparison to other objects.Peer reviewe

Cross-domain priming from mathematics to relative-clause attachment: a visual-world study in French

Human language processing must rely on a certain degree of abstraction, as we can produce and understand sentences that we have never produced or heard before. One way to establish syntactic abstraction is by investigating structural priming. Structural priming has been shown to be effective within a cognitive domain, in the present case, the linguistic domain. But does priming also work across different domains? In line with previous experiments, we investigated cross-domain structural priming from mathematical expressions to linguistic structures with respect to relative clause attachment in French (e.g., la fille du professeur qui habitait à Paris/the daughter of the teacher who lived in Paris). Testing priming in French is particularly interesting because it will extend earlier results established for English to a language where the baseline for relative clause attachment preferences is different form English: in English, relative clauses (RCs) tend to be attached to the local noun phrase (low attachment) while in French there is a preference for high attachment of relative clauses to the first noun phrase (NP). Moreover, in contrast to earlier studies, we applied an online-technique (visual world eye-tracking). Our results confirm cross-domain priming from mathematics to linguistic structures in French. Most interestingly, different from less mathematically adept participants, we found that in mathematically skilled participants, the effect emerged very early on (at the beginning of the relative clause in the speech stream) and is also present later (at the end of the relative clause). In line with previous findings, our experiment suggests that mathematics and language share aspects of syntactic structure at a very high-level of abstraction

Eye-tracking the emergence of attentional anchors in a mathematics learning tablet activity

Little is known about micro-processes by which sensorimotor interaction gives rise to conceptual development. Per embodiment theory, these micro-processes are mediated by dynamical attentional structures. Accordingly this study investigated eye-gaze behaviors during engagement in solving tablet-based bimanual manipulation tasks designed to foster proportional reasoning. Seventy-six elementary- and vocational-school students (9-15 yo) participated in individual task-based clinical interviews. Data gathered included action-logging, eye-tracking, and videography. Analyses revealed the emergence of stable eye-path gaze patterns contemporaneous with first enactments of effective manipulation and prior to verbal articulations of manipulation strategies. Characteristic gaze patterns included consistent or recurring attention to screen locations that bore non-salient stimuli or no stimuli at all yet bore invariant geometric relations to dynamical salient features. Arguably, this research validates empirically hypothetical constructs from constructivism, particularly reflective abstraction

Student Application of the Fundamental Theorem of Calculus with Graphical Representations in Mathematics and Physics

One mathematical concept frequently applied in physics is the Fundamental Theorem of Calculus (FTC). Mathematics education research on student understanding of the FTC indicates student difficulties with the FTC. Similarly, a few studies in physics education have implicitly indicated student difficulties with various facets of the FTC, such as with the definite integral and the area under the curve representation, in physics contexts. There has been no research on how students apply the FTC in graphically-based physics questions. This study investigated student understanding of the FTC and its application to graphically-based problems. Our interest spans several aspects of the FTC: student difficulties, problem-solving strategies, and visual attention. Written and interview findings revealed student difficulties common to mathematics and physics, e.g., confusion between the antiderivative difference and the function difference. Three problem-solving strategies were identified: algebraic, graphical, and integral. For a deeper analysis of problem-solving strategies, we applied the perspectives of epistemological framing (student expectations/perceptions) and epistemic games (problem-solving games). While most observed frames and epistemic games were somewhat modified versions of those previously reported, we identified one new game: the equation-based analytical game. In addition, a novel eye-tracking study was conducted to explore students’ visual attention to different parts of graphically-based FTC questions. Results indicated that students’ visual behavior was affected by the representations in the questions, such as the presence or absence of certain equation(s) and/or graphical feature(s), as well as context (math vs. physics). Because student responses seemed to be both conceptually and salient-feature driven, the results were explained using the cognitive perspectives of top-down (conceptually driven) and bottom-up (feature-driven) processes. Eye-tracking results provided support for interview findings about problem solving strategies. For many students, the absence of specific visual cues led to a particular framing of the problem that was associated with inappropriate e-games for that problem. Minor interviewer prompting often enabled students to reframe a problem and invoke relevant knowledge and strategies, suggesting that students possess knowledge of individual facets of the FTC, but this knowledge may not be elicited by a particular problem representation(s). Additionally, specific difficulties can be seen as due to inappropriate problem framing

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

A perceptual comparison of empirical and predictive region-of-interest video

When viewing multimedia presentations, a user only attends to a relatively small part of the video display at any one point in time. By shifting allocation of bandwidth from peripheral areas to those locations where a user’s gaze is more likely to rest, attentive displays can be produced. Attentive displays aim to reduce resource requirements while minimizing negative user perception—understood in this paper as not only a user’s ability to assimilate and understand information but also his/her subjective satisfaction with the video content. This paper introduces and discusses a perceptual comparison between two region-of-interest display (RoID) adaptation techniques. A RoID is an attentive display where bandwidth has been preallocated around measured or highly probable areas of user gaze. In this paper, video content was manipulated using two sources of data: empirical measured data (captured using eye-tracking technology) and predictive data (calculated from the physical characteristics of the video data). Results show that display adaptation causes significant variation in users’ understanding of specific multimedia content. Interestingly, RoID adaptation and the type of video being presented both affect user perception of video quality. Moreover, the use of frame rates less than 15 frames per second, for any video adaptation technique, caused a significant reduction in user perceived quality, suggesting that although users are aware of video quality reduction, it does impact level of information assimilation and understanding. Results also highlight that user level of enjoyment is significantly affected by the type of video yet is not as affected by the quality or type of video adaptation—an interesting implication in the field of entertainment

'There's more than meets the eye': analysing verbal protocols, gazes and sketches on external mathematical representations

When learners are asked to verbalise their thoughts about multiple mathematical representations, some researchers are left to analyse utterances based on video records of activity which may have ambiguous signifiers. They are also faced with post hoc analysis of paper-based worksheets, in which temporal order has to be guessed. In this paper, attempts to minimise such methodological problems by means of recent technologies such as eye-tracking, tablet PC screen capture, digital video cameras and the latest video analysis tools are illustrated in the context of a study of the effect of varying representational instantiations on learners' problem-solving strategies