Learning Is Moving in New Ways: The Ecological Dynamics of Mathematics Education

Whereas emerging technologies, such as touchscreen tablets, are bringing sensorimotor interaction back into mathematics learning activities, existing educational theory is not geared to inform or analyze passages from action to concept. We present case studies of tutor–student behaviors in an embodied-interaction learning environment, the Mathematical Imagery Trainer. Drawing on ecological dynamics—a blend of dynamical-systems theory and ecological psychology—we explain and demonstrate that: (a) students develop sensorimotor schemes as solutions to interaction problems; (b) each scheme is oriented on an attentional anchor—a real or imagined object, area, or other aspect or behavior of the perceptual manifold that emerges to facilitate motor-action coordination; and (c) when symbolic artifacts are introduced into the arena, they may both mediate new affordances for students’ motor-action control and shift their discourse into explicit mathematical re-visualization of the environment. Symbolic artifacts are ontological hybrids evolving from things with which you act to things with which you think. Students engaged in embodied-interaction learning activities are first attracted to symbolic artifacts as prehensible environmental features optimizing their grip on the world, yet in the course of enacting the improved control routines, the artifacts become frames of reference for establishing and articulating quantitative systems known as mathematical reasoning.     

Toward Synergizing Educational Research and Movement Sciences: a Dialogue on Learning as Developing Perception for Action

Educational Psychology Review - What could possibly be a meaningful conversation between educational researchers and movement scientists? Curiously, they have much in common. Both groups of...     

Parent Involvement,Parental Depression,and Program Satisfaction Among Low-Income Parents Participating in a Two-Generation Early Childhood Education Program

Research Findings: This study simultaneously examined parental depression and parent involvement as predictors of satisfaction with an early childhood intervention program. Parents (N = 203) of Head Start children participated in this short-term longitudinal study. Measures of parent involvement and satisfaction assessed multiple dimensions of these constructs. Nearly 40% of low-income mothers reported being sometimes or chronically depressed over the course of 1 year of the Head Start program. Compared with mothers who were never depressed, those who were sometimes depressed reported less involvement in home- and school-based activities as well as fewer interactions with their child's teacher. Never depressed parents were more likely to be satisfied with their child's teacher compared with either group of depressed mothers. Higher levels of parent involvement and parent–teacher interaction predicted optimal satisfaction with Head Start services. Practice or Policy: Implications of results for practice are considered in terms of teacher training to recognize unique needs involved in working to establish a home–school connection with mothers experiencing depression. Strategies for building community partnerships to assist with mental health needs are discussed.     

Towards an ecological-dynamics design framework for embodied-interaction conceptual learning: the case of dynamic mathematics environments

Educational technology research and development - Designers of educational modules for conceptual learning often rely on procedural frameworks to chart out interaction mechanics through which users...     

Development and Validation of a Method for Determining Tridimensional Angular Displacements with Special Applications to Ice Hockey Motions

Abstract

Tridimensional angular displacements of the skates were determined with a single cinecamera used in conjunction with two orthogonal rods attached to the rear part of the skater's boots. The method was based on the commutative law with the imposed restrictions that two of the angles were small in relation to the other one, and that a sequence of angular displacements was defined. This approach required the sole determinations of the planar coordinates of the three points delimiting the orthogonal rods. An experimental setup was designed to validate the method: the criterion angles were approximated with less than 2 degrees of error. Subsequently, the method was applied to a practical situation: the two-legged stop in ice hockey. The combination of this tridimensional method with instrumented strain gauge skates for the measurement of forces provides a new and powerful tool for performing kinetic analyses in tridimensional skating activities.     

Measurement error in short-term power testing in young people

The aim of this study was to examine the consistency or reproducibility of measuring cycling peak power in children and adults. Twenty-seven pre-pubertal girls and boys and 27 female and male physical education students (age 9.8 +/- 0.5 and 24.4 +/- 4.3 years, respectively; mean +/- s) participated in the study. All participants performed five tests over 15 days and underwent a habituation session before the study. Each test included four sprints against four different braking forces. We found that braking forces of 7.5% of body weight in children and 10% of body weight in adults were too high for most of the participants to elicit maximal cycling power. Unlike the children, the physical education students improved their performance between session 1 and session 2 (1025 +/- 219 vs 1069 +/- 243 W; P < 0.001). Therefore, to obtain reproducible measures of cycling peak power, a habituation session including a complete test protocol (i.e. warm-up plus three sprints) is highly recommended. When the protocol included three sprints in children and at least two sprints in adults, measurement of cycling peak power was found to be highly reliable (test-retest coefficient of variation approximately 3%). Finally, to avoid performance fluctuations, especially over several consecutive evaluations (e.g. longitudinal studies), it is necessary to maintain high motivation in children.     

Search behavior of dogs (Canis familiaris) in invisible displacement problems

Gagnon and Doré (1992) showed that domestic dogs are able to solve a Piagetian object permanence task called the invisible displacement problem. A toy is hidden in a container which is moved behind a screen where the toy is removed and left. Dogs make more errors in these problems than they do in visible displacement tests, in which the object is hidden directly behind the target screen. In Experiment 1, we examined components of the standard procedure of invisible displacements that may make encoding or retention of the hiding location more difficult than it is in visible displacements. In Experiment 2, we compared dogs’ performances in visible and invisible displacement problems when delays of 0, 10, and 20 sec were introduced between the object’s final disappearance and the subject’s release. The results revealed that dogs’ poorer performance in invisible displacement tests is related to the complex sequence of events that have to be encoded or remembered as well as to a difficulty in representing the position change that is signaled, but not directly perceived.     

Rethinking Intensive Quantities via Guided Mediated Abduction

Some intensive quantities, such as slope, velocity, or likelihood, are perceptually privileged in the sense that they are experienced as holistic, irreducible sensations. However, the formal expression of these quantities uses a/b analytic metrics; for example, the slope of a line is the quotient of its rise and run. Thus, whereas students' sensation of an intensive quantity could serve as a powerful resource for grounding its formal expression, accepting the mathematical form requires students to align the sensation with a new way of reasoning about the phenomenon. I offer a case analysis of a middle school student who successfully came to understand the intensive quantity of likelihood. The analysis highlights a form of reasoning called abduction and suggests that sociocognitive processes can guide and mediate students' abductive reasoning. Interpreting the child's and tutor's multimodal action through the lens of abductive inference, I demonstrate the emergence of a proportional concept as guided mediated objectification of tacit perception. This “gestalt first” process is contrasted with traditional “elements first” approaches to building proportional concepts, and I speculate on epistemic and cognitive implications of this contrast for the design and instruction of these important concepts. In particular, my approach highlights an important source of epistemic difficulty for students as they learn intensive quantities: the difficulty in shifting from intuitive perceptual conviction to mediated disciplinary analysis. My proposed conceptualization of learning can serve as an effective synthesis of traditional and reform-based mathematics instruction.     

Mobile Field‐independence and Rating as a ‘Good Teacher‘

Mobile field‐independence is described and the relationship between such cognitive style, when present in a teacher, and the pupil's designation of the person possessing such cognitive style as a ‘good teacher’ is studied. Sixty‐two ‘good’ teachers were sorted on the basis of two scales, with the extremes on the continuum identified as field‐dependent and field‐independent. A group of ‘mobiles’ (those who performed differentially with cognitive and social stimuli), was also identified. A degree of ‘goodness’ was assigned to 62 teachers by 1580 of their pupils. The results were compared for the three types of cognitive style. Tests of Pascual‐Leone's theory regarding problem‐solving strategy and Improvement Scores on the Embedded Figures Test (EFT) were also performed. The group of 10 ‘mobile’ teachers received the highest ratings as good teachers and showed the highest improvement score (the latter difference is not statistically significant). The strategy hypotheses received partial support.     

Turning Routine Exercises Into Activities that Teach Inquiry: A Practical Guide

Abstract

How can we teach inquiry? In this paper, I offer practical techniques for teaching inquiry effectively using activities built from routine textbook exercises with minimal advanced preparation, including rephrasing exercises as questions, creating activities that inspire students to make conjectures, and asking for counterexamples to reasonable, but false, conjectures.