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Title
Learning Is Moving in New Ways: The Ecological Dynamics of
Mathematics Education
Permalink
https://escholarship.org/uc/item/1xj3z1w8
Journal
Journal of the Learning Sciences, 25(2)
ISSN
1050-8406
Authors
Abrahamson, D
Sánchez-García, R
Publication Date
2016-04-02
DOI
10.1080/10508406.2016.1143370
Peer reviewed
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JOURNAL OF THE LEARNING SCIENCES, 25: 203–239, 2016
Copyright © Taylor & Francis Group, LLC
ISSN: 1050-8406 print / 1532-7809 online
DOI: 10.1080/10508406.2016.1143370
Learning Is Moving in New Ways:
The Ecological Dynamics of
Mathematics Education
Dor Abrahamson
Graduate School of Education
University of California, Berkeley
Raúl Sánchez-García
Sociology of Sport
Universidad Europea de Madrid
Whereas emerging technologies, such as touchscreen tablets, are bringing
sensorimotor interaction back into mathematics learning activities, existing educa-
tional 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 dynamic-
s—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-visu-
alization 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
Correspondence should be addressed to Dor Abrahamson, Graduate School of Education,
University of California, Berkeley, 4649 Tolman Hall, Berkeley, CA 94720-1670. E-mail: dor@
berkeley.edu
Color versions of one or more of the figures in the article can be found online at http://www.
tandfonline.com/hlns.
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204 ABRAHAMSON AND SÁNCHEZ-GARCÍA
course of enacting the improved control routines, the artifacts become frames of ref-
erence for establishing and articulating quantitative systems known as mathematical
reasoning.
Mathematics, like music, needs to be expressed in physical actions and human
interactions before its symbols can evoke the silent patterns of mathematical
ideas.—Skemp (1983, p. 288)
Rules, like birds, must live before they can be stuffed.—Ryle (1945, p. 11)
INTRODUCTION: IN SEARCH OF AN ACTION-ORIENTED THEORY OF
MATHEMATICAL ONTOGENESIS
Background and Objective: Why Educational Theory and Practice Need
an Action-Oriented Theory of Mathematics Learning
With the increasing public availability of advanced technological platforms, we
are witnessing an efflorescence of commercial products designed for interactive
learning of mathematics content. In this brave new world, users manipulate virtual
objects to complete engaging tasks and, in so doing, per the vendors, develop con-
ceptual understanding of target notions, such as arithmetic operations. Although
these electronic devices are slow to enter mainstream education, they are literally
at the fingertips of any child who has access to a tablet; a smartphone; or any
other natural user interface platform, such as Wii, Xbox Kinect, or Leap Motion.
It is understandable that this unprecedented outburst in downloadable, over-the-
counter edutainment is slow to be evaluated, let alone guided by the educational
research community (Abrahamson, 2015). It is problematic, though, that extant
theory of learning is by and large a theory of learning with paper, informed neither
by the interaction possibilities of emerging technologies nor by what these possi-
bilities could imply for mathematical epistemology and pedagogy (Papert, 2004).
In the short term, the scarcity of bold research on interactive mathematics learn-
ing impedes the formulation of empirically based progressive policies concerning
the integration of technological environments into educational institutions. In the
long term, this scarcity is accelerating misalignment between theory of learning
and emerging practices to which it should apply. As children are learning to move
in new ways, theory of learning should move in new ways, too.
A motivation behind this article is that the pedagogical quality and institutional
acceptance of action-based learning environments largely depends on developing
informed scholarly and public discourse concerning what it means to learn a math-
ematical concept and what an instructor’s role might be in this process. Thus, we
are echoing Papert’s consistent call to leverage the technological revolution as
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ECOLOGICAL DYNAMICS 205
an opportunity for deep discussion of the potentially radical changes educational
systems should undergo (Papert, 1993, 1996). Similar to Papert, we are optimistic
that technological advances in educational media bear the potential of fostering
students’ deep understanding of mathematical concepts. Complementarily, these
technological advances bear the potential of fostering researchers’ deep under-
standing of learning processes. The objective of this article is to contribute first
steps toward developing a theory of action-based mathematics learning. We take
these first steps by arguing for what we believe to be productive directions for
investigating action-based learning, namely, adopting perspectives from scientific
disciplines dedicated to the study of motor action.
We begin by introducing the empirical context and findings that have motivated
us to seek, beyond seminal theories of mathematics learning, new approaches
oriented on cognitive, physiological, material, and social factors at play in
motor-action skills development.
Empirical Context: The Mathematical Imagery Trainer for Proportion
(MIT-P)
Our argument for the added value of action-based disciplinary perspectives is sit-
uated in emerging findings from qualitative analyses of empirical data gathered
in the context of implementing an experimental design for mathematics learn-
ing, the MIT-P. In this study, volunteering study participants manually operated
an unfamiliar technological system with the task objective of bringing this sys-
tem to a prescribed goal state, namely, moving their hands in space to make
a screen green (Abrahamson & Trninic, 2011; Howison, Trninic, Reinholz, &
Abrahamson, 2011). Figures 1 and 2 offer an overview of the design.
FIGURE 1 The Mathematical Imagery Trainer for Proportion set at a 1:2 ratio, so that the
favorable sensory stimulus (a green background) is activated only when the right hand is twice
as high along the monitor as the left hand. This figure sketches out our Grade 4–6 study partic-
ipants’ paradigmatic interaction sequence toward discovering an effective operatory scheme:
while exploring, the student (a) first positions the hands incorrectly (red feedback); (b) stumbles
on a correct position (green); (c) raises hands, maintaining a fixed interval between them (red);
and (d) corrects position (green). Compare (b) and (d) to note the different vertical intervals
between the virtual objects.
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206 ABRAHAMSON AND SÁNCHEZ-GARCÍA
FIGURE 2 Mathematical Imagery Trainer for Proportion display configuration schematics,
beginning with (a) a blank screen and then featuring a set of virtual objects overlaid incre-
mentally by the facilitator onto the display: (b) cursors, (c) a grid, and (d) numerals along the
y-axis of the grid. For the purposes of this figure, the schematics are not drawn to scale. Also,
the actual device enables the tutor to flexibly calibrate the grid, numerals, and target ratio in
between trials. The full protocol includes a range of ratios as well as a ratio table.
As they attempted to determine effective bimanual choreographies for manip-
ulating the system, and still before the grid was introduced, participants discerned
within the sensorimotor interaction field latent structures affording utilities for
better satisfying the task objective. In particular, the negative space between their
hands became foregrounded as a thing that they manipulated as a means of mak-
ing the screen green—the higher they raised the interval, the bigger they made
it. Moreover, when we then introduced into the interaction system certain sym-
bolic artifacts—a grid and then numerals (see Figure 2)—the participants adopted
these screen elements as frames of action and reference. In turn, using these arti-
facts shifted the participants’ manipulation strategies into forms of engagement
closer to mathematical visualization and reasoning. For example, for a 1:2 ratio
they raised their hands sequentially, with the left hand going up 1 unit and the
right hand going up 2 units (Abrahamson, Gutiérrez, Charoenying, Negrete, &
Bumbacher, 2012; Abrahamson, Lee, Negrete, & Gutiérrez, 2014; Abrahamson
& Trninic, 2015; Abrahamson, Trninic, Gutiérrez, Huth, & Lee, 2011).
1
1
Readers are referred to earlier publications for more detail on the design rationale that led to
the development of the MIT-P, including a critical reading of previous literature on the cognition of
multiplicative concepts (Abrahamson, 2015; Abrahamson et al., 2014; Reinholz, Trninic, Howison, &
Abrahamson, 2010). The didactical principle is to support classroom teachers in implementing their
own intuitions for proportionality. Teachers (and textbooks) often introduce the concept of propor-
tional equivalence by way of presenting a situated recipe notion. Per the recipe notion of proportional
equivalence, some sensory perception of a phenomenon, such as its color, is maintained amid supple-
menting substance into the situation. Thus, the idea of equivalence in 1:2 = 2:4 might be presented as
receiving the same color of green whether one mixes 1 cup of blue paint and 2 cups of yellow paint or
2 cups of blue paint and 4 cups of yellow paint. Whether we compare ratios of paint components (color
perception), geometrically similar rectangles (aspect ratio), or food ingredients (flavor), this notion of
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