Promoting the development of embodied and integrated geometry
concepts: Gestural depictions in a computer game environment
Jonathan Michael Vitale (JMV2125@TC.Columbia.Edu)
Department of Human Development, Teachers College, 525 W. 120 th Street
New York, NY 10027 USA
Michael Swart (MIS2125@TC.Columbia.Edu)
Department of Human Development, Teachers College, 525 W. 120 th Street
New York, NY 10027 USA
John B. Black (Black@Exchange.TC.Columbia.Edu)
Department of Human Development, Teachers College, 525 W. 120th Street
New York, NY 10027 USA
Abstract
In two experiments we examined the effectiveness of a computer-based geometry learning tool,
implementing four cognitive design principles, with 3rd and 4th grade students. The four principles reflect our
hypotheses that geometry learning tasks should (1) access core knowledge, (2) incorporate diverse stimuli, (3)
promote embodied representations of higher-level concepts, and (4) prompt conceptual integration through
challenging activities. In both experiments, children in two conditions constructed a set of common four-sided
figures to fit a set of visual constraints. In experiment 1, children in the treatment condition (only) were
additionally required to verify the presence of parallel segments, congruent segments, or right angles
embedded in their figures with the assistance of animated depictions of gestures that conveyed the spatial
significance of those properties. Following training we conducted six identification tasks in which participants
attempted to discriminate two valid members of a given polygon class from four displayed polygons. In all six
instances children in the treatment condition were more likely to correctly identify both polygon class
members in a trial than children in the control condition. In experiment 2, children in the control condition
were additionally required to verify the presence of parallel segments, congruent segments, or right angles
with corresponding numerical representations of those properties. The treatment condition remained the
same. Similarly to experiment 1, significant differences were found in 5 out of 6 posttest identification tasks,
demonstrating support for our principles of embodiment and conceptual integration.
Geometry is a critical, yet often overlooked, element of mathematics education. Although the
National Council Of Teachers Of Mathematics (2000) grants geometry equal status with other domains
of mathematical instruction – including number and operations, algebra, and data analysis – U.S.
students often perform poorly on geometry items of standardized assessments, relative to other
developed nations (Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005). Eighth grade U.S. students, on
the 2003 TIMSS, in particular, scored an average of 45% of geometry questions correctly, as compared
to scores above 70% by their East Asian counterparts from Japan, Chinese Taipei, Honk Kong, Korea, and
Singapore. While underwhelming U.S. performance, in general, is due to a number of variables,
particularly SES (Berliner, 2006), the relative weakness in geometry performance by U.S. students likely
reflects inadequate emphasis in the curriculum – particularly, as concepts become more advanced.
For years, educational researchers have bemoaned the insufficient rigor of geometry curriculum that
overwhelmingly relies on static images and simplistic tasks, and, consequently, do not prepare students
for higher-level geometric reasoning (Clements, 2004; Clements & Battista, 1992). As a potential
remedy, interactive geometry software, such as Logo, was heralded as an instructional breakthrough
(Papert, 1980; Papert, Watt, DiSessa, & Weir, 1979).
Unfortunately, this early enthusiasm was
dampened by mixed or negative efficacy findings (Howe, O’Shea, & Plane, 1979; Hughes & Greenbough,
1995; Johnson, 1986; Noss & Hoyles, 1992; Simmons & Cope, 1990). However, evidence suggests that
Logo is effective when paired with a well-structured curriculum (Clements, Battista, & Sarama, 2001;
Littlefield, Delclos, Bransford, Clayton, & Franks, 1989). Regardless of specific findings, Logo – and
descendent technologies, such as NetLogo and Scratch – maintains strong theoretical support as a
primary instantiation of the view that mathematical development is best facilitated with the use of
“concrete manipulatives” (Clements, 1999) or “spatial tools” (Mix, 2009).
Based on the work of developmental and educational theorists, such as Bruner (e.g. 1966),
Montessori (e.g. 1912), and Piaget (e.g. 1970) many constructivist educators and researchers assume, as
an essential principle, that discovery-based activity with concrete manipulatives is necessary to make
abstract ideas accessible to younger children. However, like Logo, there has been a history of mixed
findings with concrete manipulatives (Ball, 1992; Fuson & Briars, 1990; Gravemeijer, 1991; Resnick &
Omanson, 1987; Thompson & Thompson, 1990). A lack of clear support suggests that continued
research, with greater emphasis on how children use the tools – in ways intended and unintended by
designers – is necessary.
Recent theory posits that spatial tools – in similarly effective physical or virtual form (Triona & Klahr,
2003) – may be more effective when introduced in a properly guided and constrained activity; thereby,
focusing learners’ attention to the critical mathematical structures of the problem space and away from
superficial or irrelevant features (Brown, McNeil, & Glenberg, 2009; McNeil & Jarvin, 2007; McNeil &
Uttal, 2009; Sarama & Clements, 2009; Uttal, O’Doherty, Newland, Hand, & DeLoache, 2009).
Consequently, designing learning tools that navigate learners along the optimal developmental
trajectory requires both detailed knowledge of the domain as well as its inherent misconceptions.
In terms of geometry instruction, extending Piaget’s stage-based theory (Piaget & Inhelder, 1956;
Piaget, Inhelder, & Szeminska, 1960), contemporary developmental researchers have made some
important discoveries regarding developmental milestones in geometry learning, particularly in regards
to the sensorimotor nature of early concepts (e.g. Clements, Swaminathan, Hannibal, & Sarama, 1999).
Yet, recent cognitive theory posits an ongoing, reciprocal relationship between the sensorimotor-based
knowledge of novices and the abstract knowledge of experts (Lakoff & Núñez, 2000), which may suggest
new ways of approaching geometry research and instructional design with more advanced learners. Our
purpose here is to detail the development of a novel geometry learning tool from an embodied
perspective and to demonstrate its effects on the skill of polygon identification, a common task on many
general mathematical assessments (e.g. the TIMSS and NAEP). We further discuss how these findings
may be implemented in a mathematics curriculum in general.
The emerging cognitive science of geometry
For years, cognitive developmental researchers, such as Piaget (1956), Bruner (1960), and their
adherents (e.g. Papert, 1980), have emphasized that learning is deeply rooted in physical activity and
interaction with the environment. Furthermore, many view the qualitative shift from sensorimotorbased knowledge to conceptual/abstract knowledge as a hallmark of conceptual development, e.g.
development of geometry concepts from highly perceptual to symbolic (Clements et al., 1999; van Hiele,
1986). Generally, with this perspective, the development of higher-level knowledge has been addressed
by information processing theorists as a matter of memory capacity (Sweller, 1988) and efficient
production systems (Anderson, Boyle, Corbett, & Lewis, 1987), best addressed by direct instruction
(Kirschner, Sweller, & Clark, 2006). Tools reflecting this view, such as cognitive tutors (Koedinger &
Anderson, 1993), take advantage of the “symbolic advantage” that occurs with complex problem solving
(Koedinger, Alibali, & Nathan, 2008). While these tools are necessary and successful in many contexts –
such as with high school students studying formal Euclidean geometry – recent emphasis on embodied
(or grounded) cognition (Barsalou, 2010; Clark, 1999) suggests that abstract concepts require an initial
grounding in perceptual-motor experience.
In particular, Lakoff and Núñez (2000) describe mathematical thinking in terms of conceptual
metaphors that connect abstract words and symbols to common physical experiences. Specifically,
“grounding metaphors” map physical objects or activities to target mathematical concepts (e.g. actions
on object collections map onto arithmetic operations).
perspective comes from studies of early numeracy.
The most compelling evidence for this
Both neuroimaging and behavioral data
demonstrate a critical association between basic numeracy skills (e.g. comparing two numbers) and
spatial processes (Dehaene, Piazza, Pinel, & Cohen, 2003; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin,
1999). According to Spelke (2000), basic numeracy is built upon core systems of knowledge, including
the ability to subitize – i.e., recognize the numerosity of a small array of objects without counting – and
approximate the numerosity of large magnitudes (Feigenson, Dehaene, & Spelke, 2004). In turn, the
strength of these spatial-numerical associations has been shown to be predictive of a wide range of
mathematical skills, including those commonly found on standardized tests (Booth & Siegler, 2006;
Halberda, Mazzocco, & Feigenson, 2008; Holloway & Ansari, 2009).
In particular, investigations of number line estimation demonstrate that while associations between
space and number emerge early, this mapping is initially inaccurate and biased towards overrepresentation of well-known, small numbers (Booth & Siegler, 2006; Siegler & Booth, 2004; Siegler &
Opfer, 2003). In a series of interventions, young children (pre-k) learned to estimate accurately by
playing a corresponding board game (Ramani & Siegler, 2008; Siegler & Ramani, 2008, 2009). This
research clearly demonstrates that engaging and accessible tools can produce robust improvements in
the underlying spatial representation of mathematical concepts.
Likewise, researchers are beginning to address geometry in terms of spatial processes in core
systems, and their development. Spelke, Lee, and Izard (2010) claim that concepts are grounded in two
core systems associated with spatial navigation and object perception. From this perspective, the
concept of a shape, e.g. rectangle, is derived from the experience of viewing rectangular objects, such as
the face of a block, or navigating along a rectangular path. The idea that initial concepts of geometric
shapes are (at least partially) rooted in our perceptual, object recognition system is both intuitively
appealing and empirically supported. Many studies of object perception and recognition have utilized
common geometric figures as stimuli (e.g. triangles, trapezoids) to investigate how concepts are
encoded in and retrieved from memory.
For example, in a study by Bomba and Siqueland (1983), 3-month-olds developed a prototypical
representation of a triangle by viewing a series of triangles, which varied randomly from the prototype,
without ever actually seeing the prototype. In other words, frequent exposure to the natural variation
of a shape may lead to the formation of an idealized conceptual representation. While this example
suggests how initial associations between mathematical concepts and perceptual experiences are
formed, as the number line example suggests, these initial mappings often diverge from expert
knowledge in important ways.
Specifically, in formal geometry, the shape-as-physical object mapping can produce misconceptions.
While a rectangle – as a mathematical concept – must contain perpendicular sides, there is an
abundance of examples in the natural environment of blocks with slightly non-perpendicular sides,
potentially contributing to the concept of a rectangle. Features other than those that formally define
the geometric class, such as symmetry (Quinlan & Humphreys, 1993), may drive identification and
classification. In a study in which participants classified a series of four-sided figures based on perceived
similarity, Behrman and Brown (1968) found that “dispersion” (irregularity), “elongation”, and
“jaggedness” emerged as the most significant factors in classification. Similar features were discovered
in subjects’ classification of U.S. state shapes (Shepard & Chipman, 1970). While, these features are
clearly important in everyday manipulation of objects, they may interfere with the learning of formal
concepts. For example, a polygon is no more or less a rhombus if it is elongated or square.
Furthermore, these real world object-based features affect identification of common polygons in
educational settings. Young children often apply informal labels, such as “slanty”, “pointy”, or “skinny”
to describe common shapes (Clements et al., 1999). In identification tasks, young children often miss
non-prototypical, but valid instances of a target shape in favor of invalid instances that share common,
holistic features with the prototype (Burger & Shaughnessy, 1986; Clements et al., 1999). For example,
while identifying rectangles many children incorrectly selected “long” (non-rectangular) parallelograms
as rectangles, while missing squares. Though one may suspect that this kind of error is isolated to young
children, Mach (1886/1959), a forebearer of gestalt psychology, demonstrated that an object would be
perceived differently – as either a square or diamond – depending upon its spatial orientation. Clearly
our initial knowledge of geometry is founded – at least, in part – on implicit natural object perception,
but requires further development to support higher-level skills.
Embodied learning in geometry instruction
The tendency for younger children to identify objects based on superficial characteristics, rather than
formally-defining properties, is not unique to geometry. Children often focus on holistic, characteristic
features while classifying common objects such as taxi cabs, islands, and robbers (Keil & Baterman,
1984). On the other hand, adults or domain experts tend to classify objects based on abstract or hidden
structure (Carey, 1985; Chi, Feltovich, & Glaser, 1981). This is often described as a general perceptual to
conceptual shift in development (Bruner, 1960; Piaget, 1952). As such, mature concepts are produced
through analytical systems that depart qualitatively from simple perceptual, association systems (Carey,
1985; Mandler, 1992; Sloman, 1996). This approach is reflected in Clements, Swaminathan, Hannibal,
and Sarama’s (1999) description of higher stages of geometry development as a synthesis of imagistic
and declarative knowledge.
Alternatively, conceptual development may be viewed as the reorganization of perceptual and motor
systems in accordance with the functional features, or affordances, of objects (Gibson, 1969; Goldstone,
1998; Goldstone, Landy, & Son, 2010; Thelen & Smith, 1994). From this perspective, what was seen as a
develomental shift from perception to conception is reframed as a perceptual realignment, in which
salient, but irrelevent features are disregarded in favor of more subtle, relevant features (Goldstone &
Barsalou, 1998; Quinn & Eimas, 1997; Schyns, Goldstone, & Thibaut, 1998). In the case of geometry,
learning about a square, for example, goes beyond assembling a verbal list of features – e.g. four right
angles, four congruent sides, etc. – to seeing the shape in terms of these features. Because the
enrichment and refinement of initial representations likely involves both perception and action systems,
we refer to this process, generally, as embodied learning.
How then might embodied learning of geometry concepts be promoted? One possibility is to simply
provide feedback on a repetitive perceptual classification task. For instance, Goldstone and colleagues
demonstrated successful perceptual learning in a categorization task, with feedback, by presenting
multiple, diverse examples (and non-examples) of a target concept over a long series of trials (Goldstone
et al., 2010; Son, Smith, & Goldstone, 2011). While this approach may be appropriate for a highlycontrolled laboratory setting, this paradigm may be unsuitable for more dynamic setttings with younger
children. Alternatively, Glenberg, Gutierrez, Levin, Japuntich, and Kaschak (2004) demonstrated that
physical interaction with a concrete model of a narrative improves comprehension more than simply re-
reading a story. Likewise, students learn challenging physics concepts better with an interactive
computer simulation than by viewing animations of the same system (Chan & Black, 2006).
While using interactive tools to promote conceptual development maintains strong theoretical
support, mixed results suggest that these tools can be misapplied. Unguided and unconstrained activity
with either a virtual or physical manipulative is likely to lead to rote proceduralization (Hiebert &
Wearne, 1986), non-reflective trial-and-error (Simmons & Cope, 1990), and little transfer (Ball, 1992).
Rather, a strong curriculum constrains the learner to confront the target concepts directly (Clements,
Battista, & Sarama, 2001).
In the case of geometry, learners must be constrained to generate polygons according to defining
features (e.g. parallel lines, congruent line segments, right angles), while permiting their figures to vary
across salient, but non-determining features (e.g. orientation, elongation). Furthermore, guidance is
necessary to provide children with initial grounding metaphors for concepts. Specifically, defining
features need to be represented in ways that are clear and accessible. While, work with indiginous
Amazonians (Dehaene & Izard, 2006) indicates that these concepts are intuitively recognized, the
mapping between multiple spatial concepts and formal mathematical concepts is often complex.
Therefore, the central challenge of a geometric learning tool is to promote the organization and
integration of multiple embodied concepts into unified, formal concepts.
Cognitive Design Principles
Given the theoretical perspective detailed above, we constructed the following four basic cognitive
design principles, attuned towards the development of geometric knowledge.
Principle 1: Provide an intuitive interface for polygon construction that taps into children’s core
knowledge of shapes.
As described above, children’s knowledge of shapes extends out of their everyday experiences
interacting with solid objects and navigating through the world (Spelke et al., 2010). Incorporating this
type of experience into a learning activity connects the geometry that children are learning in formal
settings with the geometry they are practicing in informal settings. Often, with highly symbolic,
contextually-abstract learning activities children achieve proficiency with formal mathematics without
developing a strong sense of how this knowledge might be applied to authentic settings (Lave, 1988).
On the other hand, providing children with a context that is unfamiliar and overly-contextualized, while
engaging, may limit the potential for transfer (Son & Goldstone, 2009). By incorporating simple,
recognizable objects and/or activities in the game we reinforce the relationship between formal
geometry and commonplace tasks.
As to the specific guiding metaphor for shape construction, the representation may either elicit the
core system for navigation or object-perception. However, as Spelke et al. (2010) note, these systems
may ground different aspects of geometric knowledge – i.e., navigation grounds directional sense,
object recognition grounds angle, and both ground distance. In choosing a theme that focuses solely on
one of these systems, children may find activities grounded in the other system unnecessarily
challenging.
Yet, as it happens, Logo’s application of a third-person, overhead perspective of an agent tracing a
path may represent a suitable blend between these two metaphors. If children imagine the agent’s
perspective as it traverses a path, they may access their core navigational system – providing an intuitive
sense of direction. On the other hand, if children focus on the shape that emerges as a product of the
agent’s actions, they may access their core object recognition system – providing an intuitive sense of
angle. Although our activities focus on angle, rather than direction, we chose a similar theme to provide
a robust basis for a wide range of potential activities.
Principle 2: Provide goals and constraints that promote the construction of a diverse set of valid
polygons, rather than highly idealized, prototypical polygons.
This principle is simply a matter of good curriculum design.
U.S. students’ poor performance in
geometry may be fundamentally the result of a curriculum that is “a mile wide and an inch deep”
(Schmidt & McKnight, 1998; Schmidt, McKnight, & Raizen, 2007, 1997) and devotes little attention to
the complexity of geometry. Curriculum which provides only a limited set of shape instances – likely as
a result of a limited set of physical blocks or drawing templates – encourages children to develop an
overly rigid concept of shape (e.g. a rectangle is always elongated). On the other hand, activities that
include a diverse array of stimuli facilitate the “pick-up” (Gibson, 1966) of critical information about
shapes, while disregarding superficialities.
Our approach to implementing this principle is to focus on a limited number of shapes (six) across a
greater number of construction activities (twenty-six).
Furthermore, by providing some basic
constraints within the game to prevent the construction of prototypical shapes – in some cases – we
ensure that children will be exposed to a perceptually diverse set of any specific shape.
Principle 3: Provide instruction that fosters an “embodied” understanding of formal, defining features of
polygons.
While repeated exposure to a diverse set of shapes could lead to the development of implicit higherlevel knowledge, given time restraints – particularly, considering the constructive nature of the task –
some degree of direct instruction is necessary to address higher-level concepts efficiently (Klahr &
Nigam, 2004). Yet, at this stage, we do not abandon spatially-grounded tools in favor of symbolic
procedures. While advanced geometric procedures – such as deductive proof – rely upon the use of
symbolic representations of formal properties, these properties are fundamentally spatial in nature, and
require the development of an embodied internal representation.
Fostering the development of a spatial representation can be done in a number of ways. Multimedia
representations that incorporate text and (potentially animated) images are common, effective, and
well-researched (Mayer, 2001; Moreno & Mayer, 1999). Another, more recent procedure is to direct
children’s gesture in congruence with the target representation (Broaders et al., 2007; Segal, 2011).
Given previous geometry studies’ mention of spontaneous gesture in representing common concepts
(Clements & Burns, 2000; Clements et al., 2001), there appears to be a natural relationship between our
target concepts (i.e., parallel lines, congruent line segments, and right angles) and gesture. Yet,
because of the nature of our research protocol, which limits external guidance and behavioral
constraint, enforcing the use of gesture is unfeasible. Rather, by combining multimedia elements and
gesture we developed gestural depictions, in which animated hand icons demonstrate a gesture applied
to the figure on screen. While these depictions do not ensure that the children will perform their own
gestures, or even interpret the gestures as such, they provide a base-line for instruction across all
participating children.
Principle 4: Incorporate challenges within the game that promote integration of basic concepts of
polygons with higher-level concepts regarding their properties.
By implementing Principles 1 and 2, children are responsible for the basic construction of a diverse
set of shapes. By implementing Principle 3 children are provided a means of conceptualizing higherlevel features of shapes. Therefore, children are guided to develop a grounding metaphor (Lakoff &
Núñez, 2000) for both polygons and their properties.
However, these two independent sets of
representations must be integrated, or “blended” (Fauconnier & Turner, 1998), into a new
representation. Developing an integrated representation is non-trivial, and likely requires an activity
exhibiting “desirable difficulties” (Bjork, 1994; Bjork, 2006; Bjork & Linn, 2006) – i.e. procedures which
may reduce performance during the learning task, but result in more robust learning. Specifically, in
conceptual domains, activities should promote reflection upon relationships between multiple
components of a concept (Clark & Linn, 2003; Linn, 2000; Linn, Chiu, Zhang, & McElhaney, 2010).
Promoting reflection in a learning activity is a significant design challenge. One approach may be to
simply ask children to talk about or write about their ideas. For example, Kurtz, Miao, and Gentner
(2001) asked children to write an integrated description of two visual representations of an abstract
physical concept. However, the addition of a writing task may not be relevant for all activities,
particularly games. Here we choose to incorporate a challenge that is more implicit to the goals of the
game. Specifically, children are constrained to accurately construct figures that meet both overall visual
constraints as well as specific, formal constraints (described in the next section). By navigating between
these two sets of requirements we hope to encourage reflection and development of an integrated
representation.
The digital geometry game
In this section we detail the design of a digital game that implements our four cognitive design
principles. While these principles guided the overall construction of the game, some specific choices –
not dictated by the principles – were necessary. In most cases these decisions were either arbitrary or
reflected some practical considerations given the software development tools or the projected
participant sample. For example, the choice of a robot navigation theme was made in light of our
ongoing work with Lego robotics and previous work with Logo. When potentially consequential, the
rationales for specific design decisions are explained below.
Initial construction of the shape. Given Principle 1, we chose an easily comprehended narrative
premise in which children navigate a robotic agent through an obstacle course; collecting goal objects
while avoiding danger objects (Figure 1). Placement of the goals and dangers is designed to constrain
learners to the production of a target polygon. By increasing constraint (e.g. less obstacle-free space,
more goal objects) we manipulate the difficulty of the exercise as well as the diversity of figures
produced by an individual participant from exercise to exercise (i.e. implementing Principle 2). Most
importantly, a central set of obstacles, varied between exercises, forms a schematic image of the target
polygon.
In the planning phase the player is told the name of the target polygon and simply views the array of
goals and obstacles, overlaying a grid, with the objective of planning a navigational course for his or her
agent. In this phase the player cannot construct his or her figure. The player proceeds by pressing the
“build” button when ready.
Figure 1. Screenshot of game in the planning phase. Players plan a course of motion starting at the
circle (protruding segment indicates initial heading), around the obstacles (rocks and trees), through
the goals (flags), and back to the starting circle. In this case the player is planning a parallelogram.
In the subsequent construction phase players attempt to produce the imagined path without access
to the visual layout of obstacles and goals. The grid does persist to help players coordinate between
planning and construction. A figure is built by dragging the mouse to construct lengths and rotating the
mouse about a center pivot to construct angles (Figure 2). Upon closing the figure to form a polygon,
vertices
may
be
adjusted
freely
via
“drag-and-drop”,
to
refine
the
overall
shape.
Figure 2. Screenshot of game in the construction phase. Players build a closed polygon to represent
the path of their agent. In this case the player is attempting to construct an angle of a parallelogram.
The choice to separate planning and construction, and thereby eliminate access to the visual layout
of obstacles during construction, was intended to provide a challenge for the game player. If the player
was granted a persistent view of obstacles and goals, then producing a successful figure would be a
trivial matter of tracing a figure around the obstacles and through the goals. By requiring players to
construct a figure from memory, we facilitate the internalization of an instance of the target polygon.
While advanced players may mentally encode their projected figure in terms of higher-order properties
(e.g. parallel sides, right angles, etc.), we expect that the majority of learners will simply encode their
projected figure in terms of an overall, holistic appearance. Therefore the initial construction of the
figure is simply intended to promote a basic-level conceptual representation of the target polygon.
Integrating higher-level features. At this point the player has worked with implicit constraints to
produce a polygon. While these constraints are intended to suggest the construction of a specific
polygon, we cannot ensure that the produced figure will actually be a valid instance of the target
polygon, or that the learner has attended to the defining features of the polygon. Consequently, as
Principles 3 and 4 assert, learners must be explicitly guided to develop embodied representations of
these features, and provided with means for integrated these concepts with their core representation of
the target polygon. To implement these principles we introduce a third phase in the construction
process – feature validation. In this phase learners are asked to verify that defining features of the
target polygon are exhibited by their constructed figure – including parallel lines, congruent line
segments, and right angles.
To assist in the development of embodied representations of these defining features, when selected
from a menu, a player-controlled animation depicts a gestural representation of the concept, with some
additional visual guides (Figure 3). These gestural depictions not only serve as a conceptual grounding
for defining features, but function as the feedback mechanism for the validity of defining features in the
player’s polygon. Specifically, in the course of manipulating the gestural depictions the player also
selects the sides or angles of the figure that he or she believes exhibit the target feature. If correct,
within a margin of error, a large check mark appears on screen, and the sides or angles are given
standard geometric markings indicating their property (e.g. arrowheads along parallel segments). If
incorrect, a large “x” appears on screen, and the child is given an opportunity to either try again (with
other sides or angles) or adjust the figure.
For parallel lines, players guide two hand icons, which move along the length of two displayed
parallel lines, to select the two sides of the polygon that are potentially parallel. The orientation of the
hands is automatically adjusted by the computer to match nearby or chosen sides. For congruent
segments players guide a pair of hand icons to a side of their polygon. The hands are automatically
adjusted by the computer to mark the length of the side. The learner then guides these hands, with
their distance fixed, to a second segment for comparison. For right angles, the learner guides two
hands, oriented perpendicularly, to fit a green square in the corner of the figure.
Margins of error for successful validation are kept small to ensure that this phase of the game
provides a challenge for players.
As indicated by Principle 4, we believe that the challenge in
coordinating between the physical obstacles in the construction space and the narrow spatial definitions
of defining features will prompt reflection by players. For example, in the parallelogram example shown
in Figure 1, a player will need to reflect on a way to construct a side that passes through the flag, but
maintains a slope that can be matched by its oppose side.
Finally, after successful feature validation the learner attempts to test his or her constructed path in
the obstacle course. This testing phase also grants players an additional opportunity to view the
arrangement of obstacles. After the player directs the initial heading, the robot navigates along the
constructed path. If the robot successfully navigates the obstacle course (i.e. collects all goals without
intersecting any obstacle), then the player proceeds to a new exercise. If the player is unsuccessful he
or she returns to the construction phase to adjust the figure.
Figure 4. Screenshot of game in the testing phase. Players set the initial heading and then watch as
their agent navigates the obstacle course according to their design.
Application in research. In the following two studies we detail two experiments using this geometry
learning game. The main objective of these experiments is to provide supporting evidence for our
cognitive design principles. In particular, because research on basic geometric concepts with young
children has been conducted successfully elsewhere (e.g. Sarama & Clements, 2002, 2004), we focus on
the latter two principles regarding the development of higher-level concepts. In each study we compare
students who were randomly assigned to either the full version of the software, with an embodied,
feature validation phase, or an alternate version of the software. After completing each of three sets of
unit exercises in the game, participants completed a shape identification posttest.
The primary aim of these studies is to determine how a geometry learning task that provides an
embodied representation of concepts and facilitates their integration into one’s previously existing
representation (i.e., Principles 3 and 4, respectively) will promote conceptual transfer to a simple, but
critical identification task. Furthermore, we intend to use in-game data, gathered by the game software,
to determine whether the source of differences between conditions strictly relates to the cognitive
design principles, or other, unexpected factors of the game-play experience.
Experiment 1
As an initial test of our embodied, integration paradigm we compared a group of children with the
full feature validation phase – i.e., embodied integration (EI) condition – to a group of children without
feature validation – i.e., no integration (NI) condition. Features implementing the first and second
principles – i.e., the initial construction task and the curricular sequence of exercises – were the same
between groups.
Method
Participants
Twenty-one fourth grade children were initially recruited from an after-school program located in a
low-income, predominantly Hispanic neighborhood of New York City. The children were randomly
assigned to either the embodied integration or no integration condition. The final embodied integration
condition consisted of ten children (M = 9.4 years, SD = .16, 40% female, 90% Hispanic, 10% African
American). The no integration condition also consisted of ten children (M = 9.6 years, SD = .28, 50%
female, 100% Hispanic). Two children (one from each condition) were native Spanish speakers but could
communicate sufficiently in English, and showed little difficulty understanding and completing the tasks.
Additionally, one child, originally assigned to the embodied integration condition (but not reflected in
the statistics above), was not included in the study due to prolonged absence.
Materials and Procedure
Game and curriculum. All study-related tasks were conducted in the context of a weekly afterschool robotics course. However, in some instances in which students had no other assigned classes, we
conducted multiple sessions in a week. Aside from the tools described here the children also engaged in
physical construction of Lego-based robots. This supplementary activity did not affect the structure of
this study, except to inform the visual design of the learning tool.
Children were randomly assigned to either condition at the start of the first class, thereby
determining the version of the software they would use throughout the experiment. In the embodied
integration condition children learned with the full version of the software, described above.
Specifically, children in this condition were required to validate specific defining features in their
constructions, within a small threshold of deviation, to complete each exercise. Parallel sides were
validated by computing, by the software, the absolute difference of the two selected sides’ percent
gradient (0%-horizontal, 100%-vertical, 50%-diagonal rising left-to-right, -50%-diagonal rising right-toleft). Sides within  3% gradient difference were considered parallel. Congruent sides were validated by
computing the absolute difference between segment lengths (in pixels). Sides within  23 pixel
difference (approximately 7 mm of screen size) were considered congruent. Finally, right angles were
validated by computing the difference of the degree measure of the internal angle, at the selected
vertex, from 90°. Angles 4° of 90° were considered right.
In the no integration condition children did not validate defining features of their constructed figures.
There was no explicit constraint on the defining features of the polygon, beyond those implicit in the
physical layout of obstacles and goals. Therefore, while a successful figure may resemble a trapezoid by
overall appearance, its opposite sides need not be, necessarily, parallel (Figure 8, in the discussion,
shows some examples of formally invalid, yet successful polygons constructed by NI participants).
In either condition the children completed a series of exercises across three units of study, focusing
on three defining features of polygons: parallel sides, congruent adjacent sides, and right angles,
respectively. In the first unit children completed a set of 10 construction exercises, focusing on
trapezoids and parallelograms. In the second unit children completed 6 exercises, focusing on kites and
rhombi. Finally, in the third unit children completed 6 exercises, focusing on rectangles and squares.
Each exercise was designed, via the placement of obstacles and goals, to promote construction of a path
resembling the target polygon.
Exercises within each unit generally progressed from the most
prototypical versions of the polygon to the most irregular, yet still valid, versions (see Appendices A, B,
and C for layouts of all levels).
To avoid contamination between conditions each learning session was divided into two halves, in
which the embodied integration and no integration groups alternated order from session to session.
Children not currently engaging in study activities completed homework in a separate room.
Advancement through the series of exercises occurred at the individual pace of the participant, causing
some students to finish more rapidly than others. During these learning sessions, the experimenters
(two of the authors) circulated among the children giving assistance and providing motivational support.
As an additional motivational tool, children were supplied with a “stamp certificate” in which each
completed exercise earned the child a star sticker or stamp on a personal document.
We acknowledge that, unlike traditional laboratory tools for psychological research, our learning tool
has many moving parts – all of which could affect learning. To assist in the analysis of the learning
process, the software provided a numerical record sufficient to reconstruct all operations undertaken by
the learner. For this experiment we analyze some general summary statistics regarding the difficulty of
the task and the strategies applied by children. If there are significant differences in the effort or
strategy undertaken by a child, group difference on the posttest measures will need to be addressed in
terms of alternative explanations.
Shape identification. The main assessment measure was individually administered, immediately
following a student’s completion of a unit. Each unit posttest included two subtests with 30 trials, for a
total of six subtests and 180 trials. The goal of each trial is to identify the two positive examples of a
target polygon from four simultaneously presented polygons (see Figure 5). During each trial we
recorded which figures were selected, as well as other interactions with the interface (e.g. “mouseovers”, “mouse-clicks”). To encourage more interaction, figures are initially displayed in light gray
without a border, but are darkened and highlighted with a black border when “moused-over”– providing
a clearer image of the polygon. Upon clicking on a polygon, its internal area is filled blue as a visual
indicator of currently selected figures. Figures may be selected and de-selected freely in the process of
decision making. Once two figures are selected, the participant may then click on a central button to
advance to the next trial. In some cases post-trial feedback may be provided (detailed below).
The rational for this trial structure, with four – as opposed to a simpler choice between two figures –
was to make common features of the four polygons explicit and comparable. Given the evidence that
children, without an appropriate intervention, are more likely to classify geometric figures based on
holistic similarity than defining features, this trial structure allows us to identify what feature the
participant most likely attended to and what feature he or she most likely disregarded.
Figure 5. Screenshot of shape identification task. Participants choose 2 valid examples of a polygon
(trapezoid) from four available figures. Here, two figures are currently selected (in blue), and one is
being highlighted (gray, outlined in black) through mouse interaction. The light gray, un-outlined
polygon is neither highlighted nor selected.
Six subtests assessed participants’ ability to identify trapezoids, parallelograms (following unit 1),
rhombi, isosceles triangles/trapezoids (following unit 2), rectangles and right triangles/trapezoids
(following unit 3). The triangle/trapezoid blocks were included to test whether the participant’s concept
of the target feature would extend to novel stimuli. To generate a set of stimuli – for trapezoids,
parallelograms, rhombi, and rectangles – we began with three valid members of the polygon class,
oriented in a prototypical manner (“upright”). We then created three more valid examples by rotating
each of the three original one-quarter turn. From each of these six valid polygons we created two
characteristically similar, but formally invalid polygons by slightly altering a defining feature (i.e., parallel
sides, congruent adjacent sides, or right angles) consistently across all six positive examples. For
example, in Appendix F, the positive examples of rhombi (1st and 3rd rows) were altered to become nonexamples (2nd and 4th rows) by lengthening two of four sides. The six positive examples, paired twice
with invalid derivations, produced two sets of six valid-invalid pairs (see appendix D – F, H for general
structure). From each of two sets of six valid-invalid pairs, fifteen combinations of four shapes (i.e., pairs
of pairs) were produced for a total of thirty trials.
In the case of the mixed triangle and trapezoid blocks, six valid examples of trapezoids were paired
with a single invalid derivation, while six valid examples of triangles were paired with a single invalid
derivation to produce the two sets of six valid-invalid pairs (see appendix G, I, for general structure).
Once again, from each of two sets of six valid-invalid pairs, fifteen combinations were produced for a
total of thirty trials.
An instruction page, displayed on-screen prior to identification trials, provided a written description
of the defining feature-based rules of inclusion for the target polygon class. For example, “A trapezoid is
a four-sided figure with one pair of parallel sides”, or in the case of a mixed triangle/trapezoid block, “A
shape is right if it contains at least one right angle”. Inclusion of this rule ensured that all participants
had sufficient information to identify polygons correctly. Furthermore, in the case of subtests utilizing
figures similar to those given in the learning exercises (i.e., trapezoids, parallelograms, rhombi, and
rectangles), this instruction page displayed both a valid and invalid example of the polygon. The
participants were asked to click the sides or angles corresponding to the target defining feature (parallel
sides, congruent sides, right angles) within the valid example. In the case of mixed triangle/trapezoid
blocks, which were intended to be novel visual stimuli for the given target feature, no visual example
was provided.
Because of recent work showing that some complex, productive learning tasks are often best suited
to preparing students for future learning, rather than transferring conceptual knowledge directly
(Hammer & Black, 2009; Schwartz & Bransford, 1998; Schwartz & Martin, 2004), there is a possibility
that embodied integration condition participants would show differences from no integration condition
participants in learning during the posttest (i.e., changes from performance in the first half of the test to
the second half). In this case, post-trial feedback might be necessary to trigger group differences. With
an opportunity to work with children across a number of learning units we introduced some variations in
feedback across subtests to get a sense of how directly concepts developed in the learning task
transferred to the posttest. Specifically, in the first four subtests, following learning Units 1 and 2,
participants were informed, following each trial, the number of correct polygons they had selected (0, 1,
or 2). In the final two subtests, following Unit 3, no feedback was given.
Other assessments. To measure the children’s overall mathematical ability, and to ensure that the
groups were roughly equivalent, we conducted the Woodcock Johnson III Calculation and Mathematical
Fluency subtests on each child. The test was administered individually, separately from children
performing study-related tasks. This measure was taken following the completion of the study.
Results
Woodcock Johnson. Standardized scores with Woodcock Johnson III age-normed subtests, reveal
that both groups were approximately average in performance, although bordering on high average, for
Calculation (EI: M = 107.7, SD = 9.7; NI: M = 111.0, SD = 10.0) and average in Math Fluency (Treatment
EI: M = 104.7, SD =14.0; Control NI: M = 99.1, SD = 12.7). No significant difference was found between
the two conditions on either Calculation (t[18] = 1.04, p > .1) or Math Fluency (t[18] < 1).
Shape identification. On any given identification trial, in which two out of four shapes were selected,
a participant could correctly identify 0, 1, or 2 figures. While in most testing situations one correct
would be deemed better than zero correct, here one correct suggests that the participant judged the
polygons according to the incorrect, distracter feature (e.g. choosing both “upright” figures). Therefore
we coded trials with both figures selected correctly as accurate and trials with one or zero figures
selected correctly as inaccurate. We then calculated the percentage of accurate (both-correct) trials
within each subtest. On any given trial chance performance is 1 out of 4, or 25%. Figure 6 displays the
distribution of participants’ accuracy scores across all six subtests.
Embodied integration
No integration
Embodied integration
No integration
(1) Trapezoid
(2) Parallelogram
(3) Rhombus
(4) Isosceles Triangle/Trapezoid
(5) Rectangle
(6) Right Triangle/Trapezoid
Figure 6. Distributions of % accuracy scores in experiment 1. For each of six subtests (1-6), the EI
condition is displayed on the left and the NI group is displayed on the right. The results are displayed
separately for each block. The x-axis represents the percent of trials in which a participant selected both
figures correctly. The y-axis represents the number of participants with the given level of accuracy.
We do not assume that the accuracy scores are necessarily distributed normally. For example, it may
be the case that children’s performance was skewed by ceiling/floor effects. Therefore to compare
participants between conditions directly we performed a series of non-parametric, one-tailed MannWhitney test. Significant results indicate that the distribution of EI condition participants’ accuracy
scores are shifted positively from the distribution of the NI condition participants’ accuracy scores. We
also compared conditions on mean trial duration. Because there is no reason to suspect that duration
scores should be non-normal we applied a two-tailed t-test in each of the six subtests.
Table 1
Posttest summary measures
Subtest
Condition
Percent of trials correct
Trial duration (secs.)
EI
Median
60%
Min
27%
Max
97%
Mean
12.4
SD
5.0
NI
42%
27%
83%
12.3
5.7
Parallelogram
EI
NI
78%
50%
100%
9.7
5.1
48%
7%
90%
10.0
3.3
Rhombus
EI
NI
80%
56%
97%
8.3
2.6
58%
13%
90%
11.4
2.8
EI
73%
13%
93%
9.6
2.5
NI
13%
3%
97%
9.3
4.0
EI
75%
56%
100%
8.4
3.0
NI
48%
0%
73%
8.6
2.9
Right
tri/trap
EI
NI
23%
0%
60%
11.9
5.3
8%
0%
2%
9.3
3.8
Overall
EI
NI
64%
37%
50%
19%
85%
69%
10.1
10.1
3.0
2.8
Trapezoid
Isosceles
tri/trap
Rectangle
As table 1 reveals, across all subtests the median percent of trials correct is higher for the EI
condition than the NI condition. An overall comparison of conditions, pooling all subtests items,
applying a one-tailed Mann-Whitney test reveals a significant positive shift in the distribution of EI
participants’ accuracy scores, compared to NI participants’ accuracy scores (W = 90, p < .01).
Additionally, comparisons at each subtest reveal similar advantages for EI condition participants
(Trapezoid: W = 74, p < .05; Parallelogram: W = 82, p < .01; Rhombus: W = 80.5, p < .01; Isosceles: W =
83.5, p < .05; Rectangle: W = 89, p < .01; Right: W = 72.5, p < .05).
Somewhat surprisingly, there was little difference between mean durations of conditions – overall,
pooling all subtest items, average durations were the same. In the case of rhombi a two-tailed t-test
revealed a significant difference between conditions, with participants in the EI condition completing
trials more rapidly (t[18] = -2.3, p < .05). In all other cases there were no significant differences
between conditions (Trapezoid: t(18) = .03, n.s.; Parallelogram: t(18) = -.15, n.s.; Isosceles: t(18) = .20,
n.s.; Rectangle: t(18) = -.11, n.s.; Right: t(18) = 1.3, n.s.).
To address whether participants’ accuracy improved from the first half of subtests to the second half
of subtests, and thereby indicating learning within the test itself, our approach was to perform a
repeated measures ANOVA with subtest-half as a within-subjects factor and condition as a betweensubjects factor. However, as described above, we cannot assume that all accuracy distributions were
normal.
Therefore, we performed Shapiro-Wilk’s tests of normality to determine whether the
distributions of accuracy scores – separated by condition and subtest – were significantly non-normal.
For the EI condition the isosceles triangle/trapezoid showed some departure from normality (W[10] =
.82, p < .05;), while the other subtests did not (Trapezoid: W[10] = .97, n.s.; Parallelogram: W[10] = .96,
n.s.; Rhombus: W[10] = .96, n.s.; Rectangle: W[10] = .98, p >.1; Right: W[10] = .94, n.s.). In the NI
condition the trapezoid (marginally) and isosceles triangle/trapezoid blocks showed some departure
from normality (Trapezoid: W[10] = .85, p < .1; Isosceles: W[10] = .80, p < .05) while the other subtests
did not (Parallelogram: W[10] = .93, n.s.; Rhombus: W[10] = .96, n.s.; Rectangle: W[10] = .89, n.s.; Right:
W[10] = .89, n.s.).
Given some evidence of non-normal distributions (trapezoid and isosceles trapezoid/triangle
subtests) we chose to perform ANOVAs on the four remaining, potentially normal subtests. Reflecting
the non-parametric tests, above, a significant effect of condition, favoring the EI condition, emerged
(parallelogram: F[1,18] = 8.6, p < .01; rhombus: F[1,18] = 7.2, p < .05; rectangle F[1,18] = 12.5, p < .01;
right F[1,18] = 4.7, p < .05). In the two subtests with post-trial feedback, as expected, subtest-half
proved to be significant predictor of accuracy, indicating learning from first to second subtest half
(parallelogram: F[1,18] = 20.8, p < .001; rhombus: F[1,18] = 19.8, p < .001). Surprisingly, in the rectangle
subtest – with no feedback – subtest-half was also a significant predictor of accuracy (F[1,18] = 12.5, p <
.01), while this was not the case in with the right triangle/trapezoid subtest (F[1,18] = 1.1, n.s.).
The only significant interaction between subtest-half and condition occurred on the rhombus subtest
(p < .05). Post-hoc, Bonferroni corrected t-tests reveal that this interaction is the result of a significant
difference between conditions in the first half (t[18] = 3.5, p < .01) and a non-significant difference in the
second half (t[18] = 1.1, n.s.). Therefore, feedback afforded the NI condition participants an opportunity
to “catch up” with the EI condition. In the other cases, if learning from first to second half occurred, the
change was parallel between conditions (parallelogram: F[1,18] = 1.7, n.s.; rectangle F[1,18] < 1, n.s.;
right F[1,18] = 2.0, n.s.).
Game analysis. Every child successfully completed all 26 game exercises. Due to experimenter error
some data files were lost in the second unit of study (3 participants from the control condition).
Analysis was conducted on all remaining available data.
From game-play data we sought to determine whether conditions were equivalent in terms of
difficulty and participants’ strategy. To get a sense of differences in the difficulty of the task we
computed the mean duration of each exercise. Likewise, we computed a mean count of the number of
“transforms” (i.e. vertex drags) during each exercise. The assumption for these two variables is that a
more challenging exercise will require greater time to complete and more adjustments to the shape.
To assess whether there were some general strategic differences between conditions we computed
two summary variables: mean area and relevant error of the completed figure. In regards to area, our
intention in designing exercises was to elicit the production of polygons that closely resembled the
internal set of obstacles (see appendices A-C). While this layout of internal obstacles was intended to be
suggestive of a target shape, they did not prohibit other shapes from being constructed. Rather, goal
objects and non-internal obstacles were placed to limit the variation of shapes. Yet, even within these
constraints, the possibilities of large variations remained. The area of the shape (i.e., pixels2) is an
indirect measure of how closely a participant’s figure resembled the layout of the internal set of
obstacles. We suspect that a smaller area indicates closer attention the suggested figure. Our summary
variable was calculated for each participant by averaging the area of all completed figures within a unit.
The mean relevant error assesses the degree to which the defining features of a completed figure
deviated from ideal. This statistic reflects the same computation used within the game to validate the
accuracy of defining features. In the first unit, focusing on parallel sides, we calculated the slope of
relevant sides as percent gradients. We then calculated the absolute difference between pairs of values
corresponding to opposite sides (0%: parallel – 100%: perpendicular). In the case of parallelograms,
where both pairs of opposite sides are parallel, we averaged this measure across both pairs of sides. In
the second unit, focusing on congruent adjacent sides, we calculated the difference between the length
of relevant sides (0 – 840 pixels [the screen width]). We computed this value for two pairs of sides
(kites) or all four pairs of sides (rhombi), and then averaged. Finally, in the third unit – focusing on right
angles – we calculated the absolute difference of all four angles from 90◦, and then averaged. Finally,
the summary mean was calculated for each participant by averaging the relevant error of all completed
figures within a unit. These statistics are not comparable between units.
Participants in the EI condition were explicitly constrained to keep each of these error values small
through the feature validation process. Therefore, we expect the mean relevant error of EI condition
participants to be small, necessarily. However, it may be the case that participants in the NI condition
spontaneously produce figures that adhered to the target’s defining features.
Table 2
Game summary statistics
Unit
Condition
Exercise
duration
(mins.)
Mean SD
Transform
count
Area
Mean
SD
Mean
SD
Mean
SD
2
(pixels /1000)
Relevant
error a
(1) Parallel
sides
treatment
11.4
2.5
22.0
7.2
122
6.3
0.88
0.22
control
6.7
2.3
11.2
5.7
128
7.0
8.8
2.9
(2) Congruent
adjacent sides
treatment
12.2
3.2
26.6
6.4
132
5.0
10.2
1.8
control
5.8
2.1
8.5
6.0
134
6.7
57.0
30
(3) Right
angles
treatment
10.7
5.5
26.3
9.9
128
3.0
1.1
0.34
control
3.6
1.2
6.4
1.3
128
3.5
3.0
0.60
a. Relevant error for each unit: (1) abs. difference in % gradients; (2) abs. difference in pixel
length; (3) abs. difference from 90◦
As shown in table 2, there are clear differences between conditions. Across all units, two-tailed ttests reveal that EI condition participants, on average, took longer to complete exercises (Unit 1: t[18] =
4.5, p < .001; Unit 2: t[15] = 4.6, p < .001, Unit 3: t[18] = 4.2, p < .001), and performed more
transformation operations than NI condition participants (Unit 1: t[18] = 3.7, p < .01; Unit 2: t[15] = 5.8,
p < .001, Unit 3: t[18] = 6.3, p < .001). As expected, figures produced in the EI condition had less error
than the figures produced in the NI condition (Unit 1: t[18] = -8.6, p < .001; Unit 2: t[15] = -5.0, p < .001,
Unit 3: t[18] = -8.3, p < .001). The extent of this difference is also noteworthy – figures produced in the
NI condition contained approximately ten, five, and three times as much error as treatment figures for
parallel lines, congruent sides, and right angles, respectively.
On the other hand, the differences in area between conditions were less prominent, suggesting a
roughly similar adherence to the structure of the internal obstacles. Only in the first unit was there a
small significant difference between conditions (t[18] = -2.1, p < .05). On the second and third unit the
areas of the completed figures were roughly equivalent (Unit 2: t[15] = -.85, n.s., Unit 3: t[18] = -.35, n.s).
This change is likely due to a greater use of external obstacles in Unit 2 and Unit 3 layouts, limiting the
size and variability of resulting figures.
Discussion
The results of the shape identification task demonstrate that children in the embodied integration
condition were more likely than children in the no integration condition to select polygons based on
defining features, rather than holistic visual similarity. Given this result we can conclude that the
inclusion
of
the
property
validation
phase
produced
some
change
in
the
children’s
perception/conception of polygons. Furthermore, given that the isosceles triangle/trapezoid and right
triangle/trapezoid subtests asked the children to apply their concept of congruent sides and right angles,
respectively, to new figures, this result extends beyond stimuli experienced during the learning task.
Furthermore, the embodied integration condition’s advantage was immediate, and did not accelerate
due to learning within the test. This suggests that the learning game promoted the development of
concepts that transferred intact to the assessment task. In the single case of an interaction post-trial
feedback appeared to attenuate group differences.
However, due to the differences in the in-game data we hesitate to make the stronger conclusion in
definitive support of our third and fourth cognitive design principles, regarding the embodiment and
integration of higher-level concepts. First and foremost, as the results displayed in Table 2 indicate,
embodied integration participants applied considerably more time and effort to successfully completing
each exercise.
Taking all learning units into account, on average, participants in the embodied
integration condition spent twice as much time and performed twice as many transform operations as
no integration participants.
Additionally, we observed that children in both conditions utilized gestural strategies to produce
roughly accurate initial figures, rapidly. In particular, during the planning phase many children would
place their fingers directly on the screen at imagined vertices. This technique, presumably, reduced the
need to maintain a mental image in working memory to guide the initial construction of the figure.
Often, in the case of no integration participants, this initial construction was sufficiently accurate to
successfully navigate the obstacle course. In cases where success was not immediate, figures were
often constructed in a series of tests and minor revisions, focusing on a single vertex at a time. This
strategy of local refinement likely made the attention to the shape, as a whole, unnecessary.
On the other hand, in the embodied integration condition the rough placement of fingers typically
was not sufficient to produce successful initial constructions. More so, in the case where further
refinements were required, participants could not focus on one vertex or side exclusively – as all
defining features measure reciprocal relationships between shape components – and therefore were
unable to apply the same process of iterative local refinements to produce a successful figure as no
integration participants. Rather, these participants needed to attend to multiple components of the
figure, while simultaneously coordinating with a memory representation of the obstacle layout. This
additional layer of challenge may have produced the type of desirable difficulty needed to elicit stronger
mental representations (Christina & Bjork, 1991).
Yet, even if participants in the no integration condition successfully encoded their figures into
memory, large error values suggest that these figures were often poor examples of the target polygon.
Figure 7, shown below, provides six examples of trapezoids constructed by participants from both
conditions. These constructions were chosen to demonstrate how figures produced in either condition
may look holistically similar, while differing in visually subtle, but important ways. Specifically, in each
case of a completed trapezoid from the no integration condition both pairs of sides deviate from parallel
beyond the threshold required by the embodied integration condition.
Embodied
integration
condition
No
integration
condition
Figure 7. Selected completed figures for unit 1, exercise 6. The top row displays three figures
constructed by participants in the treatment condition. The bottom row displays similar figures
constructed by participants in the control condition.
For the no integration condition, successful completion of exercises with polygons containing deviant
defining features may have reinforced the association between specific shapes and holistic
characteristics. This difference between conditions undermines an assumption of this experiment that
the exposure to a diverse set of valid polygons (i.e., Principle 2) would be held constant between
conditions. Ironically, children’s rote, non-reflective process of shape construction may have mitigated
further entrenchment of misconceptions.
While, this experiment demonstrates that children, left to a minimally constrained learning
environment, are unlikely to engage in effective learning practices, we cannot make a stronger case for
our specific principles supporting the design of the experimental EI condition. Rather, it may be the case
that by producing or viewing a greater volume of valid polygons, regardless of the instructional format,
children learn formal concepts more efficiently. In experiment 2 we seek to eliminate these differences
between conditions to test our mechanisms of embodiment and integration directly.
Experiment 2
Given the potential confounds discussed in experiment 1, we conducted a second experiment with a
strengthened control group.
Specifically, children in this new control group were constrained to
produce valid polygons, with the same precision as the treatment group, by using a numerical
mechanism for feature validation – i.e., numerical integration (NuI) condition. This additional feature
ensured that the participants in the numerical integration condition engaged in a similarly rigorous
exercise as participants in the embodied integration (EI) condition. While the results of experiment 1
allow us to conclude, tentatively, that the validation constraints produced superior performance at
posttest, experiment 2 is aimed at exploring whether these gains are specifically related to the
embodied nature of the task.
Method
Participants
A class of sixteen third grade and three fourth grade students were recruited from the same afterschool program as experiment 1.
The children were randomly assigned to either the embodied
integration or numerical integration condition. The treatment condition consisted of nine children (M =
9.3 years, SD = .76, 67% female, 100% Hispanic). The numerical integration condition consisted of ten
children (M = 9.3 years, SD = .30, 50% female, 100% Hispanic). Two children in the embodied integration
condition were native Spanish speakers, but could communicate sufficiently in English and showed little
difficulty understanding and completing the tasks. Additionally, one child, originally assigned to the
embodied integration condition (but not reflected in the statistics above), chose not to engage in
assessment materials following the first unit and was not included in further elements of the study.
Materials and Procedure
Game and curriculum. As discussed above, the strategic use of fingers for initial constructions was
effective for game purposes, but in some cases may have reduced the need to engage in mental imagery
and, consequently, the internalize concepts. Furthermore, some of the younger children in experiment
1 simply found the process of reconstructing the figure from memory overwhelming. Therefore, given
the younger sample of children in experiment 2, we assisted in initial figure construction by providing a
printed sheet of “thumbnail” images, corresponding to each exercise in the unit. With this guide
participants had continuous access to the layout of the obstacle course. Yet, by providing a (50%)
scaled-down image, participants would need to engage in some level of mental imagery to project the
illustrations to the screen. The software elements of the planning, construction, and testing phases
remained the same as in experiment 1.
In the embodied integration version of the game the validation procedure remained the same as in
experiment 1. However, in the numerical integration version of the game we introduced an additional
panel of numerical values situated below the construction space (Figure 8). This panel displays a series
of rectangular blocks with statistical measure(s) associated with each component of the constructed
figure, including sides (length, percent gradient) and vertices (degree measure of internal angle).
(a)
(b)
(c)
Figure 8. Screenshot of control version of game used in experiment 2. (a) Screenshot during
construction phase. (b) Screenshot during parallel sides validation. (c) Zoomed-in image of numerical
representation. Only “% grade” is (fully) visible during a parallel check. Likewise, only length and degree
are (fully) visible during side congruency and right angle checks, respectively. Note: 1.0 length is
equivalent to 8 grid boxes, or 23 pixels.
As a participant constructs or adjusts a figure component the corresponding numerical values reflect
this change in real-time. To validate defining features participants clicked on the boxes associated with
the target feature. For example, to validate parallel sides in the parallelogram of Figure 9 a participant
could select either the two blocks with “0% grade” (i.e., horizontal) or the two blocks with “-64% grade”
and “-65% grade.” The precision thresholds for both conditions were identical (i.e.,  3% gradient,  0.1
grid length [or 23 pixels ≈ 7 mm], and  4◦).
Shape identification. The main structure of the subtests was retained from experiment 1 – i.e.,
trapezoid, parallelogram, rhombus, isosceles triangle/trapezoid, rectangle, and right triangle/trapezoid.
Given results from experiment 1 that showed no significant advantage of post-trial feedback for either
condition, we opted to remove all post-trial feedback from these posttests. The instructions, including
text and (in some cases) figures with selectable sides or angles, remained the same from experiment 1.
While the process of constructing trials remained the same from experiment 1 some polygon stimuli
were slightly altered to maximize screen space and increase the variation between valid polygons.
These revised stimuli are provided in the appendices.
Other assessments. Like experiment 1 the children were assessed with the Woodcock Johnson III
Calculation and Math Fluency subtests, tested in random order over the course of the last three weeks
of the study.
Results
Woodcock Johnson. Standardized scores with Woodcock Johnson III age-normed subtests, reveal
that both groups were approximately average in Calculation performance (Treatment SS: M=96.1, SD =
19.8; Control SS: M=102.0, SD = 16.2) and average in Math Fluency, although bordering on low average
for the treatment group (Treatment SS: M=90.0, SD = 16.6; Control SS: M=96.0, SD =12.7). No significant
difference was found between the two conditions on either Calculation (t[17] < 1) or Math Fluency
(t[17] < 1).
Shape identification. Like experiment 1 we considered the percentage of trials in which participants
selected both shapes correctly to be the best indicator of accuracy. Additionally, like experiment 1, we
did not assume that these scores would be distributed normally. As such, we performed strictly nonparametric, one-tailed, Mann-Whitney tests to analyze participant accuracy. Also, like experiment 1,
we compared conditions for mean trial duration using two-tailed t-tests.
Table 3
Posttest summary measures
Subtest
Condition
Trapezoid
EI
Percent of trials correct
Median
37%
Min
13%
Max
90%
Trial duration (secs.)
Mean
11.6
SD
3.8
Parallelogram
Rhombus
Isosceles
tri/trap
Rectangle
Right
tri/trap
Overall
NI
20%
3%
47%
9.5
1.4
EI
27%
3%
83%
11.2
2.8
NI
10%
0%
90%
6.8
1.0
EI
NI
56%
13%
23%
0%
97%
87%
9.9
7.8
2.6
1.5
EI
39%
3%
53%
13.2
2.8
NI
7%
0%
40%
6.9
1.4
EI
NI
63%
3%
90%
8.9
1.6
5%
0%
77%
6.1
1.3
EI
23%
0%
30%
8.9
3.5
NI
8%
0%
33%
6.4
1.7
EI
NI
37%
19%
67%
10.6
1.4
11%
4%
40%
7.3
0.9
As table 3 displays, across all subtests, the median percent of trials correct is higher for the EI
condition than the NuI condition. In an overall comparison of conditions, pooling all subtests items, a
one-tailed Mann-Whitney test reveals a significant positive shift in the distribution of EI participants’
accuracy scores, compared to NuI participants’ accuracy scores (W = 76, p < .01).
Additionally,
comparisons with five of six subtests reveal similar advantages for EI condition participants (Trapezoid:
W = 68.5, p < .05; Parallelogram: W = 69.5, p < .05; Rhombus: W = 74, p < .01; Isosceles: W = 71.5, p <
.05; Rectangle: W = 73, p < .05). Only in the case of right triangles/trapezoids was there no significant
difference (W = 57, n.s.), likely reflecting the low overall scores for participants in both conditions for
this subtest (median accuracy scores at- or below-chance).
Unlike experiment 1 there were several large differences in durations between conditions. Overall,
pooling all subtest trials, two-tailed t-tests reveal that EI condition mean durations were greater than
NuI condition mean durations (t[17] = 6.2, p < .001). Additionally, comparisons within individual
subtests reveal significantly greater durations for EI condition participants in four cases (Parallelogram:
t[17] = 4.6, p < .001; Rhombus: t[17] = 2.1, p < .05; Isosceles: t[17] = 6.3, p < .001; Rectangle: t[17] =
4.2, p < .001). In the case of right triangles/trapezoids the EI condition participants showed a trend
towards longer trial duration (t[17] = 2.0, p < .1) than NuI condition participants. Finally, in the case of
trapezoids there was no significant difference between conditions (t[17] = .03, n.s.).
Game analysis. As stated in the discussion of experiment 1, large differences in learning game
performance confounded interpretation of posttest results. In experiment 2, the numerical
representation was introduced to offer a comparable alternative method to gestural depictions.
Table 4
Game summary statistics
Unit
Condition
Exercise
duration
(mins.)
Mean SD
Transform
count
Area
Mean
SD
Mean
SD
Mean
SD
(pixels2/1000)
Relevant
error a
(1) Parallel
sides
treatment
19.9
3.9
36.2
9.3
124
4.4
1.0
0.16
control
17.3
3.2
29.7
12.3
126
6.7
0.86
0.2
(2) Congruent
adjacent sides
treatment
13.9
3.9
25.1
11.8
128
2.6
12.6
8.4
control
10.2
3.2
15.2
7.9
131
6.7
7.0
2.8
(3) Right
angles
treatment
14.6
4.1
27.5
11.6
126
5.9
1.5
0.39
control
10.2
3.0
11.4
4.3
125
6.1
1.0
0.16
a. Relevant error for each unit: (1) abs. difference in % gradients; (2) abs. difference in pixel
length; (3) abs. difference from 90◦
As shown in table 2, there were some differences between conditions. Two-tailed t-tests reveal that
EI condition participants, on average, took longer to complete exercises in Unit 2 and Unit 3 (Unit 2:
t[17] = 2.3, p < .05, Unit 3: t[16] = 2.3, p < .05), but not in Unit 1 (t[17] = 1.6, n.s.). Similarly, EI condition
participants showed a trend towards performing significantly more transformation operations than NuI
condition in Unit 2 (t[17] = 2.0); performed significantly more transformation operations in Unit 3 (t[16]
= 3.9, p < .01); and performed nearly an equal number of transformation operations in Unit 1 (t[17] =
1.3, n.s.). Like experiment 1, the differences in the areas of completed figures were small; and in this
case revealed no significant differences (Unit 1: t[17] = -.86, n.s.; Unit 2: t[17] = -1.4, n.s.; Unit 3: t[16] =
.42, n.s.).
Most strikingly, unlike experiment 1 the differences between conditions in relevant error were small.
Only in Unit 2 did NuI condition participants showed a trend towards less error than EI condition
participants (t[17] = 2.0, p < .1). In the other two units there were no significant differences between
conditions (Unit 1: t[17] = 1.5, n.s.; Unit 3: t[16] = 1.5, n.s.).
Discussion
Like Experiment 1, experiment 2 showed that participants in the embodied integration condition
were more likely to identify polygons accurately than participants in the numerical integration condition.
Specifically, in one-tailed tests of group differences, embodied integration participants outperformed
numerical integration in five out six subtests. Clearly, the subtest in which both conditions performed
worst, i.e., right triangles/trapezoids, was simply too challenging for these children. Given that this
experiment was conducted on a set of end-of-year third graders, instead of beginning-to-mid-year
fourth graders, we are not surprised to discover decreased accuracy, overall. The subtle differences
between valid-invalid pairs proved taxing at times for children in both conditions. For future studies,
when addressing similarly young children, we plan to make discrepancies in defining features more
pronounced.
In addition to differences between conditions in posttest accuracy, large differences in posttest
duration emerged in experiment 2, which were not present in experiment 1. The faster response times
of participants in the numerical integration condition, in combination with their at- or below-chance
(25%) accuracy in all subtests, suggests that these children were responding quickly to figures with high
visual similarity. This echo’s Behrman’s and Brown (1968) findings, discussed earlier, that individuals are
likely to classify shapes based on salient, yet informal characteristics, such as elongation. On the other
hand, seeking differences in the defining features provided a greater perceptual challenge, and required
greater time-on-trial.
While these results favor the embodied integration manipulation, recall that in experiment 1 large
differences between conditions in the learning task tempered interpretation in terms of our cognitive
design principles. In particular, large errors in the completed figures of numerical integration
participants may have reinforced inappropriate association between polygons and characteristic
features. In contrast, in experiment 2, the game data shows a much smaller difference in the errors
between conditions, supporting the efficacy of the manipulation. In the single significant case the
numerical integration participants, on average, constructed more accurate figures than embodied
integration participants, despite their reduced posttest scores. This suggests that the attended
information (i.e., numerical representations of geometric properties) facilitated the construction of
more accurate shapes but did not seem to change their concept of the shape, as measured by the
identification task.
However, some differences in time (in unit 2 and 3) and number of constructions (in unit 2,
marginally, and 3) did emerge between conditions. Yet, as noted in experiment 1, embodied integration
participants’ exercise durations were on average, more than double, the control participants, while in
experiment 2 these differences were less than 50%. To some extent this difference in time-on-task may
reflect inherent and unavoidable differences between spatial and numerical processes. For example,
while reducing a 4% difference in gradients to a 3% difference, to meet margin of error constraints, is
perceptually challenging in the embodied task, the corresponding task is trivial with numerical values.
In many cases children in the numerical integration condition discovered numerical margins of error
explicitly, and were able to apply this knowledge while adjusting their figures. This likely reduced the
number of transform operations they needed to perform. For example, in the third learning unit –
which showed a large difference between conditions in number of transforms – participants in the
numerical integration condition could adjust a single vertex while attending to the continuouslyupdating numerical representation of an angle until it became 90°, exactly. On the other hand,
participants in the embodied integration condition received no such immediate assurances that their
angle was accurate. Rather these participants would need to commit to a particular angle, attempt to
validate it, and then apply another transformation if incorrect. Therefore, we suggest that these
differences likely reflect increased procedural encumbrance in the embodied integration condition
rather than any real difference in conceptual rigor.
General discussion
Results from experiment 1 and experiment 2 demonstrate how a computer game may promote
learning of geometric concepts. However, the goal of this study was not to simply demonstrate the
efficacy of our particular learning tool, but to study general principles of learning as applied to
geometry. The computer game simply represents one possible instantiation of a set of four cognitive
design principles. Specifically, the game (1) utilized an intuitive mechanism for initial construction, (2)
exposed children to a diverse set of valid shapes, (3) promoted embodied representations of higherlevel concepts, and (4) incorporated a challenging activity to promote integration of multiple embodied
representations.
The experiments described here assumed the validity of the first two principles (based upon prior
research) and tested the latter two principles. Our choice to test these two principles together, rather
than separately, reflects our opinion that both principles play a necessary, reciprocal role in developing
children’s knowledge. Without an integrating mechanism, even strong embodied representations are
simply disconnected – and easily forgotten – pieces of knowledge. Conversely, without embodied
representations, higher-level concepts lose their connection to intuitive representations. While these
speculations will need to be tested directly, we suggest that the results of this study, and potential
future studies, may contribute to literature regarding both (1) the role of embodiment in learning and
(2) conceptual development in geometry.
Embodiment in learning. While the term “embodied” seems to suggest that physical movement –
perhaps even whole-body movement – should be the primary source for learning, we take a broader
view of the term – perhaps best encapsulated by Barsalou "grounded" (2008) approach – in which the
notion of embodiment or grounding serves as a counterbalance to classical views of cognition that
portray abstract thinking as qualitatively dissimilar to processes in the perception and action systems.
Rather, higher-level thinking emerges from a wealth of experience with systems that interface with the
physical environment. From this perspective, embodied knowledge may be strictly perceptual, strictly
motor, or some combination of the two.
With this in mind, the focus of our embodied integration condition was to facilitate any number of
embodied representations of target concepts, which could vary across this perception-action spectrum.
While the gestural depictions were completely visual, by the end of each learning unit all children had
either spontaneously discovered the corresponding hand gestures or seen another child or
experimenter model those gestures. Yet, the frequency of and purpose behind the use of gestures
varied among children. In some cases, gestures appeared to serve as an initial reminder of the meaning
of some geometric concept, but faded as children became increasingly fluent. For some children
gestures were adopted as a tool for accurate polygon construction – i.e., by matching their hands with
components of the constructed figures. Similarly, some children, recognizing the imprecision of these
gestures, applied physical artifacts from the classroom – such as a straight edge or the corner of a sheet
of paper (for right angles) – to increase their precision in shape construction. Finally, in some cases
children showed little use of external tools or gestures.
Given the breadth of these behaviors, several questions arise regarding the proper course of
learning. First, is there a single, optimal strategy for any particular concept? Should children be taught
and expected to use appropriate gestures or are these gestures simply a poor substitute for more
precise tools? On the other hand, does the use of precise tools, and to a lesser extent gestures,
interfere with the construction of robust internal representations? In other words, will a child who
learned to measure right angles with a protractor be unable to recognize this feature in the absence of
the tool?
Although we did not track gestures explicitly, we did observe some general patterns in their
application. We noticed that children with stronger initial concepts of parallelism, congruency, and
perpendicularity more readily demonstrated corresponding gestures – at least in the beginning of a
learning unit. For struggling students, a demonstration of corresponding hand gesture, by peer or
experimenter, was often required – and many cases, multiple times to the same child. Taken together,
these observations support Beilock and Goldin-Meadow's (2010) notion of a reciprocal relationship
between conceptual thought and gesture.
In several cases, children who successfully learned to apply gesture as a tool in the learning game
applied this strategy during posttest. Yet, often these students found that the smaller size of the
posttest stimuli, relative to figures constructed during the learning task, made the accurate use of
gestures for measurement prohibitive. As such, children often resorted to a largely perceptual strategy,
with varying degrees of success.
To assess the necessity and utility of gestures directly a possible avenue for future research may lie in
emerging gesture-based technologies, such as multi-touch screens and motion-capture systems. With
these systems our research may shift from merely suggesting a gesture to enforcing the use of a specific
gesture. For example, with a motion capture system a learner might align his or her arms with opposite
sides of a figure to validate parallel lines. Similarly, with a muli-touch screen, a user might simply run
two fingers, in parallel, along the length of opposite sides.
In either case a specific gesture, requiring some form of direct instruction, becomes a necessary
element of the task. Consequently, does the invention of a gestural or non-gestural strategy – even a
sub-optimal one – facilitate conceptualization better than direct instruction? In some cases, engaging in
“productive failure” – where discovery and invention are attempted unsuccessfully – may benefit
students more than being taught a procedure directly (Kapur, 2008). By teaching students to use a
single gesture or tool, do we deprive them of an important opportunity to critically examine their own
ideas? For example, a student might benefit from constructing his or her own gesture or visual
depiction for parallel lines.
While the spontaneous strategy constructed by a learner is likely to be physical in nature, there is,
perhaps an important distinction between those strategies that internalize knowledge into embodied
representations and those that rely upon external tool use. As Martin and Schwartz (2005) discovered,
features of the environment may support performance during learning at the cost of long term
retention and transfer. Likewise, in our work here, several students, who successfully relied upon a
physical tool to produce right angles (the corner of a sheet of paper), did not succeed when their tool
was not available during testing. In these cases, failure may not reflect poor performance during the
learning task, but simply an arbitrary choice of strategy. Therefore, while the use of tools, gesture, or
fully internalized (e.g. perceptual) strategies are all valid, practical considerations dictate that learning
tools support strategies and representations which are most portable to novel activities – including
standardized assessments.
Development of mature, integrated concepts in geometry. Unless we adopt a highly amodal,
abstract interpretation of mental imagery and representation (e.g. Pylyshyn, 1973), then we must
assume that geometric concepts are at least partially built upon perceptual primitives. Yet, are these
imagistic primitives like a set of exemplar snapshots, or perhaps something more schematic in nature,
such as “perceptual symbols” (Barsalou, 1999)? In the case of number, representation appears to be
spatial in nature and independent of any particular modal system (Dehaene, 1997). Likewise, it may be
the case that mature geometric representations abstract to this same spatial level of representation –
facilitating both perception and action. For example, a child who can easily recognize a rectangle
visually likely can also draw a rectangular box or walk a rectangular path.
In the process of developing this conceptual-spatial representation exposure to a narrow range of
stimuli can lead to misconceptions. For example, the child (or adult) viewing an obliquely oriented
square may have difficulty seeing the shape as anything but a rhombus. Clearly, learning to identify
defining features is an important remedy for this misconception. Yet, does the process of learning
features fundamentally change the way one perceives a shape or is it simply a supplementary strategy,
independent of the initial recognition process? In the former case, the child learns to see the square in
terms of right angles and has little difficulty identifying a “rotated square” correctly as a square. In the
latter case, the child still sees the “rotated square” only as a rhombus (or “diamond”), but after a highly
controlled search for right angles concludes that the shape must also be a square, contrary to his or her
own intuition. In other words, how closely integrated are higher-level concepts with intuitive concepts?
We suspect that when children are first introduced to defining features, the process of recognizing a
shape by appearance and recognizing defining features are initially, loosely coupled processes, whose
association needs continued external reinforcement. However, given sufficient experience, the
representation guiding shape recognition deeply integrates these new features. So, while a child may
have to be initially reminded, repeatedly, to search for right angles to identify rectangles, eventually he
or she will simply see right angles as salient features in rectangular figures. In this process, gesture
might serve as both a means to conceptualize but as a source of informal feedback for instructors.
As a child’s concept of a shape increasingly shifts to reflect defining features we might expect to see
less explicit use of gesture. In a simple example with counting and arithmetic, children often use fingers
as a tool to support cognitive processes; yet, as they grow more competent they begin to internalize
these processes (Siegler & Robinson, 1982). In the case of a 4th grade, Logo-based geometry task,
Clements and Burns (2000) observed spontaneous gestures corresponding to angles and turns, which
the children later “curtailed” – i.e., physically scaled down –with increased expertise.
Of course gesture is only one indicator of an explicit cognitive process of coordinating multiple
geometry concepts. In general, as this explicit, deliberative process – in whatever form it takes –
becomes unnecessary, the duration of the corresponding tasks should decrease. While we cannot
compare participants in our two experiments directly, due to some differences in stimuli, we note that
for younger children (experiment 2) – who found the task more difficult, overall – trial durations were
significantly greater in the treatment condition than control condition (for 4 out of 6 subtests). For the
older children (experiment 1) this was not the case – and in fact the reverse was true in one subtest.
This suggests as a general U-shaped pattern of development in which children with a novice
understanding of shape (i.e, children in the control condition of both experiments) quickly respond
according to their intuitive, everyday-object-based concept of shape. When first beginning to learn
defining feature concepts, the process of shape identification requires multiple deliberative steps that
result in longer trial durations (EI condition in the second experiment). Finally, when children begins to
master these new concepts and integrate them into their central represenation for a shape, the
identification process becomes more rapid (EI condition in the first experiment). It is likely that the 4th
grade children had more prior experience with concepts of parallel lines, congruent sides, and right
angles then the 3rd grade children, and therefore integrated these concepts more efficiently in
experiment 1. To test this hypothesis directly, in future research, we may introduce more iterations of
the identification task in a microgenetic design.
To be clear, we do not claim that greater expertise will curtail all gesture, but simply reduce
children’s application of gesture as an explicit tool to “lighten the cognitive load” (Goldin-Meadow,
Nusbaum, Kelly, & Wagner, 2001). Gesture is likely to remain as a general product of thinking about
spatial concepts. Yet, if a natural reduction of gesture – in either abundance, necessity, or scale – is
indeed a hallmark of conceptual enrichment, should it be promoted explicitly, or allowed to emerge
according to the individual child? In the case of numerical development, typically developing children
tend to shift from finger-based strategies to verbal or mental strategies as they progress through early
primary school. In fact, delay of this shift is frequently associated with mathematical disability (Geary,
Hamson, & Hoard, 2000; Geary, Hoard, & Hamson, 1999). However, the transition to verbal and mental
strategies is often achieved through a auto-regulative process of strategic variation in which the more
efficient strategies are eventually adopted (Siegler, 1994). Attempting to enforce such curtailment with
geometric concepts may contribute to misunderstandings.
On the other hand, the gradual removal of external supports (scaffolds) is a well-reasoned form of
instruction (Vygotsky, 1978). In the case of teaching abstract concepts Goldstone and Son (2005) apply
an instructional strategy of “concreteness fading”, in which initially high context stimuli are reduced in
complexity over the learning process to promote understanding that is not bound to a specific
environment. Would a similar technique with gesture also be appropriate? Using some of the gesturebased designs, discussed above, we plan to address this issue in future studies.
Curriculum recommendations. This study reinforces the need to promote the development of
robust spatial representations with engaging, productive, and challenging activities. While this may
seem nearly self-evident, geometry instruction is often quite superficial and lacks these practices
(Clements, 2004). For example, children often engage in activities with cut-out or physical “pattern
blocks”. Though these blocks provide a potential grounding for important geometric concepts, they are
often quite limited and prototypical. As we see in experiment 2, simple exposure to valid shapes does
not entail that children are attending to and discovering critical features of the concept.
While technology can facilitate the implementation of research-based curriculum, it is not essential
for classroom instruction. Following our four cognitive design principles, children can easily draw, with
aid of a straightedge, a diverse set of polygons that fit given physical constraints – including the
“obstacle course” design applied here. Furthermore, an instructor may demonstrate hand gestures to
represent defining feature of shapes, and show how to use a precise tool, such as a protractor or ruler,
to validate the accuracy of these defining features. Providing students with the responsibility to validate
their own shapes, or those of their classmates, may even enhance this “analog” activity.
Finally, we hope that this work will encourage others within the cognitive and educational fields to
address the topic of geometry with the rigor and attention afforded to other areas of mathematics, such
as numerical development. Geometry was, of course, an essential element in classical education and
worthy of lengthy study by Piaget (Piaget et al., 1960). Geometry integrates a wide range of cognitive
processes – including numeracy, spatial sense, object recognition, navigation, and language – and
represents an ideal domain to study the development of complex mathematical concepts.
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Appendix A. Unit 1 exercise layouts: trapezoids and parallelograms. Players validate parallel sides.
Appendix B. Unit 2 exercise layouts: kites and rhombi. Players validate congruent adjacent sides.
Appendix C. Unit 3 exercise layouts: rectangles and squares. Players validate right angles and
congruent adjacent sides (for squares, only).
Appendix D. Shape identification, trapezoid stimuli, experiment 2. Top-left (outlined in green) are
valid-upright, top-right (yellow) are valid-rotated, bottom-left (orange) are valid-upright, bottomright are invalid-rotated.
Appendix E. Shape identification, parallelogram stimuli, experiment 2. Top-left (outlined in green)
are valid-upright, top-right (yellow) are valid-rotated, bottom-left (orange) are valid-upright, bottomright are invalid-rotated.
Note: Stimuli pools were constructed by combining each of six pairs to all other pairs to generate 15
combinations of 4 polygons in set “a” and 15 combinations of 4 polygons in set “b”, for a total of 30
trials. No trials included pairs from both sets. The same is true for the all of the following stimuli as
well. In the cases of isosceles or right triangle/trapezoid stimuli set “a” represents all triangles, while
set “b” represents all trapezoids.
Appendix F. Shape identification, rhombus stimuli, experiment 2. Top-left (outlined in green) are
valid-upright, top-right (yellow) are valid-rotated, bottom-left (orange) are valid-upright, bottomright are invalid-rotated.
Appendix G. Shape identification, isosceles triangle/trapezoid stimuli, experiment 2. 1st and 3rd rows
on left (outlined in green) are valid-upright, 1st and 3rd rows on right (yellow) are valid-rotated, 2nd
and 4th rows on left (orange) are valid-upright, 2nd and 4th rows on right are invalid-rotated
Appendix H. Shape identification, rectangle stimuli, experiment 2. Top-left (outlined in green) are
valid-upright, top-right (yellow) are valid-rotated, bottom-left (orange) are valid-upright, bottomright are invalid-rotated.
Appendix I. Shape identification, right triangle/trapezoid stimuli, experiment 2. 1 st and 3rd rows on
left (outlined in green) are valid-upright, 1st and 3rd rows on right (yellow) are valid-rotated, 2nd and
4th rows on left (orange) are valid-upright, 2nd and 4th rows on right are invalid-rotated