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Abstract
Many researchers and teachers advocate the use of
manipulatives, hands-on objects like base-10 blocks, to
support students’ mathematics learning. However, the
evidence supporting this practice is mixed. Results from our
prior research suggested a novel explanation for how
manipulatives help children learn. The proposal, “physically
distributed learning” (PDL), is that action with
manipulatives supports learning when it provides a way for
children to simultaneously and iteratively adapt and
interpret their environment.
Research in the area of arithmetic supports PDL, but it has
not been examined in geometry. This paper examines the
possibility that physical action may have different effects
in arithmetic and geometry tasks in the context of two
studies with kindergarten, first- and second-grade students.
In the arithmetic study, students solved addition word
problems using manipulatives and drawings. In the geometry
study, students completed a shape identification task with
manipulatives and drawings.
On addition tasks, children were more successful with
manipulatives than with pictures, replicating previous
arithmetic studies. On geometry tasks, children also
benefited from manipulation, but in different ways. Using
manipulatives and pictures led to similar overall
performance. However, all children were more likely to
rotate their paper or physical shape when this action could
help them identify the shape. In addition, when asked to
make non-triangles into triangles, children using
manipulatives (pipe cleaners) reshaped non-triangles, while
those who worked with pictures drew a prototypical triangle
(roughly equilateral) on top of these shapes. These results
suggest that while the spatial nature of geometry tasks may
change the effects of manipulatives on problem solving,
manipulation still helps children expand their
investigations in the physical environment and thereby
advance their thinking.
Introduction
Elementary classrooms employ manipulatives extensively. Many
researchers and teachers believe these hands-on objects can
help students learn mathematics concepts (NCTM, 2000; Van de
Walle, 2007). For example, teachers often use Cuisenaire
rods to teach base-10 concepts or pie-shaped pieces to teach
fraction concepts (See Figure 1).
Figure 1. Manipulatives. A. Teachers frequently use fraction
pies to teach fraction operations and meanings. B. They use
base-10 blocks to teach many whole number operations and
concepts.
However, research evidence addressing whether external
objects benefit mathematical problem solving and learning is
mixed (Ball, 1992; Chao, Stinger, & Woodward, 2000;
Thompson, 1994).
On one hand, working with manipulatives improves performance
on mathematical tasks in some cases (Chao et al., 2000). For
example, adults calculate the value of a group of coins more
quickly and accurately when they sort them (Kirsh, 1995).
Manipulating objects also helps students learn many
mathematical concepts (Behr, Wachsmuth, & Post, 1988; Fuson
& Briars, 1990). Manipulatives have been shown to support
learning in both arithmetic and geometry contexts (Glenberg,
Jaworski, Rischal, & Levin, 2006; Jaworski, 2003; Olkun,
2003; Reimer & Moyer, 2005; Sarama & Clements, 2004). For
example, in arithmetic, base-10 blocks helped second-graders
learn multi-digit addition (Fuson & Briars, 1990). In
geometry, a curriculum involving manipulating dynamic
computer-based graphs improved fourth-graders’ concepts of
two-dimensional space (Sarama, Clements, Swaminathan,
McMillen, & Gonzalez Gomez, 2003). In addition, early
elementary students who use manipulatives for extended
periods of time perform better on achievement tests
(including both arithmetic and geometry problems) than those
who do not (Sowell, 1989).
At the same time, other research suggests that concrete
objects like manipulatives do not improve problem solving
and learning. First, meta-analyses and reviews of studies
comparing student achievement in mathematics classrooms that
use manipulatives in instruction to those that do not show
no overall advantage for manipulative use (Fennema, 1972;
Sowell, 1989). Second, focused studies with individual
students illustrate some of the pitfalls of working with
concrete objects. In some cases, the more concrete an object
is, the less useful it is for solving problems (Bassok,
1989; Simons & Keil, 1995; Sloutsky, Kaminski, & Heckler,
2005). For example, in a series of studies, DeLoache, Uttal
and colleagues showed that playing with a scale model of a
room (thereby increasing the salience of the model’s
concrete properties) can impair children’s ability to use
that model as a referent to find an object hidden in an
identical full-size room (e.g., DeLoache, 2000; Uttal,
Scudder, & DeLoache, 1997).
Theories Regarding Learning with Manipulatives
These mixed results have led to the development of many
theories concerning how manipulatives help students learn
mathematics, though there is little evidence firmly
supporting any one view (Chao et al., 2000). One idea is
that exposure to multiple representations leads to better
understanding of underlying mathematical principles (Moreno
& Mayer, 1997). This view implies that the best
instructional strategy is to use multiple manipulatives to
teach mathematics concepts. Another hypothesis is that
manipulatives support a concrete to abstract shift in
conceptual understanding. Here the best manipulatives would
be those that provide an analogy for or represent abstract
concepts (Hall, 1998). In other words, useful manipulatives
have structures that mirror the semiotic systems they are
meant to represent, such that each action on a manipulative
corresponds to a semiotic action, one-to-one. Another view
is that external resources primarily help problem solvers
keep track of problem elements without wasting internal
memory resources (Cary & Carlson, 1999).
Physically Distributed Learning
In a series of studies, we investigated children’s problem
solving and learning in arithmetic tasks (fractions and
division) (Martin, Lukong, & Campbell, April, 2006; Martin &
Schwartz, 2005; Schwartz & Martin, 2006). Our results
suggested a novel explanation for how manipulatives help
children learn. We suggest that action with manipulatives
supports learning when it provides a way for children to
simultaneously and iteratively adapt and interpret their
environment. We call this process “physically distributed
learning” (PDL). In the rest of this section, we will
describe this process in more detail.
When children first attempt problems with manipulatives,
they start with some basic ideas and actions that can help
them. For example, for the problem “Sarah has three apples
and Maria has four apples. How many apples do they have
together?” a child might think, “The problem asked about how
many they both have, I need to figure out how many
altogether.” She might also try out some different actions
like moving or grouping pieces.
These actions and ideas can develop each other (or coevolve)
(Martin, 2004). As children work on these problems, they
move the materials in different ways. For example, they
might try making groups of different numbers of pieces for
the problem above. As they think about the problem, they may
decide that certain actions and configurations are more
helpful than others. For this problem, they might be
thinking about combining Sarah’s three apples and Maria’s
four apples. Consequently, they might first make a group of
three or a group of four pieces. Then they may realize that
they can combine a group of three pieces and a group of four
pieces to model the total number of apples the girls have
together.
As they solve multiple problems using manipulatives,
children’s actions and ideas could coevolve so that their
concepts about the broader mathematics topic the problems
address develop into more coherent structures (Martin et
al., April, 2006). For example, they may come to understand
addition and subtraction as related operations that involve
a part-part-whole model whose parts can be combined and
separated to compose and decompose the whole. This type of
structure can support the development of mental problem
solving strategies. In turn, practicing these mental
strategies can lead to memorizing addition and subtraction
facts (Carpenter, Fennema, Franke, Levi, & Empson, 1999).
We have conducted several studies on this theory in the area
of arithmetic (primarily with fraction and division tasks)
(Martin et al., April, 2006; Martin & Schwartz, 2005). This
research supported several aspects of the theory. These
include that students can initially solve problems better
with manipulatives than pictures, that their actions and
ideas coevolve, and that they can develop non-physical
strategies over time as they apply physical strategies to
solve problems.
Investigating The Role Of Manipulatives In Geometry
We have not examined PDL in the area of geometry. There are
several reasons that working with hands-on objects could
have different effects on geometry problem solving than on
arithmetic problem solving.
One reason is geometry tasks often require operations on
different objects and have different problem solving
outcomes than arithmetic tasks. In geometry, the objects
operated on are often visual, spatial, or concrete, whereas
in arithmetic, they are primarily numbers. Many answers for
geometry tasks involve outcomes like identifications or
measurements rather than the numerical answers arithmetic
tasks frequently involve. Manipulatives could be more
helpful in geometry than arithmetic because they are similar
to the objects operated on and resemble the outcomes of
problem solving. On the other hand, this similarity may
interfere with problem solvers’ formation of abstractions
and therefore impede their development of more advanced
concepts in the area (Goldstone & Sakamoto, 2003; Sloutsky
et al., 2005).
Another reason is that children work with spatial
representations in many elementary geometry contexts (e.g.,
shapes or computer-based coordinate grids). In this case,
using manipulatives would not require the same translation
between a verbal or numeric representation and a spatial
representation as in the case of an arithmetic problem. This
difference could make hands-on materials more useful for
geometry than arithmetic, as there is less cognitive load
involved in working with single rather than multiple
representations (Ainsworth, 2006). Conversely, translating
between representations can help children learn,
particularly in the long term (Ainsworth & VanLabeke, 2004;
Moreno & Mayer, 1997). Therefore, children may benefit more
from working with manipulatives on arithmetic tasks.
In terms of the PDL theory, a major difference is that
children’s early ideas and actions may be more similar in
geometry than in arithmetic. In geometry, physical shapes
and children’s concepts of shapes may be nearly identical,
particularly in early stages of learning. For example,
children initially identify shapes by their appearance
(e.g., it’s a circle because it looks like a frisbee)
(Clements, 2003; van Hiele, 1986). In contrast, on addition
problems, children’s ideas could be about parts and wholes
while their actions could involve moving tile pieces around.
Because ideas and actions are more similar in geometry,
coevolution and distributing work to the environment (the
processes of PDL) may operate more easily. Children may
develop their ideas and actions faster if these are more
similar. On the other hand, this very similarity may impede
development. For example, in fraction problem solving,
children transferred better when they learned about adding
fractions with manipulatives that were less like fractions
(tile pieces) than with manipulatives that were more like
fractions (pie pieces) (Martin & Schwartz, 2005). Children
may not have had to construct fraction structures mentally
with the pie pieces because they could leave that
distributed to the environment. Consequently, they learned
less.
The Current Experiments
This paper examines the possibility that physical action may
have different effects in arithmetic and geometry tasks. We
compare the results of an arithmetic experiment to that of a
geometry experiment. The studies involved kindergarten,
first- and second-grade students. In the arithmetic study,
students solved addition word problems using manipulatives
and drawings. In the geometry study, children completed a
shape identification task with manipulatives and drawings.
Our primary research questions were:
1) Do the effects of manipulatives in addition replicate the
effects found in other arithmetic tasks?
2) Are the effects of using manipulatives different in these
arithmetic and geometry tasks?
Experiment 1
In Experiment 1, kindergarteners solved addition word
problems with manipulatives and pictures. We predicted that
children would solve the problems correctly more often with
manipulatives than pictures. This result is consistent with
our prior research on fractions and division with older
students.
Methods
Participants
The participants attended an urban school that serves
children from primarily low- to middle-income families. We
randomly selected 24 students from kindergarten classes
(nine 4-year-olds and fifteen 5-year-olds). Nine of the
students were female and fifteen were male (mixed across age
group).
Materials
Manipulatives. Children solved problems with manipulatives
and pictures (See Figure 2). The manipulatives were
variously colored, 1-inch square chips. The pictures were
filled line drawings of tile pieces, and children had a
pencil to draw on the pictures. With both materials, we
presented children with a set of pieces larger than the
number they needed to solve the problems.
Figure 2. Materials Experiment 1
Problems
Children solved eight Join Result Unknown problems
(Carpenter et al., 1999). All children solved the same eight
problems. The interviewer first presented each problem with
numbers of medium difficulty. Depending on children’s
response, the interviewer gave harder or easier numbers.
Appendix A shows the problems and the number choices.
Procedure
First, the interviewer spent time in the children’s
classrooms (1-1.5 hrs per class) engaging in regular
classroom activities. This activity allowed the interviewer
to become acquainted with the children.
The one-on-one study interviews occurred in a room at the
school outside the regular classroom. The children visited
the room prior to the interview. We videotaped the
interviews. Interviews lasted approximately 25 minutes.
Children played freely with both the manipulatives and the
pictures for approximately two minutes each. Then the
interviewer asked the child to circle (with pictures) or
move (with manipulatives) an individual piece and a group of
pieces. The drawing material was first for all children.
Next, the interviewer asked the children to attempt the
eight problems in order. Half the children started with the
pictures and half with the manipulatives. For each problem,
the interviewer asked, “Can you show me how to figure out
that problem with the drawing/pieces?”
On each problem, the interviewer stated the problem using
the medium difficulty numbers and then paused. If children
initially gave the correct answer without using the pieces,
the interviewer gave the same problem again, but changed the
numbers to harder numbers. If children initially could not
make headway on the problem as evidenced by saying they had
no idea, sitting doing nothing, or taking actions that did
not progress toward a solution (e.g., scribbling all over
the paper with the pictures, or swerving pieces around with
the manipulatives), the interviewer restated the problem
with the easier numbers.
Design and Coding
We coded children’s answers as accurate or not. Each child
then received a score for number of problems correct with
manipulatives and one for number of problems correct with
pictures (each ranged from 0 – 4). We compared these scores
using a within subjects design, as all children completed
problems with both manipulatives and pictures.
Results
There was no effect of age or interaction between age and
material, so we collapsed across age to analyze the accuracy
data.
We compared the number of problems students answered
correctly using manipulatives and pictures using a repeated
measures ANOVA with material (manipulatives, pictures) as
the within subjects factor. Students were correct
significantly more often when they used manipulatives than
pictures, (Manipulatives M = 2.75, SE = .22; Pictures M =
1.25, SE = .25), F(1, 23) = 31.05, MSE = .87, p < .001.
Discussion
In Experiment 1, we found that the effects of manipulatives
in early addition learning were the same as the effects of
manipulatives for fraction and division learning for older
students. When children work with materials that they can
move, they succeed more often than when they work with
static materials.
In Experiment 2, we investigate whether these results from
arithmetic tasks will be replicated in a geometry task. The
geometry task is a shape identification task. We chose shape
identification because we wanted to start investigating
manipulatives’ effects in geometry using a very basic task.
Shape identification is one of the first tasks students
usually work on in geometry (NCTM, 2000).
Experiment 2
In this experiment, we employed the same paradigm in
geometry that we have in several arithmetic studies by
providing children physical and pictorial objects to solve
problems. Kindergarten, first-, and second-grade students
solved two tasks using manipulatives (pipe cleaner shapes)
or pictures (line drawings of the same shapes). One task was
to identify the triangles in a set of shapes (See Appendix
B). The second was to make the non-triangles into triangles.
We expect that using moveable objects will affect children’s
problem solving. This prediction is based on the prior
results with arithmetic tasks. However, these effects may be
different than those found in arithmetic. Children may not
identify shapes correctly more often with manipulatives than
with pictures. The spatial nature of both materials may make
them equally useful for this task.
However, we do expect to find effects of movement in two
situations. One situation involves the identification task.
Children often have a concept of a prototypical triangle
that is a roughly equilateral triangle with a base flat to
the ground and one vertice at the top of the shape located
such that a line dropped from the vertice would bisect the
base (e.g., triangle 5 in Appendix B) (Clements, 2003). Some
of the triangles on the shape identification task sheet
would look more similar to this prototype if they were
rotated (triangles 3, 6, 9, and 10 in Appendix B). We refer
to the orientation of a prototypical triangle as
prototypical orientation. Figure 3 illustrates a rotation
that transforms a triangle from non-prototypical to
prototypical orientation. We predict that children in both
conditions will rotate triangles originally presented in
non-prototypical orientation more often than triangles
originally presented in prototypical orientation.
Figure 3. Triangle 3 from Appendix B as presented in
non-prototypical orientation and after rotation (in
prototypical orientation).
The other situation involves the “make a triangle” task. As
shown in the arithmetic studies, children can adapt
manipulatives more easily than pictures. Therefore, children
tend to experiment with more different ways to adapt the
situation with manipulatives than with pictures. This
difference could increase the chances that children will
make more complex changes to these shapes. Referring back to
the idea of the prototypical triangle, it seems likely that
when adaptation is easier (with manipulatives), children may
be less likely to depend on the prototypical shape and may
adapt shapes in different ways to make them into triangles
(See Figure 4). These adaptations may be based on the
properties of triangles. In contrast, when adaptation is
more difficult (with pictures), children may simply attempt
to make the shapes look like prototypical triangles.
Figure 4. Responses to the “Make a Triangle” Task. A
“superimpose” response involves drawing an equilateral
triangle on top of a shape that is not a triangle. An
“adapt” response involves changing the shape so that it has
the properties of a triangle. For example, in the “adapt”
response shown here, the child straightens the curved sides
of a non-triangle to make it into a triangle. A child could
achieve the same result with the pictures by drawing the two
straight sides shown here by dashed lines.
Methods
Participants
The study occurred during the second half of the fall
semester in an urban school that serves children from
primarily low- to middle-income families. We randomly
selected 26 students from kindergarten (5- and 6-year-olds),
first grade (6- and 7-year-olds) and second grade (7- and
8-year-olds) to participate (K: N = 8; 1st N = 8; 2nd N =
10). We then randomly assigned approximately half the
students from each grade level to the manipulate condition
(N = 11) and half to the pictorial condition (N = 15).
Twelve of the students were female and fourteen were male.
Materials and Procedure
We interviewed children in a one-on-one setting outside
their regular classroom. We recorded children’s responses on
an answer sheet. We did not tape the interviews.
The shape identification task proceeded as follows. This
task is adapted from a common shape identification task
(Burger & Shaughnessey, 1986). First, the interviewer asked
the students to describe what a triangle was and to draw
one. Students had a blank sheet of paper and a pencil to
draw. Next students received a set of shapes, some of which
were triangles and some of which were not (See Appendix B).
The pictures students had a sheet of paper with the shapes
drawn on it (the sheet in Appendix B). The manipulatives
students used shapes made with pipe cleaners. The
interviewer presented these shapes on a blank sheet of
paper. The shapes were grouped and oriented the same way as
on the sheet in Appendix B.
Next the interviewer asked, “Do you see any triangles on
this page?” If the child answered affirmatively, the
interviewer asked the student to mark the shapes that were
triangles. Students circled or otherwise marked each shape
they believed to be a triangle. After students made their
selections, the interviewer asked whether there were more
triangles on the page.
Then the interviewer asked students to justify their
classification decisions. The interviewer recorded the
students’ justifications and whether students transformed
the shape (e.g., by rotating their paper or manipulative so
that the base of the triangle was horizontal).
After students gave their responses, the interviewer
followed up with the “make a triangle” task. She asked
students to change the shapes that were not triangles into
triangles. Students using the worksheet had a pencil to make
changes, and students using manipulatives could change the
pipe cleaner shapes.
Coding
We examined whether students correctly classified each
shape, the justification they gave for their decision,
whether they transformed the shape, and how they changed the
shapes on the “make a triangle” task.
A classification response was accurate if the child
correctly classified a shape as a triangle or a
non-triangle.
We coded students’ justifications as visual, attribute,
both, or other. A visual response described how the shape
looked (e.g., “It's pointy. The bottom is not pointy.”). An
attribute response included properties of the shape (e.g.,
“It has three points and three sides.”). A both response
included both visual and attribute justifications (e.g.,
“It's a pointy thing. It has three sides and three
corners.”). Other responses were rare, but an example is, “I
can't think of the name of it.”
A primary and secondary coder coded a subset (10%) of the
students’ justifications drawn randomly from the interviews.
Inter-rater agreement for the justification coding was 92%.
The primary coder subsequently coded all of the students’
justifications. The students’ justification responses were
blinded as to condition prior to all coding activities.
We coded a response as including a transformation if
students flipped, turned, rotated, or otherwise physically
moved a shape in their attempts to classify it.
Manipulatives students moved the pipe cleaner shapes.
Pictures students moved the worksheet.
We distinguished four responses to the “make a triangle”
task: incorrect, superimpose, adapt and no response. A
response was incorrect if the student did not successfully
change the non-triangle into a triangle. We coded a response
as “superimpose” if a student drew or made and placed a
roughly equilateral triangle on top of the shape. We coded a
response as “adapt” if a student changed the shape to make
it into a triangle based on the properties of triangles. For
example, with manipulatives, a child might fold in one leg
of a two-sided figure to make a three-sided figure. With
pictures, a child might draw a third side to achieve the
same result. No response meant the child said he did not
know or could not answer.
A primary and secondary coder coded a subset (10%) of the
students’ responses drawn randomly from the interviews.
Inter-rater agreement for the changes coding was 91%. The
primary coder subsequently coded all of the students’
responses. The students’ change responses were blinded as to
condition prior to all coding activities.
Results
Children in the two conditions correctly classified the
shapes at similar rates and gave similar justifications for
their decisions. However, children transformed shapes when
this action was helpful to making classification decisions.
In addition, children changed non-triangles into triangles
in more advanced ways with manipulatives than with pictures.
Accuracy
Children in both conditions correctly identified a similar
proportion of the eleven shapes (manipulate condition: M: =
.60, SE = .03; pictorial condition: M: = .56, SE = .06),
F(1, 24) = .36, MSE = .00, p = .56.
Justifications
We conducted a 3 x 2 repeated measures ANOVA on the number
of each type of justification each student gave. The within
subjects factor was type of justification (attribute,
visual, or both) and the between subjects factor was
material (manipulatives, pictures).
Children in both groups justified their decisions primarily
based on the appearance of the shapes, F(2, 48) = 23.80, MSE
= 5.98, p < .001 (See Table 1). When they gave an attribute
justification, they usually combined it with a visual
justification. There was no effect of material or
interaction between material and justification type.
Table 1. Mean Number of Each Type of Justification by
Condition
Effects of Movement
Effects of orientation. Children moved triangles that were
not in prototypical orientation (triangles 3, 6, 9, and 10)
more often than those that were (triangles 1 and 5), F(1,24)
= 25.76, MSE = .00, p < .001 (See Table 2). The pictorial
children moved the paper, and the manipulative children
moved the shape.
Children in both conditions moved these two types of shapes
a similar amount (See Table 2). There was no main effect of
material or interaction between type of triangle and
condition.
Table 2. Mean Percentage of Shapes in Each Orientation
Children Moved by Condition
“Make a triangle” task results. The second difference was
that when asked to make the non-triangles into triangles,
children in the manipulate condition usually adapted the
shape. In contrast, children in the pictorial condition were
more likely to draw a prototypical triangle on top of the
shape than those in the manipulative condition (See Figure
5). We conducted a chi-square analysis on the four possible
types of responses (incorrect, superimpose, adapt and no
response) by condition (manipulate, pictorial). The
difference between the conditions was significant, c2 (3) =
13.66, p < .01.
Figure 5. After children were told which shapes were not
triangles, they were asked to change each of these shapes
into triangles (the “Make a Triangle” task). Children who
used manipulatives were more likely to adapt the shape
(e.g., bending the pipe cleaner to make a third side to a
two sided shape). In contrast, children who drew on pictures
were more likely to draw an equilateral triangle on top of
the shape on the paper (superimpose).
General Discussion
Empirical Summary
Manipulatives helped children in the same way on this
addition task that they helped in other PDL studies in
arithmetic. Children were more successful with manipulatives
than with pictures. Children arranged and rearranged the
manipulatives, and in the process, found ways to solve the
problems. In contrast, with pictures, children usually drew
in unhelpful ways, or simply circled pieces.
Children benefited from using manipulatives in different
ways on geometry tasks than on addition tasks. Overall,
using physically manipulable triangles did not increase
students’ accuracy or lead to more advanced justifications.
However, both children who worked with manipulatives and
with drawings manipulated their materials to solve problems.
Both groups rotated their papers or physical shapes more
often when a triangle was not in prototypical orientation
than when it was. In addition, children who worked with
manipulatives adapted non-triangles, while those who worked
with pictures superimposed the prototypical triangle on the
non-triangle by drawing it on top.
Comparing the Role of Manipulatives in Arithmetic and
Geometry
Manipulatives functioned differently in some ways and
similarly in other ways in arithmetic and geometry tasks.
In terms of differences, though both groups of students
worked with spatial objects in the geometry study (as they
had in the arithmetic studies), the outcomes of the tasks
were different. In the geometry study, children’s answers
were about shapes. In the arithmetic studies, their answers
were given in terms of numbers. Therefore, in the geometry
context, children did not need to translate from a spatial
representation to a numeric representation to solve the
task. This difference had two effects. One, it removed the
effect of manipulation on all tasks except the adaptation
task. Two, it meant that children in both conditions could
use manipulation equally well to help them when they chose.
And in fact, both groups rotated shapes to help them decide
if they were triangles.
The results from the “make a triangle” task are similar to
the results found in arithmetic. Children performed better
with manipulatives than pictures. Using manipulatives may
have helped children advance their understanding of what a
triangle is because they could change the properties of the
shape more easily with the pipe cleaners than with the
pictures. Exploring shapes with different properties (e.g.,
three sides versus four sides) may have helped children
notice and start to realize the significance of these
properties for defining shapes. In contrast, with pictures
students often drew a roughly equilateral triangle on top of
the non-triangle. This superimposing may have involved
reasoning based more on the appearance of a triangle than on
its properties. Children may have primarily considered
whether the shape they drew looked like a prototypical
triangle. These results suggest that using manipulatives
allows children to adapt their environment freely and learn
by interacting with it, while drawing leads to more
pre-planned solutions.
Implications
These results are consistent with the idea of physically
distributed learning (PDL). One difference between these
results in geometry and previous studies in arithmetic was
that even though manipulatives did not lead children to
greater levels of accuracy or better justifications,
children rotated shapes more often when it was likely to
help them solve the task. This result replicates previous
findings that manipulation is useful, supporting the idea
from PDL that distributing to the environment can help
children solve problems. It extends previous findings in PDL
by showing that children are adept at choosing when to
distribute to the environment.
A similarity between these geometry results and results in
arithmetic tasks was that in certain situations
manipulatives were more helpful than pictures. The children
who used manipulatives showed more advanced reasoning on the
“make a triangle” task than those who used pictures. They
did not rely on the appearance of a prototypical triangle.
Instead they took actions that showed at least an implicit
understanding of the properties of triangles (e.g., they
made four-sided shapes into three-sided ones). Consistent
with other findings in PDL, this result suggests that people
can show more advanced reasoning when they have the
opportunity to adapt their environment freely (with dynamic
materials like manipulatives) than when they do not (with
static materials like pictures).
These early results examining the differences between the
functions of manipulatives in arithmetic and geometry tasks
are interesting, and we believe they support our idea of PDL.
However, more experiments contrasting the use of
manipulatives and pictures in geometry are needed to support
and examine these differences. Two important next steps are
to conduct studies that 1) directly compare the same
children’s performance on arithmetic and geometry tasks of
comparable difficulty, and 2) examine children’s actions
with manipulatives in geometry tasks in detail. In addition,
longer-term developmental studies of PDL in geometry could
examine how actions and ideas coevolve.
These results are also consistent with other research. Many
classroom studies have shown that children learn geometry
well with manipulatives. Reimer and Moyer (2005) showed that
working with virtual manipulatives helped children develop
conceptual and procedural knowledge in geometry, and that
virtual manipulatives are more motivating than paper and
pencil tasks. Sarama and Clements’ research (e.g., Sarama &
Clements, 2004; Sarama et al., 2003) shows that computer
manipulatives help children of different ages learn various
geometry concepts. For example, their Building Blocks
software helps children develop concepts of shape (Sarama &
Clements, 2004).
Taken together, the results of these studies and other
research suggest that manipulatives can be useful in the
classroom for geometry.
Acknowledgments
This material is based upon work supported by a Faculty
Summer Research Award and a Faculty Research Grant to the
primary author from the Office of the Dean of the College of
Education at the University of Texas at Austin. Any
opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do
not necessarily reflect the views of the University of
Texas.
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