IMAGERY
FRACTIONS
AND
CLASSIFIED
AS
LEARNING
STUDENTS
IN
DISABLED
Betsey Grobecker
Abstract. This article describes constructivist principles of
learning in geometry specific to children's imaginal anticipations in line measurement and the related area of fractions. To assist with this discussion, seven sixth-grade students classified as learning disabled (LD) and receiving some
type of special education service for mathematics were individually tested in tasks that investigated their imaginal
anticipations of space and their representations of this
understanding in the fraction symbol. Additionally, the
investigation examined the effectiveness of constructivist
teaching techniques in extending student thinking. All students perceived the most static elements of the line and
could represent a metric unit. In the related area of fractions,
they interpreted simple fractions as operational units.
However, the majority of students were unable to coordinate
and, thus, imagine the second-order nested hierarchies that
could be inferred from the line. Similarly, they had difficulty coordinating the higher-order nested relationships necessary to interpret equivalent fractions intelligently. The students benefited from instruction that questioned their way
of knowing and from manipulatives that served to support
their reflections.
Ed.D.
BETSEYGROBECKER,
The study of geometry and the related area of fractions in children with learning differences (LD) has
been minimal, making such investigation necessary
to adequately address the multiple needs of this population (Rivera, 1997). The research that has been
conducted has adhered to the information-processing model of what constitutes mathematics "disabilities." Specifically, it is generally believed that children
with average intelligence have difficulty understanding geometry and fractions due to: (a) inability to use
logical thinking without specific training; (b) visualperceptual problems that preclude seeing accurately
what is present; and (c) poor retention (Bley &
Thornton, 1995). Such disabilities result in difficulty
internalizing information and, thus, achieving at
grade-level expectancy.
Specific strategies to assist student learning consist of
structured materials to help students "see" that fractions represent parts of wholes (Bley & Thornton, 1995)
while engaging them in relevant, problem-driven
instruction using a variety of strategies (Thornton,
Langrall, & Jones, 1997). Engelmann, Carnine, and
Steely (1991) similarly argued that a variety of problems are necessary to understand geometric relationships and the fraction problems they represent.
However, these authors placed a much greater emphasis on verbal instruction. Specifically, they proposed
that students be presented with a variety of fraction
forms (e.g., 5/3, 2/3, 4/4) accompanied by specific
methods for analyzing fractions such as, "The bottom
number tells the number of parts in each group. The
top number tells the number of parts you use" (p. 294).
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These methods are referred to as "integrating big ideas"
(Grossen & Carnine, 1996). Additional teaching
methodologies in fractions include (a) identification
and integration of fundamental concepts and principles; (b) introduction of a big idea prior to its integration with another fundamental concept or principle;
(c) reasonable rates of skill introduction; (d) clear and
unambiguous teacher demonstrations; (e) efficient and
effective manipulative activities; and (f) adequate practice and review (Carnine, Jitendra, & Silbert, 1997;
Kameenui & Carnine, 1998).
In this article, I challenge the information-processing
perspective of what constitutes geometry and fractions,
the nature of difficulties in this area, and effective
instructional techniques. Specifically, I will present the
argument that spatial structures of thought are evolving, cyclical forms of mental structuring activity that
project themselves outward onto objects to create an
"image" (Kamii & Clark, 1995, 1997; Piaget & Inhelder,
1956, 1971; Piaget, Inhelder, & Szeminska, 1960;
Piaget, Marti, & Coll, 1987). In other words, through a
series of anticipatory reflections (i.e., self-regulated perceptual activity) that guide reflective thought on
objects, an image is constructed. As such, the image is
an act of representation.
Fractions are symbols that describe the image
abstracted. Consequently, they are not the source of
spatial thinking. In fact, if fractions are not supported
by the image, they become static pieces of information
devoid of self-regulated activity (Ball, 1993; Kamii &
Clark, 1995, 1997; Mack, 1993, 1998; Piaget et al.,
1960). Thus, I believe that our current emphasis on the
internalization of verbal instructions to teach fractions
(Engelmann et al., 1991; Grossen & Carnine, 1996) creates the learning "disabilities" observed by Bley and
Thornton (1995). Specifically, this instructional technique negates attention to the quality of the image that
children abstract while directing them to attend to spatial relations that are beyond what their structures of
thought are capable of coordinating and, thus, meaningfully reflecting upon.
To justify my argument, I will begin with an overview
of the developmental evolution of children's imaginal
anticipations in line measurement and the related area
of fractions as well as teaching methodologies that best
facilitate learning from this perspective. This overview
will be followed by protocols from lessons conducted
with seven sixth-grade students receiving some type of
special education service for mathematics. These protocols provide insight into the quality of spatial thought
structures of children labeled as having learning disabilities and instructional techniques that best facilitate
their growth. I will conclude with a summary of the
children's performance and the implications of their
performance on instruction.
OVERVIEW OF CONSTRUCTIVIST
PHILOSOPHY RELATED TO SPATIAL
THOUGHT AND FRACTIONS
Definition of the Imagery and Fractions
To understand the constructivist perspective of what
constitutes spatial thinking, it is necessary to rethink
the nature of the learner, the task, and the interaction
between these two forces. Of primary importance is to
define the learner as an open system of energy who
projects his or her activity (mental and emotional) into
the world to fuel growth (Dewey, 1933, 1934/1980;
Piaget, 1980, 1981, 1985). Spatial thought structures
that fuel the evolution of topological and Euclidean
space exist as evolving, expanding, biochemical feedback loops of nervous energy that serve to focus, limit,
define, and select what will be attended to as hypotheses to means-ends relationships are tested (Piaget, 1971,
1985). Thus, what is perceived is an interpretation of
what is possible based upon what we are motivated to
attend to and what our structures of spatial thought
enable us to reflectively abstract (or coordinate).
Equilibration is the regulator of activity within and
between individual and environmental energy systems.
Specifically, as schemes of spatial thought extend and
reorganize their form through a biological drive to
assimilate the world, disturbances in how reality is
defined emerge (Piaget, 1985). Such disturbances create
an emotional tension to resolve the conflict, which one
needs to be open to if schemes and the object acted
upon are to be transformed (Dewey, 1934/1980).
According to Piagetian theory, the manner in which
attention is directed outward onto objects is reflected in
the image. Specifically, an image is an internalized imitation of actions whose source is dynamic and evolving
cognitive structures. (Note that "actions" refer to organizing structures of mental activity that are projected
outward into the environment to fuel their growth.)
With specific regard to the geometric relations inherent
in line measurement, the internal image of a line that is
constructed is dependent upon: (a) the quality of
grouping relationships coordinated (subdivision); and
(b) the construction of reference systems and itineraries
involving the position of changes (change of position)
(Piaget et al., 1960). These two complementary spatial
structures become increasingly more dynamic and
complex as they expand and reorganize their form onto
higher-order levels. Thus, imagery as an act of representation has a general cognitive-structural foundation
whose purpose is to anticipate the means of realization
to changing ends and that follows a developmental
Learning Disability Quarterly
158
progression through adulthood (Clements & Battista,
1992; Dean, 1990; Kamii & Clark, 1995, 1997; Piaget &
Inhelder, 1956, 1971; Piaget et al., 1960; Wheatley &
Cobb, 1990).
There is a significant tie between the study of geometry and space and the study of fractional numbers
(Kamii & Clark, 1995; Kieren, 1993; Lamon, 1996;
Piaget et al., 1960). Specifically, fractions are symbols
that describe the image such that the abstraction of
more complex, coordinated units in line measurement
and the ability to interpret the fractions that represent
these abstractions are intertwined (Kamii & Clark,
1995; Piaget et al., 1960). Thus, to better understand
how children interpret fractions, we need to better
understand the developmental evolution of the image.
Evolution of Imagery and Fractions
Children's initial imaginal anticipations of space are
topological or static. Specifically, there is a lack of distinction between fixed sites that create a reference
point for measurement and the space taken up by movable objects. Therefore, there is an inability to coordinate intervals between fixed points and, thus, to perceive a straight line as intervals between fixed points.
To young children, a change in position of an object
along a line is simply a change in its rank order. Thus,
if a line or object is cut into two halves, each piece
becomes a separate whole in itself rather than existing
as a half of a former whole in fair-share activities
(Lamon, 1996; Piaget et al., 1960; Pothier & Sawada,
1983).
As children exercise their spatial thought structures
in activities that pose conflicts to their ways of knowing, their mental structuring activity recoordinates and
expands to the degree necessary to act on objects using
qualitative operations. Specifically, a line is coordinated
as a series of points that has a fixed starting and ending
point and that exists as a single unit of measurement.
Thus, the whole, as a unit, is conserved thereby
enabling children the powers of transitive reasoning
(Kamii & Clark, 1997; Piaget et al., 1960; Piaget,
Ackermann-Valladao, & Noschis, 1987). Transitivity is
the ability to infer a relationship from two or more relationships of equality or inequality (i.e., A = B, B = C,
therefore A = C), which children are generally able to
achieve by grade 2 (Kamii & Clark, 1997).
When each unit is assimilated as a single unit independent of the group in which the unit is nested, children assume that the number line represents the number
system as a set of discrete points in which no continuous
numbers exist between the dots represented on the line
(Ball, 1993). As a result, children at this level generally
attempt to apply rules and operations for whole numbers
rather than for fractions when interpreting fraction
symbols (Ball, 1993; Mack, 1993). For example, when
asked to compare fractions, children attend only to the
number of parts irrespective of the partitioning units
(e.g., 2/5 is more than 1/2 because 2 parts are more
than 1).
Through the ongoing engagement of reflective activity in problem-solving tasks schemes expand, thus
enabling the abstraction of a metric unit. A metric unit
is coordinated and generalized to cover transitive relations between successive portions of a given length.
Simultaneously, a change in the position of this unit as
a comparison of successive parts within each line is
made. The complete fusion between subdivision and
change of division, which makes possible the anticipation of equivalent units that can be iterated n times (i.e.,
unit iteration), occurs generally around the fourth
grade. However, while the unit is now iterated as a nested partition within the whole, this new conception of
space is limited to one-dimensional linear space (Kamii
& Clark, 1997). Because students now perceive the
necessity to nest equal-sized parts that are coordinated
with respect to each other and the whole, these continuous units are interpreted as a single number in the
form of a/b where b is not equal to zero (e.g., 1/8 of 8).
Thus, the numerator represents one unit and the
denominator another, creating a new kind of unit out of
the comparison of the two original units (Mack, 1993).
Toward the end of the middle-school years, children's structures of spatial thought evolve to the degree
necessary to coordinate their imaginal anticipations
within a comprehensive Euclidean system (i.e., reference points are coordinated within a system of multiple
perspectives or multiplicative groupings). Because spatial structures now consist of second-degree higherorder nested structures (i.e., multiplicative structures),
children become proficient at coordinating a nested
part within a part and, thus, at interpreting the meaning of equivalent fractions (e.g., 1/4 of 1/2) (Kamii &
Clark, 1995; Mack, 1998; Piaget et al., 1960). When
spatial thought structures have evolved to this level
and gained stability, objects are no longer necessary to
support the means chosen to anticipate ends, nor are
symmetrical or regular shapes a necessity. In fact, the
itinerary chosen for problem solution represents only
one of many equivalent possibilities (Piaget et al.,1960).
Because the degree of coordination in spatial thought
structures determines the quality of the image that is
abstractedwhen acting on objects, valid measures of spatial thought are those tasks that enable an observation of
children's levels of imaginal thinking as objects are acted
upon. Specifically, a representational context is necessary
in which children explore, test, reason, and argue about
mathematical ideas and tools (Ball, 1993) using diverse
materials so that fractions can stand for something
Volume 23, Spring 2000
159
(Streefland, 1993). Also significant to such investigation
is the opportunity to question and guide children's anticipatory activity. In the section that follows, lesson protocols from seven sixth-grade students who were categorized as LD are presented. The purpose of the lessons was
to investigate the quality of the students' imaginal anticipations related to line measurement and their related
ability to represent this thought in fractions.
PARTICIPANTS AND LESSON PLAN
Students
Consent forms were sent to the parents of all sixthgrade students with a history of mathematics difficulties and who were receiving some type of special education service in mathematics in a small public school
district in northern New Jersey. The seven students
who returned signed consent forms participated. Four
of these students currently had their mathematics
replaced in the resource room (2, 4, 5 & 7) and one student (1) was in a self-contained classroom for mathematics instruction. The remaining two students (3 & 6)
were in support class daily receiving remedial intervention in mathematics as needed. One student in support
class received replacement instruction in mathematics
in first through third grade. The other student was in
basic skills instruction (BSI)for mathematics in second
through fourth grade and had mathematics instruction
replaced in the resource room in grade 5. The four students in the resource room had recently worked with
fraction equivalency between quarters, halves, and
three-quarters and converting these fractions to decimals in the context of money problems. The student in
the self-contained class had not worked with fractions
during the current school year nor had the two students who were in the regular mathematics class. Six
students were in the resource room for reading and language arts; the student in the self-contained classroom
for mathematics received reading and language arts
instruction in the same classroom.
Six males and one female participated; their
mean age 11.8 (SD=2). Standardized scores in psychometric tests of intelligence (WISC-III)revealed a mean
IQ of 97 (x=100; SD=15). The reading and mathematics
grade-referenced normal-curve equivalent scores (NCE)
and SD (SD=21, x=50) are from the ComprehensiveTest
of Basic Skill - 4th edition (CTBS).This test was administered to all but one of the students with LD as part of
the regular school district assessment program approximately two and one-half months after the data were
collected and provides a general estimate of standardized performance levels. The mean NCE (SD) for
Reading Vocabulary and Comprehension were: 30 (6)
and 37 (13). The mean NCE (SD) (x = 50; SD = 21) for
Math Computation and Concepts and Applications,
respectively, were 42 (12) and 43 (9). Thus, performance was approximately one standard deviation below
the mean in reading with mathematics performance
somewhat stronger, although still below the mean.1
Tasks
Measurement task. The line measurement task consisted of sheets of unlined paper (8 1/2 x 11), each with
two lines representing a T on it (Kamii & Clark, 1997,
with modifications that follow the task description).
Although the vertical line appeared to be longer
because of a perceptual illusion, both lines of the Twere
8 inches long. The bottom of the vertical line was one
half inch from the bottom of the paper so that 2 1/2
inches remained at the top. Each of the sides of the top
of the Twas one fourth inch from the edge of the paper.
Additional tools provided for the various measurement
tasks included (a) strip of yellow tag-board 12 inches
long and 0.5 inches wide; (b) eight wooden cubed
blocks (one inch in length, width, and thickness); (c) a
pencil; (d) a toy car, one inch in length; (e) sheets of
unlined paper with four fraction problems written vertically, 1/8 + 1/8, 1/4 + 2/8, 1/2 + 2/8, and 3/4 + 2/8;
and (f) sheets of unlined paper with the same four
number problems represented as words.
A few modifications were made to the task presented
by Kamii and Clark (1997). First, in an effort to increase
task purpose, the lines were referred to as roads and a
car was placed off to the side of the "road."
Second, the Twas not presented as inverted. This was
done to help with the second part of the task, in which
the blocks were placed on the top horizontal line as a
reminder of how many blocks made up the entire road.
Placing the blocks on the top rather than the bottom
made the task less awkward for the children when
working with the blocks to solve the fraction problems.
Children's justifications (later provided) generally
showed no attention to the lines as representing the
letter T. Only one student referred to the diagram as a
T, stating, "This makes a T," after all eight blocks were
placed on the line, and commenting that they make,
"one whole road."
Third, the paper size used by Kamii and Clark (1997)
was 11 x 17 inches. The difference in paper size was an
oversight by the author. However, the lines are proportionally the same on both papers, and similar to Kamii
and Clark, the great majority of children (86%) continued to believe the vertical line was longer on their initial judgment. The remaining student judged the roads
to be the same.
Fourth, in the present study there were eight cubed,
one-inch wooden blocks rather than five plastic blocks
(1.75 inches long, 0.88 inches wide, and 0.25 inches thick)
so that appropriatefraction problems could be presented.
Learning Disability Quarterly
160
Finally, the sheets with the fraction number and
word fact problems as well as the road fractions task
were added in the current study.
Each interview examined three questions:
1. Perceptualjudgment.Presenting the child with the T,
the researcher asked, "Let'spretend that these two lines
are roads for this car to drive on. Do you think this road
(vertical line) is as long as this road (horizontal line) or
is this one (vertical) longer, or is this one (horizontal)
longer?" The car was placed off to the side of the paper.
While asking the question, the researcher traced her
fingers over the "roads."
The purpose of this problem is to provide a motivation for the child to answer subsequent questions.
2. Transitivity.With the tag-board strip in hand, the
researcher asked the child, "Can you use this to prove
(or show) that this road is longer than the other (or
whatever the child had said)?"
The question was asked to determine if the child
could demonstrate transitive reasoning (A = B, B = C,
A = C) by using a third term that was longer than the
roads being compared.
3. Unit iteration. Offering one block to the child, the
researcher asked, "Can you use this to prove (or show)
that this road is longer (or that they have the same
length)?"
The purpose of this question was to determine if the
child was able to compare the two lengths by using a
small third term as a unit to iterate.
Fraction problems. In the fraction problems, three
main areas were explored: (a) operational understanding of fractions (b) role of conflict in stretching thinking and (c) effect of past explicitly taught procedural
skills on problem solving.
1. Road fractions. After the measurement task was
completed, all eight blocks were placed on both roads (if
they were not already on them) until the student and
examiner agreed that the roads were the same length.
Eight blocks were removed from the vertical road and
two blocks were placed on the same road. The child was
asked, "If there are two blocks on the road and eight
blocks make the entire road, what part of the road do
these blocks make?" The question was repeated for four,
six and eight blocks. If the child responded incorrectly
to the six blocks, a counterexample was given, "Another
child told me that we started with 4/8 and then added
2/8. That would equal 6/8 or 3/4. What do you think
about what he or she said?" The blocks were placed on
the road as the counterexample was stated.
The purpose of this question was to determine children's operational understanding of fractions by
observing the type of nested units imagined (i.e., coordinated) from the line.
A second purpose was to investigate whether the
counterexample created conflict.
2. Number and word fractions. The students were first
given the four number fact problems previously
detailed, followed by the same four number problems
expressed in words. For both problems, the students
were asked, "Do you know what these are called? Can
you figure them out?" Individualized lessons that
examined constructivist principles of learning followed
the questions.
The purpose of these problems and the lessons that
followed was to examine: (a) the type of nested relationships imagined, which was observed in the child's
manipulations of the blocks and verbal explanations of
solutions; (b) the role of conflict in stretching thinking;
and (c) the influence of explicitly taught procedural
skills on problem solving.
LESSON PROTOCOLS
Measurement Task
In the transitivity task in which students were asked
to use the tag-board strip to test their initial judgments
of line equality in the T form, all students successfully
used the third term to compare each of the two lines.
Further, on the unit iteration task all but one student
(7) iterated one block to measure each road and then
compared the number of times the block was used on
each road. Student 7 stated that the block was "too
small" to measure the road.
Fraction Problems
Because the road fractions task was later integrated
with the number and word fractions tasks in that it
served as a source of reflections as children solved the
problems, relevant aspects of each child's performance
in these tasks will be presented together. I will begin
with a performance description of the children who
achieved at the lower levels and work upward.
On the road fractions task, only one student (2) was
unable to state that all eight blocks represented the
whole. This student stated that the eight blocks comprised "the street," suggesting that he imagined each
block as a discrete unit independent of the others rather
than inclusive of them. However, when the correct solution was provided in the counterexample (six blocks or
3/4 of the road), his response indicated that he was
stimulated to think about the unit relationships differently, "Yeah ... 'cause if you can take 2 of these away and
it would make 4 blocks left ..." This movement away from
attending to the more concrete line features could suggest that his structures are open to increased sophistication in the possibilities imagined. In fact, in the number
and word fractions that follow, he is able to coordinate
and, thus, to imagine the nested grouping relationship
Volume 23, Spring 2000
161
inherent in simple fractions as he reflects upon his
thinking activity.
(1/8 + 1/8) He wrote 2/16. "I'm going to show you
what anotherkid your age did and I want your opinion.
He said that this [one block] is 1/8 [o] and then he said
plus another [+ o] 1/8. That would equal 2/8 [= oo].
What do you think?""I guess it sounds good because
there's two of the same thing." "Isyour'sok, too?""I
don't know. Maybe."
In the second and third problems, he again added the
top and the bottom as distinct quantities (1/4 + 2/8 =
3/12; 1/2 + 2/8 = 3/10), and I again provided a correct
counterexample. A problem reduced in its complexity
(3/8 + 4/8) was given in which I asked him to show me
his thinking with the blocks. After putting on seven
blocks and stating "7/8," I asked for his justification. "I
just added 4 more on." This altered thinking suggests
that his structures may be expanded to the degree necessary to interpret simple fractions as operations (i.e.,
each of the eight blocks is coordinated as continuous
quantities that can be measured or partitioned).
However, equivalent fractions are beyond what can be
meaningfully interpreted. In fact, the recently taught
procedures to solve equivalent fractions only added to
this student's confusion.
(3/4 + 2/8) "Could I do it with the blocks?" "Ok."
Five blocks were placed on the road [oocoo] and
he wrote 5/12 as his answer. "How'dyou get 5/12?"
"just added. For the bottom I just added 8, 9, 10,
11, 12. That's 12. And then for the top 3 + 2. I
added 3 on." "Ok.I see what you did. So what part of
the road is that?" "That part of the road is 5/8 of the
road." "Why do you have 5/12 there?""Oh yeah, I
forgot that was only 8." "Ok, can you fix this?"
"Yeah, I think." "Canyou show me 3/4 of the road?"
"Three-fourths of the road?" "Yes." He added
another two blocks to the road [+ oo = Doooooo].
"Ok,so what do you have?" "Seven blocks." "Why?"
"Because I'm thinking, 75? is 3/4 of a dollar."
"Ok." "And what are you going to add to the road?"
"Add to the road? Another block?" "What does it
say to add?" "Add 8 + 4 is 12." "So what's your
answer?""Seven twelfths."
Student 3 displayed similar types of spatial constructions as student 2 in that equivalent fraction units
were beyond what could be meaningfully reflected
upon. However, the spatial structures of student 3 may
be more evolved because he consistently provided a
whole number to describe the part of the road the
blocks represented on the road fractions task. For
example, for six blocks on the road, he stated, "They
make up six of the parts of the road." When the correct suggestion was provided, he believed that both
answers were good. In the number fractions to follow,
his confusion coordinating and, thus, imagining the
second-order nested relationships, is evident.
(1/8 + 1/8) He wrote 98/, 98/ in a vertical line.
After giving him a counterexample and asking if
that answer made sense, he responded, "Yeah ...
because the roads are, if the road is 8 blocks and if
there's only 1 remaining that would be 1/8 [o]."
"Canyou show me that [1/8]?" "It would be 2 [+ o =
oo]." "So what's your answer?" "Two eighths."...
"Did you ever reduce fractions before?" "No."...
"Besides2/8, do you know what this might be called?
What part of the road?" "Two of 8? I don't
know.""Actually, it's 1/4 and this reduces to 1/4
'cause2 goes into 2, 1 time and 2 goes into 8, 4 times."
(1/4 + 2/8) He wrote 3/8 as his answer. "Can you
show me 1/4 of the road?" He took one block off
from the previous problem [oo - o = o]. "Why 1
block?""Becausethat would be 1/4. I took the other
one off because it's 2/4 of the road." (He was confused about his last answer.) "Canyou show me this
2/8 of the road?" He put on another block [+ o =
oo]."Is that 2/8?" "Yes." "Ok, what do you have
there?""Two blocks." "Is that 2/8?... What does the
problem say to do?" "One eighth or 2/8?" "Well,
here's 2/8 of the road right here. You have that. And
what are you adding to the 2/8 of the road?" "One
fourth." "So,what did we say 1/4 equals?""One part
of the road." "What's this?" (I was referring to his
answer of 1/4 on the previous problem.) "Twoeighths." "Two-eighths.So does one quarterequal 1
block?" "Yeah." "Why?" "Because the road has. I
don't know." "How many parts are there to the road
in all?" "Eight." "Ok, and so when we add 1/4, how
many eighths do we have?" "Two." "All right, did you
show me that before?""Yeah." "How?" "I put two
blocks onto the road." "Ok, can you show me the
problem one more time? How you think it through?"
Although we redid the problem two more times with
manipulatives, his inability to imagine the nested second-order relationships again overwhelmed him.
(1/2 + 2/8 = 5/8). "Two-eighths is 4 [cncc]. Add
one, you add one that would be 5 [+ o = ooDDD]."
The protocols of the four students to follow show
varying degrees of transitional behaviors between additive spatial thought structures (most noted on the road
fractions task) and the emergence of second-order nested structures (i.e., multiplicative structures) necessary
to interpret fraction equivalency. Of these four students, student 4 had the greatest difficulty coordinating these more complex nested grouping relationships.
Specifically, on the road fractions task, he stated that
four blocks were 1/2 of the road, but his justification
shows attention to additive grouping relationships,
Learning Disability Quarterly
162
"The middle." "Ok. Why the middle?" "Because 4 + 4 =
8, and so if you had another 4, it would be the whole
thing. So just one 4 would be half." When asked what
six blocks on the road represented, he stated, "toward
the end" of the road. When the countersuggestion was
given, he stated, "Oh, I thought you meant just like the
beginning, the middle, the end. That could be right too
though." "Ok,let me ask you this again [I put 2 blocks on
the road]. Let's just see if you would answer differently.If
we had just 2." "The beginning.""Thebeginning?""Yes."
On the number and word fractions task, he abstracted a part of a whole as one unit in simple fractions and
attempted to think through the second-order nested
relationships in equivalent fractions. However, these
more complex second-order nested units were too difficult for him to coordinate and, for the moment, he
returned to imaginal anticipations that nested the units
across only one level of abstraction. In fact, when first
presented with equivalent fractions, he initially pondered the possibility of adding the tops like the bottoms due to the increased level of difficulty inherent in
this problem. As with the previous student, the initial
emphasis on procedures for reducing equivalent fractions on the number fractions tasks did little to
advance his attention to the more dynamic, nested
hierarchies that can be imagined from the line.
(1/8 + 1/8) "Thissays 1/8 + 1/8. Do you thinkyou can
show me what 1/8 of the road is with those
blocks?""Maybethat?" He placed one block on the
road [n]. "Yes, why?" "'Cause there's 8 of them all
and 1." "All right, can you show me another eighth of
the road?" "Another 1/8?" "Yes,can you put that on
the road then 'cause you're adding another 1/8." "All
right [+ o = oo]." "Sowhat are you left with?" "Twoeighths." "Exactly,so can you write your answer?"He
wrote 2/8. "Great.Now let me show you that actually
that's 1/4 of the road 'cause when you reduce,2 goes
into 2 once and 2 goes into 8? "Four times." "Four
times. So that could be reducedto 1/4. Does that make
sense, that this is 2/8 or 1/4 of the road?""Yeah."
(1/4 + 2/8) "I can do the bottom ones. Do you do
the bottom ones like the same thing as the
top?" "Showme 1/4 of the road with the blocks.""One
fourth?" "Yes."He put one block on the road and
moved up along to the fourth mark made when
measuring the road [-o]. "Like right here and it
stops right here." "Ok,so it stops at the fourth line."
"Yeah." "All right, but we want 1/4 of the whole road
and not half of the road. So what would 1/4 of the road
be? Look at your first problem and see if that helps you
to figureout what 1/4 of the whole roadwould be." "He
put another block on the road although he had
puzzlement on his face [+ o = oo]. "Does that make
sense? Youput another block on there.Does 2/8 equal
1/4 of the whole road?" "I don't know. Yeah, but
there's 8 here" (he is referring to the other road).
"Right, 8 blocks make up the whole road. We know
there's 8 parts to the whole road or 8 blocks. Now, we
want 1/4 of the road. And in our firstproblem we said
that 2/8 is the same thing as 1/4." "Oh yeah, added
2 blocks to the road." "Soyou have 2/8 there. So 1/4
equals what? How many eighths?" "Two eighths."
"Two eighths. Ok. Now what are you adding?"
"Adding?" "Plus." "Two eighths plus 2/8." He
didn't know what to say or do. "Showme with the
road. You have 2/8 now and you're adding, how many
eighths areyou adding?""Eight." "Youhave 1/4 = 2/8
and then you're adding 2 more eighths." I wrote the
problem off to the side. "Right?""Yep." "So here's
your 2/8 plus." He added two more blocks [+ oo].
"Would equal 4/8 [= oooo]." "How much of the
road? What can that be reduced to?" "One half."
"Yeah, exactly. Is this 1/2 of the road?" "Yeah." "So
2/8 = 1/4 right? This is 1/4 of the road. And then
another 2/8 or another 1/4 of the road equals?"
"Equals 1/2."
Thus, consistent with the road fractions task, this student stated that four blocks on the road (4/8) is equivalent to 1/2 on the number and word fractions tasks.
However, in the problem that follows, he is not interpreting this equivalency as second-order nested units.
(Three-fourths plus two-eighths equal) He wrote
3/4 then put 3 blocks on the road [ooo]. He then
wrote 2/8 under 3/4, placed two blocks on the road,
and wrote his answer of 5/8 [+ oo = ooooo]. "Sowhy
did you have 3 blocksherefor 3/4?" "'Cause 3 for this
[3/4] and another 2 for that [2/8]. Then I added."
Similar to the performance of student 4, student 1
correctly stated that four blocks are 1/2 on the road
fractions task, but provided a justification that suggests
the abstraction of additive grouping relationships,
"Because 4 + 4 make 8. So if you only have 4 blocks,
there's 4 more and you'll have 8." She correctly stated
that 6 blocks is 3/4 of the road, but her reasoning is
weak, "It's not 1/2, it's not whole, all of it, so I guessed
3/4." In contrast to student 4, her effort is sustained to
overcome the inconsistencies in her thinking as she
attempts to coordinate the higher-order nested structures. In the last two problems on the word fractions
task, this effort pays off because she correctly solved
them without my questioning, but with the assistance
of manipulatives. However, her thinking is very fragile,
thus requiring much more reflection before her structures gain stability on this level. Another significant
observation is how this student separated fractions
used in cooking from the problems on paper on a number fraction problem.
Volume 23, Spring 2000
163
(1/2 + 2/8) She wrote 13 but then explained something different with the blocks. "One half is one of
these blocks." "One half is one of these blocks?"
"Yeah, and two-eighths is 2 of the blocks. It would
be 3/4." "Why 3/4?" "Because it's not all the way.
There's half of the way if you had one more. It
would be half the way so we have to minus like 1
and that's a half. That's not a half, that's 3/4 of the
way up." "What is that?" (I was pointing to her
answer of 1/2 on the previous problem.) "One,
three, one half." "Andshow me one-half of the blocks
on the road."She put 1 block on the road [n]. "Was
that 1/2 of the blocks on the road?""Oh no. Then it
would be. I don't know. I forget." She then put 3
more blocks on the road [+ _ = onoo]."Like that,
that's 1/2." "Why did you have 1 before?""'Cause I
was thinking 1, not 1/2." "Andwhat are you going to
add?" "One-half plus two-eighths." "Ok, can you
show me that?" "Well, 12, but not in cooking. It
wouldn't be that, 1/2. It would be 3/4 of the way.
See this [4 blocks] is half plus this [2/8]." "Ok,show
me plus this [2/8]." She added two more [+ oo =
cooooo]. "So you're adding 2 more blocks to the
road." "Yeah, but then it's not whole. You need 2
more." "So what's your answergoing to be? One half
plus 2/8?" "It would be well, if you add this problem up, it would be 13. But cooking-wise it would
be, I don't know how to say this one.""What did
you say before (on the previous task)?" "Three."
"Threewhat?" "Three quarters." "If this is 1/4, how
do you think you would write 3/4?" She wrote 1/3.
"What did you write?" "One, three." "Actually,this
is 3/4, threeover four.""Oh, all right."
(One eighth plus one eighth equal) She initially
thought that it should be a whole, but when I
asked her to show me 1/8 + 1/8, she put two blocks
[on] on the road and wrote 2/8.
(One fourth plus two-eighths equal) She was
again confused with the nested orders, but overcame her confusion as she evaluated and altered
her actions. "Here's 2/8 already [Qo] plus 1/4. One
of these is 1/4. She added one more block to the
road [+ D = ooo]. So it would equal 3/4? Half or 3/4.
"You'reteaching me what you think." "I think 3/4."
"Why 3/4?" "Because that's not half of this 8. It's
not half of 8 and it's not all of 8. It's only 3. It
would be 3 of the line of 8." "Ok. This is what the
other girl did. She said this is 1/4 Do, these two, plus
2/8 [+ oo] is what?" "Half [oooo]." "Wouldyou agree
with her or you?" "Yes,because I didn't have the 2/8.
I only put 1. So it would be half." She then went on
to solve the remaining two problems using the
blocks in a very ordered, consistent manner to
derive the correct problem solution without my
questioning to guide her reflections.
Student 5 initially displayed very strong tendencies
to direct his attention to additive groupings, particularly on the road fractions task. Specifically, when asked
what part of the road four blocks made up, he responded, "Four."For six blocks, he responded, "Two, 'cause
you have six here and then if you said like and there's
eight blocks could fit here. But there's two more spaces
left." He believed that both the countersuggestion
and his solution were good answers.
However, on the number and word fractions tasks, he
became increasingly more aware of the limitations of
the grouping relationships that he was coordinating as
he worked through the problems. As a result, he was
more consistent in directing his perceptual activity to
the second-order nested grouping relationships. In the
last three word fraction problems, he independently
resolved his conflicts as he reflected upon his imaginal
anticipations, although he did use the blocks to assist
with the relationships he was attempting to coordinate.
(1/8 + 1/8) He wrote 1/4. "Why?""One is like the
1 stays the same 'cause it's at the top and then I just
thought that like what would go into both the 8s?
And 4 'cause 4 is half of it and 4 + 4 would like
equal 8." "Can you show me with these blocks what
1/8 of the road would be?" "That would be 1/8
[o].""Why one block?""Cause you have 8 and then
1 would like 'cause, you have 8 blocks on the road
but then you have 1. You have 1 but then like add
1 and it's like 1 and 8." "Canyou add anothereighth?
That's 1/8 + 1/8. Show me with the road how you
would add another eighth." "Add another one or
something [+ n] 'cause like then you like get another 1/8 but then it comes to 2/8 [= on]. So this equals
2/8." He changed his answer. "Did you ever work
with reducingfractions?" "Yeah." "Do you remember
how to do that?" "I think 2 goes into 2 once, 2 goes
into 8, 4 times." "Thereit is, 1/4 again. You had it
right in the beginning."
(1/4 + 2/8) In this problem, he experienced conflict because he perceived a necessity for having the
bottom numbers the same to add, but didn't know
how to achieve this. "Can you show me 1/4 of the
road?" He put four blocks on the road [ooo].
"What's there?""There's 4 and like the road would
be a whole almost and be 1 whole road and 4 so
like I guess." "Well, in our first problem, what did we
say 1/4 of the road was?" "Oh, 2/8." He took off 2
blocks on the road [- oo = oo]. "Ok, so is this 1/4,
those 2 blocks?" "Yes." "Ok, and how many eighths
was it?" "Two-eighths." "Yeah,2/8. So what are you
Learning Disability Quarterly
164
going to write there?(On the sheet.) One fourthequals
how many eighths?" "Two eighths." "And what are
you going to add? Plus what?""Eight?" "How many
eighths?" "Two eighths." "Plus two more eighths."
"Equals 1/2." "Sowhat's 2/8 + 2/8?" "Thatwould be
4 and that would be 8." "Show me with the blocks
2/8 + 2/8 = 4/8." [0o + oo = oooo] "Can you reduce
that?" "Fourgoes into 4 once, 4 goes into 8 twice."
"Sowhat part of the road do you have?" "Half."Is that
half of the road?""Yes."
(One eighth plus one eighth equal) He thought
the answer would be one whole but then immediately changed his answer to 2/8 when I asked him
to show me with the blocks [oo]. He derived 1/4 by
stating, "Two goes into 2 once, 2 goes into 8, 4
times, 1/4." He computed the remaining problems
without my guiding questions but used the blocks
to guide his thinking.
Student 6 provided ordinal position numbers for all
of the problems on the road fractions task (e.g., six
blocks is "the sixth"). However, in contrast to the previous student's performance on this task, he noticed
that something in his thinking was amiss when the
counterexample was presented, "I think his is better ...
All I did is just like keep adding 2"... "Sohow is his different?""Like he used some other things. All I did was
just add them up by 2s."
On the number fractions task, he was initially confused coordinating the second-order nested relationships of equivalent fractions and considered the possibility of adding the bottoms and the tops as did the
previous student when given equivalent fractions.
However, he worked to resolve his misunderstandings
and independently abstracted the second-order nested
subunits in the last number problem on the number
fractions task. This new understanding sustained itself
in all of the word fractions tasks that followed. Note
that the verbalized procedures to assist his problemsolving were used in a concise manner and generalized.
(1/4 + 2/8) He wrote 3/12. "Why?""You add the 1
and 2 and you get 3 and then you add the 4 and
the 8 because you can't put 4 in 8 'cause it has to
be a whole thing. So you have to add the 4 and the
8." "All right. Now, I want you to show me with the
blocks what 1/4 of the road is." "One fourth?" "Yes."
"These [o]." "Ok,the two on the road, right?""Yes."
"Now, what does it say to add?" "The 8 and the 4."
"Well, you have 1/4, which you said is right here.
Right?""Yeah." "Now, what is it asking you to add?"
"Two eighths." "Ok."He put 2 more blocks on the
road [+ oo = Dooo]. "Ok,so what do you have here?"
"Four eighths." "Is that different than 3/12?"
"Yeah." "Which is right?""Foureighths." "Andwhat
do you know this can be reduced to?" "Both divide
them by 4. That would equal one over four." "Well,
think about this. Four goes into 4?" "Once." "Four
goes into 8?" "Twice. So that would equal 1/2."
... "Here'syour 1/4, which is 2/8, and here's another
1/4, which is 2/8, and you get 4/8 or?""One half."
(1/2 + 2/8) He wrote 1/4. "Why 1/4?" "On the
number 2 problem we got 4 over 8 and that
equaled 1/2. Here it's 1/2 over 2/8. So it's like the
opposite. So it would come out to 1/4." "All right,
show with the blocks 1/2 of the road." "Half of the
road is right here." "Yeah,and how many eighths is
that?" "Four [oooo]." "Four eighths. So do you
want to put 4/8 there?" (On the problem sheet.)
"All right. Now, what are you adding to that?""Two
eighths." "So what are you going to add to the road
here?""Another two [+ oo = oooooo]." "So what's
your answer going to be?" "It's going to be 6 over
8." "Ok,can you reducethat?" "Yeah, 2 can go into
both of them. So it's 2 divided by 6 = 3 and 3/4."
Finally, student 7 consistently coordinated the whole
in relation to its parts while simultaneously coordinating
the equality of parts to each other (i.e., 8 = 2(4s) = 1/2)
on all problems. For example, he stated that six blocks
equaled 75% of the road, "Becauseif two blocks equals
25% then three sets of 25 equal 75%." However, on the
first number fractions task, he became confused when he
attempted to reduce 2/8, but worked through the confusion. With the exception of one word problem in which
he needed a little guidance to facilitate his thinking, all
other problems were independently solved without the
use of manipulatives to facilitate his reflections.
(1/8 + 1/8). He wrote 2/8 and then inquired, "Do
you want me to reduce?" "Yes." "The first one is
1/6." "Why 1/6?" "Because it, hum, because it = 2/8
and 2 is hum ... "Ok, can you put 2/8 on the road?
[oo] And what did you say that equaled?"(I was referring to the previous task.) "I don't know." "What
do you think that equals. If there's 2 blocks and 8
blocks make up the whole road?""Oh, 1/4." "Sowhat
do you think that reducesto?" "One fourth."... "Both
2 and 8 can be divided by 2. Two goes into 2 once and
two goes into 4, 2 times. Make sense?" "I just forgot,
I guess. I knew that."
SUMMARY AND CONCLUSION
In this article, I argued that children's evolving spatial thought structures that direct perceptual activity
on objects determine the types of geometric relationships that are imagined (i.e., coordinated) and the
related ability to interpret fractions that represent the
image. Therefore, to understand how children are making meaning of geometric notions and fractions, we
Volume 23, Spring 2000
165
need to investigate the quality of perceptual activity
that is projected onto objects (i.e., imaginal anticipations) and to extend meaning relative to the types of
spatial relationships coordinated. To direct children to
attend to spatial relationships that their perceptual
activity cannot yet coordinate is to remove them from
their self-regulated activity and to manifest distorted
errors that come to be labeled as "disabilities."
The seven students detailed in this article demonstrated varying degrees of ability to coordinate the more
complex nested units that could be inferred from the
road, although all students were able to perceive the T
shape in its most static form. In fact, all but one could
organize space around the T form to the degree necessary to assimilate a metric unit in the measurement task.
In other words, these students could simultaneously
coordinate grouping relationships (subdivision) with a
construction of reference systems and itineraries involving position of changes (change of position).
The construction of a metric unit enables students to
interpret simple fractions as operational units (i.e., as a
continuous quantity that can be measured or partitioned, and that can be represented as a single number
in the form of a/b where b is not equal to zero). In fact,
all students demonstrated the ability to interpret simple fractions as operations, although one student initially added two tops and two bottoms as discrete units
(e.g., 1/8 + 1/8 = 2/16). The grouping relationships that
he was challenged to think about in problems that followed enabled him to solve a simple fraction problem.
However, this student, as well as five others, displayed
varying levels of difficulty coordinating the secondorder nested subunits in all fraction tasks. Specifically,
on the road fractions task, their organizations of space
were limited to the coordination of a single unit
(eighths) that is nested within a whole across only one
level of abstraction (a whole of an eighth). As a result,
their verbal explanations for the number of blocks on
the road tended to consist of a whole number, a cardinal position, or an addition problem.
Two of these six students showed marked improvement between the number and word fractions tasks with
regard to their ability to coordinate and, thus, to imagine the more complex grouping relationships inherent
in equivalent fractions (i.e., fourths nested within
halves, nested within the whole) when questions were
posed to challenge their constructions. However, they
remained dependent on manipulatives to support their
imaginal anticipations. Problem similarity most likely
influenced improved performance between the tasks to
some degree. However, their increased goal-directness in
anticipating nested units within units indicated awareness of the higher-order nested hierarchies to guide their
reflective thinking. Related to this justification for
improved performance is the observation that no student commented that the problems were the same.
One student (of the six discussed thus far) showed
evidence of spatial thought structures that were further
along in the process of reorganization onto a higherorder level. Specifically, on the road fractions task he
attempted to determine how his thinking differed from
the correct solution provided on the counterexample
for 3/4. Further, on the number fractions task, he progressed to interpreting the nested units inherent in
equivalent fractions within the number fractions task.
His thinking was sustained in the word fraction problems that followed without the use of manipulatives to
assist with his reflections.
Finally, with the exception of a small degree of initial
confusion, the remaining student imagined the second-order nested units on all fraction problems without the use of manipulatives to guide his reflections.
Because students are most successful at assimilating
nested units within portions of fractions using one half
(Mack, 1998; Piaget et al., 1960), this student will be
most appropriately challenged by introducing other
fraction quantities not divisible by two. These problems will test his understanding of rational number
(i.e., quotients that owe their existence to the ideas of
unit fractions (1/b)). However, in a rational number,
the quotient is a ratio unit (2/8 = 1/4) in which both
the numerator and the denominator are the result of a
split that comprises a single invariant unit (Behr, Harel,
Post, & Lesh, 1993; Kieren, 1993; Piaget & Inhelder,
1975). I am puzzled as to why this student did not
achieve success on the unit iteration task (line measurement) in light of this achievement. Perhaps this performance discrepancy is due to the fact that there was
no opportunity to call his reasoning into question and,
therefore, for him to reflect on his initial response on
the unit iteration task. Had I gone back and asked him
to measure the road again after the road fractions task,
he may have succeeded on this task.
Many students who could not coordinate the secondorder nested groupings inherent in equivalent fractions
showed much confusion between explicitly taught
skills to solve equivalent fractions and their ability to
reflect on their actions in an intelligent manner. This
confusion stems from the fact that explicitly taught
skills do not emphasize the relationships, operations,
and transformations that are psychologically and educationally significant to children's problem-solving
experiences (Streefland, 1993). For example, these students were unable to image 2/8 as 1/4 because their
spatial thought structures could not simultaneously
imagine the two units of eight as nested within one
unit of four quarters. Thus, they required questioning
that challenged their constructions of simple fractions.
Learning Disability Quarterly
166
If we fail to attend to the quality of children's imaginal anticipations that are derived from their spatial
thought structures, learning will be removed from its
self-organizing activity in its conception and become
static pieces of information. This disconnection in children's biologically based anticipatory activity results in
interference from rote procedures in problem-solving,
transfer (Mack, 1990), and growth in hierarchical structures of thinking (Kamii & Clark, 1995). Thus, distortions that come to be labeled as "disabilities" are not
solely in the learner, but are created in the interactive
relationship between the learner and the demands of
the instructional process.
When students' spatial thought structures were
evolved to the degree necessary to coordinate the second-order nested units inherent in equivalent fractions, they benefited from questions that challenged
them to attend to such relationships and generalized
their thinking to other problems. Also generalized to
other problems were suggested procedures to assist
them. Thus, if procedures are congruent with children's ways of ordering objects that are acted upon,
they become part of that child's reflections and will be
transformed into more complex procedures as their
thinking structures are transformed. It is significant to
note that the student who achieved at the lowest end
of the continuum in terms of the degree of expansion
in his spatial thought structures and the student who
achieved at the highest end were receiving the same
explicitly based instruction in fraction equivalency.
However, only the student at the higher end of the
continuum was flexible and coherent in applying
these procedures.
In conclusion, this article asked the reader to consider an alternative perspective regarding the development of geometry and fractions, the nature of learning problems in this area, and the teaching methodologies that mirror such assumptions. It follows that
research that examines geometric spatial thought
structures and teaching methodologies that help
evolve such structures is also necessary. In the limited
research on the quality of imaginal anticipations
when acting on geometric forms in children with LD,
delays in spatial thought structures were evident as
compared to their peers with NLD (Grobecker & De
Lisi, 2000). However, much more scientific scrutiny is
necessary to validate both the cause of geometric difficulties and the teaching methodologies that best
advance children's thinking.
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NOTES
'Additionalstandardizedtests in mathematics were not administeredbecause these tests (a) are limited in assessingfractionsand
measurement, and (b) fail to accurately capture classroom performance levels in mathematics (Parmar, Frazita, & Cawley,
1996).
Correspondence concerning this article may be addressed to:
Betsey Grobecker,3801 Winchell, #110, Kalamazoo,MI 49008.
2000 CLD OUTSTANDINGRESEARCHERAWARD
Sponsoredby the Councilfor LearningDisabilities
To promote and recognize research, the COUNCIL FOR LEARNING DISABILITIES
annually presents an award for an outstanding manuscript-length paper on learning
disabilities based on a doctoral dissertation or master's study completed within the last
five years.
The winner will receive a certificate to be presented at the Distinguished Lecture,
Saturday, October 21, 2000, during the 22nd International Conference on Learning
Disabilities in Austin, Texas. In addition, the paper will be considered for publication
in the Learning Disability Quarterly.
Six copies of the APA-style paper (25 pages) should be submitted to the Council for
Learning Disabilities, P.O. Box 40303, Overland Park, KS 66204. 913/492-8755
Deadline for submission of papers: May 1, 2000
Winners will be notified by August 15, 2000
Learning Disability Quarterly
168