THE DEVELOPMENT OF THE
PART-WHOLE AND MEASURE SUBCONSTRUCTS OF FOUR SIXTH-GRADERS DURING AN
INSTRUCTIONAL UNIT
Carol Novillis
Larson
University of
Arizona
clarson@u.arizona.edu
Abstract: Four sixth-grade students were interviewed prior to and following a six-week unit that focused on fraction concepts. The measure subconstruct of the rational number construct was tested using a number line model and an eighth-inch ruler, and the part-whole subconstruct was tested using rectangular regions. The four students’ ability to model and identify proper fractions, improper fractions, mixed numerals, and equivalent fractions with these three models are described. The students performed differently with the three models in both interviews.
The measure subconstruct is
one of the four subconstructs included
in Kieren’s (1993) analysis of the rational number construct. Behr, Post, Lesh and Silver (1983) have
included a distinct part-whole subconstruct.
Kieren (1993) and Behr et al (1983) agree that for students to
understand rational numbers, they must understand all of the interpretations of
rational numbers and the interrelationships between them. Kouba, Zawojewski and Strutchens (1997) and
Larson (1987) report that students are better able to associate fractions with
parts of geometric regions than to locate fractions on number lines. Larson (1987) and Bright et al. (1988) have
shown that locating fractions on a number line is very difficult for both
elementary and middle school students.
In past research, rulers and number lines have been considered to be
essentially the same model, with the result that few rational number research
studies have compared students’ identification of fractional parts of linear
units to locating fractions on a number line.
The purpose of this research is to explore sixth-graders’ development of
a measure subconstruct of rational numbers as represented by two different
exemplars of the measure subconstruct: the number line and the scale on an
eighth-inch ruler. Another aim is to
contrast their part-whole subconstruct with their measure subconstruct.
Design of the Study
The data presented here is
part of a study that focused on one sixth-grade teacher, Anne, teaching a
six-week unit on rational numbers at the beginning of the school year. The content addressed in 18 of the class
periods was for students to understand proper fractions, mixed numerals,
improper fractions, and equivalent fractions using area models (seven classes),
set models (seven classes), and the number line (four classes). Students used models or drawings to
represent fractions and equivalencies in groups of four students. They also participated in large group
discussions led by the teacher. No
instruction was given on fractional parts of an inch. The researcher observed and took notes during 15 of the 23 class
periods in which the unit was taught.
These classes were also audio-taped.
The focus in this paper is
on the understanding of four students in Anne’s class of the measure and
part-whole subconstructs for rational numbers prior to and following
instruction. The students were selected
by the teacher to represent the various levels of student achievement in her
classroom. The students responded to
interview tasks prior to and following instruction, all of the interviews were
audio-taped and later transcribed. Part
of each interview included tasks where the students related proper fractions,
improper fractions, mixed numerals and equivalent fractions to area models,
number lines, and an eighth-inch ruler.
Two types of tasks were used to test the area and number line models: 1)
the students were shown a shaded region (or regions) or an indicated point on a
number line scaled from 0 to 3 and asked to identify the fraction and 2) the
students were given the fraction and asked to generate the model by shading one
or more nonpartitioned rectangular regions or showing the location of the
fraction on a number line scaled from 0 to 3 on which only the points
associated with the whole numbers 0 to 3 were marked. All of the ruler tasks involved the accurate
measurement of strips of paper using an eighth-inch ruler.
Results
The results are presented
for each sixth-grade student in order to show the development of knowledge in
each student for each model.
Laura
The teacher identified Laura
as a high achieving student.
Area Tasks. Laura modeled proper
fractions with an area model and wrote proper fractions for area models in both
interviews. In the pre-interview she
was unable to identify 1 1/4 regions and when asked for a fraction she gave 5/8
rather than 5/4. When asked to model 3/2 she divided 3 by 2 and then shaded in
1 1/2. She could not give a second fraction to describe the region that was 6/8
shaded, but after a period of trial and error could shade 2/3 of a rectangle
with 6 equivalent parts. In the
post-interview she correctly responded to all area tasks similar to the ones on
the pre-interview.
Number Line Tasks. In both interviews when
given a number line with only whole numbers marked, she was able to correctly
mark where proper fractions and mixed numerals would be located. In both interviews when faced with an
improper fraction such as 5/4 she immediately divided and then correctly marked
1 1/4. In the pre-interview when asked
to indicate the number associated with a point on the number line, Laura
counted points rather than segments so that in each case she was off by one,
e.g., she labeled 3/6 as 4/7. In the
post interview she not only correctly identified the marked points on the
number line but indicated two or three names for each point, e.g., 2 4/6 was
labeled, 2 4/6, 2 2/3, 1 1/2, 9/6, a lot of names.
Ruler Tasks. On both interviews, Laura correctly measured all
strips and wrote the correct proper fractions and mixed numerals, but she never
indicated the unit, inches. In the
pre-interview when asked to indicate the unit after successfully writing 3 1/2,
she said the 3 was inches and the 1/2 was centimeters. For 2 3/8, in both interviews, she said the
2 was inches, but that she didn’t know what the little ones were. In all cases where it was possible to give
equivalent fractions, she did so without any prompting.
John
The teacher identified John
as an average achieving student.
Area Tasks. In
the pre-interview, John could successfully do all area model tasks for proper
fractions, improper fractions, and mixed numerals. He could not do the two tasks that had to do with equivalent
fractions. Given a rectangle with 6 of
8 equivalent parts shaded he could indicate 6/8 was shaded but not 3/4. Similarly when given a rectangle with 6
equivalent parts, he could not shade in 2/3 of the rectangle. In the post-interview he successfully did
all proper fractions, improper fractions, mixed numerals, and equivalent
fractions tasks.
Number Line Tasks. In the pre-interview John was
unable to correctly respond to any of the number line items. In the
post-interview, he correctly completed all tasks involving proper fractions,
mixed numerals, and improper fractions.
Ruler Tasks. In the pre-interview, John correctly indicated that
the length of a strip that was 3 1/2 inches long. He estimated to the nearest half-inch the other two strips that
were longer than an inch. In the post
interview he gave accurate measures and where appropriate equivalent fractions,
e.g., 1 2/8 inches was also 1 1/4 inches.
Anita
The teacher identified Anita
as an average achieving student.
Area Tasks. In the pre-interview Anita
could give a proper fraction, and a mixed numeral for appropriate area
models. When asked to describe 1 1/4
regions with a fraction she used 5/8.
She could not indicate that 6/8 of a region could also be described with
3/4. When asked to shade in regions to
show fractions, she could not do any of the tasks. The reason she could not show 2/5 of a rectangle was because she
could not physically make 5 equivalent parts.
In the post-interview, Anita correctly described the same model as 1 3/4
and 7/4 shaded, and she successfully modeled the improper fraction 9/4. She continued to have problems in partitioning
a sheet of paper into fifths to show 3/5, she knew she needed five equal parts
but could not do it. She was able to
indicate that 8/10 of a rectangle was shaded and then reduced to 4/5 when asked
for a second fraction but she could not relate the fraction 4/5 to the
model. Also, she could not shade 3/4 of
a rectangle partitioned into 12 equivalent parts.
Number Line Tasks. In the pre-interview, Anita
was unable to correctly respond to any of the number line items. In the post-interview, she correctly
completed all tasks involving proper fractions, mixed numerals and improper
fractions.
Ruler Tasks. In the pre-interview when measuring strips, all lengths were
incorrect. When the length was a mixed
numeral, she always had the correct whole number but the incorrect
fraction. In the post-interview she
rounded each measure to the nearest inch, e.g., a strip 2 3/8 inches long was
“about 2 inches.” In both interviews
she correctly labeled all units as inches.
Jim
The teacher identified Jim
as a low achieving student.
Area Tasks. Jim was unable to respond correctly to any of the area tasks in
the pre-interview. In the
post-interview he correctly identified 8/10 of a shaded region but could not
give the equivalent fraction 4/5. He
correctly identified 1 3/4 regions that were shaded but said a related fraction
was 7/8. When asked to model fractions
the only one he could do was 3/5.
Number Line Tasks. In the pre-interview, Jim was unable to successfully do any of
the number line tasks. In the
post-interview, he correctly completed all tasks involving proper fractions,
mixed numerals and improper fractions.
Ruler Tasks. In the pre-interview when
measuring with an eighth-inch ruler, Jim called all fractional parts of an inch
“quarters.” He labeled only the whole
number part of measures as inches, e.g., the length 2 3/8 inches was called “2
inches and 3 quarters.” In the
post-interview, he eventually identified the length of two strips in terms of
mixed numerals but only labeled the whole number part as inches. So a length of 2 3/8 inches was first called
“2 inches and 3 millimeters” and after a discussion called “2 inches and
3/8.” When asked what you could call
the 3/8, he said he didn’t know.
Discussion
All of the four students in
the study increased their knowledge and understanding of rational numbers as a
result of the instructional unit. Each
student’s profile of knowledge prior to and following instruction was specific
to that student. The four students
responded differently to the three models for fractions. The scale on a ruler and the number line
were not treated the same by the students.
The area in which they made the most progress was in associating
fractions with points on the number line.
Prior to instruction, three of the students could not associate proper
fractions, improper fractions and mixed numerals with points on the number
line. The fourth student, Laura, could
indicate a correct point given the fraction but could not correctly indicate the
fraction for a given point. About a week
after the completion of the unit all four students could correctly do all
number line tasks.
In the post-interview, Laura
and John could identify and model proper fractions, improper fractions, mixed
numerals and equivalent fractions using rectangular regions. Anita and Jim were still not proficient with
all the modeling and identifying tasks with area models. Jim could not model an
improper fraction with an area model yet he successfully modeled an improper
fraction on the number line.
Measuring to the nearest
eighth-inch with a ruler was not addressed in instruction as the teacher ran
out of time and decided that she needed to begin a unit on a new mathematical
topic. So a question to examine is: Did
the three students, John, Jim, and Anita, who could not correctly identify
fractional parts of an inch prior to instruction, improve in this area? Following the unit, Anita simply estimated
to the nearest inch, John and Jim read the scale on the ruler and identified
the correct mixed numerals. John also
indicated the length in terms of the correct number and units, inches. Jim was confused about the unit as it
related to the whole number and fractional parts of mixed numerals. Laura also showed this same confusion in
both interviews. Laura and Jim indicated
that the whole number was inches but at times that the fraction was centimeters
(Laura) or millimeters (Jim). After
questioning in the post-interview, both students said that they knew the whole
number was inches, but didn’t know what the “little ones” were. One explanation for this confusion could be
that throughout elementary textbook series, inches and centimeters are taught
one or two days apart. There is seldom
emphasis on units such as, inches and centimeters, being part of different
systems of measurement and why. Another
aspect of measurement that was apparent from the interviews is the role of
estimation in measuring. John and Anita
both estimated lengths to a specific unit.
In the pre-interview, John systematically measured to the nearest
half-inch; in the post-interview, Anita estimated to the nearest inch. The students never made estimates of this
type when trying to identify the number to correspond with points on the number
line or when associating fractions with an area model.
This research indicates that
the use of rational numbers in measurement situations needs to be taught to
show the connection between the mathematical topics of measurement and rational
numbers. This is an example of the need
for mathematical ideas to be interconnected as described in the new NCTM
Standards (National Council of Teachers of Mathematics, 2000). Students would benefit from comparing
various measurement scales to related number lines and on discussing measurement
units when fractional parts are involved.
References
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