One of the hardest math skills for students to learn is fractions.
This study used a multiple probe across participants to evaluate the
effectiveness of a mnemonic strategy called LAP Fractions on six middle
school students' ability to add and subtract fractions. Five of six
participants were able to achieve mastery, and all six students
maintained gains over a period of 6 weeks. Results are discussed in
terms of implications for research and practice.
Fractions are among the most difficult academic skills for students
to learn. Those who enter continuing education programs like the Armed
Services, often say that fractions are an area of mathematics that
caused them frustration (Riddle & Rodzwell, 2000). If one has a
learning problem, then this becomes an even more difficult skill to
Research on teaching fractions to students with disabilities has
been identified as an area in need of further research. For example, a
search of articles included in three literature reviews of math
interventions (Maccini & Hughes, 1997; Mastropieri, Scruggs, &
Shiah, 1991; Miller, Butler, & Lee, 1998) resulted in a total of
three research studies involving fractions (Kelly, Carnine, Gersten,
& Grossen, 1986; Kelly, Gersten, & Carnine, 1990; Miller &
Cooke, 1989). All three studies involved the use of videodisc
instruction for teaching fraction computational skills. More recently,
Joseph and Hunter (2001) used a cue card strategy for solving addition
and subtraction problems containing fractions. In this study, students
were given a cue card that provided prompts on how to work a fraction
problem. The cue cards asked questions such as "are the numbers of
the fractions the same on the bottom?" The cards then provided
answers: "If yes, add or subtract on top only. If no, find the
least common multiple and multiply top and bottom numbers by the same
number to get both bottom numbers equal to each other." Using this
strategy, students increased their performance in solving fraction
Many students with disabilities and those at risk for educational
failure exhibit problems with remembering academic material (Mastropieri
& Scruggs, 1998). One intervention that has been successfully used
to teach students with disabilities academics is the use of mnemonics
(Bottge, 1999; Cade & Gunter, 2002; Greene, 1999; Manalo, Bunnell,
& Stillman, 2000; Steele, 2002). A mnemonic strategy is defined as a
word, sentence, or picture device or technique for improving or
strengthening memory (Lombardi & Butera, 1998). Manalo et al. (2000)
investigated the effects of mnemonic instruction on computational skills
versus direct instruction study skills instruction, and no instruction.
The study was conducted in a middle school with 29 students who had a
learning disability in math. Twenty-three of the participants were
female and 6 were male. The students were divided into four groups. The
findings from the research were that the mnemonic instruction group made
significant improvements in their computational skills.
While research has demonstrated that mnemonic strategies have been
successful in teaching math skills, only one study was found that used a
mnemonic strategy to teach addition and subtraction of fractions (Joseph
& Hunter, 2001). Given the support for mnemonic strategies in the
literature and the lack of research on teaching fractions, the purpose
of this study was to investigate the effects of an additional mnemonic
device, LAP Fractions, on the addition and subtraction of unlike
fractions with students who demonstrate leaming problems.
The participants for this study were 6 eighth-grade students in a
special education math classroom. All students had demonstrated skill
deficits in mathematics and had not previously received any instruction
in adding or subtracting fractions. Three participants were identified
as educable mentally disabled with an IQ range of 65-70. The other 3
students had a learning disability in math with an IQ range of 96-115.
All students knew their multiplication tables and were able to divide
one digit by one digit whole numbers. Three of the students were black
and 3 were white. Standardized test scores from the Woodcock-Johnson
ranged from a grade level of 3.0 to 7.0.
Students were paired based on math ability level and compatibility
as observed by the second author. Before the study began, each pair
demonstrated their ability to work together on academic tasks. Pair one
included a black male who was identified as learning disabled. He had an
IQ of 96 and scored a 6.0 grade level on the Woodcock-Johnson math test.
The other participant was a black female who was identified as educable
mentally disabled. She had an IQ of 68 and scored a 6.0 grade level on
the Woodcock-Johnson math test. Pair two included a black female who was
labeled as educable mentally disabled. She had an IQ of 69 and scored a
5.0 grade level on the Woodcock-Johnson math test. The other participant
was a white male who was identified as leaming disabled. He had an IQ of
115 and scored a 5.0 grade level on the Woodcock-Johnson math test. Pair
three included a white male who was identified as learning disabled. He
had an IQ of 98 and scored a 4.0 grade level on the Woodcock-Johnson
math test. The other participant was a white female who was labeled
educable mentally disabled. She had an IQ of 68 and scored a 3.0 grade
level on the Woodcock-Johnson math test.
The study was conducted in a middle school in a small town located
in the southeastern part of the United States. The school had
approximately 750 students and 65 teachers. The second author, a teacher
certified in special education, conducted the study in his two math
resource classrooms. One class had six students and the other had five
The experimenter for this study was a certified special education
teacher with 22 years experience teaching students with learning
disabilities and mental disabilities. The teacher had a BA in Mental
Retardation and Learning Disabilities and was working on his
master's degree in special education.
Data Collection Procedures
This study had two dependent variables. The first dependent
variable was the percentage of steps correctly stated out of the eight
steps for LAP Fractions (called LAP Fractions strategy test). The second
dependent variable was the percent correct on a test that included 18
problems requiring addition and subtraction of fractions (called LAP
Fractions test). For the percentage of correct steps on LAP Fractions,
the instructor recorded the verbal response for LAP Fractions using a
checklist (See Table 1). A (+) symbol was used to record correct
answers. A (-) symbol was used for incorrect answers. A Type 1 fraction
was an addition or subtraction of like fractions. Type 2 was an addition
or subtraction of an unlike fraction. Type 2 denominators will divide
into each other an even number of times. Type 3 fractions were addition
and subtraction of unlike fractions. Type 3 fractions denominators will
not divide into each other an even number of times.
The second dependent variable, percentage of addition and
subtraction fraction problems worked correctly was also recorded using a
checklist. A (+) was used for problems worked correctly and (-) for
problems worked incorrectly. Students were given 18 problems, three
addition and three subtraction of each fraction type, to solve in a
9-min period. Ten 18-item tests were randomly generated. Tests were
rotated so that students could not memorize the problems.
Interobserver reliability. The primary observer in the study was
the special education teacher. A second special education teacher was
trained to collect interobserver reliability data. Interobserver
agreement on the steps for LAP Fractions and the percentage of addition
and subtraction problems worked correctly was conducted for 20% of tests
given throughout the study. Using an item-by-item analysis,
interobserver reliability was 100% for all measures.
The study was originally designed to use a multiple probe design
across three pairs of students (Cooper, Heron, & Heward, 1987).
However, before the original third pair of students began intervention,
one of the students tested out of special education. As a result, a
delayed multiple probe across pairs design had to be used (Heward, 1978;
Watson & Workman, 1981). The intervention was presented to one pair
of students at a time in a staggered fashion. After stability was
achieved in baseline, LAP Fractions was introduced to the pair of
students most in need of intervention with the other students remaining
in baseline. Once the first pair mastered the LAP Fractions strategy and
moved into LAP Fractions intervention, the next pair was introduced to
the LAP Fractions strategy instruction. This continued until all student
pairs entered the intervention phases. Once a pair achieved 89% on the
fraction test for 3 consecutive days, the pair entered maintenance.
Baseline. During baseline, students were given two tests. The first
was for the eight steps of LAP. The second was a sheet of 18 addition
and subtraction fraction problems. Baseline data were gathered until
stability was achieved.
LAP Fractions strategy instruction. The procedures used to teach
the LAP Fractions strategy began by introducing each pair of students to
the strategy (See Table 1). This was done by using an overhead
transparency describing the strategy. After introduction, students were
given an index card with the mnemonic strategy: (L) Look at the
denominator and sign. (A) Ask yourself, the question, "Will the
smallest denominator divide into the largest denominator an even number
of times?" (P) Pick your fraction type. During guided practice,
students and teacher read aloud the above steps together. Students then
were asked to read aloud the steps individually. Afterwards, the 2
students worked together to practice each letter of the LAP mnemonic
using activities designed to reinforce their learning for 30 min
sessions. The activities used were LAP Fraction Baseball and/or ZAP. LAP
Fraction Baseball was played just like a baseball game. Students picked
up a card which had a letter printed on it (i.e., L, A, or P) and stated
its meaning. If correct, the student rolled a die. If they rolled a 1 or
2 they got a single, 3 or 4 double, 5 triple, 6 out. If a student gave
an incorrect answer they also were issued an out. Students continued to
play until they got three outs. The student with most runs at the end of
the game won.
The game of Zap was played as a card game. Sixteen cards were used:
five with each letter for LAP on them and one card with the word Zap on
it. Cards were mixed and placed in the middle of the table and a timer
was set for 30 min. Students took turns drawing a card and stating the
answer for the letter on the card. If the correct answer was given, the
student kept the card. If the student drew the "Zap" card,
they returned all their cards to the pile. Students continued to play
until the timer sounded. The one with the most cards won. Once each pair
could state the LAP Fractions strategy correctly, they moved onto the
next part of LAP Fractions strategy instruction.
After learning the above LAP mnemonic, the students practiced
pointing to the bottom number or denominator of a given fraction. As
partners, they worked together on the procedure of identifying the
denominator using flash cards. This session lasted for 30 min. At the
end of each instructional session the instructor took each participant
one-on-one and gave them the eight-item LAP Fractions strategy test.
This session continued until students achieved 100% accuracy for two
After students mastered the above two steps, the students were
taught to determine if the smallest denominator would divide into the
largest denominator without a remainder. Again, working with a partner,
the students practiced dividing the smallest denominator into the
largest while playing teacher-made activities. Two teacher-made
activities were used during these sessions: Fraction Football and
Fraction Basketball. Fraction Football was a game designed for two
players. It was played like a football game. Students drew a card from a
deck of 30 different fractions problems. The students divided the
smallest denominator into the largest denominator. If correct, the
student picked a card from the yardage deck, which gave them positive
yardage. If students answered incorrectly, a down was lost. The student
with the most points at the end of the game won. Fraction basketball was
also designed for two players. Students drew from a deck of 30 different
fraction problems. They divided the smallest denominator into the
largest denominator. If correct, students got two points, and the
student with the most points at end of the game won. These sessions
lasted 30 min.
After mastery of the three steps above, the students learned and
practiced the three types of fractions with a peer partner. They
visually identified and verbally gave the fraction type. For this study,
students looked at Type 1, Type 2, and Type 3 fraction problems only.
Students practiced using the teacher-made activities described above.
These sessions lasted 30 min. Throughout the LAP Fractions strategy
instruction, at the end of each instructional session, the instructor
administered the LAP Fractions strategy test to each student
individually. LAP Fractions strategy instruction continued until both
students in a pair were able to achieve 100% accuracy for the eight
steps of the LAP strategy for two consecutive sessions.
LAP Fractions intervention. After mastering the LAP Fractions
strategy, students learned how to use the strategy to solve each
fraction type (see Table 2). Using the overhead away from the other
students' view, each pair of students was introduced to procedures
for solving Type 1 fractions. Students first practiced working Type 1
problems with the teacher at a back table using a small whiteboard.
Students then practiced working Type 1 fractions while playing one of
the teacher-made activities. This some instructional procedure was used
to teach Type 2 and Type 3 fractions. Students engaged in the
teacher-made activities for 30-min sessions, continuing with the same
fraction type until they obtained 89% accuracy. After mastering the
three fraction types in isolation, students were given 30 flashcards
containing addition and subtraction problems that included 10 Type 1,
Type 2, and Type 3 fractions. Students practiced applying LAP to these
problems by playing Baseball and Zap. Each session lasted for 30
Throughout the LAP Fractions intervention, at the end of each
instructional session, the instructor gave students an 18-item LAP
Fractions test. LAP Fractions instruction continued until the pair
obtained 89% accuracy for 3 consecutive days. After the LAP Fractions
intervention phase, students entered the maintenance phase.
Maintenance. During maintenance, students were given both the LAP
Fractions strategy test and the LAP Fractions test every 10 days for a
Procedural reliability data were gathered by the same special
education teacher who collected interobserver reliability data.
Procedural reliability data were gathered to determine the
instructor's accuracy of implementing the eight steps of the LAP
strategy and teaching students how to apply the LAP Fractions strategy
to actual addition and subtraction problems. A checklist of
instructional steps was developed and used to gather procedural
reliability data were collected twice for each phase of the study and
100% procedural reliability was maintained throughout all observed
sessions (See table 3).
After completing the study, each student was given a questionnaire
with three questions concerning the usefulness of LAP Fractions. Another
teacher, who was not involved in the study, administered the
questionnaire. This teacher read the questions aloud to students and
instructed them to circle "Yes" or "No" for their
answer. The three questions were: (1) Was LAP Fractions easy for you to
learn? (2) Are fractions easier for you to work now? (3) Do you like
working fractions using the LAP Fractions strategy?
The results of each student's performance on the LAP Strategy
and 18-item fraction test can be found in Figure 1. The results are
briefly summarized below.
LAP Fractions strategy. During baseline, Student 1 performed a mean
of 12% of the trials correctly for 3 days. During intervention, it took
Student 1 five days to reach mastery. He had a range of 62% to 100% with
a mean score of 84.8%. During maintenance, Student 1 maintained 100%
accuracy on the LAP Fractions strategy.
During baseline, Student 2 performed 12% of the trials correctly on
the LAP strategy for 3 days. During intervention, Student 2's
performance improved to 87% on day 1. After 5 days, Student 2 mastered
the intervention. She had a range of 78% to 100% with a mean score of
83.8%. During maintenance, Student 2 maintained 100% accuracy. As a
pair, it took 5 days to meet the mastery criterion of 100% for 2 days
for both students (See Figure 1).
LAP Fractions intervention. During baseline, Student 1 performed
33% of the trials correctly for 8 days. During intervention, Student
1's performance improved to 44% on day 1. After 8 days, Student 1
mastered applying LAP Fractions by maintaining at least 89% accuracy for
3 consecutive days. He had a range of 44% to 94% with a mean score of
68.5%. During maintenance, Student 1 maintained a mean score of 87% (See
During baseline, Student 2 performed 33% of the trials correctly
for 8 days. During intervention, Student 2's performance was 50% on
day 1. After 15 days, Student 2 mastered the intervention by maintaining
at least 89% accuracy for 3 consecutive days. She had a range of 50% to
100% with a mean score of 79.2%. During maintenance, Student 2
maintained a mean score of 83.3%. As a pair, it took 15 days to meet the
mastery criterion of 80% for 3 days for both students (See Figure 1).
LAP Fractions strategy. During baseline, Student 3 performed 12% of
the trials correctly for 3 days. During intervention it took Student 3
four days to reach mastery. He had a range of 50% to 100% with a mean
score of 82%. During maintenance, Student 3 maintained 100% accuracy on
the LAP Fractions strategy.
During baseline, Student 4 performed 12% of the trials correctly
for 3 days. During intervention it took Student 4 six days to reach
mastery. She had a range of 50% to 100% with a mean score of 83.7%.
During maintenance, Student 4 maintained 100% accuracy. As a pair, it
took 6 days to meet the mastery criterion of 100% for 2 days for both
students (See Figure 1).
LAP Fractions intervention. During baseline Student 3 performed 12%
to 33% of the trials correctly with a mean of 30.6% correct. During
intervention, Student 3 performed 44% of the trials correctly on day 1.
After 18 days, Student 3 mastered the intervention by maintaining at
least 89% accuracy for 3 consecutive days. He had a range of 44% to 94%
with a mean score of 66.1% correct. During maintenance, Student 3
maintained a mean score of 80.5% on the LAP Fractions test (See Figure
[FIGURE 1 OMITTED]
During baseline, Student 4 averaged 33% correct. During
intervention, Student 4 performed 33% of the trials correctly on day 1.
After 13 days, Student 4 mastered the intervention by maintaining at
least 89% accuracy for 3 consecutive days. She had a range of 33% to 89%
with a mean score of 73.7% correct. During maintenance, Student 4
maintained a score of 83.3% on the LAP Fractions test. As a pair, it
took 15 days to meet the mastery criterion of 89% for 3 days for both
students (See Figure 1).
LAP Fractions strategy. During baseline, Student 5 performed 12% of
the trials correctly for 3 days. During intervention, it took Student 5
four days to reach mastery. He had a range of 62% to 100% with a mean
score of 84.2%. During maintenance, Student 5 maintained 100% accuracy
on the LAP Strategy (See Figure 1).
During baseline, Student 6 performed 12% of the trials correctly
for 3 days. During intervention, it took student 6 four days to reach
mastery. She had a range of 50% to 100% with a mean score of 81.25%.
During maintenance, Student 6 maintained 100% accuracy on the LAP
Strategy. As a pair, it took 4 days to meet the mastery criterion of
100% for 2 days for both students (See Figure 1).
LAP Fractions intervention. During baseline, Student 5 performed
33% of the trials correctly correct for 7 days. During intervention,
Student 5's performance was 60% correct on day 1. After 17 days,
Student 5 mastered the intervention by maintaining at least 89% accuracy
for 3 consecutive days. He had a range of 60% to 94% correct with a mean
score of 82.5%. During maintenance, Student 5 maintained a mean score of
83% correct (See Figure 1).
During baseline, Student 6 performed 33% of the trials correctly
for 7 days. During intervention, Student 6's performance was 50%
correct on day 1. After 28 days, Student 6 did not master the
intervention by maintaining at least 89% accuracy for 3 consecutive
days. She had a range of 50% to 72% with a mean score of 56.7% correct
(See Figure 1). During maintenance, student 6 maintained a mean score of
55% correct (See Figure 1).
Social validity. The results for the three questions asked were as
follows. First, 100% of the students indicated LAP Fractions was easy
for them to learn. Second, 83% of the students thought fractions were
easier for them to work now. And third, 83% of the students liked
working fractions using LAP Fractions. Student 6, who did not master the
skills taught, did not agree that fractions were easier or that she
liked working fractions using LAP Fractions.
This study was designed to investigate the effects of a mnemonic
device called LAP Fractions on students' ability to solve math
problems involving addition and subtraction of fractions. Results
indicated a functional relationship between implementing LAP Fractions
and student acquisition of both the LAP Fractions strategy and their
ability to apply the strategy to adding and subtracting fractions.
Results indicated that 5 out of 6 students mastered both skills and
maintained their performance over a period of 6 weeks. While one student
did not reach mastery criteria on solving math problems involving
addition and subtraction of fractions, she did reach mastery criteria on
the LAP Fractions strategy. An item analysis of errors made by all 6
students indicated that errors were computational and not related to
applying the LAP Fractions strategy. Finally, social validity data
collected from students indicated that they liked using LAP Fractions
and that 5 of the 6 participants felt that it helped them learn to add
and subtract fractions.
The results of the present study add to the literature on teaching
math skills in a variety of ways. First, the results provide additional
support to the notion that mnemonics can help a student remember the
steps to working a math problem (Mastropieri & Scruggs, 1998).
Second, as recommended by Maccini and Hughes (1997) in their review of
mathematics interventions for adolescents with learning disabilities
from 1988 to 1995, this study included specific, detailed descriptions
of the independent variable (LAP Fractions) which should increase the
replicability of the procedures, as well as collection of maintenance
data on student performance for up to 6 weeks. Third, the results add to
the limited database of strategies for teaching students to solve
fraction problems. Previous literature reviews have noted the paucity of
research on instructional strategies involving fractions (Mastropieri et
al., 1991; Miller et al., 1998). Finally, the findings are an example of
a "low-tech" peer assisted leaming strategy. As such it also
adds to previous research using peers to teach solving math word
problems (Harper, Mallette, Maheady, & Brennen, 1993) and
calculating subtraction problems (Harper, Mallette, Maheady, Bentley,
& Moore, 1995).
The findings of the current study are limited by a number of
factors. First, as with any single subject research design, the small
number of students limits the generalizability of the results.
Systematic replication of LAP Fractions is needed to build this
generality. Second, the design was weakened by having the original third
pair of students drop out of the study and be replaced by 2 new
students. While the combination of the original two-tiered multiple
baseline and the delayed multiple baseline with the third pair did
indicate a functional relationship between the LAP strategy and solvinig
fraction problems, research is still needed to replicate the present
findings. For example, future research might investigate using different
practice games and activities with students. Students in the present
study suggested the need for more variety of games. Students also
suggested they would like to play games against other peer groups.
Future research might also be designed to investigate effects of
LAP Fractions on the performance of students without disabilities.
Research in teaching students without disabilities suggests the need for
data-based instructional strategies (Pressley & Woloshyn, 1995).
Finally, future research might be conducted to determine the effects of
LAP Fractions on the fluency with which students solved the fraction
problems. In the present study after about 8 days of using LAP
Fractions, students began working fraction problems "in their
heads." That is, students began working fraction problems without
writing down anything but the answer. While specific data were not
collected, this appeared to improve the rate at which students solved
In conclusion, the present study adds to the limited literature on
teaching students to add and subtract fractions using a student-friendly
mnemonic strategy called LAP Fractions. As a result, this study provides
further evidence that given systematic instruction, students with mild
disabilities can acquire complex math skills.
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David W. Test and Michael F. Ellis
University of North Carolina at Charlotte
Address corespondence to David Test, Special Education Program,
University of North Carolina at Charlotte, 9201, University City Blvd.,
Charlotte, NC 28223. E-mail: firstname.lastname@example.org.
Table 1 The Eight Parts of LAP Fractions
1. L-ook at the sign and denominator. Make sure the sign is addition or
subtraction. Look at the bottom numbers of your fraction. See if they
are the same or different. If they are the same, you skip down to
"Pick a fraction type," and pick Type 1. If they are different, then
you go to the "Ask yourself the question" step.
2. A-sk yourself the question. Will the smallest denominator divide into
the largest denominator an even number of times? If your answer is
yes, then you go down to "Pick a fraction type" and pick Type 2. If
your answer is no, then you go to "Pick a fraction type" and pick
3. P-ick your fraction type.
4. Type 1: 1/4 + 3/4 Bottom numbers are the same. Its sign is
addition or subtraction.
5. Type 2: 1/6 + 1/2 Bottom numbers are different and the smallest
bottom number will divide into the largest
bottom number an even number of times. Its sign
is addition or subtraction.
6. Type 3: 2/3 + 3/4 Bottom numbers are different and the smallest
bottom number will not divide into the largest
bottom number an even number. Its sign is
addition or subtraction.
7. Identify denominator Student points to denominator.
8. Divide denominator Student is shown two denominators. They divide
the smallest denominator into the largest
Table 2 Steps for LAP Fraction Intervention
Type 1 Fraction
1/5+2/5 Add or subtract your top numbers. Bottom numbers stay
Type 2 Fraction
3/8+1/4X2/2 Smallest number on the bottom will divide evenly into
the largest. Place box around smallest number on the
bottom side fractions. Ask "how many times will 4
divide into 8? Place a times sign and the answer you
get when you divide into the box.
3/8+_/_ Write fraction down that is not being changed under
original problem. Write down your sign and draw a new
fraction line beside the sign. Multiply your top
numbers in the box. Write new answer down on new top
fraction line. Multiply bottom numbers in box. Write
new answer down under new fraction line.
3/8+2/8 Add or subtract your top numbers. Bottom numbers stay
Type 3 Fraction
3/5+1/4 Smallest number on the bottom will not divide evenly
into the largest number on the bottom. Draw two new
fraction lines under original problem.
_/20+_/20 Multiply your two bottom numbers (4X5) and put this
answer as your new bottom numbers. Take your original
right bottom number (4) and multiply it by the original
top left number (3). Place this answer as your new left
top number. Take your original bottom left number (5)
and multiply it by the original top right number (1).
Place this answer as your new right top number.
12/20+5/20 Add or subtract your new top numbers. Bottom numbers
stay the same.
Table 3 Procedural Reliability Checklist
Identify Denominator ______
Divide Denominator ______
Steps for LAP:
L-ook at the sign and denominator ______
A-sk yourself the question ______
P-ick your fraction type ______
Identify fraction types:
Type 1 ______
Type 2 ______
Type 3 ______