Subject:

Mathematics
(Study and teaching)

Fractions (Study and teaching)

Disabled students (Education)

Fractions (Study and teaching)

Disabled students (Education)

Authors:

Test, David W.

Ellis, Michael F.

Ellis, Michael F.

Pub Date:

02/01/2005

Publication:

Name: Education & Treatment of Children Publisher: West Virginia University Press, University of West Virginia Audience: Professional Format: Magazine/Journal Subject: Education; Family and marriage; Social sciences Copyright: COPYRIGHT 2005 West Virginia University Press, University of West Virginia ISSN: 0748-8491

Issue:

Date: Feb, 2005 Source Volume: 28 Source Issue: 1

Product:

Product Code: 8524200 Mathematics NAICS Code: 54171 Research and Development in the Physical, Engineering, and Life
Sciences

Geographic:

Geographic Scope: United States Geographic Code: 1USA United States

Accession Number:

131753666

Full Text:

Abstract

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 learn.

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 problems.

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.

Method

Participants

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.

Setting

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 students.

Experimenter

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.

Experimental Design

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.

Procedures

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 consecutive sessions.

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 minutes.

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 6-week period.

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).

Social Validity

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?

Results

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.

Pair One

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 Figure 1).

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).

Pair Two

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 1).

[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).

Pair Three

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.

Discussion

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 problems.

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.

References

Bottge, B.A. (1999). Effects of contextualized math instruction on problem solving of average and below-average achieving students. Journal of Special Education, 33, 81-92.

Cade, T., & Gunter, P.L. (2002). Teaching students with severe emotional or behavioral disorders to use a musical mnemonic technique to solve basic division calculations. Behavioral Disorders, 27, 208-214.

Cooper, J. O., Heron, T. E., & Heward, W. L. (1987). Applied behavior analysis. Columbus, OH: Merrill-Prentice Hall.

Greene, G. (1999). Mnemonic multiplication fact instruction for students with leaming disabilities. Learning Disabilities Research & Practice, 14, 141-148.

Harper, G.F., Mallette, B., Maheady, L., Bentley, A., & Moore, J, (1995) Retention and treatment failure in classwide peer tutoring: Implications for further research. Journal of Behavioral Education, 5, 399-414.

Harper, G.F., Mallette, B., Maheady, L., & Brennan, G. (1993). Classwide student tutoring teams and direct instruction as a combined instructional program to teach generalizable strategies for mathematics word problems. Education & Treatment of Children, 16, 115-134.

Heward, W. L. (1978, May). The delayed multiple baseline design. Paper presented at the Fourth Annual Convention of the Association for Behavior Analysis, Chicago, IL.

Joseph, L. M., & Hunter, A. D. (2001). Differential application of a cue card strategy for solving fraction problems: Exploring instructional utility of the Cognitive Assessment System. Child Study Journal, 31, 123-137.

Kelly, B., Carnine, D., Gersten, R. S., & Grossen, B. (1986). The effectiveness of videodisc instruction in teaching fractions to learning disabled and remedial high school students. Journal of Special Education Technology, 8, 5-17.

Kelly, B., Gersten, R., & Carnine, D. (1990). Student error patterns as a function of curriculum design: Teaching fractions to remedial high school students and high school students with learning disabilities. Journal of Learning Disabilities, 23, 23-29.

Lombardi, T., & Butera, G. (1998). Mnemonics: Strengthening thinking skills of students with special needs. Clearing House, 71, 5, 284-290.

Maccini, P., & Hughes, C. A. (1997). Mathematics interventions for adolescents with learning disabilities. Learning Disabilities Research & Practice, 12, 186-176.

Manalo, E., Bunnell, J. K., & Stillman, J. A. (2000). The use of process mnemonics strategies in teaching students with math learning disabilities. Learning Disabilities Quarterly, 23, 137-156.

Mastropieri, M. A., & Scruggs, T. E. (1998). Enhancing school success with mnemonic strategies. Intervention in School and Clinic, 33, 201-213.

Mastropieri, M. A., Scruggs, T.E., & Shiah, S. (1991). Mathematics instruction for leaming disabled students: A review of research. Learning Disabilities Research and Practice, 6, 89-98.

Miller, S. P., Butler, F. M., & Lee, K. (1998). Validated practices for teaching mathematics to students with learning disabilities: A review of literature. Focus on Exceptional Children 31(1), 1-24.

Miller, S. C., & Cooke, N. L. (1989). Mainstreaming students with learning disabilities for videodisc math instruction. Teaching Exceptional Children, 21(3), 57-60.

Pressley, M.P., & Woloshyn, V. (1995). Cognitive strategy instruction that really improves children's academic performance (2nd ed). Cambridge, MA: Brookline Books.

Riddle, M., & Rodzwell, B. (2000). Fractions: What happens between kindergarten and the army? Teaching Children Mathematics, 7, 202-208.

Steele, M. M. (2002). Strategies for helping students who have learning disabilities in mathematics. Mathematics Teaching in the Middle School, 8(3), 140-144.

Watson, P. J., & Workman, E. A. (1981). The non-concurrent multiple baseline across individuals design: An extension of the traditional multiple baseline design. Journal of Behavior Therapy and Experimental Psychology, 12, 257-259.

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: dwtest@email.uncc.edu.

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 learn.

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 problems.

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.

Method

Participants

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.

Setting

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 students.

Experimenter

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.

Experimental Design

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.

Procedures

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 consecutive sessions.

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 minutes.

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 6-week period.

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).

Social Validity

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?

Results

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.

Pair One

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 Figure 1).

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).

Pair Two

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 1).

[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).

Pair Three

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.

Discussion

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 problems.

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.

References

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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: dwtest@email.uncc.edu.

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 Type 3. 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 denominator. Table 2 Steps for LAP Fraction Intervention Type 1 Fraction 1/5+2/5 Add or subtract your top numbers. Bottom numbers stay the same. 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 the same. 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 ______

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