(DOC) THE EFFECTS OF A MATH RACETRACK WITH TWO ELEMENTARY STUDENTS WITH LEARNING DISABILITIES
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THE EFFECTS OF A MATH RACETRACK WITH TWO ELEMENTARY STUDENTS WITH LEARNING DISABILITIES
Thomas Mclaughlin
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Abstract
A classroom intervention employing math racetracks was carried out to teach math facts to two elementary students with learning disabilities. A math racetrack is a drill and practice procedure where known and unknown facts are placed on a sheet of paper like an oval racetrack. The effectiveness of using math racetracks was evaluated with a multiple baseline design across problem sets. The results indicated that math racetracks were successful in increasing the skill sets of both participants in math. This provides a novel replication of employing a racetrack procedure that has been effective in reading, to elementary students in math. The practical implications of employing racetrack like procedures are discussed. Before a person can master difficult mathematical concepts it is important to be able to solve basic mathematical problems. Multiplication facts are a fundamental part of the primary math curriculum. It has been shown through research that students with learning disabilities often use counting strategies (e.g., finger counting and touch math) to solve basic mathematical problems (Ozaki, Williams, & McLaughlin, 1996). According to Curico (1999), Learning basic facts is not a prerequisite for solving problems, but learning the facts becomes a necessity to solve problems that are meaningful, relevant, and interesting to learners (p. 282). Mastering basic skills is necessary for students to be able to more efficiently solve difficult problems. Approaches focused around avoiding basic skills have often resulted in school and student failure (Curcio, 1999). Whole math, a form of instruction in which kids are encouraged to develop their own methods of multiplying and dividing, often producing answers that are close to correct are good enough, has been embraced enthusiastically around the country (Cheney, 1998). Many schools are now focusing on the concepts of math rather than the basics, often replacing memorization with calculators. This form of education was thought to be helpful to woman and minorities. However, in 1995, when schools were introduced to whole math test results revealed the program benefited almost no one, regardless of race or gender. The Department of Defense discovered that a year after the introduction of whole math into the Departments middle and elementary schools, the results were contrary to their expectations – 37,000 students took the Comprehensive Test of Basic Skills and scores dropped across all racial groups (Cheney, 1998; 1997). Removing emphasis from computation lowered the Department of Defense's school's scores more than anything else. It was in that area that the Department of Defense discovered a 9 point drop for third graders; 12 for fourth graders; 11 for fifth graders; 10 for sixth graders; 10 for seventh graders; and 4 for eighth graders (Cheney, 1997). Through such studies the importance
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
The International Journal of Special Education
2005, Vol 20, No.2.
THE EFFECTS OF A MATH RACETRACK WITH TWO ELEMENTARY STUDENTS
WITH LEARNING DISABILITIES
Breann R. Beveridge
Kimberly P. Weber
K. Mark Derby,
and
T. F. McLaughlin
Gonzaga University
A classroom intervention employing math racetracks was carried out to
teach math facts to two elementary students with learning disabilities. A
math racetrack is a drill and practice procedure where known and unknown
facts are placed on a sheet of paper like an oval racetrack. The effectiveness
of using math racetracks was evaluated with a multiple baseline design
across problem sets. The results indicated that math racetracks were
successful in increasing the skill sets of both participants in math. This
provides a novel replication of employing a racetrack procedure that has
been effective in reading, to elementary students in math. The practical
implications of employing racetrack like procedures are discussed.
Before a person can master difficult mathematical concepts it is important to be able to solve
basic mathematical problems. Multiplication facts are a fundamental part of the primary math
curriculum. It has been shown through research that students with learning disabilities often use
counting strategies (e.g., finger counting and touch math) to solve basic mathematical problems
(Ozaki, Williams, & McLaughlin, 1996).
According to Curico (1999), Learning basic facts is not a prerequisite for solving problems, but
learning the facts becomes a necessity to solve problems that are meaningful, relevant, and
interesting to learners (p. 282). Mastering basic skills is necessary for students to be able to
more efficiently solve difficult problems. Approaches focused around avoiding basic skills have
often resulted in school and student failure (Curcio, 1999).
Whole math, a form of instruction in which kids are encouraged to develop their own methods of
multiplying and dividing, often producing answers that are close to correct are good enough, has
been embraced enthusiastically around the country (Cheney, 1998). Many schools are now
focusing on the concepts of math rather than the basics, often replacing memorization with
calculators. This form of education was thought to be helpful to woman and minorities.
However, in 1995, when schools were introduced to whole math test results revealed the program
benefited almost no one, regardless of race or gender. The Department of Defense discovered
that a year after the introduction of whole math into the Departments middle and elementary
schools, the results were contrary to their expectations – 37,000 students took the Comprehensive
Test of Basic Skills and scores dropped across all racial groups (Cheney, 1998; 1997).
Removing emphasis from computation lowered the Department of Defense’s school’s scores
more than anything else. It was in that area that the Department of Defense discovered a 9 point
drop for third graders; 12 for fourth graders; 11 for fifth graders; 10 for sixth graders; 10 for
seventh graders; and 4 for eighth graders (Cheney, 1997). Through such studies the importance
of mastering basic skills of computation has been shown. Building fluency (i.e. improving speed),
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
as well as increasing accuracy in math should improve the likelihood of a student’s future
academic and social success (Ozaki et al., 1996).
In programs such as The Morningside Model of Generative Instruction, it has been shown to be
wise to focus the curriculum on learning basic facts before preceding to more difficult concepts.
On a course pretest of problem solving with fractions, four learners’ performance ranged from 3
to 7 problems correct out of 14. When tested a second time with significant component skills
present that were nonexistent at the time of the pretest, the student’s performance was
dramatically improved. These repertoires included the elements of problem solving with whole
numbers and fractional computation, learner performance on a second test ranged from 13 to 14
correct out of 14 (Johnson & Layng, 1994).
To improve the fluency and skill sets of children with disabilities, we have employed a self drill
and practice procedure called copy, cover, and compare (McLaughlin & Skinner, 1996; Skinner,
McLaughlin, & Logan, 1997). This procedure allows the student to self-tutor and engage in error
correction in a very straight forward and simple manner. Copy, cover, compare has been
implemented with a large number of children and across a wide range of academic behavior
(McLaughlin, Weber, & Barretto, 2004; Skinner et al., 1997).
Another procedure, which has been used to improve the accuracy and fluency that students read
Grade Two Priority Words in isolation, is the reading racetrack. With the implementation of the
reading racetracks there was an immediate improvement for the number of words read correctly.
The reading racetrack intervention was effective in improving frequency of words read correctly,
and eliminating almost all errors. This method was shown to be easy to implement and manage
(Anthony, Rinaldi, Hern, & McLaughlin, 1997). The reading racetrack also proved to be
effective on the fluency of see-to-say words in isolation by a student with learning disabilities in
another study by Rinaldi and McLaughlin (1996). The use of the reading racetrack in the study
by Rinaldi and McLaughlin (1996) also appeared to positively effect the participants reading
fluency while reading orally during the participant’s regular reading group. Recent replications
of copy, cover, and compare as well as reading racetracks have found that either of these
procedures can improve the basic skills of students with and without disabilities (Conley, Derby,
Roberts-Quinn, Weber, & McLaughlin, 2004; Falk, Band, & McLaughlin, 2002; Stone,
McLaughlin, & Weber, 2002).
The purpose of this study was to replicate what has already been done with reading racetrack,
but with the basic multiplication and division facts. The procedure aimed to determine whether
the procedure could be used to teach multiplication facts to children with learning disabilities and
whether or not the procedure could be successfully invoked in a school setting?
Method
Participants and Setting
The participants were two boys, one in third grade, Mike, and the other in sixth grade, Jason
(names have been changed for confidentiality). Each participant attended math class and at least
one other class in the resource room. All of the participants had been diagnosed with a learning
disability. The experimenter has used these participants because the teacher suggested these
students due to their low skills in math.
The setting was a resource room located in a large Department of Defense elementary school in
the Pacific Northwest. The classroom had up to three certified teachers in the room at one time,
each teacher with their own students in a different corner or space of the room. The teacher that
the experimenter worked with had three students in the room while sessions were completed.
The amount of students there at one time, changed part way through the project, so the
experimenter then had each student individually on a different day. The participants were in the
room for 25 to 30 minutes for each math session the experimenter was there. For many of the
math periods the experimenter would test one session per period, towards the end of the project
two sessions may be tested in one period. During the later part of the study there were a
maximum of six other children in the room, these children were separated from the participants
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
by a four-foot high bookshelf. The noise level was very low, the children were used to having to
be quiet with other students in the same room, and the participants and the experimenter easily
ignored the little noise there was. The project was part of an ongoing research program
employing students from Gonzaga University and documenting part of the NCATE and the
Washington State Standard. This Standard requires teacher candidates provide documentation of
their ability to positively affect student behavior (McLaughlin, Williams, Williams, Derby,
Weber, Bjordahl, 1999).
Materials
The materials requires were a math racetrack, a testing sheet with all the multiplication or
division problems, a testing sheet with the seven unknown problems, a point record sheet, and
two timers. The experimenter used a token reward system. The token system used points that
were traded in for edible treats (M & M’s, Mini Oreo Cookies, and Starbursts, etc.)
Table 1. Multiplication Problems for Jason’s Baseline and Math racetracks.
Multiplication Facts
(0-7) (8-9) (10-12)
Set One 3x8; 7x9 8x7; 9x2; 9x9 12x3; 12x8
Set Two 7x6; 7x8 8x9; 9x8 12x5; 12x6; 12x9
Set Three 7x7 8x8; 9x6; 9x7 12x2; 12x4; 12x7
Table 2. Division Problems for Mike’s Baseline and Math racetrack.
Division Facts
(0-7) (8-9) (10-12)
Set One 243; 186 248; 728; 819 11011; 9612
Set Two 324; 404 369; 639; 729 6012; 12012
Set Three 486; 606; 357; 847 408; 648 11010
Dependent Variable
The dependent variable was the total number of math problems the participants completed correct
during each session. For example a correct answer would be 5x6=30, an incorrect answer would
be 5x6=25. The total number of math problems available to complete during each session was
seven during baseline and twenty on during intervention. The time it took for each participant to
complete the math racetrack each session was also recorded.
Data Collection and Interobserver Agreement
There were two different data collection systems used. First a permanent product data collection
method using event recording of each math problem answered correct was measured with a tally.
The participants were given sheets with math problems on it and told to complete them to the
best of their ability. The math sheets were scored and recorded by the experimenter, data
collection sheets. Second, a duration recording system was used to time the length of time it took
the participant to complete the math racetrack. This was recorded starting after the experimenter
said start the first problem to when the participant completed the last problem on the math
racetrack.
The mean agreement for Mike’s data was 98.5% (range 95% - 100%) and the mean agreement
for Jason’s data was 98.5% (range 95% - 100 %). All of the data collection sessions were
employed to determine interobserver agreement. This data were collected by making copies of
the completed math sheets and two individuals scoring a different copy with out any marking.
Design
The design used was a multiple baseline across behaviors (Kazdin, 1982). Baseline was
employed with the problems the participants did not answer correctly from a sheet with all the
multiplication or division problems. Baseline was conducted for at least three days for the first
set of math problems, at least five days for the second set of math problems, and at least seven
days for the third set of math problems. The math racetrack was implemented for 3 to 7 days in a
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
staggered fashion different sets of math facts (multiplication facts for Jason and division facts for
Mike).
Procedures and Experimental Conditions
Baseline. During the baseline, each participant was given a sheet with every multiplication or
division problem ranging from 0x0=n to 12x12=n and 1/1=n to 12/120=n, this was tested once
and had a total of either 143 multiplication problems or 140 division problems. The
multiplication facts were single or double digit factors multiplied by a single or double-digit fact,
resulting in a single or double-digit product. The division facts were single, double or triple digit
factors divided by single, double or triple digit factors. Resulting in a single, double or triple
digit factor. No corrective feedback provided during baseline. Each participant was asked to try
their best to answer each problem; they were also told if they did not feel they could answer a
problem that it was all right. The experimenter explained to the participants that the point was
just to try to see what they could do and how smart they were. The participants were tested with
seven multiplication or division facts for at least 3, 5, or 7 data points for baseline. The math
facts were given to the participants in a test with seven multiplication or division facts 0x0=N to
12x12=N or 1/1=N to 120/12=N, the last step in baseline was to time each participant on the
racetrack with all problems they had shown to know and get correct on the sheet with every
multiplication or division fact ranging from 0x0=N to 12x12=N or 1/1=N to 120/12=N. The
points for the token reward system were earned when the participant would improve their time on
the math racetrack or after they had truly shown they worked to the best of their ability.
Math racetrack. During the math racetrack the participants were given the racetrack with a mix
of seven problems they did not know and the rest were problems they knew totaling 28 problems.
Each participant had three different racetracks with the same 21 problems they knew and seven
different problems they did not know. The problems were arranged in an order that presented
two to four problems they knew and one they did not know. This is very similar to what we have
done with sight word vocabulary reading racetracks (Falk et al., 2003; Rinaldi, Sells, &
McLaughlin, 1997).
During the racetrack the participants were timed while completing the racetrack, a maximum of
five seconds was given to the participants to complete each problem. If a problem was answered
incorrectly, or the participant was unable to answer in the five-second allotment, then the correct
answer was provided to the participant. After the racetrack was completed, the participant
reviewed the answers to all problems. The racetrack was used until the participant showed an
increase in the number of problems they answered correctly. During the math racetrack a token
system was also employed. Participants were timed with the racetrack containing problems that
were unknown. The following times the participants were timed to see if they increased their
original time or their goal time. On subsequent timings if the participant increased their time or
were able to show a good effort, they received a small bag of candy of their choice (i.e. M & M’s,
Mini-Oreos, Mini-Snickers or Starbursts). Each participant was shown the record sheet used to
keep track of their points, and adding their points at the end of each session.
Results
The outcomes below show the graph results of baseline and math racetrack for each of the
participants. Although the participants preformed better on some days than others, there was a
definite upward trend concerning the amount of problems answered correctly during math
racetrack. During the course of the project the experimenter would give a brief break of
approximately one minute between each set of data, but frequently Mike and Jason would request
to keep working, asking What’s next.
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
Baseline Racetrack
Set One
1 2 3 4 5 6 7 8 9 10
Set Two
1 2 3 4 5 6 7 8 9 10
Set Three
1 2 3 4 5 6 7 8 9 10
SESSIONS
SESSION
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
Baseline Racetrack
Set One
1 2 3 4 5 6 7 8 9 10
Set Two
1 2 3 4 5 6 7 8 9 10
Set Three
1 2 3 4 5 6 7 8 9 10
SESSIONS
During baseline for Mike, he performed well on a few of the problems, but overall his
performance and number of problems answered correctly was lower than the numbers of
problems answered correctly during the intervention phase using the racetrack. For set one, Mike
averaged 3.3 with a range of 2 to 4 problems correct for baseline. For set two, Mike averaged 5.2
with a range of 3 to 7 problems correct for baseline. For set three, Mike averaged 5.1 with a
range of 1 to 7 problems correct for baseline. For set one, Mike averaged 6.8 with a range of 6 to
7 problems correct for the math racetrack. For set two, Mike averaged 6.8 with a range of 6 to 7
problems correct for the math racetrack. For set three, Mike averaged 7 with a range of 7 to 7
63
INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
problems correct for the math racetrack. During baseline for Jason, he started to perform better
as time went one, but overall his performance and number of problems answered correctly was
lower than the numbers of problems answered correctly during the intervention phase using the
racetrack. For set one, Jason averaged 1.25 with a range of 1 to 2 problems correct for baseline.
For set two, Jason averaged 1.3 with a range of 0 to 3 problems correct for baseline. For set
three, Jason averaged 2.75 with a range of 0 to 5 problems correct for baseline. For set one,
Jason averaged 5.2 with a range of 4 to 7 problems correct for the math racetrack. For set two,
Jason averaged 5.75 with a range of 5 to 7 problems correct for the math racetrack. For set three,
Jason averaged 6 with a range of 5 to 7 problems correct for the math racetrack.
Discussion
A racetrack procedure was proven effective in improving the fluency of see-to-say words in
isolation by a student with learning disabilities in a study by Rinaldi and McLaughlin (Rinaldi,
McLaughlin, 1996). In addition this replicates our additional work in reading with children with
and without disabilities (Rinaldi et al., 1995) and our more recent work with elementary school
students with learning disabilities (Falk et al., 2003). In this study, we attempted to use a
racetrack procedure similar to that used in a study by Rinaldi and McLaughlin to improve two
students with learning disabilities fluency in and ability to complete multiplication or division
facts. The math racetrack procedure resulted in an increase in the correct responses across all
multiplication or division sets targeted.
While the Math Racetrack was shown to be effective (an increase in the problems answered
correctly after intervention), there were weaknesses found during the study. First, no long-term
data available for this study due to the completion of student teaching by the first author. It was
difficult to implement because the regular special education teacher was also working with the
participants during the rest of the week on other tasks and skills. Unfortunately, towards the end
of the study the first author had to carry out as much as three sessions in one class period due to
time constraints and the ending of student teaching. It may be possible in the future to increase
the length of data collection and analysis by having students spend a minimum of one day per
week in their respective special education classroom settings during their methods courses during
the first eight weeks of the semester. Due to high stakes testing (WASL and WAAS) in the State
of Washington, we are implementing such a change this upcoming spring semester.
The present study does provide some additional evidence that a racetracks procedure can be
adapted to math. The outcomes also lend support for the use of various easily implemented
classroom procedures such as copy, cover, and compare (Conley et al., 2004), add-a-word
spelling (Schermerhorn & McLaughlin, 1997), and tutoring techniques (Malone & McLaughlin,
1997). These procedures can improve the academic performance of a wide variety of children
and youth in their basic skills and allow for active student responding (ASR) on the part of the
students in the classroom (Heward, 2004). At this time we are exploring the use of copy, cover,
and compare with handwriting and spelling and further uses of a racetracks procedure in these
and other academic skill areas (Derby, Mortenson, Conley, & McLaughlin, 2000).
References
Anthony, C., Rinaldi, L., Hern, C., & McLaughlin, T.F. (1997). Reading racetracks: A direct
replication and analysis with three elementary school students. Journal of Precision Teaching
and Celebration, 14(2), 31-36.
Cheney, L.V. (1997, August). Dialogue: creative math, or just fuzzy math? [Math Instruction].
Message posted to http://www.danenet.wicip.org/ncs/forumcheneymath.htm.
Cheney, L.V. (1998, February). Whole hog for whole math. The Wall Street Journal. Retrieved
March 27, 2003, from http://www.junkscience.com.news/wholemat.htm.
Conley, C. M., Derby, K. M., Roberts-Gwinn, M., Weber, K. M., & McLaughlin, T. F. (2004). An
analysis of initial and maintenance of sight words following picture matching and copy, cover,
and compare teaching methods. Journal of Applied Behavior Analysis, 37, 339-350.
Curico, F. R. (1999). Dispelling myths about reform in school mathematics. Mathematics:
Teaching in the Middle School, 4, 282-284.
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INTERNATIONAL JOURNAL OF SPECIAL EDUCATION Vol 20 No.2
Derby, K. M., Mortenson, S., Conley, C., & McLaughlin, T. F. (2000, October). Using
racetracks to teach multiplication facts. Symposium presented at the Northwest Association for
Behavior Analysis, Leavenworth, WA.
Falk, M., Band, M., & McLaughlin, T. F. (2003). The effects of reading racetracks and flashcards
on sight word vocabulary of three third grade students with a specific learning disability: A
further replication and analysis. International Journal of Special Education, 18(2), 51-57.
Heward, W. L. (2004). Teaching exceptional children: An introduction to special education.
Upper Saddle River, NJ: Prentice-Hall/Merrill
Johnson, K. R., & Layng, T. V. (1994). The Morning Side Model of generative instruction. In R.
Gardner III, D. M. Sainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshelman, & T. A.
Grossi (Eds.). Behavior analysis in education: Focus on measurably superior instruction (pp.
173-197). Pacific Grove, CA: Brooks/Cole.
Kazdin, A. E. (1982). Single case research designs: Methods for clinical and applied settings.
New York: Oxford University Press.
McLaughlin, T. F., & Skinner, C. H. (1996). Improving academic performance through self-
management: Cover, copy, and compare. Intervention in School and Clinic, 32, 113-118.
McLaughlin, T. F., Weber, K. P., & Barretto, A. (2004). Spelling: Academic interventions. In T.
S. Watson & C. H. Skinner (Eds.), Encyclopedia of school psychology (pp. 317-320). New
York: Kluwer Academic/Plenum Publishers.
McLaughlin, T. F., Williams, B. F., Williams, R. L., Peck, S. M., Derby, K. M., Bjordahl, J. M., &
Weber, K. M. (1999). Behavioral training for teachers in special education: The Gonzaga
University program. Behavioral Interventions, 14, 83-134.
Malone, R., & McLaughlin, T. F. (1997). The effects of reciprocal peer tutoring with a group
contingency on quiz performance in vocabulary with seventh and eighth grade students.
Behavioral Interventions, 12, 27-40.
Ozaki, C., Williams, R.L., & McLaughlin, T.F. (1996). Effects of a see/copy/cover/write/and
compare drill and practice procedure for multiplication facts mastery with a sixth grade student
with learning disabilities. B.C. Journal of Special Education, 20, 65-74.
Rinaldi, L. & McLaughlin, T.F. (1996). The effects of reading racetracks on the fluency of see-
to-say words in isolation by a student with learning disabilities. Journal of Precision
Rinaldi, L., Sells, D., & McLaughlin, T. F. (1997). The effects of reading racetracks on sight
word acquisition of elementary students. Journal of Behavioral Education, 7, 219-234.
Schermerhorn, P., K., & McLaughlin, T. F. (1997). Effects of the add-a-word spelling program
on test accuracy, grades, and retention of spelling words with fifth and sixth grade regular
education students. Child & Family Behavior Therapy, 19(1), 23-35.
Skinner, C. H., McLaughlin, T. F., & Logan, P. (1997). Cover, copy, and compare: A self-
managed academic intervention across skills, students, and settings. Journal of Behavioral
Education, 7, 295-306.
Stone, S., McLaughlin, T. F., & Weber, K. P. (2002). The use and evaluation of copy, cover, and
compare with rewards and a flash cards procedure with rewards on division math facts mastery
with a fourth grade girl in a home setting. International Journal of Special Education, 17(2), 82-
91.
65
January 14, 2021
Thomas Mclaughlin
Gonzaga University, Faculty Member
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