Education and Training in Autism and Developmental Disabilities, 2012, 47(3), 345–358
© Division on Autism and Developmental Disabilities
Cognitive Strategy Instruction for Functional Mathematical
Skill: Effects for Young Adults with Intellectual Disability
Youjia Hua, Benjamin S. T. Morgan, Erica R. Kaldenberg, and Minkowan Goo
The University of Iowa
Abstract: This study assessed the effectiveness of a three-step cognitive strategy (TIP) for calculating tip and
total bill for young adults with intellectual disability. In the context of pre- and posttest nonequivalent-groups
design, 10 students from a postsecondary education program for individuals with disabilities participated in the
study. A teacher delivered six lessons to students in the experimental group using the working instructional
model for teaching learning strategies. Results indicate that the experimental group outperformed the comparison group on items that assessed the ability to calculate tip and total bill. Students from the experimental
group also generalized the procedural knowledge to tasks that required using percent values in different contexts.
Four of the students from the experimental group maintained the use of the strategy 8 weeks after the
intervention.
Self-determination, the ability to control one’s
own life without relying on caregivers, is an
important educational outcome for individuals with disabilities as they transition to adult
living (Field & Hoffman, 2002; Wehmeyer,
Agran, & Hughes, 1998). Self-determination is
often associated with the success achieved in
the areas of living, employment, and education (Halpern, 1993). One set of skills that is
necessary to fulfill the expectation of independent functioning in these areas is financial
autonomy including the ability to earn, budget, and spend money (Browder & Grasso,
1999). Unfortunately, individuals with disabilities often have difficulties with money management skills (Yan, Grasso, Dipipi-Hoy, & Jitendra, 2005). These difficulties, in turn,
affect all aspects of adult life. For example, the
National Longitudinal Transition Study (Wagner et al., 1991) examined the financial management activities performed by young adults
with learning disabilities who had been out of
the schools for up to two years and found that
very few participants had checking accounts
(8.1%), a credit card of their own (8.1%), or
Correspondence concerning this article should
be addressed to Youjia Hua, Department of Teaching and Learning, N256 Lindquist Center, The University of Iowa, Iowa City, IA 52242. Email: [email protected]
other investments (0.4%). The results of the
study indicate that young adults with learning
disabilities are not engaged in some of the
essential money management activities that
most adults will need.
Researchers suggest that one important reason why individuals with disabilities have limited opportunities to reach financial autonomy is that they lack essential functional
mathematical skills (Patton, Cronin, & Bassett, 1998). Mathematical deficiencies are as
common as difficulties in reading for individuals with disabilities (Deshler, Ellis, & Lenz,
1996). Researchers found that students with
learning difficulties tend to acquire mathematical skills at a slower rate and have difficulties with skill retention and generalization
(Miller & Mercer, 1997). Limited mathematic
literacy for students with learning difficulties
appears in elementary school and persists
throughout their adulthood (e.g., Kirby &
Becker, 1988). As a result, social and economic independence of adults with disabilities is negatively affected (Lerner, 1993). In
order to prepare these individuals for the demand and challenges of adult life, postsecondary mathematical curriculum should focus on
the skills that will be used on the job, at home,
and in the community (Deshler et al., 1996;
Patton et al., 1998).
In addition to a functional skills oriented
Functional Mathematical Skill
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345
mathematics curriculum, postsecondary learners also need evidence-based instruction.
Analysis of reviews of literature suggests that
young adults with learning difficulties benefit
from teacher-directed explicit instruction
(Deshler et al., 1996). Explicit instruction is
characterized by beginning the teaching sequence with a review of the prerequisite skills
followed by teacher modeling and guided and
independent practice (Archer & Hughes,
2010). During instruction, the teacher presents information in small steps, uses full range
of examples and nonexamples, elicits frequent student responses, and provides immediate feedback (e.g., Miller & Hudson, 2007).
The effectiveness of explicit instruction for
students with learning difficulties has been
well documented. For example, Swanson and
Hoskyn (2001) conducted a meta-analysis of
180 published intervention studies and found
that interventions that included components
of explicit instruction resulted in larger effect
sizes than those in the comparing conditions.
Likewise, the research synthesis by Baker, Gersten, and Lee (2002) indicates that interventions using teacher directed explicit instruction had a positive, moderately strong effect
on the mathematics performance of lowachieving students.
On the other hand, researchers found that
using explicit step-by-step structured instruction is not sufficient for students with disabilities when teaching mathematical skills to
solve real-life problems as successful problem
solving requires learners to use effective strategies that utilize both cognitive and metacognitive process (Montague, 1997; Reid & Lienemann, 2006). Strategies are often defined as a
series of sequenced procedures that allow an
individual to complete a task using the awareness and action of planning, implementing,
and evaluating the process and outcome
(Reid & Lienemann, 2006). Unfortunately,
students with disabilities have difficulties selecting and applying appropriate strategies to
solve problems because they have a limited
awareness of the potential usefulness of the
strategies for a given task. Even when they
have an idea of the appropriate strategy to
use, they use it ineffectively because they often
take too much effort to retrieve procedures,
operations, and basic mathematic facts, leav-
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TABLE 1
Three-Step TIP Strategy
T:
I:
P:
Take a look at the total bill and enter it on the
calculator.
Identify the tip by multiplying the total by 15%.
Plus the total and find out how much to pay.
ing little cognitive resources to process new
information (Maccini & Hughes, 1997).
Research in the area of cognitive strategy
suggest that students with learning difficulties
can benefit from strategy instruction focusing
on mathematical word problems (e.g., Case,
Harris, & Graham, 1992; Montague, 1992)
and computation problems (e.g., Brown &
Frank, 1990; Rivera & Smith, 1988). However,
research in the area of functional mathematical skill acquisition for postsecondary students with disabilities is sparse and unable to
guide practice. Recent reviews (Browder &
Grasso, 1999; Yan et al., 2005) on teaching
money skills for students with disabilities highlighted the needs for additional research that
include (a) learners at postsecondary level
with academic skills and (b) interventions that
address more complex money management
skills that require using mathematical computation skills. While the utility of strategy instruction has yet to be proven in the area of
functional mathematical skill acquisition it
seems to be a logical intervention in this domain for young adults with intellectual disability.
The purpose of this study was to assess the
effectiveness of a three-step strategy (TIP) for
calculating tip and total bill for learners with
disabilities in a postsecondary education program. The TIP strategy includes three essential steps that can be used to calculate tip and
total bill in a variety of contexts (e.g., restaurant, hotel, cab). Table 1 presents the three
steps of the TIP strategy. The mnemonic device (TIP) may serve as a natural cue to
prompt learners to use the strategy in the
applied setting. The study extends the literature by examining the utility of strategy instruction as a means for teaching a functional
mathematical skill to students with disabilities
at postsecondary level. Specifically, we sought
to answer the following research questions:
Education and Training in Autism and Developmental Disabilities-September 2012
TABLE 2
Participants Demographic Data by Experimental Condition
Variable
Experimental Group
Student One
Student Two
Student Three
Student Four
Student Five
Comparison Group
Student One
Student Two
Student Three
Student Four
Student Five
Gender
Age
Ethnicity
Disability
IQ
Female
Female
Male
Female
Female
21
22
22
18
18
Caucasian
Caucasian
Caucasian
Caucasian
Caucasian
Severe LD, ADHD
MR
ASD
ASD
ASD
82
65
74
77
80
Female
Male
Female
Female
Male
24
22
20
21
23
Caucasian
Caucasian
Caucasian
Caucasian
Caucasian
ASD, ADHD
ASD
MR, ADHD
Severe LD
ASD
65
72
63
92
69
Note. IQ ⫽ full scale intelligence quotient scores reported for the Wechsler Intelligence Scale for children.
1. Can young adults with disabilities acquire
and apply the TIP strategy to calculate tip
and total bill?
2. Will the learners generalize the procedures of the TIP strategy to solve functional mathematical problems involving
percentage values?
Method
Participants and Setting
Participants were students enrolled in two
classes from a post-secondary education program for young adults with learning and
intellectual disability at a Midwestern university. The program provided an integrated
collegiate experience including academic
coursework, career development, student life,
and community life. The program academic
coordinator suggested that some students
needed additional help with functional mathematical skills in the area of calculating tips
and total bill. At the time of the study, all
participants took a life skill class as part of the
program curriculum. The class covered a variety of skills including social interactions, domestic skills, pre-vocational and vocational
skills, as well as daily living skills. Students did
not receive any instruction on how to calculate tip and total bill from the courses offered
by the program before the intervention.
We did not get permission from the pro-
gram to randomly assign the students to
groups as the two classes had different schedules and vocational placement throughout the
day. Instead we used a pre- and posttest nonequivalent-groups design to examine the effects of the intervention on acquisition of the
tip and bill calculation skill of the participants.
To ensure that the participants possessed the
prerequisite skills and had difficulties with
tip and total bill calculation, we required the
students to meet the following inclusion criteria. First, students were able to complete the
mathematical computation tasks accurately
using a calculator. Second, students must
have completed the word problems requiring
them to calculate the tip and the total bill
on the pretest with accuracy lower than
50%. Five students from one class met the
inclusion criteria and participated in the study
as an experimental group. We also enrolled
five students who met the inclusion criteria
from the other class as a comparison group.
Table 2 presents a detailed description of
each student, including age, gender, ethnicity,
IQ score, and disability category. We conducted an analysis of variance (ANOVA) to
evaluate comparability of the two groups before the intervention as this research design
does not control for interactions between subject selection and other factors including
school history and maturation (Campbell &
Stanley, 1966). The ANOVA tests showed that
there were no significant differences between
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347
TABLE 3
Examples of Types of Problems Used on Pretest, Posttest, and Maintenance Check
Problem type
Target item
Example
Mike had a dinner at a restaurant and the bill was $35.63. He paid the bill with
additional 15% for the tip.
(a) How much was the tip?
(b) How much did he pay in total?
Generalization Item
Different context
The price for a can of soda is $2.99 and the sales tax is 6%.
(a) How much is tax?
(b) How much total will you pay for the soda?
Different operation The book is on sale for 20% off. The price for the book you want to buy is $20.00.
(a) How much money will you save from the sales?
(b) How much is the book now?
Different value
Your bill for your dinner is $127.56 and you really enjoyed the service. You want to
give the waiter a 20% tip.
(a) How much is the tip?
(b) How much is the total you will pay?
the two groups of students with regard to their
pretest (F [1, 8] ⫽ .20, p ⫽ .67), IQ scores (F
[1, 8] ⫽ .32, p ⫽ .58), or age (F [1, 8] ⫽ 2.42,
p ⫽ .16).
A program instructor who had two years
of experience teaching functional life skills
delivered the strategy instruction to the
experimental group three times a week
(i.e., Monday, Wednesday, and Friday) during
the regularly scheduled class time. Each
lesson lasted for approximately 30 –35 minutes. The total duration of the intervention
was two weeks (i.e., six lessons). During the
intervention, students in the comparison
group attended their regularly scheduled
classes.
Materials
We developed three equivalent probes for the
pretest, posttest, and maintenance check.
Each probe contained 16 items including
(a) 10 target items that assessed students’
ability to calculate tip and total bill in a variety
of situations (e.g., restaurant, delivery, taxi
fare) and (b) six generalization items that
assessed students’ ability to solve problems
using percentages in a different context
(e.g., calculate sales tax), requiring different
operation (e.g., calculate sales price), or using
different percentage values. Table 3 presents
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examples of each type of problem on the
three tests. Each target and generalization
item on the probes contained two questions
that were related to the two steps of the problem solution. Therefore, the total available
number of problems from the target items was
20 and 12 from the generalization items. The
order of the different types of items on the
tests was randomized (Case et al., 1992). We
also developed additional worksheets that
only contained target items for guided and
independent practice during the intervention.
General Procedure
The instructor of the course delivered the
intervention to the experimental group using
the scripted lesson plans developed by the
researcher. Each scripted lesson plan included a step-by-step protocol, teacher wording, examples, student worksheets, and overhead transparencies. Before each lesson, the
researcher and the teacher went over the lesson plan together and simulated the teaching
process using the script.
Each lesson followed the structure of an
explicit instruction lesson. For example, the
teacher began each lesson with a review of the
previous lesson and a description of the learning objectives and expectations. Following the
Education and Training in Autism and Developmental Disabilities-September 2012
opening of the lesson, the teacher modeled
skills using clear, consistent and concise language. Then the students practiced the skill
with teacher prompts. After the students successfully completed the tasks, the teacher
gradually faded prompts and the students
practiced the skills independently. Before the
conclusion of the lesson, the teacher reviewed
the skills and gave a preview of the upcoming
lessons. The teacher also embedded some critical presentation techniques throughout the
instruction including eliciting frequent unison responses, monitoring student responses,
providing feedback, and maintaining appropriate pace of the lessons.
TIP Strategy Instructional Procedure
The development of the TIP strategy intervention was based on the features of the working
instructional model for teaching learning
strategies described by Deshler and colleagues
(1996). We chose this model because it addresses the learning needs of students with
disabilities and its effectiveness has been well
documented (Montague & Dietz, 2009). The
TIP strategy contained six instructional stages
including pretest and making commitments,
describing the strategy, modeling the strategy,
verbal elaboration and rehearsal, controlled
practice and feedback, and advanced practice
and feedback.
Stage 1. The purpose of this stage was to
(a) review the pretest results, (b) provide a
rationale for learning the strategy, and (c)
obtain commitment from the students to
learn the new strategy. At the beginning of the
lesson, the teacher communicated the pretest
results to the students and compared students’ performances to the mastery criteria.
The teacher also asked the students to identify
the examples and situation, in which they
used ineffective tip calculation. The teacher
and the students then discussed the purpose
and importance of learning a new strategy and
its relevance to their ability to live independently. At the end of the first lesson, the
teacher made a commitment to the students
to teach the strategy effectively and prompted
the students to commit to learning the strategy with effort and time.
Stage 2. The purpose of the lesson at this
stage was to describe the TIP strategy. First,
the teacher described the utility of the strategy. For example, the teacher and the students discussed where and when the TIP strategy should be used. The teacher also
described situations in which TIP strategy was
not appropriate. After the initial overview of
the strategy, the teacher described why and
how each step of the strategy was used while
emphasizing the importance of using selfinstruction to regulate the use of the strategy
(e.g., “What we say to ourselves as we take
each step will help us use the strategy and
solve the problem.”). At the end of the lesson,
the teacher discussed how to remember each
step of the strategy by using the acronym
TIP and asked the students to make their own
cue cards containing each step of the TIP
strategy.
Stage 3. The purpose of this lesson was for
the teacher to model the use of TIP strategy.
The teacher first demonstrated the cognitive
processes and acts required to carry out each
step of the strategy while “thinking aloud”
using the following types of statements: (a)
problem definition (“The problem asks me
how much I tip and how much I will pay in
total.”); (b) strategy use (“It says tip. It means
I can use TIP strategy to calculate the tip and
bill. How do I spell TIP?”); (c) self instruction
(“Where do I find the total of the bill? Find
the number that says ‘total’.”); and (d) self
monitoring (“I am not done yet. I will need to
figure out how much in total I have to pay.”).
Following initial modeling, the teacher involved the students to perform each step of
the strategy by eliciting frequent choral response. The students were prompted to (a)
self-talk using the actual words they would say
to use the strategy themselves (e.g., “The second step is the second letter of the tip. Everyone all together, what is the second letter of
tip?”) and (b) carry out the physical act of
each step (e.g., “Yes. The total is ___ and you
enter it on the calculator. Now look at your
calculator. All together, tell me the number
on your calculator”). At the end of the lesson,
the teacher told the students that in order to
make the strategy work they need to practice
the new strategy and complete each step
much faster.
Stage 4. The purpose of the lesson at this
stage was to ensure that each student understood and memorized each step of the TIP
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strategy. The teacher asked the students to
explain the purpose of the TIP strategy and
provided examples of when the strategy can
be used. The teacher then checked student
comprehension by asking them to verbally describe in their own words what they would do
during each step. Following verbal elaboration, the teacher used “rapid-fire practice” to
promote student memorization of each step
of the strategy (Deshler et al., 1996). Using
this activity, the teacher asked the students
to name the steps of the TIP strategy in succession individually. As students became
more fluent with the verbalization of each
step, the teacher gradually increased the difficulty of the activity by increasing the speed
and calling on students randomly. The
teacher also gradually erased words from each
step on the board until all three steps were
invisible. The teacher reminded the students
to use the letters in TIP to help them remember the steps. After the majority of the students memorized the steps, the teacher asked
the students to rehearse the steps with a partner. The teacher conducted an individual oral
quiz at the end of the class. To reach the
mastery criterion, the student must have (a)
described the purpose of the TIP strategy and
(b) named and explained each of the three
steps in order without referring to a cue card
or receiving additional prompts. If a student
did not reach the 100% accuracy criterion on
the oral quiz, the teacher provided additional
feedback and asked the student to return to
the seat and practice it with a partner. The
teacher used a verbal practice checklist to
track students’ attempts at passing the oral
quiz.
Stage 5. The purpose of the lesson was to
ensure that students mastered the use of the
TIP strategy with guided practice. During the
guided practice, the teacher and the students
collaboratively completed two problems. Initially, the teacher told the students to use the
TIP strategy by saying “You see the word tip in
the question. That will help you remember
the TIP strategy and its three steps. Each letter
represents one step. You need to tell yourselves each step so that you will know how to
find the answers.” At the beginning of the
second problem the teacher only reminded
the students of the strategy by asking “What
strategy will you use and tell yourself what the
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first step is and complete it.” The teacher also
frequently checked the accuracy of each step
by eliciting choral responses (e.g., “Tell me,
what’s the number on your calculator?”). After successful completion of the two problems
collaboratively, the teacher provided the students with an opportunity to practice the skill
independently. As the students worked on the
four problems independently, the teacher
monitored student performance by walking
around the room, provided feedback to the
students, and prompted the students to use
think-aloud process. At the end of the class,
the teacher collected and graded students’
independent work. All students completed
the four problems on the worksheets independently with a minimum accuracy of 95% or
above.
Stage 6. The purpose of the learning stage
was to provide students with additional opportunities to practice the strategy independently. As the students worked on the independent worksheets containing 10 problems
requiring calculating tip and total bill the
teacher continued to monitor student performances of the tasks. At the conclusion of the
lesson, the teacher collected their work and
praised their performance.
Dependent Variables
The dependent variables were total number of
problems answered correctly on the target
items (i.e., items that required calculating
tip and total bill) and the generalization
items. We also analyzed the total number of
procedural errors (i.e., no answer, guessing,
and wrong operation) students made as identification of mathematical error patterns can
help educators improve instruction and student learning (Case et al., 1992; Montague,
1992).
Data Collection
Students in both experimental and comparison groups took the test before the intervention (pretest) and immediately following the
intervention (posttest). Students in the experimental group also took a maintenance test 8
weeks after completion of the intervention.
To ensure assessment integrity, the researcher
developed a scripted protocol to administer
Education and Training in Autism and Developmental Disabilities-September 2012
TABLE 4
Total Number of Items Answered Correctly by Students in the Comparison and Experimental Group
Comparison
Experimental
Variable
Pretest
M(SD)
Posttest
M(SD)
Pretest
M(SD)
Posttest
M(SD)
Maintenance
M(SD)
Target
Generalization
.00 (.00)
.20 (.45)
.20 (.45)
.40 (.89)
.40 (.89)
.00 (.00)
19.60 (.89)
10.20 (1.30)
16.80 (6.61)
7.60 (5.32)
the tests. At the beginning of each test, the
teacher asked the students to complete the
tasks with the following statement: “This is not
a test or an assignment. It will not count towards your grades. Try your best to answer
these questions. If you don’t know how to
answer the question, make an ‘X’ on it and go
to the next one. You have 15 minutes to work
on these problems. It is okay if you don’t
finish all the questions in 15 minutes. If you
finish early, raise your hand and I will collect
your worksheets. If you need a calculator,
please raise your hand and I will give you
one.” At the end of the 15 minutes, the
teacher stopped the test and collected students’ answer sheets. A graduate student
graded the probes.
Reliability and Procedural Integrity
The researcher developed procedural checklists for individual lessons based on the
scripted lesson plans. The researcher trained
a graduate student to observe the lessons and
check the treatment integrity data using the
procedural checklists. The procedural integrity was calculated by the total number of steps
completed by the teacher divided by the total
number of available steps on the procedural
checklist. We collected treatment integrity
data for all of the intervention sessions and it
was 100% across the sessions. One graduate
student who was not involved in the data collection and was blind to the participants in
both comparison and treatment groups conducted the interobserver reliability checks for
all of the probes independently. Interobserver
agreement was calculated by total number of
agreements divided by agreements plus disagreements multiplied by 100%. Given the
objective nature of the answers, the mean
agreement for total number of questions answered correctly on the target items were 99%
(range between 92% to 100%) and 100% on
the generalization items.
Results
Table 4 presents the means and standard deviations of total number of questions answered correctly by the students on the pretest, posttest, and maintenance probe.
Students in the comparison group scored an
average of 0 on the target items on the pretest
and .20 on the posttest with a gain score of .20.
With regard to the generalization items (Table 5), students in the comparison group
scored an average of .20 on the pretest and .40
on the posttest with a gain score of .20. Students in the experimental group scored an
average of .40 on the target items on the
pretest and 19.60 on the posttest with a gain
score of 19.20. With regard to the generalization items, students in the experimental
TABLE 5
Analysis of Gain Scores on the Target and
Generalization Items between the Comparison and
Experimental Group
Comparison Experimental
Mean
M (SD)
M (SD)
Difference
Variable
Target Items
Gain Score
Generalization
Items Gain
Score
.20 (.45)
19.20 (.45)
.20 (1.10) 10.20 (1.30)
19.00*
10.00*
* p ⬍ .001
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Figure 1. Total number of procedural errors (represented by bars) on the target items committed by students
(number above bars) from experimental and comparison group.
group scored an average of 0 on the pretest
and 10.20 on the posttest with a gain score of
10.20. A comparison of the gain scores between the comparison and experimental
group using an ANOVA indicated that the
differences on both target (F [1, 8] ⫽ 1289.29,
p ⬍ .001) and generalization items (F [1, 8] ⫽
172.41, p ⬍ .001) were statistically significant.
In addition, students from the experimental
group answered an average of 16.80 target
items and 7.60 generalization items correctly
on the maintenance probe.
We also analyzed the types of errors participants made on both target and generalization
items (see Figure 1 and 2). All five students in
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the comparison group made procedural errors on the target and generalization items,
accounting for 100% of the total errors on the
pre- and posttest. Similar error patterns occurred on both types of items on the pretest
by all five students in the experimental group,
accounting for 98% and 100% of the total
errors on the target and generalization items.
During posttest, students in the experimental
group did not have any procedural errors on
the target items and four of them had a procedural error on one of the generalization
items. In addition, four students from the
experimental group completed the target
items on the maintenance probe without mak-
Education and Training in Autism and Developmental Disabilities-September 2012
Figure 2. Total number of procedural errors (represented by bars) on the generalization items committed by
students (number above bars) from experimental and comparison group.
ing any procedural errors. With regard to
their completion of the generalization item
on the maintenance probe, three students
made errors due to lack of procedural knowledge.
Discussion
Learning how to calculate tip is a useful skill
for young adults with disabilities. The TIP
strategy addressed a common but important
problem that young adults with disabilities
were frequently encountering in postsecondary settings. At the time of the study, all of the
participants were living on a university campus
independently for the first time. Greater independence and change in their support network may have presented significant challenges for these students who lacked necessary
functional money management skills in postsecondary settings (Hughes & Smith, 1990).
For example, two of the most frequent ineffective tipping approaches described by the
students at the beginning stages of strategy
instruction were guessing and giving fixed
amount of money as tip regardless of the total
bill (e.g., always gave two dollars as a tip). All
of the students acknowledged that the ineffec-
Functional Mathematical Skill
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353
tive strategies were impediments to their inclusion and functioning in the community
and workplace.
Participants’ lack of effective strategies to
solve this type of functional mathematical
problem was verified by the results of the pretest. Initially, all of the errors committed by
the participants were due to lack of procedural knowledge; in other words they did not
know the steps involved in calculating tip and
total bill. The utility of the TIP strategy became evident as learning the strategy resulted
in improved performance and corresponding
reduction in the number of errors due to lack
of an effective strategy (Case et al., 1992; Montague, 2001). Four of the students from the
experimental group also maintained the use
of strategy 8 weeks after the intervention.
More impressively, students in the experimental group generalized the procedural knowledge they gained from the TIP strategy to
tasks that required using percentage values in
different contexts, evidenced by significant reduction of the procedural errors on the test
immediately following the intervention. In addition, three of the students maintained the
skill generalization after the intervention.
Therefore, this study extended the utility of
strategy instruction in the area of functional
mathematical skill with postsecondary students who had intellectual disability.
Several features of the TIP strategy and instruction contributed to the effectiveness of
the intervention in the study. First, the content and the design of the TIP strategy
bridged the gap between the demands of the
functional mathematical skills in the applied
settings and the skill deficit of the students
with disabilities. Successful tipping requires
an individual to use an effective strategy that
utilizes both cognitive and metacognitive
process. For example, the individual will first
recognize the situation when a tip calculation
is necessary, then determine the steps and
operations involved in the computation, and
complete the procedures while retrieving
and applying the knowledge of basic math
facts. Simply teaching young adults with intellectual disability the three steps involved
in the tip and total bill calculation may not
result in acquisition and functional use of
the procedures as participants in the study
lacked an effective strategy to guide the exe-
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cution of this process that is highly metacognitive. For example, error analysis of the pretest indicated that all of the errors on the
pretest were due to lack of procedural knowledge, suggesting that the participants had difficulties with selecting task-appropriate strategies and coordinating the procedures. In
order to meet the demands of the functional
skills, we developed the TIP strategy that addressed both cognitive and metacognitive
learning needs of the learners in the study.
The mnemonic device (TIP) served as a natural cue to activate learners’ awareness of the
strategy and helped the learners store and
retrieve information from the long-term memory. Each step of the strategy then prompted
learners to take both overt action and use
cognitive processes (e.g., problem definition,
self-instruction, self-monitoring) to facilitate
the execution of the procedures to calculate
tip and total bill.
Second, the strategy development model
and the instructional procedures we used in
the study also contributed to the student
gains. We delivered the strategy instruction
following the working instructional model for
teaching learning strategies developed by
Deshler and colleagues (1996). This strategy
development model addressed both motivational and cognitive characteristics of students
with learning difficulties. Given the long history of difficulties in the area of mathematics,
motivation is an important factor that contributes to the successful acquisition of new skills
for adults with disabilities (Deshler, Schumaker, & Lenz, 1984). In order to encourage
the learners to be involved in and take ownership of the strategy, we made learning more
relevant to their personal goals. At the beginning of the intervention, the teacher discussed
the utility and benefits of the TIP strategy with
the students. Throughout instruction, the
teacher also emphasized that independent
money management skill was a result of personal effort in learning and the use of the
strategy. In addition to motivation, modeling
was the other critical component of the strategy instruction (Montague & Dietz, 2009).
During modeling, we emphasized the covert
processes involved in the successful application of the TIP strategy using “think aloud.”
Teacher’s verbalization of the thought processes gave learners the opportunities to ob-
Education and Training in Autism and Developmental Disabilities-September 2012
serve both cognitive (e.g., self instruction,
paraphrase) and metacognitive processes
(e.g., analyze the task, develop a plan, evaluate
the outcome) required to carry out each step
of the strategy, thus improving learners’
knowledge of the procedure and reason for
each individual step of the strategy (Reid &
Lienemann, 2006).
Similar to findings from previous research,
all the students in the experimental group
generalized the procedural knowledge they
gained from the TIP strategy to solve functional mathematical problems that involved
using percentage values in a different context
immediately following the intervention (Case
et al., 1992; Hutchinson, 1993; Jitendra &
Hoff, 1996). Researchers found that requiring
students to perform the strategy fluently increased the likelihood that the learners will
successfully generalize the skill to the untaught context (Schmidt, Deshler, Schumaker, & Alley, 1989). When a learner can
perform a strategy with automaticity the individual will be able to decrease the cognitive
requirement with regard to the details of the
solution and instead allocate more cognitive
resources to identify the similarity between
trained and novel tasks, thus fostering generalization (Fuchs & Fuchs, 2003). In the current study, we provided students with frequent
opportunities to respond to the tasks that facilitated understanding and memorization of
individual steps and application of the strategy. Moreover, the teacher monitored learners’ responses and provided feedback to ensure that learners performed the skills with
speed and accuracy before proceeding to the
next stage of learning. On the other hand, the
results of the study pointed to the necessity of
programming for generalization in order for
young adults with disabilities to transfer and
adapt the strategy to solve for problems that
require similar solutions. For example, four of
the participants from the experimental group
had a procedural error on the posttest that
assessed their ability to solve the problem using similar procedures with different operation (i.e., calculate sales price based on the
discount rate). Instead of using subtraction,
the participants added the discounted value to
the original price of the product. Their rigid
execution of the strategy without modification
supported research findings that learners with
learning difficulties are less likely to use the
cognitive mechanism appropriately to generalize and adapt the strategies to solve for tasks
that vary in complexity and purpose (Montague, 2008).
Limitation and Future Research
Results of this study must be interpreted
within the context of its limitations. First, we
used a quasi-experimental design in which we
chose students from an intact class as the experimental group and a relatively comparable
group of students from the other class as a
comparison sample. Although the pretest differences and participants’ demographics between the two groups were below the recommended one-half standard deviation, it is still
possible that some unknown variables that differentiate the two groups other than the intervention were responsible for the observed
effects (Gersten, Baker, & Lloyd, 2000). Researchers also need to determine whether
other individuals with intellectual disability
would benefit from this cognitive strategy as
young adults with intellectual disability have
large discrepancies of needs and proficiency
levels in mathematical and functional skills
(Patton et al., 1998). In addition, replicating
studies with small samples allows researchers
to discover the causal relationship between
the intervention and the dependent variables
(Gersten et al., 2000; Gersten, Fuchs, Coyne,
Greenwood, & Innocenti, 2005).
Second, the results of the study point to the
need of programming for maintenance as
part of the strategy instruction. Although all of
the participants in the experimental group
improved their accuracy of all of the tasks on
the posttest, the accuracy of target (student 1)
and generalization items (student 1 and 3) on
the maintenance probe was much lower than
the posttest, suggesting that the learners did
not maintain the skill of using TIP strategy
over time. Therefore, providing ongoing practice opportunities that facilitate maintenance
of the skills for learners with disabilities may
be a critical component of the strategy instruction to promote long-term use of the strategy
(Deshler et al., 1996).
Third, we did not assess learners’ ability to
perform the task in natural setting (e.g., cal-
Functional Mathematical Skill
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355
culate tip and total bill after a meal in a restaurant). Researchers recommend that providing opportunities to use the strategy in real
context is a key component to ensure that
learners have mastered functional use of
money (Browder, Spooner, Ahlgrim-Delzell,
Harris, & Wakeman, 2008). Future researchers need to include both training and probing
of functional skills for generalization to real
life situations using simulations, role play, and
training in multiple settings. On the other
hand, although we did not directly assess
learners’ ability to perform the task in real life
situations three students from the experimental group reported that they applied the TIP
strategy in the restaurant and calculated the
tip and total bill using the calculator from
their cell phones. As an assistive technology to
compensate for the difficulties with mathematic computation, the use of cell phone
calculator may be more appropriate for young
adults with disabilities because of its portability, compatibility, and social appropriateness
(Raskind, 1998). With an increasing availability and use of cell phone, teaching learners to
use a cell phone calculator may be a logical
extension of the TIP strategy we implemented
in the study.
centage values in a variety of contexts. It is
possible that educators can enhance the utility
and impact of the TIP strategy by including
procedures that promote the use of strategy
beyond the context in which it was taught and
teach learners how to modify the components
of the strategy to meet the new and different
task demand (Deshler et al., 1996; Fuchs &
Fuchs, 2003; Maccini, McNaughton, & Ruhl,
1999; Montague, 1997; Montague & Dietz,
2009).
Practical Implication
References
Teaching functional living skills will prepare
young adults with disabilities to be successful
in daily living, places of employment, and education (Alwell & Cobb, 2009). Such instruction
will improve their quality of life and opportunities that may result in self-determination
(Carter, Lane, Pierson, & Stang, 2008). Compared to the typical duration and intensity level
of teaching functional life skills for young adults
with disabilities (i.e., three to four months at an
intensity of a few times per week; Alwell & Cobb,
2009), the TIP strategy required only six sessions
with a total duration of approximately three
hours. In addition, the teacher expressed satisfaction with the strategy and instructional procedures. He indicated that he wanted to use the
TIP strategy instruction procedures in the upcoming year.
The results of the study also indicate that
the TIP strategy instruction may be a promising strategy to teach learners with disabilities
to solve for problems that required using per-
Alwell, M., & Cobb, B. (2009). Functional life skills
curricular interventions for youth with disabilities: A systematic review. Career Development for
Exceptional Individuals, 32, 82.
Archer, A. L., & Hughes, C. (2010). Explicit instruction: Effective and efficient teaching. New York: Guildford Press.
Baker, S., Gersten, R., & Lee, D. (2002). A synthesis
of empirical research on teaching mathematics to
low-achieving students. Elementary School Journal,
103, 51.
Browder, D. M., & Grasso, E. (1999). Teaching
money skills to individuals with mental retardation: A research review with practical applications. Remedial & Special Education, 20, 297.
Browder, D. M., Spooner, F., Ahlgrim-Delzell, L.,
Harris, A. A., & Wakeman, S. Y. (2008). A metaanalysis on teaching matehmatics to students with
significant cognitive disabilities. Exceptional Children, 74, 407– 432.
Brown, D., & Frank, A. R. (1990). Let me do it! -Self-monitoring in solving arithmetic problems.
Education & Treatment of Children, 13, 239 –248.
Campbell, D. T., & Stanley, J. C. (1966). Experimental
356
/
Conclusion
Previous research has demonstrated the effectiveness of cognitive strategy instruction in
academic areas for students with learning
disabilities. This study extended the research
by utilizing the strategy instruction in the
area of functional mathematical and problem solving skills with postsecondary students
with severe learning disabilities and mild
mental retardation. Researchers should continue this line of research for young adults
with intellectual disability with a focus on generalization and a more flexible use of the
strategy.
Education and Training in Autism and Developmental Disabilities-September 2012
and quasi-experimental designs for research. Chicago:
Rand McNally.
Carter, E. W., Lane, K., Pierson, M. R., & Stang, K. K.
(2008). Promoting self-determination for transition-age youth: Views of high school general and
special educators. Exceptional Children, 75, 55–70.
Case, L. P., Harris, K. R., & Graham, S. (1992).
Improving the mathematical problem-solving
skills of students with learning disabilities. Journal
of Special Education, 26, 1–19.
Deshler, D. D., Ellis, A. K., & Lenz, B. K. (1996).
Teaching adolescents with learning disabilities: Strategies and methods. Denver, CO: Love Publishing.
Deshler, D. D., Schumaker, J. B., & Lenz, B. K.
(1984). Academic and cognitive interventions for
LD adolescents: Part I. Journal of Learning Disabilities, 17, 108 –117.
Field, S., & Hoffman, A. (2002). Preparing youth to
exercise self-determination. Journal of Disability
Policy Studies Special Issue: Self-determination, 13,
113–118.
Fuchs, D., & Fuchs, L. S. (2003). Enhancing the
mathematical problem solving of students with
mathematics disabilities. In H. L. Swanson (Ed.),
Handbook of learning disabilities. New York: Guildford Press.
Gersten, R., Baker, S., & Lloyd, J. W. (2000). Designing high-quality research in special education: Group experimental design. Journal of Special
Education, 34, 2–18.
Gersten, R., Fuchs, L. S., Coyne, M., Greenwood, C.,
& Innocenti, M. S. (2005). Quality indicators for
group experimental and quasi-experimental research in special education. Exceptional Children,
71, 149.
Halpern, A. S. (1993). Quality of life as a conceptual
framework for evaluating transition outcomes. Exceptional Children, 59, 486 – 498.
Hughes, C. A., & Smith, J. O. (1990). Cognitive and
academic performance of college students with
learning disabilities: A synthesis of literature.
Learning Disability Quarterly, 13, 66 –79.
Hutchinson, N. L. (1993). Effects of cognitive strategy instruction on algebra problems solving of
adolescents with learning disabilities. Learning
Disability Quarterly, 16, 34 – 63.
Jitendra, A. K., & Hoff, K. (1996). The effects of
schema-based instruction on the mathematical
word-problem-solving performance of students
with learning disabilities. Journal of Learning Disabilities, 29, 422– 431.
Kirby, J. R., & Becker, L. D. (1988). Cognitive components of learning problems in arithmetic. Remedial & Special Education, 95, 7–16.
Lerner, J. (1993). Learning disabilities: Theories, diagnosis, and teaching strategies (6th ed.). Boston:
Houghton Mifflin.
Maccini, P., & Hughes, C. A. (1997). Mathematics
interventions for adolescents with learning disabilities. Learning Disabilities Research & Practice,
12, 168 –176.
Maccini, P., McNaughton, D., & Ruhl, K. L. (1999).
Algebra instruction for students with learning disabilities: Implications from a research review.
Learning Disability Quarterly, 22, 113–126.
Miller, S. P., & Hudson, P. J. (2007). Using evidence-based practices to build mathematics competence related to conceptual, procedural, and
declarative knowledge. Learning Disabilities Research & Practice, 22, 47–57.
Miller, S. P., & Mercer, C. D. (1997). Educational
aspects of mathematics disabilities. Journal of
Learning Disabilities, 30, 47–56.
Montague, M. (1992). The effects of cognitive and
metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning
Disabilities, 25, 230 –248.
Montague, M. (1997). Cognitive strategy instruction
in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 164 –177.
Montague, M. (2001). The effects of cognitive and
metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning
Disabilities, 25, 230 –248.
Montague, M. (2008). Self-regulation strategies to
improve mathematical problem solving for students with learning disabilities. Learning Disability
Quarterly, 31, 37– 44.
Montague, M., & Dietz, S. (2009). Evaluating the
evidence base for cognitive strategy instruction
and mathematical problem solving. Exceptional
Children, 75, 285–302.
Patton, J. P., Cronin, M. E., & Bassett, D. S. (1998).
Math literacy: The real-life math demands of
adulthood. In S. A. Vogel & S. Reder (Eds.),
Learning disabilities, literacy, and adult education.
Baltimore, MD: Paul H. Brookes Publishing.
Raskind, M. H. (1998). Literacy for adults with
learning disabilities through assistive technology.
In S. A. Vogel & S. Reder (Eds.), Learning disabilities, literacy, and adult education. Baltimore, MD:
Paul H. Brookes.
Reid, R. C., & Lienemann, T. O. (Eds.). (2006).
Strategy instruction for students with learning disabilities. New York: Guildford Press.
Rivera, D., & Smith, D. D. (1988). Using a demonstration strategy to teach midschool students with
learning disabilities how to compute long division. Journal of Learning Disabilities, 21, 77– 81.
Schmidt, J. L., Deshler, D. D., Schumaker, J. B., &
Alley, G. R. (1989). Effects of generalization instruction on the written language performance of
adolescents with learning disabilities in the main-
Functional Mathematical Skill
/
357
stream classroom. Journal of Reading, Writing, and
Learning Disabilities, 4, 291–311.
Swanson, H. L., & Hoskyn, M. (2001). Instructing
adolescents with learning disabilities: A component and composite analysis. Learning Disabilities
Research & Practice, 16, 109 –119.
Wagner, M., Newman, L., D’Amico, R., Jay, E. D.,
Butler-Nalin, P., Marder, C., & Cox, R. (1991).
Youth with disabilities: How are they doing? Menlo
Park, CA: SRI International.
Wehmeyer, M. L., Agran, M., & Hughes, C. (1998).
358
/
Teaching self-determination to students with disabilities. Baltimore: Brookes.
Yan, P. X., Grasso, E., Dipipi-Hoy, C., & Jitendra, A.
(2005). The effects of purchasing skill Instruction for individuals with developmental disabilities: A meta-analysis. Exceptional Children, 71,
379 – 400.
Received: 18 May 2011
Initial Acceptance: 21 July 2011
Final Acceptance: 28 September 2011
Education and Training in Autism and Developmental Disabilities-September 2012