Model-Drawing Strategy
to Solve Word Problems
for Students with LD
Dr. Olga Jerman
The Frostig Center
IARLD Conference
Miami, Florida
January 14-16, 2010
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Abstract
The study examined the effectiveness
of using model-drawing methodology
to solve problems for a group of high
school students. The 30-week
intervention used a single-subject
design to teach an 8-step modeldrawing approach for solving problems
with fractions and percentages. The
results showed improvement in
solution accuracy.
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Word-problem Solving and LD
Word problem-solving is an area of
difficulty and frustration for a considerable
number of students, and this, to a great
extent, could be attributed to a large
number of cognitive processes involved in
successful problem completion. It is an
especially difficult area for those students
who are identified with learning disabilities
(LD).
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Recently, a considerable amount of work
has been done to examine the sources of
difficulties in problem-solving, predictors
of success, and the best practices and
programs aimed at helping struggling
learners to better problem-solve.
Research findings indicate that the
reduction of demands on the working
memory system (WM) seems to be highly
beneficial. Different ways to minimize
these demands on the WM system have
been tested (e.g. use of visual support via
pictures, diagrams & schemas, and use of
cognitive strategies).
Purpose of the Study
An 8-step model-drawing technique is
intended to enhance the conceptual
understanding of the problem at task and to
reduce the amount of information to be held in
working memory, which, consequently, would
lead to the increased chances of solving
problems correctly. Although the approach was
found to be successful for a regular student
population (typically-achieving kids), no
studies, to the author’s knowledge, have
examined the effectiveness of this methodology
for students with learning disabilities.
Therefore, the primary purpose of this study
was to assess the usefulness of Singapore
model drawing technique for students with LD.
Model Drawing Strategy
 8 Steps of Model drawing
1.
2.
3.
4.
5.
6.
7.
8.
Read the problem
Decide who is involved
Decide what is involved
Draw unit bars
Read each sentence
Put the question mark
Work computation
Answer the question
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Example:
Word Problems with Percentage
40% of the school students went to the
National History Museum for a field trip.
20% of students went to the zoo. 50%
of the remaining students went to a
farm. Only 60 students didn’t have a
field trip and stayed at school. How
many students are there in this school?
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Solution
Step 1: Draw a unit bar and divide it into 10 equal parts
50% of remaining
40%
Museum
Total students = ?
20%
Zoo
60
school
Farm
100% remaining students
One unit bar = ?
1) 60 : 2 = 30
2) 30 x 10 = 300
Answer: There are 300 students in the school
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Example: Fraction Problems
a) Rosie baked 63 cookies. 3/7 of them were
chocolate chip cookies and the rest were
sugar cookies. How many sugar cookies did
Rosie bake?
1
2
3
4
5
6
7
63
63 : 7 = 9 (one unit bar equals 9)
9 x 4 = 36 (sugar cookies)
63 : 7 = 9 (one unit bar equals 9)
3 x 9 = 27 (chocolate chip cookies)
63 – 27 = 36 (sugar cookies)
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Example: Fraction Problems
b) 5/8 of the students in my class are boys.
1/5 of the boys have black hair. If 40 boys
don’t have black hair, how many students
are in my class in all?
1
2
3
4
5
6
7
8
1)
1
5/8 - boys
2
3
4
3/8 - girls
5
5 units - boys
2)
1/5 – boys with black hair
1
3)
2
3
Or
4/5 without black hair
4
40
40 : 4 = 10 (one unit bar) =>
10 x 8 = 80 (students in the class)
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Method
• 5 students (2 control)
 2 girls & 3 boys (mean age 16-1)
 10th grade
• 30 weeks intervention
• 20 weeks for fraction problems, 10
weeks percent problems
• Treatment fidelity 73%
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Scores and Progress of a Control Student #1
Intervention 1
Fractions
Baseline
110
Intervention 2
Fractions
No Intervention
Intervention 3
Percentiles
100
90
No Intervention
R____
80
60
40
30
M=20 20
10
Follow-up Percentiles
Follow-up Fractions
50
33
29
30
27
28
25
26
22
23
24
a
24
b
20
21
18
19
16
17
14
15
12
13
9
10
11
8
7
6
5
4
3
2
1
0
31
32
Scores
70
Weeks
Accuracy Points
Accuracy Percentage
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Scores and Progress of a Control Student #2
110
No Intervention
Intervention 3
Percentiles
100
90
80
Scores
70
60
50
40
Follow-up Percentiles
Baseline
Intervention 2
Fractions
Follow-up Fractions
Intervention 1
Fractions
No Intervention
E____
30
M=21.33
20
10
33
31
32
29
30
27
28
25
26
23
24
a
24
b
21
22
19
20
17
18
15
16
13
14
11
12
9
10
8
7
6
5
4
3
2
1
0
Weeks
Accuracy Points
Accuracy Percentage
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Scores and Progress of a Tx student #1
100
90
80
Follow-up Percentiles
110
Intervention 3
Percentiles
Intervention 2
Fractions
No Intervention
Follow-up Fractions
Intervention 1
Fractions
Baseline
No Intervention
C______
Scores
70
60
50
40
30
20
33
31
32
29
30
27
28
25
26
22
23
24
a
24
b
20
21
18
19
16
17
14
15
12
13
9
10
11
8
7
6
5
4
3
2
1
10
M=1.25
0
Weeks
Accuracy Points
Accuracy Percentage
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Scores and Progress of a Tx student #2
No Intervention
Intervention 3
Percentiles
100
90
80
Follow-up Percentiles
Baseline
110
Intervention 2
Fractions
Follow-up Fractions
Intervention 1
Fractions
No Intervention
J____
Scores
70
60
50
40
30
20
10
33
31
32
29
30
27
28
25
26
23
24
a
24
b
21
22
19
20
17
18
15
16
13
14
11
12
9
10
8
7
6
5
4
3
2
0
1
M=1
Weeks
Accuracy Points
Accuracy Percentage
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No Intervention
Intervention 3
Percentiles
100
90
80
Follow-up Fractions
Baseline
110
Intervention 2
Fractions
No Intervention
O____
Intervention 1
Fractions
Follow-up
Scores and Progress of a Tx student #3
Scores
70
60
50
40
30
20
33
31
32
29
30
27
28
25
26
22
23
24
a
24
b
20
21
18
19
16
17
14
15
12
13
9
10
11
8
7
6
5
4
3
2
1
10
M=2
0
Weeks
Accuracy Points
Accuracy Percentage
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Conclusion
• Model-drawing strategy can be an effective
alternative method of teaching fraction and
percent problems to students with LD;
• Although the training yielded improvement,
it took longer for the students to learn the
technique than initially planned;
• Students’ performance remained higher than
their pre-intervention scores, though it
slightly declined at the 4-week follow-up;
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Implications
The current results have important theoretical and
practical considerations. Because of the abstract
nature and complex calculation processes
involved, word problems with percent and
fractions are especially hard to tackle for students
with LD. The model-drawing approach gives
students a more concrete method in
comprehending and solving word problems in
order to get past their language difficulties. By
drawing out what they are reading, the students
are creating a concrete visual application of the
problem. This helps them to manipulate the
numbers more easily.
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Implications
(cont.)
The word problem instruction could also be
applied in different ways: either in the largegroup format or as part of differentiated
instruction. The model drawing gives students a
clear procedure for comprehending and
executing problems. As students understand
each level of a problem, the problem of the day
or of the lesson can eventually be taught at
grade level.
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References
•
Jitendra, A. K., Griffin, C. C., McGoey, K., Gardill, M. C., Bhat, P., & Riley, T. (1998).
Effects of mathematical word problem-solving by students at risk or with mild
disabilities. Journal of Educational Research, 91, 345-355.
•
Marshall, S. P. (1995). Schemas in problem solving, Cambridge University Press.
•
Montague, M. Self-Regulation strategies for better math performance in middle school.
(In M Montague and A Jitendra 2006, pp. 86-106).
•
Newcombe, N. S., Ambady, N., Eccles, J., et al (2009). Psychology’s Role in mathematics
and Science Education. American Psychologist, 64, 6, 538-551.
•
Powell, S. R., Fuchs, L. S., Fuchs, D., Cirino, P. T., & Fletcher, J. M. (2009). Do wordproblem features affect problem difficulty as a function of students’ mathematics
difficulty with and without reading difficulty? Journal of Learning Disabilities, 42, 99111.
•
Swanson, H. L. & Beebe-Frankenberger, M. (2004). The relationship between working
memory and mathematical problem solving in children at risk and not at risk for serious
math difficulties. Journal of Educational Psychology, 96, 471-491.
•
Xin, Y. P., Wiles, B., & Lin, Y. (2008). Teaching conceptual model-based word problem
story grammar to enhance mathematics problem solving. The Journal of Special
Education, 42, 163-178.