Model-Drawing Strategy
to Solve Word Problems
for Students with LD
Olga Jerman and Jacqueline Knight
The Frostig Center
www.frostig.org
DISCES CEC
Riga, Latvia
July 11- 14, 2010
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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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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 8step model-drawing approach for
solving problems with fractions and
percentages.
• The results showed improvement in
solution accuracy.
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Word-problem Solving and LD
• difficult and frustrating
• cognitive processes involved in successful
problem completion.
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• Research findings indicate that the
reduction of demands on the working
memory system (WM) seems to be highly
beneficial.
• Different ways to minimize demands:
 use of visual support via pictures, diagrams &
schemas
 use of cognitive strategies
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Purpose of the Study
• An 8-step model-drawing technique is
intended
 to enhance the conceptual understanding
of the problem at task
 to reduce the amount of information to
be held in working memory
• No prior studies done with students with
learning disabilities
• Primary purpose of this study-to assess the
usefulness of Singapore model drawing
technique for students with LD
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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
20%
Zoo
Farm
60
school
?
Total students = ?
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)
Rosie baked 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)
There were 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
Theoretical and Practical Considerations
 Due to their abstract nature, 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.