Math Problem Solving Through Graphic Representations Discussion Week 7 Forum – Different Graphical Representations: This week’s lecture introduced you to v

Math Problem Solving Through Graphic Representations Discussion Week 7 Forum – Different Graphical Representations: This week’s lecture introduced you to various types of graphical representations used in elementary mathematics. This week’s reading displayed different graphical organizers that could be used in answering word problems, similar to those used in English classes.

Which representation do you think could be better understood by your students and why?

What is the benefit of having the students organize the data or information from word problems into these representations?

Read your peers’ answers and post a response

Peer Response

Tara

Which representation do you think could be better understood by your students and why?

The graphical representation that I think could better understood by my pre-k students would be the pictograph or a bar graph. I feel that we could use the pictograph just like in the example given in this week’s lecture “favorite color chat” but instead of using crayons to represent how many students favor that color I would allow my students to place a picture of their face to and I would use a crayon to represent the colors. A bar graph will be a great tool in a pre-k classroom to help with counting lessons. For example, we had a lesson where we used rainbow gold fish to make a bar chart to show the amount of each color fish that each student had in their cup. My students really enjoyed this activity because they were allowed to eat their fish after we finished the lesson and they were able to count all pf their fish before doing so.

What is the benefit of having the students organize the data or information from word problems into these representations?

Some of the benefits for having students organize the data or information from word problems into theses representations would be to provide students with a simple, visual way to interpret data and to draw conclusions about mathematical relationships, such as more than, less than and grouping. By providing visual graphs it can provide students with clues that words and equations may not provide. Teaching Students Math Problem-Solving Through
Graphic Representations
TEACHING Exceptional Children, Vol. 34, No. 4, pp. 34-38. Copyright 2002 CEC.
Asha Jitendra
Lisa is an 8-year-old third grader who
receives special education services in a
learning support classroom for mathematics and reading. She has little confidence in her problem-solving ability
and has a phobia for mathematics.
When asked to solve word problems,
Lisa typically grabs all the numbers in
the problem and starts to add rather
than pause to understand the nature of
the problem she has to solve. Many students who take Lisa’s approach to problem-solving—often students with learning disabilities and those at risk for
math failure—become unsuccessful
problem-solvers.
But here’s the good news: We can
use graphic representations to teach students with learning disabilities to be
effective problem-solvers (see box,
“What Is a Graphic Representational
Technique?”). Recent studies have indicated that problem-solving instruction
(e.g., graphic representational strategy)
that emphasizes conceptual understanding can significantly help children
with learning disabilities meet the challenges of the general education classrooms (Xin & Jitendra, 1999).
This article describes the graphic
representational technique, shows how
to use the strategy for solving word
problems, and discusses how to assess
students’ problem-solving performance.
Teaching Problem-Solving
Using the Graphic
Representational Strategy
In addressing “big ideas” in mathematical problem-solving, we need to focus
on carefully chosen problems. Here, we
look at three different problem types,
“change,” “group,” and “compare,” that
characterize most addition and subtraction problems presented in commercial
basal mathematics programs (Marshall,
1990). Using graphic representations,
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students can develop their problemsolving skills, but also integrate addition
and subtraction concepts. The essential
elements of the graphic representation
strategy are problem schemata identification and representation and problem
solution.
Phase 1. Problem Schemata
Identification and Representation
The goal of schema identification and
representation instruction, a prerequisite to understanding and organizing
information for later problem solution,
is to facilitate conceptual understanding. Problem schemata identification
involves recognizing the problem pattern (e.g., “change” or “group”), or
semantic structure of the problem,
whereas problem representation refers
to translating a problem from words
into a meaningful graphic representation.
The key aspects of this instruction
are for students to
• Identify the separate features of each
problem type (problems involving a
change, a grouping, or a comparison—change, group, and compare).
• Organize and represent the relevant
information in the story situation
using schematic diagrams (see Figure
1).
Begin Phase 1 by providing students
with story situations (see Figure 1) of
each problem type. These situations
present all quantities as known before
students actually learn to solve word
To guide students in
memorizing the rules,
provide them with a
self-instructional sheet.
What Is a Graphic
Representational Technique?
Effective word problem-solving requires the
learner “to create a representation of the problem that mediates solution” (Goldman, 1989, p.
45). A primary characteristic of a graphic representational strategy that distinguishes it from
other approaches is the use of schematic diagrams that allow students to organize information in the problem to facilitate problem translation and solution. The external representation
(e.g., diagrams) may serve to reduce a learner’s
cognitive processing load and make available
mental resources for engaging in problem analysis and solution.
For example, the problem, “John has 3 marbles; Tim gives him 6 more marbles; how many
marbles does John have now?” illustrates a
beginning set, a “change” set, and an ending set
requiring the selection of the change diagram.
Based on the representation of key sets (i.e.,
beginning, change, and ending sets) in the problem, the problem-solver can select the appropriate arithmetic operation to find the quantity of
the unknown set (i.e., ending) in the problem
representation.
In contrast, though other instructional
strategies also may include diagrams, the
emphasis is less on identifying the semantic relations in the problem necessary for effective problem solution. In sum, the graphic representational strategy is a viable approach for enhancing
students’ conceptual knowledge about problemsolving, because it emphasizes instruction that
goes beyond the mastery of algorithms used to
perform operations to focus more on the semantic structure of problems.
problems. Introduce problem-schema
analysis using explicit modeling with
several examples of story situations.
During guided practice, use frequent
student exchanges to facilitate the identification of critical elements of the
story. The following examples describe
how to teach students to discern the
features that govern each of the three
different story situations:
• ”Change” story situation. Consider
the following story situation involving a change:
John had 47 baseball cards in
his collection. He lost 15 of
them when his family moved
from Florida to New York. Now
John has 32 baseball cards.
Teach students that the “change”
schema relation starts with a “beginning set” (e.g., 47 baseball cards) in
which the object identity (e.g., baseball
cards) and the amount of the object are
defined (e.g., 47). Then a change (loses
15 baseball cards) occurs to the beginning set that results in an “ending set”
(32 baseball cards) in which the new
amount (i.e., 32) is defined. The ending
set is less than the beginning set,
because the word “lost” in the story situation implies that the change resulted
in fewer baseball cards than the amount
John initially had. Students learn that in
this situation, both the beginning and
ending sets could not occur at the same
time; either there were 47 baseball cards
or 32 baseball cards. Based on the position of the beginning, change, and ending sets in the story situation, students
learn to infer the passage of time from
past to present.
• ”Group” story situation. Here is a
sample story situation involving a
group:
Tim has 54 fruit trees in his
orchard; 39 are apple trees, and
the remaining 15 are peach
trees.
Here, teach students that unlike the
“change” relation, the passage of time is
irrelevant to this “group” situation. The
story did not say that Tim first had 39
apple trees and then planted 15 peach
trees.
In addition, the “group” situation
does not involve any change of object
amounts. Instead, the group situation
involves understanding part-whole relationships and knowing that the whole
(i.e., superordinate) is equal to the sum
of its parts (subordinates). Students
learn that the subordinate categories
(apple trees and peach trees) should
Figure 1. Sample Story Situations and Schemata Diagrams for (a)
Change, (b) Group, and (c) Compare Problem Type
Change Story Situation
John had 47 baseball cards in his collection. He lost 15 of them when his
family moved from Florida to New York. Now John has 32 baseball cards.
Change set
15
Baseball cards
32
Baesball cards
47
Baseball cards
Beginning set
Ending set
Group Story Situation
Tim has 54 fruit trees in his orchard. 39 are apple trees, and the remaining
15 are peach trees.
Apple
trees
Peach
trees
Apple &
peach trees
39
15
54
Smaller sets
Larger set
Compare Story Situation
Mitch has 43 CDs and Anne has 70. Anne has 27 more CDs than Mitch.
Anne
Mitch
70 CDs
27 CDs
43 CDs
Compared set
Difference set
Referent set
Source: From Schemas in Problem Solving (p. 135) by S. P. Marshall, 1995, New York:
Cambridge University Press. Copyright 1995 by Cambridge University Press.
Representation adapted by permission.
have semantic ties to the superordinate
category (trees).
For example, students learn the common attributes of the three elements,
“apple trees,” “peach trees,” and
“trees,” and the relationships among
them in the story situation. Although
“apple trees” and “peach trees” are
instances of trees with some similar
attributes, they are distinct in that an
apple tree is not the same as a peach
tree.
• ”Compare” story situation. Here is a
story situation involving a comparison, or a “compare” relation:
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Mitch has 43 music CDs and
Anne has 70. Anne has 27 more
music CDs than Mitch.
Teach students to focus on the presence of two sets (Mitch and Anne and
their music records), each associated
with a smaller or larger value (i.e., 43
and 70) and both having the same unit
of measure (e.g., music CDs). One set
serves as the comparison set (Anne’s
music CDs) and the other as the referent
set (Mitch’s music CDs). Typically, students learn to identify the compared
and referent sets and determine the larger set by examining the comparison
statement (i.e., Anne has 27 more music
CDs than Mitch). Students focus on the
more than or less than concepts to compare the two sets and identify the difference in value between the two sets.
For each problem type, the teacher or
students read the story situation and
map features of the situation onto the
appropriate
schematic
diagrams.
Initially, worksheets should include
story situations of a single problem
type. When students learn to identify
and map subsequent problem types,
worksheets should gradually include all
three problem types. This phase should
continue until students are able to accurately discern the three different problem types.
Figure 2. Sample Procedure for Solving (a) Change, (b) Group,
and (c) Compare Problems
Change Problem
A balloon man had some balloons. Then 14 balloons blew away and the man
now has 29 balloons. How many balloons did the man begin with?
Change set
14
balloons
29
balloons
?
balloons
Beginning set
T
Ending set
Total is not known, so add.
29 + 14 = 43
The man began with 43 balloons.
Group Problem
Jenny saw 25 birds on a camping trip. She saw 17 sparrows and some owls.
How many owls did Jenny see on the camping trip?
Sparrows
Owls
Birds
17
?
25
Phase 2. Problem Solution
Problem solution includes the selection
and application of appropriate mathematical operations based on the problem representation. The key aspects of
problem solution instruction are for students (a) to plan to solve the problem
by identifying the action procedure
(e.g., counting, adding, subtraction)
and sequence of steps; and (b) to carry
Using graphic
representations to
emphasize conceptual
understanding can help
children with learning
disabilities in inclusive
classrooms.
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Smaller sets
Larger set
T
Total is known, so subtract.
25 – 17 = 8
Jenny saw 8 owls on the camping trip.
Compare Problem
Barbara is 37 years old. Cindy is 7 years older than Barbara. How old is
Cindy?
Cindy
? years
Compared set
T
Barbara
7 years
Difference set
Total is not known, so add.
37 + 7 = 44
Cindy is 44 years old.
37 years
Referent set
out the operation identified in the planning step to solve the problem.
Once students display adequate
knowledge of problem schemata, teach
them to solve word problems. Begin this
phase with a review of the problem
schemata within the context of real
word problems, rather than the story situations shown in Phase 1. Through a
question-and-answer session, prompt
students to identify the problem type
and the key features of the problem.
Then ask them to map the information
onto the relevant diagram and flag the
missing information in the problem,
using a question mark. During this
phase, address any misconceptions with
explicit feedback and additional modeling.
After students identify and map the
information onto diagrams, they learn
to plan to solve the problem. They begin
by finding the “whole” or total amount
in the word problem by focusing on the
specific information provided in the
text, as follows:
• In a “change” problem, students learn
whether the problem ends up with
more or less than the beginning
amount. Teach students that when
the problem ends up with more, the
ending amount represents the total. If
the problem ends up with less, however, the beginning amount is the
total.
• For the “group” problem, the larger
group object always represents the
total, because the smaller groups
combine to form the larger group.
• To find the total amount in the “compare” problem, students determine
whether the referent or the compared
set represents the higher value. Teach
students to examine the comparison,
Figure 3. Rules to Identify the Total and the Operation for (a)
Change, (b) Group, and (c) Compare Problems
Finding the Total
a. “Change” Problem
If the problem ends up with more than it started with, then the
ending set is the total. If the problem ends up with less than it
started with, then the beginning set is the total.
b. “Group” Problem
The larger group set is always the total.
c. “Compare” Problem
The larger set (compared or referent) in the comparison or difference statement is the total.
Figuring Out the Operation (Addition or Subtraction)
When the total is unknown, add to find the total. When the total is
known, subtract to find the other amount.
Figure 4. Representational Strategy
Problem Schemata Identification and Representation
1. Find the problem pattern
(a) Read the problem carefully
(b) Ask whether the problem is a change, group, or compare problem type
2. Organize and represent the information in the problem using schemata
diagrams
(a) Map the known information (object identity and object amount) onto
the schema diagram
(b) Flag the unknown information using a question mark
Problem Solution
3. Plan to solve the problem
(a) Find the object identity that represents the “whole” or total amount
and write a “T” for total under the set
(b) Select an arithmetic operation based on the known and unknown
information (i.e., “When the total is not known, add to find the total;
when the total is known, subtract to find the part.”)
4. Solve the problem
Schematic diagrams
allow students to
organize information in
the problem to
facilitate problem
translation and
solution.
(a) Add or subtract
(b) Check if the answer makes sense
(c) Write the whole answer
or “difference” statement, to find the
set that represents the total amount in
the “compare” problem. In each
instance, students learn to write T for
total under the specific set that represents the total (see Figure 2).
Next, students select an arithmetic
operation based on which part of the
problem situation is unknown and
which of the critical elements in the
problem structure represents the total or
whole. Finally, they solve the problem
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using a generalizable rule, “When the
total is not known, add to find the total;
when the total is known, subtract to
find the part or other amount,” which is
based on the part-whole concept (see
Figure 2). To guide students in memorizing the rules, provide them with a
self-instructional sheet with written
rules for identifying the total and determining the operation needed to solve
the problem (see Figure 3). The note
sheet can serve as a scaffold until students can independently verbalize the
rules. The self-instructional note sheet
and diagrams serve to improve students’ understanding and memory of
information in word problems. However, once students are proficient in
using the schemata diagrams, fade or
remove them and encourage them to
represent the problem using their own
diagrams. Figure 4 presents the instructional sequence of the representational
strategy.
Evaluating Problem-Solving
Performance
When teaching the graphic representational strategy, ask students to complete
word problem tests at the end of each
instructional session to check their
understanding and to evaluate whether
they have mastered the strategy steps.
Examine students’ completed tests not
only for correct solutions but also for
strategy use (e.g., draw a diagram, map
information onto the diagram, plan to
solve the problem by figuring out the
operation to use and the sequence of
steps) and provide them with additional
instruction, as needed, before they
move to the next problem type.
Initially, assess students’ performance in solving one problem type. Later,
when they complete instruction in the
use of the strategy steps for all problem
types, present tests that include all three
problem types. In addition, evaluate
how well students maintain the skill
and knowledge after the strategy lessons are over; and discover how well
they have transferred the skills to novel
and complex problems.
Student Success
My colleagues and I have successfully
taught the graphic representational
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strategy to students with disabilities in
elementary and middle schools in our
research investigations (Jitendra et al.,
1998; Jitendra & Hoff, 1996; Jitendra,
Hoff, & Beck, 1999). We found dramatic
improvements in students’ problemsolving scores. In addition, they maintained the skills for as long as 4 weeks,
and students were able to use the strategy in novel and complex problems. In
these studies, we also found that the
performance of students with learning
disabilities either approached or surpassed that of a sample of students
without disabilities.
Students also reported positive attitudes regarding the strategy instruction.
For example, students’ responses about
what they liked the most about the
strategy were as follows: “Learning to
solve problems,” “Getting the correct
answers,” “It’s easier to understand, it
gave me clues,” “I liked the problems,”
“It was fun,” “They were cool.” When
asked if they would recommend this
strategy to a friend, they all answered in
the affirmative. One student even
reported that she felt she could go to
college and become a teacher, which
she was not sure she would be able to
do before participating in our study.
Further, this student was so pleased
with her success that she became
involved in tutoring a younger student
in a general education classroom using
the strategy.
Teachers of students in our studies
also have indicated transfer of effective
problem-solving skills into their classrooms. One teacher noted that the student who evidenced math phobia before
the intervention showed increased confidence in her math performance and
was more consistent in labeling her
work on problem-solving tasks.
Final Thoughts
This article shows that we can teach
students with learning disabilities to use
effective problem-solving skills. We
should not have to wait until students
with learning disabilities master easier
material (e.g., computation skills)
before presenting them with problemsolving opportunities.
References
Goldman, S. (1989). Strategy instruction in
mathematics.
Learning
Disability
Quarterly, 12, 43-55.
Jitendra, A. K., Griffin, C., McGoey, K.,
Gardill, 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, 345-356.
Jitendra, A. K., & Hoff, K. (1996). The effects
of schema-based instruction on mathematical word problem solving performance of students with learning disabilities.
Journal of Learning Disabilities, 29, 422431.
Jitendra, A. K., Hoff, K., & Beck, M. (1999).
Teaching middle school students with
learning disabilities to solve multistep
word problems using a schema-based
approach.
Remedial
and
Special
Education, 20(1), 50-64.
Marshall, S. (1990). The assessment of
schema knowledge for arithmetic story
prob…
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