Article 4: Mathematical Word Problem Solving in Third-Grade ...

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Mathematical Word Problem Solving in
Third-Grade Classrooms
ASHA K. JITENDRA CYNTHIA C. GRIFFIN ANDRIA DEATLINE-BUCHMAN
Lehigh University University of Florida Easton School District
EDWARD SCZESNIAK
Easton School District
ABSTRACT The authors conducted design or classroom
experiments (R. Gersten, S. Baker, & J. W. Lloyd, 2000) at
2 sites (Pennsylvania and Florida) to test the effectiveness of
schema-based instruction (SBI) prior to conducting formal
experimental studies. Results of Study 1 conducted in 2 3rdgrade,
low-ability classrooms and 1 special education classroom
indicated mean score improvements from pretest to
posttest on word problem solving and computation fluency
measures. In addition, student perceptions of SBI according to
a strategy satisfaction questionnaire revealed SBI as effective
in helping solve word problems. Results of Study 2, which
included a heterogeneous (high-, average-, and low-achieving)
sample of 3rd graders, also revealed student improvement on
the word problem solving and computation fluency measures.
However, the outcomes were not as positive in Study 2 as in
Study 1. Lessons learned from the 2 studies are discussed
with regard to teaching and learning mathematical word problem
solving for different groups of students.
Keywords: elementary grade students, mathematics instruction,
schema-based instruction, word problem solving
Current education reforms support challenging
learning standards and school accountability. In
mathematics education, the emphasis is on the
development of conceptual understanding and reasoning
over memorization and rote learning (Goldsmith & Mark,
1999; Hiebert et al., 1996; National Research Council,
2001). Mathematical problem solving is a central theme in
the Principles and Standards for School Mathematics (National
Council of Teachers of Mathematics [NCTM], 2000).
Although the current emphasis is on solving complex
authentic problems situated in everyday contexts, story
problems that range from simple to complex represent “the
most common form of problem-solving” assignment in
school mathematics curricula (Jonassen, 2003, p. 267).
Story problems pose difficulties for many elementary
students because of the complexity of the solution process
(Jonassen, 2003; Lucangeli, Tressoldi, & Cendron, 1998;
Schurter, 2002). Mathematics tasks that involve storycontext
problems are much more challenging than are nocontext
problems (Cummins, Kintsch, Reusser, &
283
Weimer, 1988; Mayer, Lewis, & Hegarty, 1992; Nathan,
Long, & Alibali, 2002). Although children may know
procedures for solving no-context problems, solving story
problems requires them to integrate several cognitive
processes that are difficult for children with an insufficient
knowledge base or limited working memory capacity.
When solving story problems, for example, children
need to (a) understand the language and factual information
in the problem, (b) translate the problem with relevant
information to create an adequate mental representation,
(c) devise and monitor a solution plan, and (d)
execute adequate procedural calculations (Desoete, Roeyers,
& De Clercq, 2003; Mayer, 1999). In short, solving
word problems relates closely to comprehension of the
relations and goals within the problem (e.g., Briars &
Larkin, 1984; Cummins et al.; De Corte, Verschaffel, &
De Win, 1985; Kintsch & Greeno, 1985; Riley, Greeno,
& Heller, 1983).
Story problems are critical for helping children connect
different meanings, interpretations, and relationships to
mathematics operations (Van de Walle, 2004). How then
do educators enhance students’ mathematical word problem-solving
skill Traditional textbook problem-solving
instruction has not effectively improved the learning of students
at risk for mathematics difficulties. Many mathematics
textbooks are organized so that the same procedure
(e.g., subtraction) is used to solve all problems on a page.
As a result, students do not have the opportunity to discriminate
among problems that require different solution
strategies. Furthermore, traditional instruction teaches students
to use keywords (e.g., in all suggests addition, left suggests
subtraction, share suggests division; Lester, Garofalo,
& Kroll, 1989, p. 84) that are limiting. According to Van
de Walle, “Key words are misleading,” “Many problems do
not have key words,” and “Key words send a terribly wrong
message about doing math” (p. 152). Thus, this approach
Address correspondence to Asha K. Jitendra, Lehigh University,
College of Education, 315 A Iacocca Hall, 111 Research Drive, Bethlehem
PA 18015. (E-mail: akj2@lehigh.edu)
Copyright © 2007 Heldref Publications

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ignores the meaning and structure of the problem and fails
to develop reasoning and making sense of problem situations
(Van de Walle).
In contrast, current mathematics reform efforts advocate
that teachers act as facilitators to help students construct
their own understandings of mathematical concepts and
relationships (e.g., developing the abilities of inquiry, problem
solving, and mathematics connections). However, the
literature is inconclusive about the benefits of that approach
for all learners (Baxter, Woodward, Voorhies, & Wong,
2002). As the range of skills that students bring to the classroom
increases, the challenge for teachers is synthesizing
knowledge regarding mathematics content and processes,
student learning, effective instruction models, and appropriate
classroom opportunities and experiences (Ma, 1999).
Learners who lack sufficient prior knowledge may need more
supportive instructional strategies in which the teacher scaffolds
information processing by supplying a greater degree of
instructional facilitation during the learning process.
To promote elementary students’ mathematical problemsolving
skills, we relied on models for understanding and
assessing children’s solutions for addition and subtraction
word problems derived from schema theories of cognitive
psychology (Briars & Larkin, 1984; Carpenter & Moser,
1984; Kintsch & Greeno, 1985; Riley, Greeno, & Heller,
1983). On the basis of those models, we designed an intervention
(i.e., schema-based instruction [SBI]) for students
with high-incidence disabilities and for those at risk for
failure in mathematics (e.g., Jitendra et al., 1998; Jitendra,
Hoff, & Beck, 1999). Contrary to earlier investigations of
SBI that provided individual or small-group instruction by
researchers in rooms adjacent to the students’ classrooms,
we deemed it critical to broaden the learning environment
to a typical classroom context wherein the classroom
teacher provided all instruction. In addition, we added selfmonitoring
to strategy use because teaching students to
self-regulate their learning has an added positive effect on
their mathematics problem-solving performance (L. S.
Fuchs et al., 2003; Schunk, 1998; Verschaffel et al., 1999).
Theoretical Framework
Results of a number of research studies on solving addition
and subtraction word problems have shown that
semantic or mathematical structure of problems (i.e., specific
characteristics of the problem and the semantic relationships
among the various problem features) is much more
relevant than is syntax (i.e., how a problem is worded; Carpenter,
Hiebert & Moser, 1983; Carpenter & Moser, 1984).
Notwithstanding the ease or difficulty of the syntactic
structure of problems, students who lack a well-developed
semantic structure use a “bottom-up or text-driven
approach to comprehend a problem statement” (Kameenui
& Griffin, 1989, p. 581). Given the critical role of semantic
structure in problem solution, Carpenter and Moser postulated
a classification of addition and subtraction wordproblem
types that include change, combine, compare, and
equalize. Of those problem types, change, combine, and compare
problems are characteristic of most addition and subtraction
word problems presented in elementary mathematics
textbooks, indicating the need for research focusing
on these problem types. Table 1 shows the three different
problem types and their characteristics. Change problems
usually begin with an initial quantity and a direct or
implied action that causes either an increase or decrease in
that quantity. The three sets of information in a change
problem are the beginning, change, and ending. In the
change situation, the object identities (e.g., video games)
for beginning, change, and ending are the same (see Figure
1). In contrast, combine or group problems involve two distinct
groups or subsets that combine to form a new group or
set. Group problems require an understanding of part-partwhole
relations. The relation between a particular set and
its two distinct subsets is static (i.e., no action is implied).
Compare problems involve the comparison of two disjoint
sets (compared and referent); the emphasis is on the static
relation between the two sets (see Table 1). The three sets
of information in a compare problem are the compared, referent,
and difference. For each problem type, there are
three items of information; the position of the unknown in
these problems may be any one of the three items, which
can be found if the other two items are given.
Also critical to successful mathematics problem solving
is domain-specific knowledge (conceptual and procedural;
e.g., Hegarty, Mayer, & Monk, 1995). An important aspect
of domain-specific concept knowledge is problem comprehension/representation,
which involves translating the
text of the problem into a semantic representation on the
TABLE 1. Semantic Structure of Addition and Subtraction Problems
Problem characteristics
Relation
Problem type Component Object identity between sets
Change Beginning, change, ending Same Dynamic
Combine or group Part, part, whole Different Static
Compare Compared, referent, difference Different Static

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Change: Jane had 4 video games. Then her mother gave her 3 more video games for her birthday. Jane now has 7 video games.
+ 3
video games
Change
4
video games
7
video games
Beginning
Ending
Group: Sixty-eight students at Hillcrest Elementary took part in the school play. There were 22 third graders, 19 fourth graders, and
27 fifth graders in the school play.
3rd
graders
4th
graders
5th
graders
All students (3rd,
4th & 5th graders)
22
19
27
68
Small Group
or Parts
Large Group
or Whole (Sum)
Compare: Joe is 8 years older than Jill. Jill is age 7 years and Joe is age 15 years.
Joe
15
years
Jill
7
years
8
years
Bigger Smaller Difference
FIGURE 1. Schematic diagrams for change, group, and compare problem situations. Change diagram adapted from “Schemas in
problem solving,” by S. P. Marshall, 1995, p. 133. Copyright 1995 by Cambridge University Press. Adapted with permission of
the author.
basis of an understanding of the problem structure.
Although procedural knowledge (e.g., knowing the series of
sequential steps used to solve routine mathematical tasks,
such as adding 27 + 19 as well as knowing mathematical
symbolism, such as +, =, and >) is also important, it “is
extremely limited unless it is connected to a conceptual
knowledge base” (Prawat, 1989, p. 10). In fact, a lack of
understanding of the problem situation may result in a solu-

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tion plan that is immaturely developed. Successful problem
solvers can translate and integrate information in the problem
into a coherent mental representation that mediates
problem solution (Mayer, 1999; Mayer & Hegarty, 1996).
However, many students have difficulty with problem comprehension
and solution and would benefit from instruction
in constructing a model to represent the situation in
the text, followed by solution planning based on the model
(Hegarty et al., 1995). Evidently, schema-based instruction
(SBI), with its emphasis on semantic structure and problem
representation, is one solution to advancing students’
mathematical problem-solving skills.
The goal of SBI is to help students establish and expand
on the domain knowledge in which schemata are the central
focus. A schema is a general description of a group of
problems that share a common underlying structure requiring
similar solutions (Chen, 1999; Gick & Holyoak, 1983).
According to Marshall (1995), schemata “capture both the
patterns of relationships as well as their linkages to operations”
(Marshall, 1995, p. 67). SBI analyzes explicitly the
problem schema (e.g., the part-part whole) and the links
pertaining to how different elements of the schema are
related (e.g., parts make up the whole). Understanding
these links is crucial in selecting appropriate operations
needed for problem solution. For example, if the whole in
the problem is unknown, adding the parts is necessary to
solve for the whole; if one of the parts is unknown, subtracting
the part(s) from the whole is needed to solve for
the unknown part. An important difference between SBI
and other instructional approaches is that only SBI emphasizes
integrating the various pieces of factual information
essential for problem solving. Although factual details are
important, they should not be the central focus of instruction
and learning. In short, SBI allows students to approach
the problem by focusing on the underlying problem structure,
thus facilitating conceptual understanding and adequate
word-problem-solving skills (Marshall).
Purpose
We conducted design or classroom experiments concurrently
at two different geographical sites. Our research team
collaborated with teachers to conduct a series of word problem
solving teaching sessions using SBI with a small sample
of students to understand what works, prior to conducting a
formal experiment (Cobb, Confrey, diSessa, Lehrer, &
Schauble, 2003). Design experiments involve in-depth
study of an innovation in a small scale to understand not
only how responsive an intervention is to the dynamics of
the classroom but also to understand “the nature of effective
adaptations for students with disabilities” (Gersten et al.,
2000, p. 29). Specifically, a design experiment is aimed at
improving the connections between teaching and learning
through the process of continuous cycles of “design, enactment,
analysis, and redesign” of lessons in authentic classrooms
(Cobb et al., 2003; The Design-Based Research Collective,
2003). Researchers use the processes of iteration
and feedback loops to empirically refine the innovation
(e.g., curriculum product, tasks, scaffolds) and design an
effective and efficient intervention for use in experimental
or quasi-experimental studies (Gersten et al., 2000).
The learning environment in our design experiments
included a typical classroom context rather than an exceptional
instructional setting (i.e., special rooms outside the
regular classroom), wherein the teacher conducted all
instruction. The purposes of the design experiments were
twofold. First, we investigated the effects of SBI on the
acquisition of skills for solving mathematical word problems
by third graders and explored the differential effects of
SBI for different groups of children (e.g., children with
learning disabilities; low, average, and high achievers). Second,
we examined the influence of word problem-solving
instruction on the acquisition of computational skills given
the role that word problems play in the development of
number operations (Van de Walle, 2004).
The basic methods were similar across both studies. In
Study 1, we addressed the issue of the efficacy of SBI for a
homogeneous sample of students from low-ability mathematics
classrooms and potential differences between students
with learning disabilities and students without disabilities
who were low achievers. Study 2 focused on the
efficacy of SBI for a heterogeneous sample of third graders,
including students differing in initial mathematics gradelevel
achievement.
DESIGN STUDY 1
Method
Participants
Participants included 40 third-grade students who
attended an elementary school in a suburban school district
in Pennsylvania serving 472 students in Grades K–5.
Approximately 15% of the student population was African
American, Hispanic, or Asian; 17% were economically disadvantaged;
and 5% were English language learners (ELLs).
Third-grade students were grouped according to ability levels
(two low- and two high-ability classrooms). Teachers in
the two low-ability third-grade classrooms (Classrooms 1
and 2), as well as the special education teacher (Classroom
3), participated to learn about innovative approaches to
improving students’ mathematical problem-solving performance.
The final sample, however, comprised 38 students
(20 boys and 18 girls) because 2 students moved out of the
school district before completing the study. The mean
chronological age of students was 102.60 months (range =
91 to 119 months; SD = 5.54). Twenty-eight students (74%)
were Caucasian, 3 (8%) were African American, 6 (16%)
were Hispanic, and 1 was Middle Eastern (3%).
The sample included 9 students (6 boys and 3 girls) with
learning disabilities (LD). While 3 of the students with LD
received instruction in the general education classrooms,

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the remaining 6 students were instructed in the learning
support classroom. In addition, we designated 9 of the 38
students as low achievers (LA) on the basis of their score
(below the 35th percentile) on the Computation and Concepts
and Applications subtests of the TerraNova mathematics
achievement test (CTB/McGraw-Hill, 2001).
Tables 2 and 3 show a summary of participant demographic
information by classroom and group status.
Separate one-way analyses of variance (ANOVAs) indicated
no significant differences between classrooms on the
mathematics subtests of the TerraNova: Concepts and
Application, F(2, 35) = 0.38, ns, and Computation, F(2,
35) = 0.80, ns; Total Reading subtest, F(2, 35) = 1.03, ns.
However, there was a significant difference between classrooms
on chronological age, F(2, 35) = 15.99, p < .000.
The mean age of special education students in Classroom 3
was greater than that of students in Classrooms 1 and 2,
which we expected given that the special education students
were in the third and fourth grades. Chi-square
analyses revealed no significant between-classroom differences
on gender, χ 2 (2, N = 38) = 2.57, ns, and race, χ 2 (6,
N = 38) = 8.68, ns.
Similarly, separate one-way ANOVAs indicated no significant
between-group (LD and LA) differences on the Concepts
and Application subtest of the TerraNova, F(1, 16) =
2.84, ns, and on the Total Reading subtest, F(1, 16) = 1.96,
ns. However, there was a significant between-group difference
on the Computation subtest, F(1, 16) = 5.53, p < .05.
The mean score for students with LD was greater than that
for LA students. In addition, there was a significant betweengroup
difference on chronological age, F(1, 16) = 10.73, p <
.01. The mean age of LD students was greater than was that
of LA students. Chi-square analyses revealed no significant
between-group differences on gender, χ 2 (1, N = 18) = .90,
ns, and race, χ 2 (2, N = 18) = 1.62, ns.
Three female teachers participated in Study 1, each
teaching in one classroom. The two general education
teachers were certified in elementary education and each
had more than 25 years of teaching experience; the special
education teacher was certified in special education and
had 19 years of teaching experience. Teachers attended 1
hr of inservice training on ways to implement the intervention;
they received ongoing support from two research
assistants (doctoral students in special education) throughout
the study.
Procedures
In all classrooms, teachers taught mathematics five times
a week for 50 min with the district-adopted basal text,
Heath Mathematics Connections (Manfre, Moser, Lobato, &
Morrow, 1994). Teachers taught the word problem-solving
unit during regularly scheduled mathematics instruction for
15 weeks. They taught students how to solve one-step addition
and subtraction problems involving change, group,
and compare problems for 30 min daily, 3 days per week.
Instruction was scripted to ensure consistency of information
and included an instructional paradigm of teacher
modeling with think-alouds, followed by guided practice,
paired partner work, independent practice, and homework.
In addition, instruction during guided practice emphasized
frequent teacher–student exchanges to facilitate problem
solving. Word problem-solving instruction involved two
phases: (a) problem schema and (b) problem solution. We
developed a strategy checklist to help scaffold student
learning. The checklist included the following six steps: (a)
read and retell the problem to discover the problem type:
(b) underline and map important information in the word
problem onto the schematic diagram, (c) decide whether to
add or subtract to solve the problem, (d) write the mathematics
sentence and solve it, (e) write the complete answer,
and (f) check the answer.
Schema for Problem-Solving Instruction
During the problem-solving phase, students received
story situations that did not contain any unknown information.
Instruction focused on identifying the problem
schema for each of the three problem types (change, group,
compare) and on representing the features of the story situation
with schematic diagrams (see Figure 1). That is, students
learned to interpret and elaborate on the main features
of the story situation and map the details of the story
onto the schema diagram. Teachers introduced each problem
type successively and cumulatively reviewed them to
help students discern the three problem types.
Change story situation. Students learned to identify the
problem type by using story situations such as, “Jane had
four video games. Then her mother gave her three more
video games for her birthday. Jane now has seven video
games.” Using Step 1 of the strategy checklist, students
identified the story situation as change because it initially
involved four video games, then an action occurred that
increased this quantity by three, which resulted in a total
of seven video games. Also, the object identity of the
beginning, change, and ending is video games. As such,
this story situation is considered a change problem
schema. For Step 2, teachers prompted students to use the
corresponding diagram (see Figure 1) to organize or represent
the information. That step involved identifying and
writing the object identity or label (e.g., video games) for
the three items of information (i.e., beginning, change,
and ending) in the change diagram. Students then read
the story to find the quantities associated with the beginning,
change, and ending and wrote them in the diagram.
Next, students summarized the information in the story
with the completed diagram and checked the accuracy of
the representation.
Group story situation. Students identified the group story
problem type by using story situations such as, “Sixty-eight
students at Hillcrest Elementary took part in the school
play. There were 22 third graders, 19 fourth graders, and 27

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TABLE 2. Student Demographic Information and Performance by Classroom: Site 1
Classroom 1 Classroom 2 Classroom 3
Variable n % M SD n % M SD n % M SD ES
Age (in months) 99.69 3.39 102.44 3.74 110.83 5.40
Gender
Male 6 37.50 10 62.50 4 66.67
Female 10 62.50 6 37.50 2 33.33
Race
EA 10 62.50 13 81.25 5 83.33
AA 2 12.50 1 6.25 0 0.00
Hispanic 4 25.00 2 12.50 0 0.00
Other 0 0.00 0 0.00 1 16.67
Spec. ed. student 1 2 6
TerraNova
C & A 567.62 21.23 575.79 29.31 569.50 20.86
Computation 540.38 10.78 531.21 28.16 541.33 19.39
Total reading 612.15 15.64 607.79 27.55 594.17 36.96
WPS-CRT (25/50)
Pretest 21.38 7.62 16.69 7.17 16.92 5.18 C1 vs. C2: +0.63
C1 vs. C3: +0.63
C2 vs. C3: −0.03
Posttest 29.03 7.02 25.47 7.66 24.83 5.03 C1 vs. C2: +0.49
C1 vs. C3: +0.64
C2 vs. C3: +0.09
Growth 7.66 6.95 8.78 7.49 7.92 5.91 C1 vs. C2: −0.16
C1 vs. C3: −0.04
C2 vs. C3: +0.12
WPS-F (16/32)
Pretest 12.31 5.26 9.38 5.73 11.00 5.36 C1 vs. C2: +0.53
C1 vs. C3: +0.25
C2 vs. C3: −0.29
Posttest 19.91 5.47 16.34 6.00 17.84 5.73 C1 vs. C2: +0.62
C1 vs. C3: +0.37
C2 vs. C3: −0.25

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Growth 7.59 4.92 6.97 4.73 7.13 5.07 C1 vs. C2: +0.13
C1 vs. C3: +0.09
C2 vs. C3: −0.03
Computation a (25/43)
Pretest 7.81 4.90 6.50 3.69 12.83 3.76 C1 vs. C2: +0.30
C1 vs. C3: −1.08
C2 vs. C3: −1.71
Posttest 35.69 4.87 23.63 8.05 26.83 8.89 C1 vs. C2: +1.81
C1 vs. C3: +1.45
C2 vs. C3: −0.39
Growth 27.88 6.89 17.13 7.14 14.00 6.88 C1 vs. C2: +1.49
C1 vs. C3: +1.96
C2 vs. C3: +0.43
Strategy satisfaction
questionnaire (5/5)
Enjoyed 3.94 0.68 3.31 1.35 4.67 0.52 C1 vs. C2: +0.59
C1 vs. C3: −1.13
C2 vs. C3: −1.14
Diagram 3.81 0.83 3.87 1.20 4.67 0.82 C1 vs. C2: −0.06
C1 vs. C3: −1.04
C2 vs. C3: −0.72
Help solve 4.20 1.03 3.67 1.03 5.00 0.00 C1 vs. C2: +0.52
C1 vs. C3: −0.90
C2 vs. C3: −1.49
Recommend 3.56 1.32 3.88 1.31 4.83 0.41 C1 vs. C2: −0.24
C1 vs. C3: −1.09
C2 vs. C3: −0.82
Continue 3.81 1.47 3.69 0.95 4.33 0.82 C1 vs. C2: +0.10
C1 vs. C3: −0.39
C2 vs. C3: −0.70
Total 19.13 0.87 18.75 0.87 23.50 1.41 C1 vs. C2: +0.44
C1 vs. C3: −4.24
C2 vs. C3: −4.60
Note. Parenthetical numbers separated by slash are total number of test items and total possible score for each pretest and posttest. EA = European American; AA = African American; Spec. ed.= special
education; C & A = Concepts and Application mathematics subtest; WPS-CRT = word problem solving criterion test; WPS-F = word problem solving fluency measure; C = classroom; ES = effect size.
a Digits correct.

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TABLE 3. Student Demographic Characteristics and Mathematics Performance by Group: Site 1
Group
LD (n = 9) LA (n = 9)
Variable n % M SD n % M SD ES
Age (in months) 108.44 6.50 100.22 3.80
Gender
Male 6 66.67 4 44.44
Female 3 33.33 5 55.56
Race
EA 6 66.67 8 88.89
AA 0 0.00 0 0.00
Hispanic 2 22.22 1 11.11
Other 1 11.11 0 0.00
IQ
Full scale 94.78 6.92
Verbal 92.17 6.82
Performance 95.00 7.75
TerraNova
C & A 567.00 18.46 552.29 14.81
Comp. 537.50 18.24 511.71 24.18
Total reading 588.63 33.40 607.14 10.61
WPS-CRT (25/50)
Pretest 14.83 5.40 18.22 8.34 −0.48
Posttest 22.61 5.61 25.83 7.64 −0.48
Growth 3.57 4.20 4.83 7.43 0.03
WPS-F (16/32)
Pretest 14.29 5.71 13.94 5.19 0.06
Posttest 17.50 4.58 18.78 6.91 −0.22
Growth 8.89 2.40 9.72 4.01 −0.21
Computation a (25/43)
Pretest 10.44 4.67 4.00 1.73 +1.83
Posttest 25.56 8.09 25.78 11.95 −0.02
Growth 15.12 6.85 21.78 11.95 −0.56
Strat. Quest. (5/5)
Enjoyed 4.56 0.73 3.44 1.13 +1.18
Diagram helps 4.44 1.13 4.00 0.87 +0.44
Help solve 4.89 0.33 3.44 1.01 +1.93
Recommend 4.56 1.01 3.89 1.05 +0.65
Continue 4.44 0.73 3.00 1.32 +1.35
Total 22.89 3.26 17.78 2.73 +1.70
Note. Parenthetical numbers separated by slashes are total number of test items and total possible score. LD = learning disability; LA = low achieving;
EA = European American; AA = African American; C & A = Concepts and Application mathematics subtest; Comp. = Computation mathematics
subtest; WPS-CRT = word problem solving criterion test; WPS-F = word problem solving fluency measure; ES = effect size.
a
Digits correct.
fifth graders in the school play.” Using Step 1 of the strategy
checklist, students learned that because the story
described a situation in which three small groups (third-,
fourth-, and fifth-grade students) combine to form a large
group (all students in the play), it is a story situation of the
group problem type. For Step 2, teachers prompted students
to use the group diagram (see Figure 1) to represent the
information. Instruction involved identifying the three
small groups and the large group and writing the group
names in the diagram. Students then read the story to find
the quantities associated with each group and wrote them
in the diagram. Next, students summarized the information
in the story with the completed diagram and checked the
accuracy of the representation.
Compare story situation. Students identified the story
comparison problem type by using story situations such as,
“Joe is 15 years old. He is 8 years older than Jill. Jill is 7
years old.” Using Step 1 of the strategy checklist, students
identified the story situation as compare because it required
a comparison of Joe’s age to Jill’s age. For Step 2, students
used the corresponding diagram (see Figure 1) to organize
or represent the information. That diagram involved reading
the comparison sentence (He is 8 years older than Jill)
to identify the two sets that were compared in the story—

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determining the identity of the large (Joe’s age) and small
(Jill’s age) sets—labeling them in the diagram, and writing
the difference amount in the diagram. Students then read
the story to find the quantities associated with the two sets
and wrote them in the diagram. Next, students summarized
the information in the story with the completed diagram
and checked the accuracy of the representation.
Problem-Solution Instruction
During the problem solution phase that followed problem
schema instruction, students solved problems with
unknowns by using either addition or subtraction; SBI in
this phase included six steps.
Change problems. Teachers prompted students to identify
and represent the problem (i.e., Steps 1 and 2) with the
change schematic diagram similar to the problem schema
instruction phase. The only difference was that students
used a question mark to represent the unknown quantity in
the diagram. Step 3 involved selecting the appropriate
operation and transforming the information in the diagram
into a number sentence. That is, students learned that they
needed to add the parts if the whole was unknown and subtract
for the part when the whole was known. Instruction
emphasized that when the change action caused an
increase, the ending quantity represented the whole; when
the change action involved a decrease, the beginning
quantity was the whole. For Step 4, students had to write
the mathematics sentence and solve for the unknown using
the operation identified in the previous step. Step 5
prompted students to write a complete answer. Finally, for
Step 6, students had to check the reasonableness of their
answer and ensure the accuracy of the representation and
computation.
Group problems. Students learned to identify and represent
the problem (i.e., Steps 1 and 2) with the group
schematic diagram as in the problem schema instruction
phase. For Step 3, when selecting the operation to solve for
the unknown quantity in group problems, students learned
that the large group represents the whole and the small
groups are the parts that make up the whole. Steps 4
through 6 were identical to those described for the change
problem.
Compare problems. Students identified and represented
the problem (i.e., Steps 1 and 2) using the compare
schematic diagram as in the problem schema instruction
phase. For Step 3, when selecting the operation to solve for
the unknown quantity in compare problems, students
learned that the larger set is the big number or whole,
whereas the smaller set and difference are the parts that
make up the larger set. Steps 4 through 6 were identical to
those described for the change and group problems.
When instructors taught students to solve word problems
using SBI, only one type of story situation or word
problem with the corresponding schema diagram initially
appeared on student worksheets following the instruction
of that problem type (e.g., change problem). After students
learned how to map the features onto schematic diagrams
or solve change and group problems, teachers presented
story situations or word problems with both types, along
with a discussion of the samenesses and differences between
the change and group problems. Later, when students completed
instruction for change, group, and compare problem
types, teachers distributed worksheets with word problems
that included all problem types.
We measured fidelity of treatment with a checklist of
critical instructional steps. The checklist included salient
instructional features (e.g., providing clear instructions,
reading word problems aloud, modeling the strategy application,
providing guided practice). Two graduate students
in special education collected fidelity data as they observed
one of the instructors for approximately 30% of the teaching
sessions. Treatment fidelity, estimated as the percentage
of steps completed correctly by the instructor, was 93%
(range = 85–100%) across the 3 teachers.
Measures and Data Collection
Two research assistants administered and scored the
mathematical word problem-solving tests and computation
tests with scripted directions and answer keys. All data
were collected in a whole-class arrangement.
Word problem-solving criterion referenced test (WPS-CRT).
To assess student growth on third-grade addition and subtraction
word problems, students completed the WPS-CRT
prior to and at the end of the intervention. The test consisted
of 25 one-step and two-step addition and subtraction
word problems. Part I of the CRT comprised 9 one-step
word problems derived from the Test of Mathematical
Achievement (Brown, Cronin, & McEntire, 1994). Of the
9 problems, 5 items included distracters. Part II of the WPS-
CRT included 16 addition and subtraction word problems
selected from five commonly used third-grade mathematics
textbooks. The items consisted of 12 one-step and 4 twostep
problems that met the semantic criteria for change,
group, and compare problem types. In addition, 2 of the
problems included distracters. The design of the measure
included the three problem types and both operations.
Word problems on Parts I and II of the WPS-CRT
required applying simple (e.g., single-digit numbers) to
complex computation skills (e.g., three- and four-digit
numbers; regrouping). Students had 50 min to complete
the test. Directions for administering the word problemsolving
test required students to show their complete work
and to write the answer and label. Scoring involved assigning
one point for the correct number model and one point
for correct answer and label for a possible total score of 2
points for each item. Cronbach’s alpha for each of the
pretest and posttest measures was 0.84. Interscorer agreement
assessed by two research assistants independently
scoring 30% of the protocols was .92 at pretreatment and
.97 at postreatment.

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Word problem-solving fluency (WPS-F) measure. We
developed six forms to represent items in Part II of the
WPS-CRT but modified them to include fewer problems,
less advanced computation (one- and two-digit numbers
only), and no problems with distracters to address the
timed nature of the task. The problems for each WPS-fluency
probe differed with respect to numbers, context, and
position of problems, which were random. Given that each
probe included only half the number of problems on the
WPS-CRT, the probes were not identical and differed with
respect to the unknown quantity to be solved (e.g., result,
beginning, compared). We covered all possible combinations
of problem types and operations in two probes rather
than one probe. To control for the difficulty of the probes,
we designed odd- and even-numbered probes to be parallel
forms. Teachers administered the probes every 3 weeks to
monitor students’ progress in solving word problems. Students
had 10 min to complete eight problems. For the purpose
of this study, we used the combined score of the first
two probes and the combined score of the last two probes
in the data analysis. Cronbach’s alphas for the two aggregated
probes were 0.83 and 0.80, respectively. Interscorer
agreement was .93 at pretreatment and .98 at postreatment.
Basic mathematics computation fluency measure (Fuchs,
Hamlett, & Fuchs, 1998). We monitored student progress
toward proficiency on third-grade mathematics computation
curriculum prior to and at the completion of the intervention
with basic mathematics computation probes. Students
had 3 min to complete 25 problems, with a maximum
score of 43 correct digits. We scored the performance on the
computation probes as the total number of correct digits,
which provided credit for correct segments of responses.
That assessment system is known to have adequate reliability
and validity (see Fuchs, Fuchs, Hamlett, & Allinder,
1989). Interscorer agreement was .99 at pretreatment and
1.00 at postreatment.
Strategy satisfaction questionnaire. We developed and
administered a strategy-satisfaction questionnaire following
the intervention to provide information about student perceptions
regarding the problem-solving strategy intervention
(i.e., SBI). The questionnaire included five items that
required students to rate whether they (a) enjoyed the
strategy, (b) found the diagrams helpful in understanding
and solving problems, (c) improved their problem-solving
skills, (d) would recommend using the strategy with other
students, and (e) would continue to use it to solve word
problems in the classroom. Ratings for the Likert-type
items on the questionnaire ranged from a high score of 5
(strongly agree) to a low score of 1 (strongly disagree). Cronbach’s
alpha for the questionnaire was 70.
Data Analysis
The unit of analysis was each student’s individual score
(N = 16 for the general education classrooms; n = 6 for the
special education classroom) rather than the classroom
because of sample limitations. On the WPS-CRT, WPS-F,
and computation pretest measures, we conducted separate
one-way, between-subjects (classroom or group) analysis of
variance (ANOVA) to examine initial classroom or group
(LD and LA) comparability. On the WPS-CRT, WPS-F,
and computation pretest and posttest scores, we conducted
a one between-subjects (classroom or group), one withinsubjects
(time: pretest vs. posttest) ANOVA. We used the
Fisher least significance difference (LSD) post-hoc procedure
to evaluate any pairwise comparisons for significant
effects for the full sample. If a lack of classroom or group
comparability on pretreatment measures occurred, we conducted
a one-factor ANOVA on the change scores, with
classroom or group as the between-subjects factor. (We
used change scores for analyzing the two-wave data because
their interpretation is straightforward.) In addition, we
analyzed scores from the strategy questionnaire with multivariate
analysis of variance (MANOVA) with teacher or
group as the between-subjects factor. To estimate the practical
significance of effects for classroom and group, we
computed effect sizes (ESs) by subtracting the difference
between the posttest means, then dividing by the pooled
standard deviation of the posttest. On the computation
measure, we calculated improvement effects by dividing
the difference between the improvement means by the
pooled standard deviation of the improvement divided by
the square root of 2(1– rxy; Glass, McGraw, & Smith,
1981).
Results
Table 2 shows the means and standard deviations for all
measures for the full sample; Table 3 shows means and standard
deviations for the measures by group (LD and LA).
Pretreatment Differences Among Classrooms on
Mathematics Performance
Results indicated that differences between classrooms on
the WPS-CRT, F(2), 35) = 1.95, ns, and WPS-F, F(2, 35 =
0.99, ns, were not significant prior to the study.
On the computation pretest, however, there was a significant
effect for classrooms, F(2, 35) = 4.87, p < .05. LSD
follow-up tests indicated that mean scores for Classroom 3
were significantly different than those for Classroom 1 (p <
.05) and Classroom 2 (p < .01); Classroom 3 > Classroom
1 > Classroom 2. However, the difference between Classrooms
1 and 2 was not significant. We found large effect
sizes of 1.08 and 1.71 for Classroom 3 when compared with
Classrooms 1 and 2, respectively.
Posttreatment Differences Among Classrooms on
Mathematics Performance
Results of the repeated-measures ANOVA applied to the
pretest and posttest scores demonstrated a significant main

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effect for time on the WPS-CRT, F(1, 35) = 40.90, p < .000,
and WPS-F, F(1, 35) = 12.56, p < .01. Effect sizes for time
at posttest when compared with pretest were moderate for
the WPS-CRT (ES = 0.77) and large for the WPS-F (ES =
0.83). There was no significant main effect for classroom or
interaction between classroom and time. ANOVA values
for the classroom main effect on WPS-CRT and WPS-F
were, F(2, 35) = 2.14, ns, and F(2, 35) = 0.11, ns, respectively.
The corresponding values for the interaction effect
on the two measures were, F(2, 35) = 0.11, ns, and F(2, 35)
= 0.64, ns.
On the computation measure, results of an ANOVA on
change scores indicated a significant effect for classrooms,
F(2, 35) = 13.19, p < .000. Follow-up tests indicated that
the differences between Classrooms 1 and 2 and between
Classrooms 1 and 3 only were significant (p < .01). (Mean
posttest scores for Classroom 1 > Classroom 2 > Classroom
3.) We found large effects of 1.49 and 1.96 for Classroom 1
when compared with Classrooms 2 and 3, respectively.
Posttreatment Differences Between Classrooms on
the Strategy Satisfaction Questionnaire
Results of the MANOVA applied to the Strategy Satisfaction
Questionnaire posttreatment scores revealed no significant
differences among classrooms, Wilks’s lambda =
.69, approximate F(2, 35) = 1.29, ns.
Pretreatment Differences Between Groups (LD and LA) on
Mathematics Performance
Results indicated that differences between groups on the
WPS-CRT, F(1, 16) = 1.05, ns, and WPS-F, F(1, 16) =
0.36, ns, were not significant prior to the study.
On the computation pretest, however, there was a significant
main effect for group, F(1, 16) = 15.09, p < .01.
The mean computation score for LD students was higher
than that for LA students, with a large effect size of 1.83 for
LD students.
Posttreatment Differences Between Groups on
Mathematics Performance LD and LA Sample
Results of repeated-measures ANOVA applied to the
pretest and posttest mathematics test scores demonstrated a
significant main effect for time on the WPS-CRT, F(1, 16)
= 26.94, p < .000, and WPS-F, F(1, 16) = 8.56, p < .01.
Effect sizes for time at posttest when compared with pretest
were large for the WPS-CRT (ES = 1.12) and moderate for
the WPS-F (ES = 0.76). There were no significant main
effects for group or interaction between group and time.
ANOVA values for the main effect of group on WPS-CRT
and WPS-F were F(1, 16) = 1.32, ns, and F(1, 16) 0.96, ns,
respectively. The corresponding values for the interaction
effect on the two measures were, F(1, 16) = 0.00, ns, and
F(1, 16) = 0.89, ns. On the computation measure, results of
an ANOVA on change scores indicated the lack of a significant
main effect for group, F(1, 16) = 2.11, ns.
Posttreatment Differences Between Groups on
the Strategy Satisfaction Questionnaire
On the strategy satisfaction posttreatment scores, we
found significant differences between groups, Wilks’s lambda
= .43, approximate F(1, 16) = 3.16, p < .05. Results of
separate univariate ANOVAs indicated significant differences
between groups regarding total strategy satisfaction
scores, F(1, 16) = 13.02, p < .01, enjoying the strategy, F(1,
16) = 6.15, p < .05, usefulness of the strategy in solving
word problems, F(1, 16) = 16.49, p < .01, and continuing
to use the strategy in solving word problems, F(1, 16) =
8.24, p < .01. Specifically, LD students rated the strategy
higher than did LA students.
Discussion
Results must be interpreted in light of two serious limitations.
First, the full sample (N = 38) and group sample
sizes (n = 9 each for LD and LA students) were small. Second,
the design was unbalanced; classroom sizes ranged
from 16 to 6 students. As such, the findings indicate only
preliminary evidence regarding the effectiveness of SBI.
Within the constraints of the limitations, results from
Study 1 provide evidence that SBI led to improvements in
word problem-solving performance for the three classrooms.
Although the overall treatment fidelity was high,
teaching styles varied concerning teachers’ adherence to
the scripted curriculum (i.e., read verbatim or used their
own explanations). The high level of treatment fidelity
finding suggests that SBI accounted for improved student
learning, which is encouraging.
Furthermore, SBI was effective in enhancing the word
problem-solving performance of students with LD, whether
they received instruction in general education mathematics
classrooms or in a special education classroom. When
we separated outcomes on the word problem-solving measures
for students with LD and their LA peers, the effects of
the word problem-solving curriculum on students’ performance
was comparable for both groups. Those results are
notable because teachers implemented the treatment in a
whole-class format and taught students to solve only onestep
problems, although students were tested on one-step
and two-step problems. The findings support and extend
previous research regarding the effectiveness of SBI in solving
arithmetic word problems (e.g., Fuchs, Fuchs, Finelli,
Courey, & Hamlett, 2004; Jitendra et al., 1998; Zawaiza &
Gerber, 1993).
Our results indicated that classrooms and groups were
not comparable on their computational skills prior to the
study and that computational skill improvement varied as
a function of classroom and group. In general, students with
LD at pretreatment demonstrated better computational

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skills than did the other students at pretreatment. We
expected that result given that many of those students were
older students receiving instruction in a special education
classroom in which the focus of instruction was on the
acquisition of basic skills. However, the general finding
that computation improvement was evident for all students
was encouraging because opportunities for word problem
solving facilitate computation skills. The effect size for
posttest over pretest was large for the entire sample (ES =
2.98) and for the LD and LA students (ES = 2.16). Evidently,
having students solve story problems enhanced the
development of number operations or computational skills.
The use of schematic diagrams, which provided meaning to
number sentences, had an added value in promoting computation
(Van De Walle, 2004).
Finally, the positive evaluation of SBI by students in the
study seemed to play a role in enhancing mathematics performance
as in several previous investigations (e.g., Case,
Harris, & Graham, 1992; Jitendra et al., 1999). Students
with learning disabilities especially were more enthused
about SBI than were their peers with regard to treatment
acceptability and benefits. Wood, Frank, and Wacker
(1998) stated that “Student preference is an important factor,
because students are not as likely to exhibit effort over
time with strategies that they do not like or do not feel are
helpful” (p. 336).
Overall, the study helped us learn about effective ways to
enhance the problem-solving curriculum as well as facilitate
teacher implementation and student learning. Teachers
became conversant about modifying the curriculum
only when they had completed the problem-solving unit
on teaching change and group problems. The lessons that
we learned from our observations and teacher input (prior
to instruction on compare problems) during SBI implementation
allowed us to make several modifications that
we organized in the following paragraphs according to curriculum,
teacher, and student enhancements.
Curriculum
Although Marshall (1995) discussed the importance of
presenting problem schema instruction in concert with the
three problem types followed by problem-solution instruction,
teachers raised concerns that problem solution was
seemingly removed from problem schema instruction for
each problem type. Therefore, we revised the curriculum
such that instruction for the change problem type began
with problem schema instruction, followed by problem
solution instruction. We used that same sequence for group
and compare problem types.
We redesigned the self-monitoring strategy checklist to
include four steps, and used an acronym, FOPS (Find the
problem type, Organize information in the problem using
the schema diagram, Plan to solve the problem, and Solve
the Problem), to help students remember the steps. Furthermore,
we elaborated on each step to align with
domain-specific knowledge consistent with SBI. For example,
to find the problem type (Step 1), we prompted students
to examine information pertaining to each set
(beginning, change, and ending) in the problem (e.g.,
change).
Given that the compare problem type was difficult for
many students during the problem schema phase, we collaborated
with teachers to develop a compare structure that
was coherent to third-grade students. For example, we
focused on three sets (bigger, smaller, and difference) of
information and eliminated reference to difficult terms
(e.g., compared, referent) when discussing the problem. In
addition, we added oral exercises prior to written work to
emphasize the critical features and relations in the problem
to ensure that students were familiar with the information
needed for problem comprehension, a key aspect of SBI.
We found that teachers were spending considerable
amounts of instructional time explaining unfamiliar terms
to students who did not have the necessary experiential
background, which detracted from the focus on problem
solving. Therefore, we modified the textbook problems to
meet the needs of the students. For example, unknown
words such as “alpaca” were replaced by more familiar terms
(sheep).
An important issue that emerged was teacher accountability
for student performance on statewide testing. Teachers
believed that the use of word problems presented only
in text format would not adequately prepare students to
generalize to word problems on the state mathematics test.
Therefore, revisions to the problem-solving curriculum
content entailed inclusion of items that presented information
in tables, graphs, and pictographs. Also, because the
state test required the use of mathematical vocabulary (e.g.,
addend) and emphasized written communication, we modified
our instruction to include key mathematical terms and
provided students with practice in writing explanations for
how the problem was solved.
Furthermore, teachers noted that the time required to
implement SBI was unrealistic given the need to cover
other topics in the school curriculum. In the redesign of the
problem-solving curriculum, we reduced problem schema
instruction for each problem type from three 30-min
lessons to one 50-min lesson and developed four lessons for
each problem type that addressed problem solution instruction.
In addition, we incorporated fading of schematic diagrams
to ensure that students were able to apply learned
solution procedures independent of diagrams. Also, we
eliminated homework problems from the curriculum
because of teacher concerns regarding students’ inconsistent
completion of homework.
Teachers
Our observations revealed that teachers need ongoing
support during the initial implementation of a newly developed
intervention. Although we provided teaching scripts

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to ensure consistency in implementing the critical content,
we suggested that teachers use them as a framework for
instructional implementation. However, our observations
indicated that Teacher 1 read the script verbatim and followed
it in its entirety, whereas Teacher 2 followed the
script inconsistently and required reminders to adhere to
the relevant information needed to promote problem solving.
In contrast, Teacher 3 (special education teacher)
familiarized herself with the script and used her own explanations
and elaborations to implement the intervention
with ease. Obviously, general education teachers in this
study needed more support to implement the intervention.
Also, we noticed that some students in one classroom were
struggling during partner work and had to wait until wholeclass
discussion to get corrective feedback, indicating the
need for providing teachers with explicit guidelines about
direct monitoring student work.
In addition, we found that the general education teachers
seemed more at ease in communicating their concerns
to the two research assistants who supported them during
the course of the project than to the primary researcher,
which raised the issue of how and with whom communication
should be facilitated.
Students
Our observations indicated that for many students, especially
those with LD, scaffolding instruction (modeling, use
of schematic diagrams and checklists) was critical as they
learned to apply the strategy. Explicit modeling and explanations
using several examples enhanced their problemsolving
skills. Following teacher-led instruction, we noticed
that several students with LD, as well as low-achieving students,
were enthused and participated actively, as evidenced
by their raised hands in response to teacher questioning.
DESIGN STUDY 2
Method
Participants
Students in Study 2 were 56 third-grade students in two
heterogeneous classrooms in a parochial school located in
a small city in Florida that served 570 students in Grades
pre-K–8. Approximately 20% of the student population
was African American, Hispanic, or Asian. The total sample
included 27 boys and 29 girls. The mean chronological
age of the students was 108 months (range = 96 to 113
months). Forty-four of the students (78%) were Caucasian,
5 (8%) were African American, 6 (10%) were Hispanic,
and 1 was Asian American (2%). Although not formally
identified at the school, the two third-grade classroom
teachers reported that 9 of the 56 third-grade students had
either LD, attention deficit disorder, or were considered
low achievers. On the basis of scores from the Problem
Solving and Data Interpretation (PSDI) mathematics subtest
of the Iowa Test of Basic Skills (ITBS), we designated
each student’s initial mathematics achievement status as
low performing (LO; below the 34th percentile), average
performing (AV; between the 35th and 66th percentiles),
or high performing (HI; above the 66th percentile). Tables
4 and 5 summarize participant demographic information by
the two classrooms and by student type (LO, AV, and HI
students).
Separate one-way analysis of variances (ANOVAs) indicated
no significant differences between classrooms on the
mathematics and reading subtests of the ITBS, Problem
Solving and Data Interpretation, F(1, 54) = 0.11, ns; Computation,
F(1, 54) = 0.04, ns; Vocabulary, F(1, 54) = 0.35,
ns; and Comprehension, F(1, 54) = 0.01, ns. In addition,
there was a lack of a significant difference between classrooms
on chronological age, F(1, 54) = 0.31, ns. Chi-square
analyses revealed no significant between-classroom differences
on gender, χ 2 (1, N = 56) = .27, ns, and ethnicity,
χ 2 (3, N = 56) = 1.96, ns.
Similarly, separate one-way ANOVAs indicated significant
differences between student type (i.e., LO, AV, and
HI) on the PSDI, F(2, 53) = 239.81, p < .000; computation,
F(2, 53) = 9.498, p < .000; vocabulary, F(2, 53) =
17.80, p < .000; and comprehension, F(2, 53) = 25.54, p <
.000. For all analyses, LSD post-hoc tests revealed that the
differences between the three groups were significant.
(Performance of HI students > AV students > LO students.)
In addition, there was no significant between-student
type difference on chronological age, F(2, 53) = 0.95,
ns. Chi-square analyses revealed no significant betweenstudent
type differences on gender, χ 2 (2, N = 56) = 0.27,
ns, or race χ 2 (6, N = 56) = 1.96, ns.
The two general education teachers who participated in
Study 2 were women; one teacher held a master’s degree
with 10 years’ teaching experience, and the other teacher,
a bachelor’s degree with 30 years’ experience. The teachers
attended 1 hr of inservice training on ways to implement
the intervention; they received support from one of the
researchers periodically throughout the study.
Procedures and Measures
Procedures and measures were similar to those used in
Study 1. In this section, we report only how components of
the two studies differed. First, although students received the
same amount of instructional time as those in Study 1,
teachers began Study 2 later in the school year. Second,
teachers in Study 2 used the textbook series, “Mathematics—The
Path to Math Success! Grade 3” by Silver, Burdett,
and Ginn (as cited in Fennell et al., 1998) for daily mathematics
instruction. Third, instructional procedures differed
from Study 1 with regard to compare instruction. On the
basis of student difficulty with the compare problem type,
teachers in Study 1 used the revised compare diagram and
instruction. However, teachers in Study 2 decided to implement
the original compare instruction and determine how

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TABLE 4. Student Demographic Information and Mathematics Performance by Classroom: Site 2
Classroom 1 Classroom 2
Variable n % M SD n % M SD ES
Age (in months) 106.69 4.35 107.26 3.21
Gender
Male 13 44.82 14 51.85
Female 16 55.17 13 48.15
Race
EA 23 79.31 21 77.77
AA 3 10.34 2 7.40
Hispanic 2 6.90 4 14.81
Asian 1 0.03 0 0.00
ITBS
Mathematics subtests
PSDI 554.01 114.55 565.99 100.64
Computation 550.07 99.79 543.67 84.14
Reading subtests
Vocabulary 568.22 95.34 582.00 85.95
Comprehension 558.04 95.86 563.14 91.86
WPS-CRT (25/50)
Pretest 30.05 10.61 25.50 7.65 +0.24
Posttest 29.98 12.01 28.15 7.18 +0.09
Growth −0.07 7.96 2.65 6.82 −0.18
WPS-F (16/32)
Pretest 19.76 8.45 19.67 6.40 +0.01
Posttest 22.59 8.50 21.17 3.97 +0.10
Growth 2.83 5.03 1.50 4.74 +0.13
Computation a (25/43)
Pretest 31.79 5.55 33.48 7.50 −0.12
Posttest 33.45 5.88 36.22 6.57 −0.21
Growth 1.66 4.04 2.74 2.65 −0.15
Note. EA = European American; AA = African American; ITBS = Iowa Test of Basic Skills (Houghton Mifflin, 1999); PSDI = problem solving
and data interpretation; WPS-CRT = word problem solving criterion test; WPS-F = word problem solving fluency measure; ES = effect size. Parenthetical
numbers separated by slashes are total number of test items and total possible score.
a
Digits correct.
their students, who were predominately average and high
performing, would respond. Fourth, in Study 2, we did not
use the student strategy satisfaction questionnaire administered
in Study 1 because of time constraints at the end of the
school year. Treatment fidelity was 97.5% (range =
91–100%) across the two teachers.
Results
Table 4 shows the means and standard deviations for all
measures for the full sample. Table 5 displays means and
standard deviations for the measures by group (LD and LA).
Pretreatment Differences Between Classrooms on
Mathematics Performance
Results indicated that differences between classrooms on
the WPS-CRT, F(1, 54) = 3.35, ns, WPS-F, F(1, 54) = 0.00,
ns, and computation pretest scores, F(1, 54) = 0.92 ns, were
not significant prior to the study.
Posttreatment Differences Between Classrooms on
Mathematics Performance
Results of ANOVA applied to the pretest and posttest
data yielded no significant classroom by time of testing
interaction for WPS-CRT, F(1, 54) = 1.39, ns, indicating
that classroom had no effect on changes in correct responses
from pretest to posttest.
However, the analyses yielded a significant main effect
for time of testing for WPS-F, F(1, 54) = 10.93, p < .00, and
for computation measures, F(1, 54) = 22.81, p < .000, indicating
improvement from pretest to posttest for the entire
sample of students regardless of classroom condition (average
for WPS-F = 2.19; SD = 4.90; computation = 2.18, SD
= 3.45). Effect sizes for time at posttest when compared
with pretest were small to moderate for the WPS-F (ES =
0.31) and computation measures (ES = 0.34). The main
effect for time of testing on the WPS-CRT was not significant,
F(1, 54) = 1.68, p = .20. We found no significant
main effect for classroom for all three tests: WPS-CRT, F(1,

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54) = 1.80, ns, for WPS-F, F(1, 54) = .18, ns, and for computation,
F(1, 54) = 1.84, ns.
Pretreatment Differences Between Groups (LO, AV, and HI)
on Mathematics Performance
Results indicated that, as expected, differences between
groups on the WPS-CRT, F(2, 53) = 5.68, p < .000 and computation,
F(2, 53) = 5.68, p < .01, were significant prior to
the study.
LSD follow-up tests indicated that mean scores for HI
were significantly different than were those for AV (p <
.00) and LO (p < .00) on the WPS-CRT (HI > AV > LO).
However, the difference between LO and AV was not significant.
We found large effect sizes of 1.14 and 1.15 for HI
when compared with AV and LO, respectively. For the
WPS-F measure, follow-up tests indicated that the mean
scores for HI were significantly different than those for AV
(p < .000) and LO (p < .000; HI > AV > LO). In addition,
the difference between LO and AV was significant (p <
.05), favoring the AV group. On the WPS-F, we found large
effect sizes of 1.26 for HI compared with AV, 2.51 for HI
compared with LO, and 0.94 for AV compared with LO.
Finally, follow-up tests on the computation measure
revealed that mean scores for HI were significantly different
than were those for LO (p < .00; (HI > LO), but not
between HI and AV (p = .15). The difference between LO
and AV on the computation measure was also significant (p
= .05); the AV group scored higher. We found large effect
sizes of 1.67 and 0.91 for HI compared with the LO, and for
AV compared with LO, respectively.
Results of ANOVA applied to the WPS-CRT pretest
and posttest data yielded no significant group by time of
testing interaction, F(2, 53) = 1.19, ns. That finding indicated
that group status (HI, AV, LO) had no effect on
changes in correct scores from pretest to posttest. In addition,
there was no significant main effect for time, F(1, 53)
= 0.26, ns, indicating lack of improvement from pretest to
posttest for the entire sample of students, regardless of
group status. However, the analysis yielded a significant
main effect for group, F(2, 53) = 12.45, p < .000. The LSD
follow-up test indicated that WPS-CRT mean scores for HI
students were significantly different from AV (p < .000)
and LO (p < .000). We found effect sizes of 1.05 and 1.48
for HI students when compared with AV and LO students,
respectively, on the WPS-CRT. The difference between LO
and AV, however, was not significant (p = .12).
Results for WPS-F indicated the following significant
effects: group by time of testing interaction, F(2, 53) =
3.19, p < .05, time of testing, F(1, 53) = 17.37, p < .000,
and group status, F(2, 53) = 16.72 p < .000. Because the
group by time of testing interaction supercedes main
effects, we conducted follow-up tests only for the significant
interaction. Follow-up tests indicated that from
pretest to posttest, averaged across all groups, the LO students
improved reliably more than did the HI students (p =
.02); the trend of improvement for LO students compared
with AV students approached significance (p = .07); the
growth of the AV and HI students was comparable (average
growth for LO students = 5.61; SD = 5.81; AV students =
2.13, SD = 4.04; HI students = 0.96, SD = 4.88). The effect
size for LA growth compared with AV growth was 0.76; for
LA compared with HI, 0.91; for AV compared with HI,
0.26.
The analysis of computation data yielded no significant
group by time of testing interaction, F(1, 54) = 0.36, ns,
indicating that group status (HI, AV, LO) had no effect on
changes in correct scores from pretest to posttest. However,
results indicated a significant main effect for time, F(1,
53) = 18.84, p < .000, indicating improvement from pretest
to posttest for the entire sample of students, regardless of
group status (M = 2.19; SD = 4.90). The effect size for time
from posttest to pretest was 0.34. In addition, there was a
significant main effect for group, F(2, 53) = 5.74, p < .01.
The follow-up test for group indicated that mean scores for
HI students were significantly different than that for LO
students (p < .00) but comparable to those for AV students
(p = .20). In addition, the difference between LO and AV
students was significant (p < .05). We found effect sizes of
1.39 and 0.29 for HI students when compared with LO students,
and for AV students compared with LO students,
respectively.
Discussion
Students in Study 2 made small-to-moderate improvements
in their word problem-solving and computation performance
despite unforeseen problems that occurred during
this field-based research. The benefits of a SBI were particularly
apparent for the low-performing students in the two
heterogeneously grouped third-grade classrooms. As in
Study 1, third-grade teachers delivered instruction in
whole-class arrangement without special instructional
adaptations for low-performing students. Despite the lack
of individualized instruction, those students showed the
greatest amount of growth from pretest to posttest on two
of the three measures. Conversely, the results indicated
that the high-performing students made the least amount
of improvement. The findings support previous studies
regarding the importance of explicit instruction for struggling
learners (e.g., Swanson, 1999) and its triviality for
high performers. Even young children with high intelligence
are capable of using strategies spontaneously to perform
given tasks without explicit instruction (Cho & Ahn,
2003).
Results for the WPS-CRT, in particular, were disappointing.
Our hypothesis that there would be a significant
effect for time from pretest to the posttest was not realized.
The lack of improvement over time may be attributed to
several reasons. First, the length of the WPS-CRT task
(i.e., 25 one-step and two-step word problems) in conjunction
with administration of the posttest only 2 weeks before

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TABLE 5. Student Demographic Characteristics and Mathematics Performance by Mathematics Grade-Level Status: Site 2
Mathematics grade-level status
High (n = 24) Average (n = 23) Low (n = 9)
Variable n % M SD n % M SD n % M SD ES
Age (in months) 107.14 2.88 107.23 4.88 106.00 3.67
Gender
Male 14 58.33 8 34.78 6 66.67
Female 10 41.67 15 65.22 3 33.33
Race
EA 20 83.33 15 65.22 8 88.89
AA 1 4.17 4 17.39 1 11.11
Hispanic 2 8.33 4 17.39 0 0.00
Asian 1 4.17 0 0.00 0 0.00
ITBS
Mathematics
PSDI 660.47 62.74 517.47 21.68 466.92 67.67
Computation 598.40 96.41 524.66 60.49 399.45 44.81
Reading
Vocabulary 639.00 57.42 542.54 70.77 486.54 93.62
Comprehension 631.13 67.90 524.23 70.48 464.86 60.66
WPS-CRT (25/50)
Pretest 33.27 8.17 24.50 7.26 22.00 11.18 LO vs. AV: −0.29
LO vs. HI: −1.25
AV vs. HI: −1.13
Posttest 34.67 5.64 26.89 8.90 19.89 12.97 LO vs. AV: −0.69
LO vs. HI: −1.81
AV vs. HI: −1.05

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Growth 1.40 6.10 2.39 8.52 −2.11 7.93 LO vs. AV: −0.54
LO vs. HI: −0.53
AV vs. HI: +0.13
WPS-F (16/32)
Pretest 24.85 5.49 17.35 6.42 12.06 4.66 LO vs. AV: −0.88
LO vs. HI: −2.42
AV vs. HI: −1.26
Posttest 25.81 3.89 19.48 6.09 17.67 8.91 LO vs. AV: −0.26
LO vs. HI: −1.45
AV vs. HI: −1.25
Growth 0.96 4.88 2.13 4.04 5.61 5.81 LO vs. AV: +0.76
LO vs. HI: +0.91
AV vs. HI: +0.26
Computation a (25/43)
Pretest 34.96 5.35 32.35 7.25 27.00 4.06 LO vs. AV: −0.82
LO vs. HI: −1.58
AV vs. HI: −0.41
Posttest 36.67 5.40 34.91 6.82 29.44 4.50 LO vs. AV: −0.87
LO vs. HI: −1.39
AV vs. HI: −0.29
Growth 1.71 2.94 2.57 4.02 2.44 3.40 LO vs. AV: −0.03
LO vs. HI: +0.24
AV vs. HI: +0.24
Note. Parenthetical numbers separated by slashes are total number of test items and total possible score. EA = European American; AA = African American; ITBS = Iowa Test of Basic Skills (Houghton
Mifflin, 1999); PSDI = Problem solving and data interpretation; HI = high; AV-average; LO = low; WPS-CRT = word problem solving criterion test; WPS-F = word problem solving fluency measure;
ES = effect size.
a Digits correct.
* p = .05;
** p = .01;
*** p = .001.

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school ended for the summer may not have been motivating
for students, in spite of explicit instructions to do their
best. Teachers reported students’ less-than-determined
attempts to complete the test. As such, that problem could
have negatively affected student performance on the WPS-
CRT posttest, which is not uncommon to intervention
research (Dunst & Trivette, 1994).
Another plausible explanation for a lack of improvement
was that the high-performing students may have
known how to solve third-grade word problems before the
study began and did not require extensive strategy instruction.
A closer look at the data suggests that that may be the
case. Fifteen of the high-performing students scored close
to an accepted criterion level of 80% correct before the
study began, indicating possible ceiling effects (M = 39.91,
SD = 3.99; range = 34 to 47 on the 50-point WPS-CRT).
As such, pretest to posttest improvement was not evident
for the students, and ceiling effects limited the ability to
ferret out differences among higher scoring students.
Results on the WPS-F measures were more encouraging.
All students improved their word problem-solving performance
on the shorter, 8-item, probes from pretest to
posttest. The LO group made the most progress on this
measure, as evidenced by the large effect size of 0.79 when
compared with effect sizes of 0.34 and 0.20 for the AV and
HI groups, respectively. Given that the measure included
easier items than did the WPS-CRT, was shorter in length,
and required only 10 min to complete each probe, thereby
reducing fatigue, it may have positively influenced the
results of the WPS-F.
The pretest-to-posttest results for the computation measure
revealed a pattern similar to those for the WPS-F. All
groups improved significantly over time; however, the LO
group demonstrated the largest gains from pretest to
posttest, with a medium effect size of .57 when compared
with small-to-moderate effect sizes of .36 and .32 for the
AV and HI students, respectively. Again, word problem
solving seems to play a role in the development of number
operations, such as computational skills (Van de Walle,
2004), particularly for low-performing students.
As in Study 1, we learned ways to enhance curriculum,
instruction, and measures. For example, whereas teachers
from both sites were concerned with the length of the
intervention, Study 2 teachers were less concerned about
coverage of other topics. Instead, teachers from Study 2
reported the need to reduce what they considered redundancy
in the curriculum as well as highly directive instruction
because the majority of their students (i.e., average
and high performers) occasionally appeared unmotivated
during the instructional sessions. Considerable evidence
shows that pacing curriculum and instruction to match the
needs of the student is one way to ensure that the needs of
highly able students are addressed (Rogers, 2002).
Attempts to replicate SBI in heterogeneous classrooms may
necessitate changes to differentiate the instruction for
mixed-ability groupings (e.g., Tomlinson, 1995).
Our observations in Study 2, like those in Study 1, indicated
that LO students benefited from the explicit modeling,
guided practice, use of schematic diagrams, and checklists
that are characterized by SBI. We observed that LO
students tended to participate more than was typical during
SBI instruction, and their teachers stated that instruction
had clear benefits for them. However, as a group, the performance
of LO students on the three measures was inconsistent
and highly variable. They performed well on the
WPS-F and computation measure but not on the WPS-
CRT. On the WPS-CRT (pretest: M = 22.00, SD = 11.18;
posttest: M = 19.89, SD = 12.97), performance of LO students
was highly variable, as indicated by large standard
deviations. That result did not occur for the WPS-F
(pretest: M = 12.06, SD = 4.66; posttest: M = 17.67, SD =
8.91) or computation test (pretest: M = 27.00, SD = 4.06;
posttest: M = 29.44, SD = 4.50). The length and difficulty
of the WPS-CRT appeared to affect the performance of the
LO group most directly; variability decreased when the
measures were shorter or less challenging.
Finally, although the two teachers who participated in
Study 2 reported that they had few problems implementing
SBI instruction and had high levels of treatment fidelity
(M = 98%), in hindsight, the teachers may have had more
to reveal had they received our consistent support. Providing
teachers with more frequent opportunities to converse
with us may have allowed for instructional adaptations that
addressed varying student needs. Instruction that meets the
needs of heterogeneous groups of students is not easily
designed, and, consequently, may require more structured,
collaborative efforts between teachers and researchers.
General Discussion
Design studies have the potential to allow researchers to
develop in-depth understandings of teaching and learning
in classroom environments, that in turn, can assist them in
the design of experimental or quasi-experimental studies
(Gersten et al., 2000). Specific examples of those benefits
include enhanced knowledge about (a) instructional design
features of an intervention, (b) student and teacher
responses to the intervention, (c) measures used, and (d)
the role of the researcher in facilitating the study. Like pilot
studies, design experiments allow researchers to determine
the feasibility of an experiment before implementing it on
a larger scale and under more carefully controlled conditions
(Gersten, in press).
We undoubtedly learned much about SBI for promoting
mathematics word problem solving by conducting the two
design studies reported here. Although we may have a few
unanswered questions, neither the knowledge gained nor
the uncertainties revealed would have occurred without
carrying out the investigative work of these studies. The
most notable insights revealed to us from conversations
with the teachers and from our own observations included
the benefits of strategy fading to promote independent stu-

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dent use of SBI. Encouraging independent, self-regulated
use of strategies has been supported in the reading literature
for years and is perceived as an important long-term goal of
strategy instruction (Pressley et al., 1992). We suggest that
supporting the independent use of strategies is an important
aspect of mathematics word problem-solving instruction
as well.
We also realized the need for changes in the compare
problem instruction by collaborating first with the teachers
at Site 1, and later with teachers at Site 2. We observed
that the vocabulary words and compare schema diagram
were difficult and confusing for students at all achievement
levels. Having the opportunity to alter academic interventions
on the basis of student and teacher input is a powerful
outcome of design studies and will strengthen the
instruction when examined in subsequent studies. We also
learned about the importance of providing extensive support
to teachers during the study implementation. We are
convinced that the frequent and consistent support provided
to teachers in Study 1 allowed their students to perform
better. Alternatively, if we had offered more support in
Study 2, the teachers may have shared additional comments
with us about the implementation of the study.
Those comments could have prompted changes in the
instruction, which may have better addressed the academic
needs of the students and improved their performance.
Furthermore, Study 2 helped reveal the complexities of
working in heterogeneously grouped classrooms. With the
strengthened Least Restrictive Environment (LRE) provision
in 1997 mandating that students with disabilities have
access to, and make progress in, the general education curriculum
(U.S. Department of Education, 1997), more students
with disabilities may be part of general education
classrooms. Consequently, teachers will be increasingly
challenged to create instruction that addresses the needs of
students in academically heterogeneous classrooms. The
method of doing that well is still unclear to us. However,
we conjecture that some aspects of SBI will likely benefit
high-, average-, and low-performing students differentially,
and others may improve their performance as a group.
Low-performing students in both studies most likely benefited
from direct, explicit, and systematic instruction.
High-performing students may have enjoyed the pairedlearning
opportunities provided during SBI. Research that
spans more than a decade suggests that the learning of
high-achieving students, such as those in Study 2, is facilitated
through peer interaction in learning (e.g., Swing &
Peterson, 1982; Webb, 1991; Webb, Troper, & Fall, 1995).
The use of schema-based diagrams may have contributed to
the word problem-solving and computation improvement
among many of the participating students, regardless of
ability level. Using what we learned from the design studies
will allow us to strengthen the instruction before it is
examined in future research studies. In the interim, preliminary
findings reveal the potential of SBI in mathematics
word problem solving with third-grade students.
NOTES
This research was supported by Grant H324D010024 from the U.S.
Department of Education, Office of Special Education Programs.
The authors thank Dr. Russell Gersten for his insightful feedback during
the study. A version of this article was presented at the 2004 annual
meeting of the Council for Exceptional Children in New Orleans,
Louisiana.
REFERENCES
Baxter, J., Woodward, J., Voorhies, J., & Wong, J. (2002). We talk about it,
but do they get it Learning Disabilities: Research & Practice, 17, 173–185.
Briars, D. J., & Larkin, J. H. (1984). An integrated model of skill in solving
elementary word problems. Cognition and Instruction, 1, 245–296.
Brown, V., Cronin, M. E., & McEntire, E. (1994). TOMA-2 Test of Mathematical
Abilities. Austin, TX: Pro-Ed.
Carpenter, T. P., Hiebert, J., & Moser, J. M. (1983). The effect of instruction
on children’s solutions of addition and subtraction word problems.
Educational Studies in Mathematics, 14, 55–72.
Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and
subtraction concepts in grades one through three. Journal for Research in
Mathematics Education, 15, 179–202.
Case, L. P., Harris, K. R., & Graham, S. (1992). Improving the mathematical
problem-solving skills of students with learning disabilities:
Self-regulated strategy development. The Journal of Special Education,
26, 1–19.
Chen, Z. (1999). Schema induction in children’s analogical problem solving.
Journal of Educational Psychology, 91, 703–715.
Cho, S., & Ahn, D. (2003). Strategy acquisition and maintenance of gifted
and nongifted young children. Exceptional Children, 69, 497–505.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003).
Design experiments in educational research. Educational Researcher,
32(1), 9–13.
CTB/McGraw-Hill. (2001). TerraNova. Monterey, CA: Author.
Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role
of understanding in solving word problems. Cognitive Pyschology, 20,
405–438.
De Corte, E., Verschaffel, L., & De Win, L. (1985). Influence of rewording
verbal problems on children’s problem representations and solutions.
Journal of Educational Psychology, 77, 460–470.
The Design-Based Research Collective. (2003). Design-based research:
An emerging paradigm for educational inquiry. Educational Researcher,
32, 5–8.
Desoete, A., Roeyers, H., & De Clercq, A. (2003). Can offline metacognition
enhance mathematical problem solving Journal of Educational
Psychology, 95(1), 188–200.
Dunst, C. J., & Trivette, C. M. (1994). Methodological considerations
and strategies for studying the long-term effects of early intervention. In
S. L. Friedman & H. Carl Haywood (Eds.), Developmental follow-up:
Concepts, domains, and methods (pp. 277–313). New York: Academic
Press.
Fennell, F., Ferrini-Mundy, J., Ginsburg, H. O., Greenes, C., Murphy, S. J.,
& Tate, W. (1998). Mathematics: The path to math success! Grade 3.
Needham, MA: Silver Burdett Ginn.
Fuchs, L. S., Fuchs, D., Finelli, R., Courey, S. J., & Hamlett, C. L. (2004).
Expanding schema-based transfer instruction to help third graders solve
real-life mathematical problems. American Educational Research Journal,
41, 419–445.
Fuchs, L. S., Fuchs, D., Hamlett, C. L., & Allinder, R. (1989). The reliability
and validity of skills analysis within curriculum-based measurement.
Diagnostique, 14, 203–221.
Fuchs, L. S., Fuchs, D., Prentice, K., Burch, M., Hamlett, C. L., & Owen,
R., et al. (2003). Enhancing third-grade students’ mathematical problem
solving with self-regulated learning strategies. Journal of Educational
Psychology, 95, 306–315.
Fuchs, L. S., Hamlett, C. L., & Fuchs, D. (1998). Monitoring basic skills
progress: Basic math computational manual (2nd ed.). Austin, TX: Pro-Ed.
Gersten, R. (in press). Behind the scenes of an intervention research
study. Learning Disabilities Research and Practice.
Gersten, R., Baker, S., & Lloyd, J. W. (2000). Designing high-quality
research in special education: Group experimental designs. Journal of
Special Education, 34, 2–19.
Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical

283-302 Jitendra M_J 07 5/8/07 10:57 AM Page 302
302 The Journal of Educational Research
transfer. Cognitive Psychology, 15, 1–38.
Glass, G. V., McGraw, B., & Smith, M. L. (1981). Meta-analysis in social
research. Beverly Hills, CA: Sage.
Goldsmith, L. T., & Mark, J. (1999). What is a standards-based mathematics
curriculum Educational Leadership, 57, 40–44.
Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of
arithmetic word problems: A comparison of successful and unsuccessful
problem solvers. Journal of Educational Psychology, 87, 18–32.
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray,
H., et al. (1996). Problem solving as a basis for reform in curriculum and
instruction: The case of mathematics. Educational Researcher, 25(4),
12–21.
Houghton Mifflin/Riverside Publishing. (1999). Iowa Test of Basic Skills
(ITBS). Itasca, IL: Author.
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. The Journal of Educational
Research, 91, 345–355.
Jitendra, A. K., Hoff, K., & Beck, M. M. (1999). Teaching middle school
students with learning disabilities to solve word problems using a
schema-based approach. Remedial and Special Education, 20, 50–64.
Jonassen, D. H. (2003). Designing research-based instruction for story
problems. Educational Psychology Review, 15, 267–296.
Kameenui, E. J., & Griffin, C. C. (1989). The national crisis in verbal
problem solving in mathematics: A proposal for examining the role of
basal mathematics programs. Elementary School Journal, 89, 575–593.
Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word
arithmetic problems. Psychological Review, 92, 109–129.
Lester, F. K., Garofalo, J., & Kroll, D. L. (1989). Self-confidence, interest,
beliefs, and metacognition: Key influences on problem-solving behavior.
In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical
problem solving (pp. 75–88). New York: Springer-Verlag.
Lucangeli, D., Tressoldi, P. E., & Cendron, M. (1998). Cognitive and
metacognitive abilities involved in the solution of mathematical word
problems: Validation of a comprehensive model. Contemporary Educational
Psychology, 23, 257–275.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teacher’s
understanding of fundamental mathematics in China and United States.
Mahwah, NJ: Erlbaum.
Manfre, E., Moser, J. M., Lobato, J. E., & Morrow, L. (1994). Heath mathematics
connections. Washington, DC: Heath.
Marshall, S. P. (1995). Schemas in problem solving. New York: Cambridge
University Press.
Mayer, R. E. (1999). The promise of educational psychology: Vol. I. Learning
in the content areas. Upper Saddle River, NJ: Merrill Prentice Hall.
Mayer, R. E., & Hegarty, M. (1996). The process of understanding mathematics
problems. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature
of mathematical thinking (pp. 29–53). Mahwah, NJ: Erlbaum.
Mayer, R. E., Lewis, A. B., & Hegarty, M. (1992). Mathematical misunderstandings:
Qualitative reasoning about quantitative problems. In J. I.
D. Campbell (Ed.), The nature and origins of mathematical skills (pp.
137–154). Amsterdam: Elsevier.
Nathan, M. J., Long, S. D., & Alibali, M. W. (2002). The symbolic precedence
view of mathematical development: A corpus analysis of the
rhetorical structure of textbooks. Discourse Processes, 33, 1–21.
National Council of Teachers of Mathematics. (2000). Principles and standards
for school mathematics. Reston, VA: Author.
National Research Council. (2001). Adding it up: Helping children learn
mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Mathematics
Learning Study Committee, Center for Education, Division of
Behavioral and Social Sciences and Education. Washington, DC:
National Academy Press.
Prawat, R. S. (1989). Promoting access to knowledge, strategy, and disposition
in students: A research synthesis. Review of Educational Research,
59, 1–41.
Pressley, M., El-Dinary, P., Gaskins, I., Schuder, T., Bergman, J., & Almasi,
J., et al. (1992). Beyond direct explanation: Transactional instruction of
reading comprehension strategies. Elementary School Journal, 92,
513–556.
Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s
problem–solving ability in arithmetic. In H. P. Ginsburg (Ed.),
The development of mathematical thinking (pp. 153–196). New York: Academic
Press.
Rogers, K. (2002). Re-forming gifted education: Matching the program to the
child. Scottsdale, AZ: Great Potential Press.
Schunk, D. H., (1998). Teaching elementary students to self-regulate
practice of mathematical skills with modeling. In D. H. Schunk & B. J.
Zimmerman (Eds.), Self-regulated learning: From teaching to self-reflective
practice (pp. 137–159). New York: Guilford.
Schurter, W. A. (2002). Comprehension monitoring: An aid to mathematical
problem solving. Journal of Developmental Education, 26(2), 22–33.
Swing, S. R., & Peterson, P. L. (1982). The relationship of student ability
and small group interaction to student achievement. American Educational
Research Journal, 19, 259–274.
Swanson, H. L. (1999). Instructional components that predict treatment
outcomes for students with learning disabilities: Support for a combined
strategy and direct instruction model. Learning Disabilities Research and
Practice, 14, 129–140.
Tomlinson, C. (1995). How to differentiate instruction in mixed-ability classrooms.
Alexandria, VA: Association for Supervision and Curriculum
Development.
U.S. Department of Education. (1997). Amendments to the IDEA, P.L.
105-17. Retrieved February 27, 2007, from http://www.ed.gov/offices/
OSERS/Policy/IDEA/the_law.html
Van de Walle, J. A. (2004). Elementary and middle school mathematics:
Teaching developmentally (5th ed.). Boston: Allyn & Bacon.
Verschaffel, L., De Corte, E., Lassure, S., Van Vaerenbergh, G., Bogaerts,
H., & Ratinckx, E. (1999). Learning to solve mathematical application
problems: A design experiment with fifth graders. Mathematical Learning
and Thinking, 1(3).
Webb, N. M. (1991). Task-related verbal interaction and mathematics
learning in small groups. Journal for Research in Mathematics Education,
22, 366–389.
Webb, N. M., Troper, J., & Fall, J. R. (1995). Constructive activity and
learning in cooperative small groups. Journal of Educational Psychology,
87, 406–423.
Wood, D. K., Frank, A. R., & Wacker, D. P. (1998). Teaching multiplication
facts to students with learning disabilities. Journal of Applied Behavior
Analysis, 31, 323–338.
Zawaiza, T. B. W., & Gerber, M. M. (1993). Effects of explicit instruction
on community college students with learning disabilities. Learning Disabilities
Quarterly, 16, 64–79.

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Among Those Present
STEVEN B. SHELDON is an associate research scientist,
Center on School, Family, and Community
Partnerships, Johns Hopkins University. He studies the
development of family and community involvement
programs in schools, along with the impact of these
programs on student outcomes. He also conducts
research into the influences of parent involvement,
including parent beliefs and social relations, and school
outreach.
KERRY HOLMES is an assistant professor of elementary
education, School of Education, the University
of Mississippi. Her research interests focus on
closing the reading achievement gap through sound
early childhood reading practices and the motivation
of reluctant readers. SARAH POWELL is an honor
student in the School of Education at the same university.
She graduated with honors from the Sally
McDonnell Barksdale Honors College. STACY
HOLMES is an associate professor of engineering at
the University of Mississippi. He is interested in
bringing quantitative techniques to the study of reading
acquisition. EMILY WITT is a fourth-grade
teacher at Oxford Elementary School. She graduated
with a master’s degree in elementary education from
the University of Mississippi in 2005.
ASHA K. JITENDRA is professor, College of Education,
Lehigh University. Her areas of research interests
include mathematics instruction and assessment.
Other interests include reading interventions, textbook
analysis, and curriculum-based assessment.
CYNTHIA C. GRIFFIN is an associate professor,
Department of Special Education, University of Florida.
She is interested in beginning special educators
and inclusive practice in mathematics and reading.
ANDRIA DEATLINE-BUCHMAN is a special
education consultant, Easton Area School District,
Pennsylvania. She provides academic support to elementary,
middle, and high school special education
students. Her areas of research interests include writing
instruction and curriculum-based assessment.
EDWARD SCZESNIAK is also a special education
consultant with the Easton Area School district. He
provides behavioral support to elementary, middle,
and high school special education students by conducting
functional behavioral assessments and developing
behavioral intervention plans. He is interested
in mathematics instruction and assessment.
HERBERT WARE is associated with George Mason
University, Fairfax, Virginia. As a former public school
teacher, research coordinator, and administrator, his
interests lie with the application of quantitative methods
to the study of school climate. He is particularly
interested in the use of national data sets as a way of
studying the relationship of school climate to student
achievement. ANASTASIA KITSANTAS is an associate
professor, College of Education and Human
Development, also at George Mason University. Her
research interests focus on social cognitive processes,
self-regulated learning, and motivational beliefs.
KELLIE J. ANDERSON is a school psychologist, Student
Services Center, Anne Arundel County Public
Schools, Maryland. Her areas of interest include family
and school collaboration, crisis intervention, and adolescent
development. KATHLEEN M. MINKE is a
professor, School Psychology Program, University of
Delaware. Her research interests include family–school
collaboration, self-concept, and professional issues in
school psychology.

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