Teaching Problem Solving Using Strategic Instruction, Concrete-Representational-Abstract
Sequence, and Schema-Based Instruction
Margaret M. Flores, Ph.D., BCBA-D, Vanessa M. Hinton, Ph.D., Jessica H. Milton, M.Ed., & Alexcia J. Moore, M.Ed.
Bradley J. Kaffar, Ph.D.
Auburn University
St. Cloud State University
Rationale
Understanding problems and persisting in solving them is the first of the eight standards for
mathematical practice that are listed in the Common Core State Standards Initiative (CCSI, 2010).
Teaching problem solving is a process that is integral to mathematics; it is more than simply
extracting numbers from a word problem (Griffin & Jitendra, 2009). Woodward et al. (2012)
recommend: (a) use problems during instruction that are both routine and non-routine and consider the
students’ knowledge of mathematical content; (b) model how to monitor and reflect on the problem
solving process; (c) teach how to use visual representations in problem solving; (d) expose students to
a variety of strategies; and (e) teach students to articulate their problem solving processes.
These recommendations should be included in mathematics interventions for students who struggle
with mathematics. Case, Harris, and Graham (1992) examined effects of a self-regulated strategy
intervention and found a functional relation between the intervention and problem solving behaviors.
Cassel and Reid (1996) successfully incorporated the use of concrete-representational-abstract
sequence (CRA) and explicit instruction to teach problem solving. Jitendra and Hoff (1996) explored
strategy-based instruction in a new way by classifying different types of word problems with graphic
representations or schema-based instruction. A functional relation was demonstrated between the
schema strategy and problem solving performance. Then, Jitendra et al. (1998) compared schemabased instruction to problem solving instruction using a basal curriculum and found the elementary
students with disabilities who received schema-based instruction performed better on generalization
tasks. Jitendra, DiPipi, and Perron-Jones (2002) extended the research related to schema-based
instruction from the elementary level to the middle school level. Jitendra, George, Sood, and Price
(2010) effectively implemented schema-based instruction with students with emotional or behavioral
disorders. In a randomized controlled trial, schema-based instruction was found to result in greater
gains when compared to instruction using a standards-based curriculum (Jitendra et al., 2014).
There is much evidence that schema-based instruction is effective in improving the problem solving
performance of students with disabilities and students who are at risk for mathematics failure.
However, the focus of schema-based instruction is through drawings and pictures. Other effective
mathematics inventions for students who struggle begin instruction using manipulative objects in
order to develop conceptual understanding through hands-on activities such as the concreterepresentational-abstract sequence (Morin & Miller, 1998; Miller & Kaffar, 2011; Mancl, Miller, &
Kennedy, 2012; CRA). With the exception of Cassel and Reid (1996), there is little published research
regarding its effects on the completion of word problems specifically. Therefore, the purpose of this
study was to combine schema-based instruction and the CRA sequence to provide a problem solving
intervention for students receiving tiered interventions.
Dustin B. Mancl, Ph.D.
Clark County School District
Setting
The study took place at a rural elementary school in the Southeastern U.S. Students received 20 minutes of
instruction during an after-school program for 4 days per week.
Materials
Assessment materials. The assessment materials included 20 different probes created by the researchers. Each
probe had four word problems with extraneous information and were change (join and separate), part-part-whole,
and compare. The problems were written so that they were culturally relevant to the students.
Instructional materials. There were different materials used to implement four phases of instruction: teaching
problem types, concrete, representational, and abstract instruction. First, students discriminated between problem
types and used schema diagrams. Materials were sheets of paper with complete (4 + 2 = 6) and incomplete (4 +
___ = 6) problems and pre-printed schema diagrams. At the concrete and representational levels, materials were
sheets of paper with a word problem, and array of schema diagrams, and the problem solving strategy (FAST)
written out with directions for each step: a) Find what you are solving for; b) Ask, “What are the parts of the
problem?” (cross out the parts that are not needed and underline the parts needed to solve), c) Set up the numbers
(act out and draw the problem), and d) Tie down the sign (decide what operation and solve). Concrete instruction
included materials used to act out the problem, but these were not used at the representational level.
Procedures
Assessment procedures. The researcher gave each student a probe sheet, told him/her that there was no time limit,
and asked him/her to solve the problems. At four points in time, a researcher who was not part of the
implementation of the intervention interviewed each student one-on-one and asked each student to explain how he
or she solved problems within a previous probe.
Instructional procedures. During each phase of instruction, the instructor used explicit instruction. Teaching
problem types involved labeling the problem type, showing how each sentence could be drawn using numbers and
diagrams as well as mathematical symbols. Concrete instruction involved solving the three types of problems that
had extraneous information and they were solved using the FAST Strategy, physical materials, and use of materials
through acting out. Representational instruction involved solving problems using the FAST Strategy with physical
diagrams. Abstract level instruction involved solving problems using the FAST Strategy without pre-printed
diagram choices.
F ind what you are solving for
- Look at the question. What are you solving for?
A sk “What are the parts of the problem?”
- Cross out parts that are not needed. Underline the parts needed to solve.
S et up the numbers
more than
less than
- Pick diagram and draw the problem.
T ie down the sign
Methods
- Use the diagram to decide which operation and solve.
Participants
Student
Age
Grade
Carla
9
3
Tim
Trey
10
9
3
3
Cultural
Background
African
American
White
African
American
Cognitive
Ability a
91
Computation
Achievement b
83
88
113
75
80
a = standard score reported in most recent special education evaluation or re-evaluation
b = standard score Operations subtest Key Math 3 Diagnostic Assessment (Connolly, 2007 )
Treatment Integrity, Inter-observer Agreement, Social Validity
Treatment integrity data were collected for 90% of all sessions. Integrity was calculated at 95%. All assessment
probes were checked for inter-observer agreement and agreement was 100% across all students. Social validity
questionnaires indicated that students thought the intervention was helpful and their teachers recognized
improvement.
Research Design
A multiple-probe across students design was used to investigate the relation between problem solving and the
intervention. Qualitative methods were used to analyze the students’ explanation of their thinking processes.
Results
Percent Word Problems Correct
Percent Word Problems Correct
Percent Word Problems Correct
Results for Carla. Tim, & Trey
Baseline
Effect Size:
CRA-SIM Intervention
Carla
Baseline
CRA-SIM Intervention
Tau-U was calculated for all students as well as overall. The effect size for each student was calculated
using Tau-U. Non-overlapping data points between phases are combined with an analysis of trend
within each of the intervention phases. Tau-U also accounts for any trend within baseline (Parker,
Vannest, Davis, & Sauber, 2011). There were no significant trends for any of the students within
baseline phases. In comparing Carla’s baseline and intervention phases, a strong effect was indicated
(Tau-U = 0.91). For Tim, there was a strong effect between baseline and intervention phases (Tau-U =
1.0). In comparing Trey’s baseline and intervention performance, a strong effect was indicated (Tau-U
= 0.90). The researchers found a strong overall effect for the study (Tau-U = 0.94).
Qualitative Results:
Tim
Baseline
CRA-SIM Intervention
Trey
Carla’s descriptions of her problem solving showed an increased ability to make sense of the problem.
Carla shared the meaning of the numbers in the problem and could explain how she determined which
information was extraneous. She still struggled with computational errors, but could recognize what to
do in the problems. There were two instances when she explained her solution strategy, realized she
made a mistake, and explained why she made the error. Tim did not enjoy the interview process. He
groaned when told about the interview and began with very short responses. Tim began to identify
extraneous information. He began to use the descriptions of the stories in the problem to explain
determination of the operation. Tim made meaning of the problem and examined how the information
could help him find the solution. Beginning the intervention phase, Trey struggled to determine
extraneous information and showed signs of using key words without meaning. However, during the
second interview, Trey showed improvement in reading the word problem for meaning. During the final
interview, he discussed the content of the problem and explained how information related or did not
relate to the question.
Discussion
The purpose of this study was to investigate the effects of an intervention that combined the CRA instructional sequence and schema-based instruction using the
FAST Strategy for problem solving. A functional relation was demonstrated between problem solving accuracy and the intervention. Qualitative results showed
changes in students’ understanding of problems and systematic thinking processes used when solving problems. The results of this study are consistent with previous
research in which CRA and the FAST Strategy were implemented as well as results of schema-based interventions (Cassel & Reid, 1996; Jitendra, DiPipi, & PerronJones, 2002; Jitendra et al., 2013; Jitendra et al., 2010; Jitendra & Hoff, 1996; Jitendra, Sczesniak, & Deatline-Buchman, 2005). The information gleaned from the
interview illuminated students’ progress in reading problems for meaning, rather than following a procedure without examining the entire situation. The FAST
Strategy provided a series of steps to follow, rather than students’ previous impulsive approach of associating one or two words in the problem with an operation and
using all numbers within the problem to arrive at the solution. The CRA sequence assisted students in comprehending the language within the word problems. At the
concrete level, students identified extraneous as well as needed information more readily. The process of acting out the problem and physically manipulating objects
brought the words within the problem to life. The students observed the physical joining or separating of objects which made identifying the parts of the problem
easier. The combination of concrete and representational instruction and the use of schematic maps assisted students in organizing the parts of the number sentence
and the needed operation.