1. 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.
Auburn University
Bradley J. Kaffar, Ph.D.
St. Cloud State University
Dustin B. Mancl, Ph.D.
Clark County School District
Rationale
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.
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 schema-based 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 concrete-representational-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.
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.
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.
Methods
Participants
Student
Age
Grade
Cultural Background
Cognitive Ability a
Computation Achievement b
Carla 9 3
African American 91 83
Tim 10
White 88 75
Trey
113 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)

2. Results Effect Size: Qualitative Results: Discussion
Results for Carla. Tim, & Trey
Effect Size:
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:
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, & Perron-Jones, 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.