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Teaching Problem Solving to Students Receiving Tiered Interventions Using the Concrete-Representational-Abstract Sequence and Schema-Based Instruction

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Title: Teaching Problem Solving to Students Receiving Tiered Interventions Using the Concrete-Representational-Abstract Sequence and Schema-Based Instruction
Language: English
Authors: Flores, Margaret M., Hinton, Vanessa M., Burton, Megan E.
Source: Preventing School Failure. 2016 60(4):345-355.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 11
Publication Date: 2016
Document Type: Journal Articles
Reports - Research
Education Level: Intermediate Grades
Grade 3
Primary Education
Elementary Education
Early Childhood Education
Descriptors: Problem Solving, Word Problems (Mathematics), Intervention, Teaching Methods, Mathematics Instruction, Computation, Schemata (Cognition), Mathematical Logic, Grade 3, Elementary School Mathematics, Program Effectiveness, Mathematics Skills, Graphs
DOI: 10.1080/1045988X.2016.1164117
ISSN: 1045-988X
Abstract: Mathematical word problems are the most common form of mathematics problem solving implemented in K-12 schools. Identifying key words is a frequent strategy taught in classrooms in which students struggle with problem solving and show low success rates in mathematics. Researchers show that using the concrete-representational-abstract (CRA) sequence with explicit instruction improves students' computational skills. Researchers also show that schema-based instruction increases students' problem-solving performance. This study combined CRA and schema-based instruction for three students receiving tertiary interventions for mathematics instruction. A functional relation was found for the three students' problem-solving performance. The researchers also interviewed each student to gain a more complete picture of students' thinking. Results and implications are discussed.
Abstractor: As Provided
Number of References: 27
Entry Date: 2016
Accession Number: EJ1110089
Database: ERIC
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  Value: <anid>AN0117948699;psf01oct.16;2019Mar15.12:12;v2.2.500</anid> <title id="AN0117948699-1">Teaching Problem Solving to Students Receiving Tiered Interventions Using the Concrete-Representational-Abstract Sequence and Schema-Based Instruction. </title> <p>Mathematical word problems are the most common form of mathematics problem solving implemented in K–12 schools. Identifying key words is a frequent strategy taught in classrooms in which students struggle with problem solving and show low success rates in mathematics. Researchers show that using the concrete-representational-abstract (CRA) sequence with explicit instruction improves students' computational skills. Researchers also show that schema-based instruction increases students' problem-solving performance. This study combined CRA and schema-based instruction for three students receiving tertiary interventions for mathematics instruction. A functional relation was found for the three students' problem-solving performance. The researchers also interviewed each student to gain a more complete picture of students' thinking. Results and implications are discussed.</p> <p>Keywords: concrete-representational-abstract sequence; elementary; problem-solving strategy; word problems</p> <p>"Making sense of problems and persevering in solving them," is the first of the eight standards for mathematical practice that are listed in the Common Core State Standards Initiative (CCSI, [<reflink idref="bib5" id="ref1">5</reflink>], p. 6). Although problem solving does not always include word problems, they are the most common form of problem solving seen in K–12 schools (Jonassen, [<reflink idref="bib20" id="ref2">20</reflink>]). Problem solving involves ignoring superfluous information, designing a strategy to solve the problem, completing steps required to solve the problem, using equations to represent the word problem, and applying computational procedures (Jitendra et al., [<reflink idref="bib16" id="ref3">16</reflink>]). Teaching problem solving is a process that is integral to mathematics; it is more than simply extracting numbers from a word problem (Griffin & Jitendra, [<reflink idref="bib10" id="ref4">10</reflink>]; National Research Council, 2001; National Council of Teachers of Mathematics, 2014). Woodward et al. ([<reflink idref="bib29" id="ref5">29</reflink>]) provide recommendations for effective problem-solving instruction that are especially relevant for students who struggle. Teachers should model overt and covert problem-solving and thinking processes. Students need to be taught how to use visual representations as well as learn how to articulate their problem-solving and thinking processes.</p> <p>Teachers use a variety of strategies to support students in solving word problems. The most popular strategy reported by 70 elementary teachers in a study by Bruun ([<reflink idref="bib2" id="ref6">2</reflink>]) was circling, underlining, or highlighting key information in the problem. The second most popular strategy taught by elementary teachers according to Bruun's study was identifying key words. Furthermore, Jonassen ([<reflink idref="bib20" id="ref7">20</reflink>]) found the strategy of identifying key words for problem solving prevalent among classrooms in which students struggle with problem solving and have low success rates. Key-word strategies can be problematic for students because a narrow focus on key words can be misleading, encourage removal of meaning from the problem, and may not be applicable when problems are written with particular words and phrases (Van de Walle, Karp, & Bay-Williams, [<reflink idref="bib27" id="ref8">27</reflink>]). In contrast to identifying key words, Jonassen ([<reflink idref="bib20" id="ref9">20</reflink>]) notes the importance of constructing a conceptual model of the problem before solving it. Students need to understand that there are various problem types in order to make sense of the problem structure and create conceptual models (Blessing & Ross, [<reflink idref="bib1" id="ref10">1</reflink>]; Jonassen). Intervention research related to word problems has the following components in common: (a) explicit instruction, (b) comprehension of language within the problem, and (c) strategy use (Case Harris, & Graham, [<reflink idref="bib3" id="ref11">3</reflink>]; Fuchs et al., [<reflink idref="bib9" id="ref12">9</reflink>]; Cassel & Reid, [<reflink idref="bib4" id="ref13">4</reflink>]; Jitendra & Hoff, [<reflink idref="bib17" id="ref14">17</reflink>]; Wilson & Sindelar, [<reflink idref="bib28" id="ref15">28</reflink>]). The aforementioned components assist students in representation and eventual solving of word problems.</p> <p>Wilson and Sindelar ([<reflink idref="bib28" id="ref16">28</reflink>]) taught students to differentiate between types of word problems, rather than focusing on specific operations. The researchers built upon prior research that suggested problem-solving instruction must include sequencing, adequate practice, cognitive strategies, explicit instruction, and generalization techniques (Darch, Camine, & Gersten, [<reflink idref="bib7" id="ref17">7</reflink>]; Fleischner & O'Loughlin, [<reflink idref="bib8" id="ref18">8</reflink>]). The students learned a general classification strategy and were provided explicit instruction in how to solve each type of problem. Students with learning disabilities' performance improved as a result of direct instruction and sequencing practice.</p> <p>Case, Harris, and Graham ([<reflink idref="bib3" id="ref19">3</reflink>]) focused on a common student error found in problem solving, students choosing the wrong operation. The researchers examined effects of a self-regulated strategy intervention on the problem-solving performance of fifth and sixth graders with specific learning disabilities (SLD). The students learned to read the problem aloud, look for words that would assist in understanding the problem, draw pictures about the story, write the equation, and write the answer. The researchers found a functional relation between the intervention and problem-solving behaviors.</p> <p>Cassel and Reid ([<reflink idref="bib4" id="ref20">4</reflink>]) incorporated the use of the concrete-representational-abstract sequence (CRA) and explicit instruction for word problem solving to support third- and fourth-grade students who received special education services for mild intellectual disabilities. The CRA sequence involves instruction in computation using manipulative objects at the concrete level. Representational level instruction includes computation instruction using drawings. The abstract level involves computation using numbers only, commonly with the aid of a mnemonic strategy. After instruction at the concrete and representational levels, students used a strategy for computing problems without the aid of objects or drawings: (a) discover the sign, (b) read the problem, (c) answer or draw and check, and (d) write the answer (DRAW). Once the students were competent in computation, they were taught to solve word problems using the following steps: (a) Find what you are solving for; (b) Ask, "What are the parts of the problem; (c) Set up the numbers; (d) Tie down the sign (FAST). Next, students followed steps to complete the operation using DRAW. The steps of FAST DRAW were similar to those found in many classrooms observed by Jonassen ([<reflink idref="bib20" id="ref21">20</reflink>]) and Bruun ([<reflink idref="bib2" id="ref22">2</reflink>]). The strategy was effective in increasing the number of correct problems and the students maintained their performance eight weeks after instruction.</p> <p>Other intervention problem-solving strategies include more support for students such as the use of drawings and graphic organizers to assist students in understanding and completing word problems. Jitendra and Hoff ([<reflink idref="bib17" id="ref23">17</reflink>]) explored strategy-based instruction in a new way by classifying various types of word problems with graphic representations or schema-based instruction. In this study, third- and fourth-grade students translated and represented story problems using various schema maps and solving the problems. A functional relation was demonstrated between the schema strategy and problem-solving performance.</p> <p>Jitendra et al. ([<reflink idref="bib16" id="ref24">16</reflink>]) compared schema-based instruction to problem-solving instruction using a basal curriculum. The students in the schema-based strategy group received instruction in various types of story problems. This involved representation of each problem type using a graphic organizer or schematic map that helped students understand the problem situation, identify the needed operation, and determine the solution. The students in the basal group received instruction in a general approach to problem solving that involved finding appropriate information, planning, and computation of the solution. The students who received schema-based instruction demonstrated greater improvement and generalization.</p> <p>Jitendra, DiPipi, and Perron-Jones ([<reflink idref="bib13" id="ref25">13</reflink>]) extended the research related to schema-based instruction from the elementary level to the middle school level. Within a different setting, the effects were assessed using more complex problem types such as multiple-step problems requiring multiplication and division. The researchers demonstrated a functional relation between schema-based instruction and problem solving. Another extension of the research involved implementation with students with emotional behavior disorders in a self-contained setting (Jitendra, George, Sood, & Price, [<reflink idref="bib15" id="ref26">15</reflink>]). Using schema-based instruction, the students demonstrated gains in problem-solving performance.</p> <p>The most recent research regarding schema-based instruction included small group tutoring for third-grade students at risk for mathematics difficulties (Jitendra et al., [<reflink idref="bib14" id="ref27">14</reflink>]). In a randomized controlled trial, schema-based instruction was compared to instruction using a standards-based curriculum. Schema-based instruction resulted in greater gains in student performance on standardized assessments and word problem performance.</p> <p>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. The focus of schema-based instruction is drawings and pictures. Other mathematics interventions for students who struggle begin instruction using manipulative objects in order to develop conceptual understanding through hands-on activities. One such instructional approach is the concrete-representational-abstract sequence (CRA) that has been shown to be an effective mathematics intervention for computation (Mancl, Miller, & Kennedy, [<reflink idref="bib22" id="ref28">22</reflink>]; Miller & Kaffar, [<reflink idref="bib23" id="ref29">23</reflink>]). The CRA sequence involves the use of manipulative objects and drawings before instruction using just numbers. With the exception of Cassel and Reid ([<reflink idref="bib4" id="ref30">4</reflink>]), there is little published research regarding its effects with respect to word problems specifically. The combination of CRA and schema-based instruction may provide a more explicit intervention, especially when translating words into equations. The purpose of this study was to combine schema-based instruction and the CRA sequence to provide tiered intervention for students at risk for failure.</p> <hd id="AN0117948699-2">Method</hd> <p></p> <hd id="AN0117948699-3">Setting</hd> <p>The researchers conducted the study in an intermediate elementary school (grades three through five) located in a rural school district in the southeastern United States. The students received instruction four days per week for 20 minutes during an afterschool care program provided by the school. The intervention was conducted in a small classroom used for pull-out intervention during the school day. Student interviews occurred in an empty classroom nearby.</p> <hd id="AN0117948699-4">Participants</hd> <p>The participants, Tim, Carla, and Trey, were third-grade students receiving tertiary interventions in mathematics. Inclusion in the study required parent permission to participate and demonstrated proficiency in addition and subtraction with regrouping (defined as writing 20 correct digits on a two-minute timed probe) and reading proficiency at or above the second-grade level. Students demonstrated a need for the intervention by scoring at or below 25% correct on a problem-solving probe that included extraneous information and one-step computation involving addition or subtraction with regrouping. The students were alert and engaged in each of the intervention lessons even though the study occurred at the end of their school day. The students' characteristics are summarized in Table 1.</p> <p>Table 1. Student Demographic Information</p> <p> <ephtml> <table><thead><tr><td>Student</td><td>Age</td><td>Grade</td><td>Cultural background</td><td>Cognitive ability (IQ)<xref ref-type="fn" rid="t1fn0001" /></td><td>Mathematics applications achievement<xref ref-type="fn" rid="t1fn0002" /></td></tr></thead><tbody><tr><td>Carla</td><td char=".">9</td><td char=".">3</td><td>African American</td><td char=".">91</td><td char=".">83</td></tr><tr><td>Tim</td><td char=".">10</td><td char=".">3</td><td>White</td><td char=".">88</td><td char=".">75</td></tr><tr><td>Trey</td><td char=".">9</td><td char=".">3</td><td>African American</td><td char=".">113</td><td char=".">85</td></tr></tbody></table> </ephtml> </p> <ulist> <item>10001 Standard Score <emph>Kaufman Brief Intelligence Test, 2nd Edition</emph> (Kaufman & Kaufman, 2004).</item> <item>10002 Standard Score for Applications <emph>Key Math 3 Diagnostic Assessment</emph> (Connolly, 2007).</item> </ulist> <hd id="AN0117948699-5">Materials</hd> <p></p> <hd id="AN0117948699-6">Assessment Materials</hd> <p>The assessment materials included 20 different probes created by the researchers. Each was a sheet that had four problems with extraneous information printed using 14-point font and space for written answers. The types of problems were: change (join and separate), part-part-whole, and compare. The problem content included the students' names, names of their teachers and classmates, and familiar locations in order to be culturally relevant.</p> <hd id="AN0117948699-7">Instructional Materials</hd> <p>There were four phases of instruction: teaching problem types, concrete instruction with word problems, representational instruction with word problems, and abstract instruction with word problems. The materials for teaching problem types provided just enough information for the problem type to be described and differentiated from others. The purpose of the materials was to show the ways in which amounts can be joined and separated. The materials were sheets of paper with short sentences and included complete (4 + 2 = 6) and incomplete (4 + ___ = 6) problems. These materials served the purpose of teaching differences in problem structure. For example, change problems included sentences such as: <emph>There were 5 and 2 were added, now there are 7</emph>; <emph>there were 8 and 2 were taken way, now there are ___</emph>. Examples of compare problems were sentences such as the following: <emph>5 is 2 fewer than 7; 3 is 4 less than ___</emph>. Example of part-part-whole sentences involved the following: <emph>there are 6 red blocks and 3 green blocks, 9 blocks in all; there are 7 students, 4 are girls and ___ are boys</emph>.</p> <p>The materials for teaching word problems at the concrete level were sheets of paper with a word problem requiring one-step computation with extraneous information. The sheet included the problem-solving strategy (FAST) written out with directions for each step. The procedures for instruction at the concrete level involved acting out the problem, so materials for acting out the problem were included. An example of a concrete-level lesson sheet is included in Figure 1.</p> <p>Graph: Fig. 1. Instruction in problem type.</p> <p>The materials used for representational instruction were written using the same format with the following exceptions: (a) there was no prompt to act out; (b) there were diagrams for problem type and the student chose the appropriate diagram for solving the problem; and (c) there was a prompt to draw the problem. Physical materials were not included since problem solving involved drawings rather than acting. The materials for abstract-level instruction followed a similar format as representational materials, but diagrams were not printed on the sheets.</p> <hd id="AN0117948699-8">Procedures</hd> <p></p> <hd id="AN0117948699-9">Assessment Procedures</hd> <p>The students completed all written probes using the same procedures; the researcher gave students a probe sheet, told them that there was no time limit, and asked them to solve the problems. During intervention, students completed probes prior to daily instruction in order to assess the learning that occurred in the previous day's lesson. Assessment of progress also included qualitative data in the form of semistructured student interviews. At four points in time, a researcher who was not part of the implementation of the intervention interviewed each student one-on-one for approximately 15 minutes. The researcher recorded the students' voices using a digital recorder and took observational notes based on mannerisms, facial expressions, and other behaviors that would not be captured on the audio recording. The researcher began with an open-ended question and then asked probing questions. The researcher asked the students how they knew what to do and what each number in the word problem meant. At times, the researcher asked students to draw pictures to show the meaning of their process. These qualitative data paired with the quantitative data of the probes serve as a concurrent triangulation method design (Hossler & Vesper, [<reflink idref="bib12" id="ref31">12</reflink>]).</p> <hd id="AN0117948699-10">Instructional Procedures</hd> <p>During each of the four phases, the instructor used the explicit instruction format by providing an advance organizer (verbally described the task), demonstration (the instructor modeled the process), guided practice (the instructor and the student used the process trading turns), independent practice (the student used the process without teacher assistance), and a post-organizer (reviewed task). The first phase involved teaching problem types. The second phase was concrete-level instruction in which the three problem types were solved using the FAST strategy along with physical materials and acting out the problem. The third phase of instruction was the representational phase in which the FAST strategy and diagrams were used to solve problems. The final phase of instruction was the abstract level in which problems were solved using the FAST strategy without other visual aids.</p> <hd id="AN0117948699-11">Teaching Problem Types</hd> <p>The researchers used materials with simple sentences and single digits to teach identification of problem type. The students learned how each sentence could be drawn using numbers and diagrams as well as written using equations. In addition, the student learned which amount in the sentence was the largest. Examples are shown in Figure 1.</p> <p>The researcher explained how a sentence represented a particular problem type (part-part-whole, comparison, and change). The sentence included all parts needed to solve a problem (There were five and three were added; now there are eight). The instructor drew a diagram corresponding to the sentence and circled the portion of the diagram that represented the largest number. Next, the researcher wrote the complete equation using numbers (3 + 2 = 5) and demonstrated how the equation would be solved if certain numbers were missing. The same process was used for sentences with missing parts (e.g., six students; four are girls, __ are boys).</p> <hd id="AN0117948699-12">Concrete Instruction</hd> <p>Concrete instruction involved solving the three types of problems that included extraneous information using the FAST strategy, physical materials, and acting out. An example of a part-part-whole problem involved an irrelevant number of students eating snacks, two different kinds (e.g., vanilla cookies and chocolate chip cookies); the number of vanilla cookies and the total amount of cookies was known and the number of chocolate chip cookies was unknown. The first step in FAST is to "find what you are solving for." The student was directed to the question at the end of the problem. Next, the strategy directs the student to, "Ask, <emph>what are the parts of the problem?</emph>" The instructor and student went to the problem and matched materials to the words within the problem, discussing the problem situation. Then, they underlined the parts of the problem related to the question and crossed out the parts that were not needed. With the physical objects (a bag of vanilla cookies and chocolate chip cookies), the teacher and student acted out and drew the problem using diagrams learned during the first instructional phase; this was the third strategy step, "set up the numbers." The teacher asked when there would be the most cookies, when the two kinds were combined or when they were separated? This identified the big number in the problem. The smaller numbers were identified as the amounts representing vanilla and chocolate chip cookies. The problem diagram was drawn. With the total amount of cookies given, the instructor and student physically separated the chocolate chip from vanilla. It was emphasized that vanilla cookies were taken away from the total to find the amount of chocolate chip cookies, connecting the physical process of separation and subtraction. The physical process demonstrated the need for subtraction when the big number is known and a small number is missing. If the big number was missing, the operation was addition. The last step of the strategy, "tie down the sign," involved identifying the operation, writing the equation, and obtaining the answer. Example problems are in Figure 2.</p> <p>Graph: Fig. 2. Concrete level lesson sheet.</p> <p>An example of a compare problem involved students receiving pencils and markers for a project and two students receiving various amounts of markers. The difference amount and the amount given to one student were known; the amount given to the other student was unknown. An example of a change problem involved a teacher opening a new box of markers with an unknown number in the box; the amounts given and remaining in the box were known.</p> <hd id="AN0117948699-13">Representational Instruction</hd> <p>Representational instruction involved solving the three types of problems, including extraneous information using the FAST strategy with schema diagrams. The first step in FAST is to, "find what you are solving for." The student was directed to the question at the end of the problem. Next, the strategy directs the student to, "Ask, <emph>what are the parts of the problem?</emph>" The instructor and student underlined the parts of the problem related to the question and crossed out the unneeded parts. They determined the appropriate diagram for the problem type and drew the problem; this was the third strategy step, "set up the numbers." The teacher and student determined which number within the diagram represented the largest amount. The last step of the strategy, "tie down the sign," involved identifying the operation. If the big number was missing, the operation was addition. If the big number was present and a small number was missing, the operation was subtraction. Last, they wrote the equation and answer. Examples of representational level problems are included in Figure 3.</p> <p>Graph: Fig. 3. Completed representational lesson problems.</p> <hd id="AN0117948699-14">Abstract Instruction</hd> <p>Abstract-level instruction involved solving the three types of problems with extraneous information using the FAST strategy. The FAST strategy was written on lesson sheets. Problem solving required determination of the problem type without representational assistance. No visual aids were provided to assist with drawing the problem when setting up numbers or determining the operation.</p> <hd id="AN0117948699-15">Treatment Integrity</hd> <p>Treatment integrity was measured using live observations of instruction and a checklist of instructor behaviors. A second researcher observed instruction for 90% of the lessons. The observation checklist consisted of a list of behaviors and response columns in which the observer responded "yes" that the instructor demonstrated the behavior and "no" the instructor did not demonstrate the behavior. Treatment integrity for the study was 95%.</p> <hd id="AN0117948699-16">Inter-Observer Agreement</hd> <p>Assessment probes were checked by two researchers. Inter-observer agreement was calculated by adding the item agreements and dividing the total by the sum of the item disagreements and item agreements. Inter-observer agreement for Carla was 100%. Inter-observer agreement for Tim was 100%. Inter-observer agreement for Trey was 100%. With regard to qualitative data, two researchers checked the recordings, student notes, and notes of the interviews. Each separately took notes and drew conclusions. Conclusions and statements were verified for accuracy of representation. Inter-observer agreement was 100%.</p> <hd id="AN0117948699-17">Social Validity</hd> <p>Social validity was assessed using student and teacher surveys as well as samples of student work, showing lack of problem-solving skills. The students' work samples collected prior to intervention showed that they did not solve word problems at a level appropriate to their grade with accuracy less than 25%. Prior to intervention, the students' teachers reported that problem-solving intervention was needed. The students stated that they did not like solving word problems. They reported that they would like to know easier ways to solve word problems. After the intervention, the students and their teachers reported that their problem solving improved. Two students said that they liked using objects and drawings to solve problems and all reported that the drawings and objects were helpful.</p> <hd id="AN0117948699-18">Results</hd> <p>The data were analyzed using visual inspection of the students' graphs, which are shown in Figure 4. The effects of the intervention were measured based on changes in level between baseline and intervention, immediacy of effect after phase change, number of probes to the criterion, and amount of overlap between baseline and intervention phases. The researchers used the qualitative data gained from interviews to capture the students' thinking.</p> <p>Graph: Fig. 4. Results for Carla, Tim, and Trey.</p> <hd id="AN0117948699-19">Baseline</hd> <p>All of the students began the study with scores of 25% or below on problem-solving probes. Carla's baseline began with scores of 25%, but decreased to scores of zero and remained stable across three probes prior to intervention. Tim's baseline scores were stable across all baseline probes with scores of zero on each. Trey's baseline scores were more variable, but were stable across the last four probes with 25% on each and a baseline level of 11%.</p> <p>When interviewed about baseline probes, all students had difficulties with extraneous information. The students' reasoning for why a number was not needed was faulty. For example, Tim did not use the number 85 because he said that it was too big, not because it was the name of an interstate highway used by the trucks described within the problem. For other problems, students ignored extraneous information, using all of the numerals that appeared in the problem (32-5-13). Across students, explanations about choosing an operation included key-word strategies. Carla reported that "how many are left" indicated addition and that "how many more" indicated subtraction. Trey reported that "how many are left" indicated subtraction and the equation he used to solve that particular problem included three numbers (32-5-13 = 10). It appeared that the students did not understand the situations within the word problems. Across problems, their approach involved finding key words, using those words to choose an operation, and selecting numbers to form equations without logical reasons.</p> <hd id="AN0117948699-20">Intervention</hd> <p>There was an immediate change in performance after the first intervention probe. Carla's performance was increasing, but began to decrease after the third probe. Her performance was variable across most of the intervention, although only one data point overlapped with baseline. The level of Carla's data path was 75%, ranging from zero to 100%. Carla met the criterion for mastery after 16 probes.</p> <p>Carla's descriptions of her problem solving showed an increased ability to make sense of the problem. Carla identified extraneous information and told why it was not important. In the third and fourth interview, she described problem types. She still struggled with computational errors, but could choose operations logically. There were two instances when she realized an error in her solution strategy. Carla self-corrected and explained why she made the error.</p> <p>There was an immediate change in Tim's performance after the first intervention probe. Tim's data path during intervention was increasing with little variability. Tim's data path had a level of 70%, ranging from 25% to 100%. There were not overlapping data points. Tim met the criterion for mastery after 10 probes. Tim maintained his performance four weeks after the intervention with a score of 100%.</p> <p>Tim did not enjoy the interview process. He groaned when told about the interview and began with short responses. When probed, Tim began to identify extraneous information and over time, he crossed out the information in the problem. He began to use the descriptions of the stories in the problem to explain how he determined the operation. Although Tim did not identify the type of problems, it was clear he was making meaning of the problem and examining how the information could help him find the solution.</p> <p>There was an immediate change in Trey's performance after the first intervention probe. Trey's data path during intervention increased slowly. The level of Trey's data path was 69% with a range from 25% to 100%. There were two intervention data points (17%) that overlapped with baseline. Trey met the criterion for mastery after 12 probes. Four weeks after intervention, Trey's maintenance score was 75%.</p> <p>Beginning the intervention phase, Trey struggled to determine the extraneous information and continued to subtract three numbers, such as 2-4-5. During the second interview, Trey showed improvement in reading the word problem for meaning and identified extraneous information. In the third interview, he described the meaning of each number. During the final interview, he clearly explained how information in the problem related or did not relate to the question.</p> <hd id="AN0117948699-21">Effect Size</hd> <p>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).</p> <hd id="AN0117948699-22">Discussion</hd> <p>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 problem-solving strategy. All three of the students improved their problem-solving performance and achieved mastery as defined as three probes with scores of 100% correct. This is consistent with previous research associated with CRA, the FAST strategy, and schema-based interventions (Cassel & Reid, [<reflink idref="bib4" id="ref32">4</reflink>]; Jitendra, DiPipi, & Perron-Jones, [<reflink idref="bib13" id="ref33">13</reflink>]; Jitendra, George, Sood, & Price, [<reflink idref="bib15" id="ref34">15</reflink>]; Jitendra & Hoff, [<reflink idref="bib17" id="ref35">17</reflink>]; Jitendra et al., [<reflink idref="bib18" id="ref36">18</reflink>]; Jitendra, Sczesniak, & Deatline-Buchman, [<reflink idref="bib19" id="ref37">19</reflink>]). The information gleaned from the interview illuminated students' progress in reading problems for meaning.</p> <p>The students began the study with a preference for a key-word strategy, choosing operations based on the presence of specific words. For example, the phrase, "in all" led students to add and the phrase, "less than" led the students to subtract. In addition, the students included all numbers written in the problem within their solution number sentence. An example problem follows. Three students ate chocolate and vanilla cookies. The students ate four chocolate cookies and some vanilla cookies. The students ate six cookies in all. How many vanilla cookies did the students eat? Following previous error patterns, a common solution was 3 + 4 + 6 = 13. The students relied on key words and failed to use the entire problem. Instruction at the concrete level encouraged students to use all of the words within the problem in order to act out the scenario.</p> <p>The FAST strategy provided a series of steps to follow, rather than their 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 objects which made identifying the parts of the problem easier. The combination of concrete instruction and the use of schematic maps assisted students in organizing the parts of the number sentence and choosing the operation. When students moved to the representational level, the schematic maps remained, providing the means to organize information within word problems. Once the students mastered problem solving during representational lessons, the students progressed to using FAST without visual aids.</p> <p>The students progressed to mastery slowly throughout the intervention and Carla's data path was highly variable. During lessons, the students solved problems with physical and visual aids, but the assessment probes involved only written problems. During instructional lessons, the students completed independent practice with 100% accuracy. Their performance on probes the following day continued to be 50% or less. Previous error patterns and the dependence on key words continued throughout the probes during the concrete and representational phases. This might be explained by the lack of supports incorporated into the assessment task. The students' performance did not consistently change until the abstract phase of instruction; during this phase, perhaps the physical processes used during previous phases became cognitive. Thinking aloud without visual aids was modeled and guided and this may have been the critical difference in their learning. The concrete and representational phases provided the foundation for the cognitive processes that were learned during the abstract phase.</p> <hd id="AN0117948699-23">Limitations and Future Research</hd> <p>This study is limited because the instructor was a researcher rather than the students' classroom teacher. It is likely that treatment integrity was influenced by researcher implementation. Future research should include implementation by classroom teachers within intervention settings as well as professional development in the intervention. This would better inform researchers regarding the feasibility and effectiveness of the intervention in authentic settings. Another limitation is the research design, multiple-probe across students; this design prevents comparison of this intervention to others and results cannot be generalized beyond the current students in the study. Within single-case design, generalization is shown with replication of results across studies (Horner, Carr, Halle, McGee, Odom, & Wolery, 2005); therefore, additional research is needed to draw conclusions regarding generalization. With regard to comparison, it is not known whether the current combination of CRA and schema-based instruction is as or more effective than other problem solving interventions. Future research might include implementation of the current intervention and another intervention with a larger group so that effects can be compared and conclusions related to generalization might be made.</p> <hd id="AN0117948699-24">Implications and Conclusions</hd> <p>The current study's findings are significant in that they demonstrated the effectiveness of the combination of two evidence-based interventions for students who demonstrated poor problem-solving performance with persistent error patterns. The intervention was relatively short and required few physical resources. This has implications for its use as a tertiary intervention because implementation time is consistent with time periods available during regular school days devoted to differentiated instruction. In addition, the intervention was successful as an intensive intervention with small teacher-to-student ratio and explicit instruction; significant changes might be necessary if implemented with larger instructional groups. The students' progress in problem-solving processes can be applied across situations and built upon as problems increase in complexity. Additional research is needed to address previously discussed limitations related to implementation and comparison to other problem-solving interventions. This study extends research and provides additional evidence that may lead to further refinement of problem-solving interventions.</p> <ref id="AN0117948699-25"> <title> References </title> <blist> <bibl id="bib1" idref="ref10" type="bt">1</bibl> <bibtext> Blessing, S. B., & Ross, B. H. (1996). Content effects in problem categorization and problem solving. Journal of Experimental Psychology Learning, Memory, & Cognition, 22(3), 792–810.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref6" type="bt">2</bibl> <bibtext> Bruun, F. (2013). Elementary teachers' perspectives of mathematics problem solving strategies. Mathematics Educator, 23(1), 45–59.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref11" type="bt">3</bibl> <bibtext> 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.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref13" type="bt">4</bibl> <bibtext> Cassel, J., & Reid, R. (1996). Use of self-regulated strategy intervention to improve word problem solving skills of student with mild disabilities. Journal of Behavioral Education, 6, 153–172.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref1" type="bt">5</bibl> <bibtext> Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics. Retrieved from <ulink href="http://www.corestandards.org/assets/CCSSI%5fMath%20Standards.pdf">http://www.corestandards.org/assets/CCSSI%5fMath%20Standards.pdf</ulink></bibtext> </blist> <blist> <bibl id="bib6" type="bt">6</bibl> <bibtext> Connolly, A. J. (2007). KeyMath (3rd ed.). Minneapolis, MN: Pearson.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref17" type="bt">7</bibl> <bibtext> Darch, C., Camine, D., & Gersten, R. (1984). Explicit instruction in mathematical problem solving. Journal of Educational Research, 4, 155–165.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref18" type="bt">8</bibl> <bibtext> Fleischner, J., & O'Loughlin, M. (1985). Solving story problems: Implications of research for teaching the learning disabled. In J. Cawley (Ed.), Cognitive strategies and mathematics for the learning disabled (pp. 163–182). Rockville, MD: Aspen.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref12" type="bt">9</bibl> <bibtext> Fuchs, L. S., Fuchs, D., Prentice, K., Burch, M., Hamlett, C. L., Owen, R., & Schroeter, K. (2003) Enhancing third-grade students' mathematical problem solving with self-regulated learning strategies. Journal of Educational Psychology, 95, 306–315.</bibtext> </blist> <blist> <bibtext> Griffin, C., & Jitendra, A. (2009). Word problem-solving instruction in inclusive third-grade mathematics classrooms. Journal of Educational Research, 102, 187–202.</bibtext> </blist> <blist> <bibtext> Horner, R. H., Carr, E. G., Halle, J., McGee, G., Odom, S., & Wolery, M. (2005). The use of single-subject research to identify evidence-based practice in special education. Exceptional Children, 2, 165–179.</bibtext> </blist> <blist> <bibtext> Hossler, D., & Vesper, N. (1993). An exploratory study of the factors associated with parental savings for postsecondary education. Journal of Higher Education, 64(2), 140–165.</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., DiPipi, C. M., & Perron-Jones, N. (2002). An exploratory study of schema-based word problem solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. The Journal of Special Education, 36(1), 23–38.</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., Dupuis, D. N., Rodriguez, M. C., Zaslofsky, A. F., Slater, S., Cozine-Corroy, K., & Church, C. (2014). A randomized controlled trial of the impact of schema-based instruction on mathematical outcomes for third-grade students with mathematics difficulties. The Elementary School Journal, 114, 252–276.</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., George, M. P., Sood, S., & Price, K. (2010). Schema-based instruction: Facilitating mathematical word problem solving for students with emotional behavioral disorders. Preventing School Failure, 54, 145–151. doi:10.1080/10459880903493104</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., Griffin, C. C., McGoey, K., Gardill, M. C., Bhat, P., & Riley, T. (1998). Effects of mathematical word problem solving by students at risk or with mild disabilities. Journal of Educational Research, 91, 345–355.</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., & Hoff, K. (1996). The effects of schema-based instruction on the mathematical word-problem solving performance of students with learning disabilities. Journal of Learning Disabilities, 29, 422–431.</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., Rodriguez, M., Kanive, R., Huang, J., Church, C., Corroy, K. A., & Zaslofsky, A. (2013). Impact of small group tutoring interventions on the mathematical problem solving of third-grade students with mathematics difficulties. Learning Disability Quarterly, 36(1), 21–35.</bibtext> </blist> <blist> <bibtext> Jitendra, A. K., Sczesniak, E., & Deatline-Buchman, A. (2005). An exploratory validation of curriculum-based mathematical word problem-solving tasks as indicators of mathematics proficiency for third graders. School Psychology Review, 34, 358–371.</bibtext> </blist> <blist> <bibtext> Jonassen, D. H. (2003) Designing research-based instruction for story problems. Educational Psychology Review, 15, 267–296.</bibtext> </blist> <blist> <bibtext> Kaufman, A. S., & Kaufman, N. L (2004). Kaufman Brief Intelligence Test (2nd ed.). Minneapolis, MN: Pearson.</bibtext> </blist> <blist> <bibtext> Mancl, D. B., Miller, S. P., & Kennedy, M. (2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research and Practice, 27(4), 152–166.</bibtext> </blist> <blist> <bibtext> Miller, S. P., & Kaffar, B. J. (2011). Developing addition with regrouping competence among second graders with mathematics difficulties. Investigations in Mathematics Learning, 4(1), 24–49.</bibtext> </blist> <blist> <bibtext> National Council of Teachers of Mathematics. (2014). Principles to actions: Executive summary. Retrieved from <ulink href="http://www.nctm.org/uploadedFiles/Standards%5fand%5fFocal%5fPoints/Principles%5fto%5fAction/PtAExecutiveSummary.pdf">http://www.nctm.org/uploadedFiles/Standards%5fand%5fFocal%5fPoints/Principles%5fto%5fAction/PtAExecutiveSummary.pdf</ulink> on October 1, 2014.</bibtext> </blist> <blist> <bibtext> National Research Council. (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), A report from the NRC. Washington, DC: National Academies Press.</bibtext> </blist> <blist> <bibtext> Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B. (2011). Combining nonoverlap and trend for single-case research: Tau-U. Behavior Therapy, 42, 284–299.</bibtext> </blist> <blist> <bibtext> Van de Walle, J., Karp, K. S., & Bay-Williams, J. M. (2009). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Allyn and Bacon.</bibtext> </blist> <blist> <bibtext> Wilson, C. L., & Sindelar, P. T. (1991). Direct instruction in math word problems: Students with learning disabilities. Exceptional Children, 57, 512–519.</bibtext> </blist> <blist> <bibtext> Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012–4055). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from <ulink href="http://ies.ed.gov/ncee/wwc/publications%5freviews.aspx#pubsearch/">http://ies.ed.gov/ncee/wwc/publications%5freviews.aspx#pubsearch/</ulink>.</bibtext> </blist> </ref> <aug> <p>By Margaret M. Flores; Vanessa M. Hinton and Megan E. Burton</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Author Notes Margaret M. Flores is an associate professor in the Department of Special Education, Rehabilitation, and Counseling at Auburn University. Her research interests include mathematics interventions for elementary students with disabilities or who receive tertiary interventions.</p> <p>Vanessa M. Hinton is an assistant clinical professor in the Department of Special Education, Rehabilitation, and Counseling at Auburn University. Her research interests include early number mathematics interventions for elementary students with disabilities or who receive tertiary interventions.</p> <p>Megan E. Burton is an associate professor in the Department of Curriculum and Teaching at Auburn University. Her research interests include elementary mathematics instruction and teacher preparation.</p> </aug> <nolink nlid="nl1" bibid="bib20" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib16" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib10" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib29" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib27" firstref="ref8"></nolink> <nolink nlid="nl6" bibid="bib17" firstref="ref14"></nolink> <nolink nlid="nl7" bibid="bib28" firstref="ref15"></nolink> <nolink nlid="nl8" bibid="bib13" firstref="ref25"></nolink> <nolink nlid="nl9" bibid="bib15" firstref="ref26"></nolink> <nolink nlid="nl10" bibid="bib14" firstref="ref27"></nolink> <nolink nlid="nl11" bibid="bib22" firstref="ref28"></nolink> <nolink nlid="nl12" bibid="bib23" firstref="ref29"></nolink> <nolink nlid="nl13" bibid="bib12" firstref="ref31"></nolink> <nolink nlid="nl14" bibid="bib18" firstref="ref36"></nolink> <nolink nlid="nl15" bibid="bib19" firstref="ref37"></nolink>
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  Data: Teaching Problem Solving to Students Receiving Tiered Interventions Using the Concrete-Representational-Abstract Sequence and Schema-Based Instruction
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  Data: Routledge. Available from: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
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  Data: <searchLink fieldCode="DE" term="%22Problem+Solving%22">Problem Solving</searchLink><br /><searchLink fieldCode="DE" term="%22Word+Problems+%28Mathematics%29%22">Word Problems (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Intervention%22">Intervention</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Schemata+%28Cognition%29%22">Schemata (Cognition)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Logic%22">Mathematical Logic</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+3%22">Grade 3</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Mathematics%22">Elementary School Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Program+Effectiveness%22">Program Effectiveness</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Graphs%22">Graphs</searchLink>
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  Data: Mathematical word problems are the most common form of mathematics problem solving implemented in K-12 schools. Identifying key words is a frequent strategy taught in classrooms in which students struggle with problem solving and show low success rates in mathematics. Researchers show that using the concrete-representational-abstract (CRA) sequence with explicit instruction improves students' computational skills. Researchers also show that schema-based instruction increases students' problem-solving performance. This study combined CRA and schema-based instruction for three students receiving tertiary interventions for mathematics instruction. A functional relation was found for the three students' problem-solving performance. The researchers also interviewed each student to gain a more complete picture of students' thinking. Results and implications are discussed.
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