A multiple-probe-across-participants design included baseline, treatment, generalization, and maintenance. During treatment, students received schema strategy training in problem schemata (conceptual understanding) and problem solution (procedural understanding). Results indicated that the schema-based strategy was effective in substantially increasing the number of correctly solved multiplication and division word problems for all 4 participants. Maintenance of strategy effects was evident for 10, 5 1/2, and 2 1/2 weeks following the termination of instruction for Sara, Tony, and Percy, respectively. In addition, the effects of instruction generalized to novel word problems for all 4 participants.
The importance of mathematics literacy and problem solving has been emphasized by researchers (e.g., De Corte, Greer, & Verschaffel, 1996; Goldman, Hasselbring, & the Cognition and Technology Group at Vanderbilt, 1997; Patton, Cronin, Bassett, & Koppel, 1997) and in national reports (e.g., National Council of Teachers of Mathematics [NCTM], 2000; National Education Goals Panel, 1997). Although instruction that emphasizes reflective thinking and reasoning is considered by many to be critical to mathematics reform efforts (Baroody & Hume, 1991; Bottge, 1999; Hofmeister, 1993; Montague, 1997a), procedures that encourage memorization and the completion of lengthy worksheets requiring rote practice are common in many classrooms (Parmar & Cawley, 199 1). Mathematics instruction in special education, particularly, has been characterized to a large extent by its emphasis on rote memorization of facts and computational skills, rather than on developing important concepts and applying mathematics to real-world problem situations (Baroody & Hume, 1991; Bottge, 1999; Parmar, Cawley, & Miller, 1994; Woodward & Montague, 2000).
Many researchers argue that highly procedural instruction (meaningless drill and practice of computation facts) may sustain the characterization of students with learning disabilities as passive learners and fail to fill the gaps in their conceptual understanding of the core concepts and principles underlying mathematical thinking (Baroody & Hume, 1991; Parmar et al., 1994; Torgesen, 1982; Woodward & Montague, 2000). It is not surprising, then, that many students with learning disabilities have difficulty with higher level mathematics skills, such as solving word problems (Xin & Jitendra, 1999). Parmar, Cawley, and Frazita (1996) compared the performance of students with and without disabilities on arithmetic word problems involving all four operations and problems that contained direct or indirect statements, extraneous information, and one-step or two-step problems. Results indicated that the students with disabilities performed at significantly lower levels than the students without disabilities on all problem types. The students with disabilities experienced considerable difficulty with problem representation or identifying relevant information, along with difficulties in reading, computation, and identifying operations.
One plausible deficiency of traditional mathematics instruction is its failure to make explicit the key aspects of domain knowledge needed for problem solving. Emerging views in special education indicate the importance of explicit instruction and practice in domain-specific problem solving (Hofmeister, 1993; Mercer & Miller, 1992; Parmar et al., 1996), with an increased emphasis on the domain heuristic of graphically representing word problems (Jitendra, Hoff, & Beck, 1999; Xin & Jitendra, 1999). Domain-specific knowledge encompasses both conceptual and procedural knowledge. Conceptual knowledge refers to the hierarchical network of knowledge and its corresponding relationships (Hiebert & Lefevre, 1986); procedural knowledge results from the organization of conceptual knowledge into action units (Anderson, 1989). Knowledge organization and pattern recognition are key aspects of conceptual knowledge (Silver & Marshall, 1990). For example, mathematical problem solving requires the ability to organize problems (e.g., distance-rate-time problems, interest problems, discount problems) by structural similarity (e.g., generalized rate problems). During problem solving, all problem-relevant knowledge is accessible only when the knowledge is adequately organized in memory by a suitable cognitive structure (i.e., problem schemata). Problem schemata are elements of knowledge that are closely linked with each other within the knowledge base (Chi, Glaser, & Rees, 1982). Knowledge of the mathematical structure of problems, in turn, can facilitate activation of the relevant schemata or patterns that would guide problem representation, which is necessary for solving problems.
Clearly, providing quality instruction that emphasizes both problem representation and problem solution is deemed important to successful problem solving (Fraivillig, Murphy, & Fuson, 1999; Fuson & Willis, 1989). Problem representation involves translating a problem from words into a meaningful representation. This could include a “combination of something written on paper, something existing in the form of physical objects and a carefully constructed arrangement of an idea in one’s mind” (Janvier, 1987, p. 68). Problem solution refers to the selection and application of appropriate mathematical operations based on the representation. It involves both solution planning and execution of mathematical operations.
Mathematical problem-solving instruction should not only emphasize conceptual knowledge of the operations but also facilitate “a highly integrated understanding of the operations and the many different but related meanings these operations take on in real contexts” (Van de Walle, 1998, p. 117). The big ideas for developing meanings for the operations should, for example, show that addition and subtraction are connected and that multiplication and division are related. In the context of solving story problems, for example, models or diagrams can be used to represent the information in a problem and to figure out what operation is needed to solve the problem (Van de Walle, 1998).
Most models for understanding and assessing children’s solution of problems are generally derived from cognitive psychology (Briars & Larkin, 1984; Carpenter & Moser, 1984; Fennema, Carpenter, & Peterson, 1989; Kintsch & Greeno, 1985; Riley, Greeno, & Heller, 1983). These models of problem solving emphasize the importance of the problem’s semantic characteristics (Silver & Marshall, 1990). As students develop knowledge in a domain (e.g., mathematics), the knowledge structure eventually takes on the form of schema mapping of relationships. Schema as a knowledge structure serves the function of knowledge organization. According to Marshall (1995), schemata are the basis for understanding and the appropriate mechanism for the problem solver to use to “capture both the patterns of relationships as well as their linkages to operations” (p. 67). A distinctive feature of schemata is that when one piece of information is retrieved from memory during problem solving, other connected pieces of information will be activated. Problem schemata pertaining to a wide range of problems involving all four operations include ” change,” “group,”” compare,” “vary,” and “restate” (Marshall, Pribe, & Smith, 1987). These problem types dominate word problems typically found in the elementary and middle grades (Van de Walle, 1998).
Recent reviews provide empirical support for schemabased word-problem-solving instruction that emphasizes conceptual understanding (Jitendra & Xin, 1997; Xin & Jitendra, 1999). The schema-based representational strategy, with its focus on schemata (i.e., problem pattern or structure) identification, is known to benefit both students with learning disabilities (elementary, middle, and high school) and students at risk for math failure (Hutchinson, 1993; Jitendra & Hoff, 1996; Jitendra et al., 1999; Jitendra et al., 1998; Zawaiza & Gerber, 1993). Strategies (e.g., schema- based instruction) that entail “looking systematically for patterns, are very close to content curriculum goals” (NCTM, 1998, p. 4). A primary characteristic of a schema-based strategy that distinguishes it from other approaches is the use of schemata diagrams to map important information and highlight semantic relations in the problem to facilitate problem translation and solution. Other strategy-training procedures (e.g., cognitive and metacognitive) also may include diagrams, but the emphasis is less on identifying the semantic relations in a problem and more on problem solving heuristic procedures that lead to its solution.
Although the use of schema strategy in teaching students with learning disabilities is promising, most schema- based research studies reported in the literature have focused on teaching addition and subtraction word problems (i.e., change, group, compare; e.g., Jitendra et al., 1998; Jitendra & Hoff, 1996; Jitendra et al., 1999) or algebra word problems (Hutchinson, 1993; Maccini & Hughes, 2000). However, children typically experience significant problems with multiplicative (includes all problems involving multiplication and division structure) rather than additive (includes all problems involving addition and subtraction structure) situations (Van de Walle, 1998). Also, the shift in focus during middle school to the more complex relations found in “vary” or “equal groups” and “restate” (i.e., “multiplicative comparison”) multiplication and division word problems makes it necessary to teach these problem types. Although Greer (1992) discusses two other types of multiplication and division problems (i.e., combinations, and products of measures), these types of problems do not receive much attention in school (Van de Walle, 1998). The study by Zawaiza and Gerber (1993) is the only one that has investigated the effectiveness of the schema strategy in solving multiplicative comparison-type word problems. However, that study did not address “vary” problems, which are also prevalent in elementary and middle school (Van de Walle, 1998).
In summary, the schema strategy is seen as a viable approach for teaching students with learning disabilities to solve addition and subtraction word problems. However, research on teaching middle school students with learning disabilities to solve multiplication and division word problems using schema-based instruction is lacking. Therefore, the purpose of this exploratory study was to examine the effectiveness of the schema strategy in solving multiplication and division problems. The present study extends the existing body of research regarding the applicability of the schema strategy to promote word-problem-solving skills in middle school students with learning disabilities (e.g., Jitendra et al., 1999). Specifically, the following research questions were posed: (a) Is a schema- based instructional strategy effective in teaching one-step multiplication and division word-problem solving to middle school students with learning disabilities who are low-performing in mathematics? (b) Will the students maintain the acquired word-problem-solving skills? (c) Will the students generalize the word-problem-solving skills to novel problems, including multistep problems? To address these questions, we used a multiple-probe-across- participants design. However, this single-subject design may not be well suited to determining the specific aspects of the treatment to which to attribute effects. In light of this limitation, results from this study should be viewed as preliminary.
Method
Participants
Participants included four eighth-grade students with learning disabilities (e.g., one girl and three boys) attending a suburban middle school in the northeastern United States. Participant selection was based on several criteria. First, participants had been previously identified as learning disabled by meeting Pennsylvania State criteria for learning disabilities, which stipulate that a child must demonstrate (a) a chronic condition of presumed neurological origin that selectively interferes with the development, integration, or demonstration of language or nonverbal abilities; (b) a severe discrepancy between achievement and intellectual ability in one or more of several areas (i.e., oral expression, listening comprehension, written expression, basic reading skills, reading comprehension, mathematics calculation, and mathematics reasoning), which is not correctable without special education and related services; (c) specific deficits in receptive and expressive language and deficiencies in initiating or sustaining attention, impulsivity, and other specific conceptual and thinking difficulties; (d) normal or above-normal intelligence; and (e) learning problems that are not due primarily to other disabling conditions or environmental, cultural, or economic disadvantage. This determination was made by a full assessment and comprehensive report by a certified school psychologist. A summary of participating students’ characteristics is presented in Table 1. It must be noted that although the sample in this study was identified as learning disabled, whether these students had learning disabilities in mathematics is questionable according to the conventional cutoff scores for a mathematics disability (a discrepancy of at least 1 standard deviation between their scores on a standardized test of mathematics and their full scale score on an intelligence test). *Only Percy and possibly Tony would fit this definition of having a learning disability in mathematics (i.e., a discrepancy existed between their IQs and each of the math composite and numerical operations scores on the WIAT, but their math reasoning scores were in the average range).
Second, a teacher interview indicated that these students were experiencing significant cant difficulties with mathematical problem solving, an area that was specifically targeted for instruction on each student’s Individualized Education Program goals. However, the teacher reported that all students had successfully passed a criterion test of mathematics computation skills involving all four operations. In addition, each student had to complete a sample of six one-step multiplication and division word problems similar to the criterion tests used in the study. An examination of word problems completed (see Table 1) indicated that all four students experienced significant difficulties solving them; they had not reached the mastery (less than 50%) needed to solve more complex word problems.
In this study, the special education teacher conducted all testing and instruction in the learning support classroom. She was certified to teach students with mental and physical disabilities and had 12 years’ teaching experience. At the time of the study, the teacher was completing her master’s degree in special education.
Setting
All four students were included in general education classrooms but received special education services in a learning support classroom for mathematics and other subjects (e.g., reading, English, study skills). Each participant sat at a table across from the teacher when receiving instruction in the study. During this time, the other students in the classroom worked independently on different skills, while the classroom instructional aide supervised and provided assistance as needed. Three of the participants in the study were present in the classroom during the same period. The seating of these participants was arranged to place them away from one another, at opposite ends of the room, While one participant received instruction, the others received direct instruction from the classroom aide to prevent incidental learning. The fourth participant had received mathematics earlier in the day. During independent practice and the completion of tests, each student sat at his or her assigned desk and worked alone.
| Variable | Students | |||
|---|---|---|---|---|
| Sara | Tony | Percy | Andy | |
| Gender | Girl | Boy | Boy | Girl |
| Ethnicity | Caucasian | Caucasian | African American | Caucasian |
| Age | 13-10 | 13-8 | 13-5 | 13-7 |
| Grade | 8 | 8 | 8 | 8 |
| Classification | LD | LD | LD | LD, ADHD |
| SES (a) | Medium | Low | High | Medium |
| Years in special ed. | 4 | 4 | <1 | <1 |
| Learning support classroom placement | Math, reading, English | Math, English | Math, study skills | Math, reading, English, study skills |
| % in general ed. class | 62.5 | 75 | 85 | 62.5 |
| IQ(b) | ||||
| Full Scale | 95 | 103 | 101 | 89 |
| Verbal | 101 | 107 | 98 | 93 |
| Performance | 90 | 98 | 106 | 86 |
| Achievement (c) | ||||
| Math composite | 87 | 83 | 79 | 76 |
| Math reasoning | 93 | 93 | 80 | 84 |
| Numerical operations | 84 | 77 | 84 | 76 |
| Composite reading | 81 | 97 | 85 | 91 |
| Composite writing | 87 | 94 | 92 | 78 |
| Protest | ||||
| One-step word probs. | 50% | 33% | 33% | 16% |
Note. LD = learning disability; ADHD = attention-deficit/hyperactivity disorder. (a) Based on parents’ profession. (b) Wechsler Intelligence Scale for Children-III (Wechsler, 199 1). (c) Standard scores for subtests of the Wechsler Individual Achievement Test (Wechsler, 1992). |
||||
Dependent measures
Word-problem tests.
Each of the series of tests constructed for the study consisted of six one-step multiplication and division word problems involving two different problem types (i.e., vary and multiplicative comparison) based on Marshall’s (1995) and Van de Walle’s (1998) word-problem classification system. However, we did not include multiplicative comparison problems in which the comparison amount (based on one set’s being a particular multiple or part of the other set) is unknown, because these types of problems rarely occur in textbooks. Vary and multiplicative comparison problems in this investigation were selected from the SRA Spectrum Math (Richard, 1997) textbook used in the classroom (see Table 2 for examples of each problem type). After consultations with the classroom teacher, we modified word problems from the text to include names of participants and -their peers in the classroom and familiar contexts to make them interesting to students. The order of problem types within each test was determined randomly.
In addition, to assess response generalization of the strategy, we developed a separate test that consisted of 12 one- step and multistep multiplication and division word problems. To assess near transfer, we included three each of one-step vary and multiplicative comparison word problems that were similar in structure to those used in the study but differed in con-, text and the position of the unknown (see Table 2). To assess far transfer, six multistep problems (see Table 3) of the type in which students did not receive instruction were included. The order of one-step and multistep problems within the test was randomly ordered. For all tests, two problems to a page were typed on 8 1/2-inch by 11-inch unlined paper, and each problem included a workspace and a line for writing the entire answer.
Tests were scored by counting the number of problems answered correctly. For each step of a problem, I point was assigned for a correct answer, whereas an incorrect answer was awarded a score of 0. As such, the total possible points for one-step problems ranged from 0 to 1, whereas scores for multistep problems ranged from 0 to 3, depending on the number of steps.
| Instructional and testing problems | Generalization problems |
|---|---|
Vary
|
Vary
|
Multiplicative comparison
|
Multiplicative comparison
|
Strategy questionnaire.
Students were administered a strategy questionnaire to complete at the end of the investigation. The questionnaire contained both Likert-type and open-ended questions that provided information on each student’s perception of the strategy’s effectiveness and his or her satisfaction with it. Ratings ranged from a high of 5 to a low score of I with respect to the usefulness of the strategy and its specific components (e.g., using diagrams, mapping information onto diagrams). Additionally, students were asked whether they would continue to use the strategy and recommend it to other students. The two open-ended questions required students to report what they liked the most and least about solving word problems.
The classroom teacher also completed a questionnaire that contained both Likert-type and open-ended questions. Her questionnaire was designed to assess her overall level of satisfaction with the schema-based instruction in terms of its effectiveness for her students, ease of use, efficiency, flexibility, and generalizability. Additionally, the teacher was asked whether she would continue to use the strategy and recommend it to other teachers. The two open-ended questions asked the teacher to list aspects of the strategy that were most beneficial and note any changes she believed would enhance the strategy’s effectiveness.
Strategy usage.
All completed test worksheets were examined to determine the extent to which students effectively used the schema strategy. We determined whether students used the strategy to (a) identify the problem type (problem schemata) by drawing a picture and (b) develop a plan (action schemata) by setting up the mathematics sentence(s) for one-step and multistep problems prior to solving (strategy knowledge) them.
Intervention materials
Materials included scripted lessons for teaching word problems, strategy diagram sheets, and numerous practice problems designed for this phase of the study. In addition, story situations that did not involve any unknown information were developed for use in teaching students to discern the two different problem types (vary and multiplicative comparison). Worksheets with story situations included problem schemata diagrams (e.g., Marshall, 1995; Marshall, Barthuli, Brewer, & Rose, 1989). Additional materials included note sheets with key features of the two problem types.
| Problem type | Example |
|---|---|
| Vary/vary |
|
| Vary/MC |
|
| Change/vary |
|
| Change/vary/MC |
|
Note. MC = multiplicative comparison. |
|
Teacher training
Before the study began, the first and second authors met with the teacher to discuss the training procedures for experimental conditions. The teacher was provided with instructional materials (i.e., scripts, worksheets, and tests) and participated in a 1-hr training session. In this session, the teacher was informed that during baseline, generalization, and maintenance conditions, she could read the problems to students or praise them for their efforts, but that she could not assist the students in any other way during tests. Because the teacher was experienced and knew how to implement explicit instructional strategies, intervention training consisted of going over the key elements of the teaching scripts. The teacher was provided with the opportunity to read the scripts and clarify questions prior to implementing the first intervention lesson. In addition, treatment integrity was measured using a checklist (see the Treatment Integrity section).
Experimental design
A multiple-probe-across-participants design was used to evaluate the effects of the schema strategy on the mathematical word-problem-solving performance of four middle school students who were low-performing in math. A functional relationship between the intervention and word-problem solving was demonstrated because each student’s performance remained stable or displayed a contratherapeutic trend during the baseline condition and increased only after the intervention was applied. Given that the fourth student was identified late in the school year for special education services, the baseline data for this student are limited.
The experimental phases included baseline, instruction, response generalization, and maintenance. During baseline, each test assessed word-problem-solving performance on both problem types. In addition, generalization to novel problems was assessed once during baseline. Next, participants were introduced to the strategy one at a time, beginning with training in problem schemata. Once mastery (100% correct for two sessions) in identifying and representing problem schemata for both problem types was achieved, the schemabased strategy was introduced to teach word problems. Again, a criterion of 100% correct for two sessions in solving each of vary and multiplicative comparison problem types was required. Following instruction in each problem type, word-problem-solving performance was assessed on both problem types. The intervention with the second student was then implemented. The same sequence continued with the third and fourth students. Each student also completed a generalization probe after completion of the intervention. The design ended with a maintenance condition for all students.
Procedure
Baseline testing.
Baseline testing was conducted using the word-problem test for each participant. A different version of the test was administered for each session. Participants were given as much time as needed and were instructed to do their best, show all their work in the space provided, and write the complete answer on the line at the end of the problem. None of the participants required more than 20 minutes to complete each test. Participants were encouraged to call on the teacher if they had difficulty reading a word; on no occasion did any of the students require assistance with reading. During baseline testing, the special education classroom teacher provided praise only for completing the tests.
Intervention.
All instructional procedures were implemented using scripted lessons. Each instructional session lasted 35 to 40 min. In this study, schema-training procedures were criterion based and required students to obtain 100% correct on two sessions prior to progressing to the next problem type. Instructional components included explicit strategy modeling, interactive discussion, guided practice, monitoring and corrective feedback, and independent practice (Rosenshine, 1986; Rosenshine & Stevens, 1984). Schema-based strategy instruction to solve vary and multiplicative comparison problems was presented in two phases (problem schemata identification and problem solution). The easier problem type, vary, was introduced first, followed by multiplicative comparison. The latter problem type is deemed to be more difficult cult because it can involve both a prealgebra relation and an arithmetic relation (Marshall et al., 1989). On average, problem schemata identification and problem solution training lasted 6 and 12 sessions, respectively.
Problem schemata identification condition.
Instruction began with the problem schemata identification training, a prerequisite to understanding and organizing information for later problem solution. In this phase, students were provided with worksheets that included story situations only with problem schemata diagrams (see Figure 1); the diagrams were used for instruction and student work. The teacher demonstrated the problem schemata analysis using several examples. Examples of story situations for the two problem types were presented to help students recognize and understand the key features and relations of the problem schemata (Marshall, 1995). For example, the problem analysis for the vary story situation “A car travels 25 miles on a gallon of gas; it can travel 75 miles on 3 gallons of gas” had students identify several key features. They included (a) a constant per unit (e.g., I gallon of gas) or unit ratio value that was explicitly stated or implied by the story wording; (b) four quantities (i.e., 25, 1, 75, and 3), two of which were subject-units and two of which were object-units; (c) the association (e.g., goes) that paired each subject-object (gallon of gas and miles) unit; and (d) an if-then relationship (“If a car on I gallon of gas goes 25 miles, then a car on 3 gallons of gas goes 75 miles”). Constraints of the vary relation required that each of the subject-units’ and object-units’ identities be expressed using the same measures (gallon of gas and miles) and that the associations (goes) between the if-then statements be identical.
In the multiplicative comparison story situation”Linda answered 5 problems correctly, and Cindy correctly answered 15 problems. Linda correctly answered 1/3 as many problems as Cindy”—instruction focused on the presence of compared (problems correctly answered by Linda) and referent (problems correctly answered by Cindy) sets and their relative sizes (5 and 15). In addition, instruction emphasized the part-whole relationship. That is, the comparison or relational statement (i.e., Linda correctly answered 1/3 as many problems as Cindy) described one set as the multiple or part (1/3) of the other set. Identifying the comparison statement also helped the student readily recognize the compared and referent sets.
In general, the problem schemata instruction employed teacher-led demonstration and modeling, along with frequent student exchanges, to identify critical elements of problem schemata and map them onto the relevant schemata diagrams. Before mapping the information, the student was taught to underline sentences in story situations that indicated the unit set and relational statement in vary and multiplicative comparison story situations, respectively. The underlining served as a memory aid to help students identify and retrieve the essential elements in the problem. At the end of the training session, students independently completed a worksheet containing six story situations by reading the story situation and mapping the information onto the diagrams. Initially, worksheets included story situations of a single problem type (vary or multiplicative comparison). When students learned to correctly identify and map the two problem types, worksheets, included both story situations. Problem schemata identification instruction continued until the student was able to distinguish between the two different problem schemata.
Problem solution condition.
This phase began with a review of each problem schemata, but in the context of word problems rather than story situations. Teacher-led demonstrations and a facilitative questioning procedure allowed students to identify and map critical elements of the specific problem onto the schemata diagrams. Additionally, the strategy mapping instruction required flagging the missing element in the problem with a question mark. During this phase, any misconceptions about problem schema constraints were consistently addressed with explicit feedback and additional modeling, and instruction was provided when needed.
Instruction then proceeded to representing the given information in the diagram as a mathematical sentence prior to solving it. For example, using the completed vary schema diagram for the first problem in Table 2, the student sets up the math sentence as follows:
9 computers |
27 students | |
–––––- |
= |
–––––- |
1 computer |
?students |
Next, the student was taught to use the equivalent fraction rule (i.e., multiply or divide the top and bottom numbers by the same nonzero number to get an equivalent fraction) to solve the problem. In some instances, instruction had to be broken down into more steps to apply the equivalent fraction rule. For example, following scaffolded instruction using a simple problem (6 x ? = 12), students were questioned as follows: 9 x ? = 27; therefore, 1 x 3 = ? Finally, instruction required the students to reason whether the answer made sense and to check their answers using cross multiplication. In contrast, instruction for the multiplicative comparison problem focused on first identifying whether the unknown represented the compared or referent set. Students were taught to examine the relational statement in the problem to identify whether the compared or referent set, was a multiple or part of the other set, and then multiply or divide accordingly. In this investigation, to solve for the unknown in the first multiplicative comparison problem presented in Table 2, we had students use the prealgebra. relation. That is, they set up the math sentence as follows based on the information given in the problem: 1/4 x ? = 20. Finally, they solved for ? (i.e., ? = 20 x 4) to complete the problem. As with vary problems, the student checked the reasonableness of the answer.
To assist students in remembering the key features of each problem type, a note sheet with the essential elements by problem type was provided. The notesheet was used as a scaffold while the students completed problems during practice trials, until students could independently verbalize them. At the end of each session, students completed a worksheet containing word problems. Initially, students worked on only one type of problem; later, when they had completed instruction in the use of the strategy steps for both problem types, worksheet with word problems that included both problem types were presented for independent practice. Upon completion, the worksheet was checked and appropriate feedback provided.
It must be noted that the diagrams were eventually faded and the final independent review of both problem types required students to complete the word problems without the aid of diagrams. Upon completion of instruction in and mastery of each problem type, the student completed six-item word-problem tests similar to those used in baseline using the same procedures.
Vary. A car travels 25 miles on a gallon of gas. It can travel 75 miles on 3 gallons of gas.
View sample diagramsMultiplicative comparison. Linda answered 5 problems correctly, and Cindy correctly answered 15 problems. Linda correctly answered 1/3 as many problems as Cindy.
View sample diagramsNote. From Schemas in Problem Solving (p. 135) by S. P. Marshall, 1995, New York: Cambridge University Press. Copyright 1995 by Cambridge University Press. Adapted with the permission of Cambridge University Press.
Generalization and maintenance.
Students completed a generalization test of novel word problems before and after the intervention. To assess maintenance of the strategy effects, all students were administered tests at different points in time (e.g., at the end of Weeks 4, 8, 9, and 10 following instruction for Sara). Procedures for administering the generalization and maintenance tests were identical to those in the baseline and postinstructional test conditions.
Observation system and interobserver agreement
Word-problem tests.
The classroom teacher and the second author conducted interobserver agreement checks on the word-problem tests. The classroom teacher rated each test using answer keys, while the second author independently scored all tests. Agreement was defined as both raters’ recording that the same problem was answered correctly or incorrectly. Interscorer agreement was computed by dividing the number of agreements by the number of agreements plus disagreements and then multiplying by 100%. Interscorer agreement was 100% across students for all experimental phases.
Treatment integrity.
During the instructional sessions in which each of the strategy steps (problem schemata and problem solution) was taught to a participant, the second author and a graduate student in school psychology independently collected treatment integrity data for approximately 30% of the lessons. The observers were given identical observer checklists on which were listed 10 critical parts of the lesson (e.g., providing clear instructions, having students read word problems aloud). The observers independently completed checklists by marking the parts of the lesson implemented. The mean interobserver agreement across the lessons observed was 100%.
Results
Figure 2 presents the number correct of word problems during the baseline, intervention, postintervention, and maintenance conditions. In general, results indicate improved word-problem-solving performance for all four participants following schema instruction on one-step multiplication and division word problems. The participants also maintained their word-problem-solving performance following termination of the intervention. Figure 3 presents the percentage correct of word problems during the pretreatment and posttreatment generalization conditions. Again, high levels of performance (100%) on generalization word problems after instruction were evident for all four participants. The following section describes the results of word-problem-solving performance for the participants.
Baseline performance
During baseline, the mean number of correct word problems for participants was 41%. Overall, the average performance on word-problem-solving tests for Sara, Tony, Percy, and Andy was 2.7, 3.0, 2.2, and 1.8, respectively. Sara’s performance during baseline was relatively stable. Although her performance improved slightly from Test 1 (33% correct) to Test 2 (50% correct) and remained the same on Test 3, the low scores (mean correct = 44%) indicated a need for intervention. In contrast, Tony’s performance was extremely variable, and a decreasing trend was evident during baseline. Given Tony’s inconsistent and low performance (mean correct = 50%), continued difficulty in mathematics classes as reported by his teacher, and the need to prepare for school-wide standardized testing, a decision was made to begin the intervention. Similar to Sara’s, Percy’s performance was stable and low (mean correct = 37%) during baseline. His scores increased from Test 1 to Test 2 and then plateaued, plausibly due to practice effects. Baseline scores for Andy were relatively stable and low (mean correct = 29%). His highest and lowest scores during baseline were 50% and 0%, respectively. Because his teacher reported that Andy’s overall performance in mathematics was well below grade level, it was deemed important to begin instruction on multiplication and division word problems.
Intervention
Problem schemata performance.
Instruction in identifying and describing the features of problem schemata resulted in Sara’s improving her independent performance on vary story situations from 67% to 100%. Her performance on multiplicative comparison story situations was 100%. Sara readily acquired the problem schemata for vary and multiplicative comparison story situations in a total of six sessions. Problem schemata intervention for Tony began when Sara began to show an increasing trend during the problem solution intervention phase. Tony’s independent performance on each problem type was 100%, and he acquired the problem schemata for both problem types in seven sessions. The remaining two participants, Percy and Andy, also learned to correctly identify and describe the features of the two problem schemata with 100% accuracy. This phase lasted five sessions each for Percy and Andy.
Problem solution performance.
Following schema strategy instruction to solve each problem type in isolation, the students completed word-problem tests that included both problem types. On average, it took Sara, Tony, Percy, and Andy 13, 13, 11, and 11 sessions, respectively, to acquire word-problem-solving skills.
Level 1 of problem solution-vary.
During the vary intervention phase, Sara scored 100% on all independent work following teacher-led instruction. After instruction in vary problems, Sara’s average performance on tests that assessed both problem types was 58%. A decreasing trend in data was evident, and an examination of each test indicated that Sara correctly completed all vary problems and that the errors involved multiplicative comparison problem types only. Tony also completed vary problems on independent worksheets with 100% accuracy. His performance on tests following instruction in vary problems was 58%. Again, all of the errors involved multiplicative comparison problems.
Percy correctly completed 100% of the vary problems on independent worksheets following teacher-led instruction. Although he scored an average of 92% correct on word-problem-solving tests containing both problem types, his performance on vary problems was 100% correct. The one problem that Percy completed incorrectly was a multiplicative comparison problem. While Andy’s mean performance on tests following instruction in vary problems was 75% correct, he scored 100% on all vary problems. Overall, instruction in solving vary problems was not sufficient for solving multiplicative comparison word problems for Sara and Tony, whereas the other two participants were able to generalize the use of schema diagrams and word-problem-solving skill to solve the untaught problem type.
Level 2 of problem solution-multiplicative comparison.
In general, all participants demonstrated the ability to independently discriminate between vary and multiplicative comparison problem types and use the correct schema diagram with 100% accuracy on independent worksheets and tests following teacher-led instruction.
Maintenance
Maintenance tests given 4 and 8 weeks after the intervention indicated that Sara maintained her high level of performance (100%). A 9-week follow-up check showed that although Sara’s performance dropped slightly (83%), it was much higher than her baseline performance (mean correct = 44%). A 10-week follow-up check indicated an increase in performance to 100%. Tony completed maintenance tests at 1, 2, 31/2, 5, and 51/2 weeks following the intervention. Similar to Sara’s, his performance dropped to 83% on the third probe. However, Tony scored 100% on a follow-up check administered 5 and 51/2 weeks later, indicating maintenance of word-problem-solving skill. On I - and 2 1/2-week follow-up checks, Percy scored 100%, demonstrating maintenance of the learned information. Maintenance data were not available for Andy due to the ending of the school year.
Figure 2. The number of word problems correct during baseline, intervention, and maintenance conditions for the four participants.
View Figure 2Note. T1 = baseline tests; PS = problem schemata training; T2 = tests following instruction on vary problems; T3 = tests following instruction on multiplicative comparison problems; 11 = instruction on vary problems; 12 = instruction on multiplicative comparison problems.
Generalization
Pretreatment generalization scores across participants were low (mean correct = 37%). The mean scores for Sara, Tony, Percy, and Andy were 44%,39%,44%, and 28%, respectively. Following the intervention, all participants scored 100%, indicating that they were able to generalize the strategy to solve novel word problems. An examination of one-step vary problems on the generalization test indicated that with the exception of Tony, the participants’ pretreatment performance was high: Sara, Percy, and Andy, scored 67%, 100%, and 100%, respectively. Posttreatment performance for each of the four participants was 100%. Tony’s performance on vary problems increased from 33% during pretreatment to 100% during posttreatment, indicating generalization of the word-problem-solving skill.
FIGURE 3. The percentage correct of word problems during the pretreatment and posttreatment generalization conditions by the four participants.
View Figure 3
Although pretreatment generalization scores on both multiplicative comparison and multistep problems were low (less than 40%) for all four participants, their performance on these problem types substantially improved (100%) during posttreatment. It must be noted that multistep problems were not directly taught in the study, yet students were able to complete them with 100% accuracy after the intervention, indicating that the strategy usage generalized not only to novel one-step problems but also to multistep problems.
Strategy Use
Table 4 presents the percentage of time students displayed overt use of the strategy steps (i.e., drawing diagrams and writing the number sentence) when completing the tests during each phase of the study. Strategy steps that entailed writing the operation and doing the computation were not examined, as students had to do this to complete each problem.
Visual inspection of the data in Table 4 reveals that other than Sara, none of the participants drew diagrams to represent the information in the word problems during baseline. However, the percentage of time that Sara drew diagrams during baseline was low (20%). In contrast, all participants wrote the number sentence during baseline. The mean percentage for writing the number sentence for Sara, Tony, Percy, and Andy was 29, 100, 58, and 33, respectively. On tests following instruction on vary and multiplicative comparison word problems, all students consistently increased their use of diagrams. Both Sara and Tony continued to draw diagrams during the maintenance phase (100%). whereas Percy’s use of diagrams decreased from 100% following the intervention to 75% during maintenance (but was still higher than during baseline [58%]). Maintenance data were not available for Andy.
Pretreatment generalization data indicate that with the exception of Percy, the participants did not use diagrams prior to instruction. After the intervention, Sara used diagrams for a majority of the problems (83%). The other participants correctly represented word problems by drawing diagrams 100% of the time on the posttreatment generalization test.
In general, students used diagrams more when solving vary than when solving multiplicative comparison problems. When students attempted to draw and map diagrams for multiplicative comparison problems following instruction on vary problems, it seemed that some did not generalize the use of diagramming learned in solving vary problems. Although Sara and Tony both developed diagrams for the untaught problems, their representations were not consistently correct. For example, they attempted to use the vary diagram to represent the multiplication comparison problem, which seemed to interfere with correctly solving the problem. Once students were instructed on multiplicative comparison problems, the frequency of accurately drawing and mapping diagrams increased.
When students’ worksheets were examined for the strategy step of writing the number sentence, it appeared that they were more likely to write the number sentence than to draw diagrams during baseline, which was further maintained during and following instruction. For Tony, the mean percentage of writing the number sentence was 100% for all phases (baseline, instruction, maintenance, and generalization) of the study. However, about half of his number sentences written during baseline (52%) and following instruction on vary problems (50%) were incorrect. Sara, Percy, and Andy consistently showed an increase in writing the number sentence from baseline to postinstruction on vary and multiplicative comparison problems. In addition, Sara and Percy maintained (100%) the strategy usage during the maintenance phase. (Maintenance data were unavailable for Andy.) Sara, Percy, and Andy also demonstrated an increase of 100%, 100%, 41 %, respectively, in writing the number sentence from pretreatment to posttreatment during generalization.
Strategy Questionnaire Interviews
Results of the strategy questionnaire indicated that all students found the strategy in general, and drawing and mapping information onto diagrams in particular, to be most helpful in understanding and solving the word problems (M = 5.0). The overall mean ratings for strategy satisfaction (i.e., continue to use the strategy and recommend the strategy) were 5, 4.5, 4.2, and 5, for Sara, Tony, Percy, and Andy, respectively.
Student comments about the strategy indicated that they liked solving the word problems. Their answers varied from “It made it easier for me to solve problems” to “It helps me in every day life,” and “Helped me learn something I could never learn before.” Student responses about what they least liked included “Nothing,” “Too many word problems drive me crazy,” “It can get a bit confusing,” and “Doing the math.”
The teacher ratings for strategy effectiveness, efficiency, ease of use, flexibility, application, and generalizability were 5, 49 5, 5, 5, and 5, respectively. Regarding efficiency, which received a score of 4, the teacher commented that “it was worth the investment.” The teacher responded that the strategy was helpful because it was visual and because explicit application of key components of word problems allowed for student self-instruction. She noted that the “systematic practice led to strong independence over time” for her students. When asked to recommend ways to facilitate word-problem solving for students with disabilities included in general education classrooms, she noted that the operational procedures for the vary problem solution should be based on students’ proficiency level in mathematics. For example, some students were able to readily follow the algebraic process, which was used to check the final answer derived using the equivalent fraction rule, whereas for others (e.g., Sara and Tony) it was more difficult. These students needed extensive practice to be able to use the algebraic process, but once they learned it, they used it more frequently to solve the problem.
| Generalization | ||||||
|---|---|---|---|---|---|---|
| Condition | Baseline | Level 1(a) | Level 2(b) | Maintenance | Pretreatment | Posttreatment |
| Sara | ||||||
| Draw diagrams | 20 | 92 | 100 | 100 | 0 | 83 |
| Write number sentence | 22 | 58 | 100 | 96 | 0 | 100 |
| Tony | ||||||
| Draw diagrams | 0 | 42 | 100 | 100 | 0 | 100 |
| Write number sentence | 100 | 100 | 100 | 100 | 100 | 100 |
| Percy | ||||||
| Draw diagrams | 0 | 100 | 100 | 75 | 33 | 100 |
| Write number sentence | 58 | 100 | 100 | 100 | 0 | 100 |
| Andy | ||||||
| Draw diagrams | 0 | 50 | 100 | na | 0 | 100 |
| Write number sentence | 33 | 83 | 100 | na | 17 | 58 |
Note. -Percentages were computed by dividing the number of times the strategy step was written by the total number of possible times; na = not available. (a) Tests completed following instruction in using the strategy with vary word problems. (b) Tests completed following instruction in using the strategy with multiplicative word problems. |
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Instructors’ notes and observations
All students stated that they enjoyed participating in the study and believed that learning the strategy was helpful. Sara commented that she would be able to transfer the learned information to real-world activities. Both Sara and Tony demonstrated generalization of the learned skill to complete unfamiliar word problems on a school-wide standardized state test administered during the study. The teacher reported that Tony repeatedly classified word problems as either vary or multiplicative comparison on several occasions during the test by stating, “Hey, this is a vary problem, I can do this.” During the study, all four participants were highly cooperative and engaged in appropriate behavior. For example, Andy, a student diagnosed with attention- deficit/hyperactivity disorder, was physically active during instruction, and it was difficult for him to remain on task, but he managed to complete all work with minimal prompting. Students in the study were functioning at different levels in terms of computational fluency, and the varied prompts helped them to successfully solve word problems. For example, at the onset of the study, Tony used a multiplication chart as a scaffold to assist in computing multiplication problems. However, by the end of the study, he seemed to gain confidence and used the chart less and less. In general, students expressed surprise at their ability to accurately complete the word problems before the end of the study.
Discussion
Although results of this exploratory study are encouraging, the nature of the single-subject design used in this investigation indicates that caution is called for in interpreting the findings. Results of the study seem to indicate that middle school students with learning disabilities who are low-performing in mathematics can be taught to effectively apply schema-based instruction to correctly solve multiplication and division word problems. In this study, the four participants’ performance substantially improved after they received instruction. Replication of the effects of the schema-based instruction occurred across participants, extending the findings of previous research on the effectiveness of schema-based instruction in teaching mathematical word-problem solving (Jitendra et al., 1998; Jitendra & Hoff, 1996; Jitendra et al., 1999; Marshall, 1995). It should be noted that each of the participants experienced success by acquiring the word-problem-solving skill in a reasonable amount of time (12 sessions). They were able to discuss key features of each problem type, verbally explain what the word problem was asking, and draw a diagram of the relationship present in the problem. This result suggests that practice with the schema strategy helped students to develop a conceptual understanding of the core concepts, which is considered important to problem solving (Baroody & Hume, 1991; Cawley & Parmar, 1992; Woodward & Montague, 2000). In addition, the positive benefits may be attributed to the personalized contexts during acquisition learning. It may be the case that, as in the study by Davis-Dorsey, Ross, and Morrison (1991). personalization made the problems more motivating, made it easier to construct a meaningful conceptual representation to connect the problem information and solution strategies, and made successful encoding and retrieval more likely.
The schema-based instruction was also associated with maintenance of the high level of postinstructional. performance during follow-up probes several weeks after the intervention was terminated. This result supports and extends the findings described by Jitendra and Hoff (1996) and Jitendra et al. (1999). It is encouraging to note that for one of the students (Sara), the effects of the schema strategy were maintained for 10 weeks-longer than that reported in the literature. This is an interesting finding given that students with learning disabilities often experience difficulty with long-term retention of skills.
Also encouraging is that generalization to novel (one-step vary and multiplicative comparison) and untrained word problems (i.e., multistep) occurred for all students following instruction in solving one-step multiplication and division problems. This is an exciting finding given the severity of the students learning difficulties. Furthermore, because of the unique design of this study, whereby students were taught to apply the strategy to one problem type at a time, it was possible to determine if learning to use the strategy on one problem type (vary) generalized to the other problem type (multiplicative comparison). For two students (Percy and Andy), a generalized effect was seen on multiplicative comparison problems after learning to apply the strategy first to vary word problems. Similar results were found in the Jitendra et al. (1999) study for addition and subtraction problems, and in the study by Hutchinson (1993) , for algebra word problems by students with learning disabilities. One plausible explanation for the generalized effects on performance in solving untrained word problems (e.g., multistep) not targeted for instruction in this investigation is that schema-based instruction, with its emphasis on conceptual understanding, allowed students to successfully encode and apply the learned schemata to represent and solve multistep problems, which is consistent with Butterfield and Nelson’s (1989) cognitive theory of elements and mechanisms of transfer.
The participants seemed to be more enthusiastic about solving word problems during and after the implementation of the instruction than during baseline. The students’ and special education teacher’s positive feelings toward the strategy and teaching procedures seemed to contribute to the students’ improved performance and task behavior, as in several previous investigations (e.g., Case, Harris, & Graham, 1992; Jitendra, et al., 1999). The teacher indicated that participating students were enthused about the strategy and spontaneously applied it when completing word problems on the standardized state test. As noted by Wood, Frank, and Wacker (1998), “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). One participant in the present study (Sara) commented that the strategy was easy to use and applicable to everyday life. Strategies that are connected with real-world situations are important in, promoting skill acquisition and generalization (Bottge, 1999). Furthermore, the participating teacher believed that the strategy was helpful as an introduction to prealgebra, which is required in most college curricula and is often a difficult area for college students with learning disabilities (Maccini & Hughes, 2000). Finally, the social validity of the schemabased instructional approach was enhanced because no external investigators were present in the classroom during this investigation.
Several limitations of this study call for caution in interpreting the findings. First, the small number of participants limits the generalizability of results to other student populations (e.g., students with behavior disorders). Second, the range of problems addressed in this study was limited to vary and multiplicative comparison problem types; future research should examine how students would do on a varied problem set. A third limitation is that instruction occurred individually, which can be time consuming and personnel intensive. Future research should examine whether the effects found in the present study generalize to small instructional groups. However, recent research findings (Jitendra et al., 1998) provide preliminary evidence regarding the strategy’s applicability to larger groups of students. Fourth, given the trend toward inclusionary practices, future research should address transferability of this strategy to general education classrooms. Finally, the use of one teacher limits generalizability. It would be worth exploring how effectively other teachers would use this intervention.
In addition, one of the major concerns in this investigation relates to the appropriateness of the sample selection. The four participants in this study had been diagnosed as having a learning disability and were receiving services in a learning support classroom for mathematics. However, Sara’s and Tony’s scores in mathematical reasoning were within 1 standard deviation of the mean, thereby raising the issue as to whether they were truly mathematics disabled. Also, employing the discrepancy criterion would mean that only Percy qualified as being learning disabled in mathematics. Perhaps the team’s decision to identify the four participants as needing special education services in mathematics was a function of these students’ performance in relation to others’ in this high-functioning school district. For example, the team must consider whether, with appropriate modifications and support in the general education classroom, the student evidences academic problems. Both Percy and Andy received mathematics instruction in the learning support classroom immediately upon being diagnosed as learning disabled. In contrast, Sara and Tony were not receiving special education services in mathematics when they were first diagnosed as learning disabled. However, they were eventually placed in the learning support classroom for mathematics instruction because they were not able to keep up with their peers in the general education classroom. In sum, the variation in criteria used by schools and researchers presents problems in terms of accurately identifying the sample, a common struggle that researchers encounter when conducting applied research in the classroom.
It is also the case that the single-subject design employed in this investigation does not help clarify whether the study findings are attributable to the specific schema-based nature of the instruction or to the generally carefully designed instruction and increased focus on the two problem types. Therefore, further research is needed to determine whether schema-based instruction is necessary to promote these outcomes. This would entail using a group-design study to compare and evaluate the relative efficacy and cost efficiency of the schema diagram strategy, intervention procedures that employ manipulatives, and other empirically validated strategies (e.g., cognitive-metacognitive strategy) described in the literature (Jitendra & Xin, 1997). It must be noted that the tests and worksheets employed in this investigation were designed to match students’ interests, based on a list of preferred items provided by the teacher. One suggested extension of the present research would involve employing tasks that reflect the varied situations that students typically encounter in real life. For example, asking students to calculate percentages on sale items or figure out how much tip to give a waitress based on the cost of a meal would be extremely valuable in teaching students functional mathematical skills. Therefore, using schema-based instruction to teach functional academics is an area to further explore, because even though students with learning disabilities have the ability to complete community routines (e.g., paying rent, shopping for groceries and clothing), they often struggle with parts of those routines (e.g., money usage, calculation, budgeting; Patton et al., 1997).
Implications for practice
The findings from this study have several implications for practice. First, the schema-based intervention, with its emphasis on conceptual understanding, helped students with learning disabilities not only acquire word-problem-solving skills but also maintain the taught skills. Therefore, results of the study highlight the effectiveness of strategy instruction for addressing mathematical difficulties evidenced by students with learning disabilities (Montague, 1995, 1997b). Second, the results of this study suggest that teaching students to identify the relationships present in each word problem promotes generalization to other, untaught skills (e.g., multistep problems). Students with learning disabilities should receive instruction that teaches them to understand the key features of problems prior to solving them. Third, the effectiveness of the strategy when implemented by the classroom teacher may indicate the importance of researchers’ collaborating with practitioners to adapt instruction to meet students’ individual needs. Involving the classroom teacher in the implementation of this study was important because the teacher is now more likely to invest effort in continuing to use a strategy that had beneficial effects for her students.
Asha Jitendra and Caroline M. DiPipi, Lehigh University Nora Perron-Jones, Salisbury School District
The Journal of Special Education Vol. 36/NO. 1/2002/pp. 23-38
The Journal of Special Education Vol. 36/NO. 1/2002/pp. 23-38