Improving the Fraction Word Problem Solving of Students With Mathematics Learning Disabilities

Abstract

Students with mathematics learning disabilities (MLD) have a weak understanding of fraction concepts and skills, which are foundations of algebra. Such students might benefit from computer-assisted instruction that utilizes evidence-based instructional components (cognitive strategies, feedback, virtual manipulatives). As a pilot study using a multiple baseline design, with multiple probes, this study investigated the effects of Fun Fraction, a multi-component computer-assisted instructional program, on word problem solving with fractions abilities of three middle school students with MLD. Mixed findings were observed on word problem solving performance from the baseline to intervention phase after students received instruction through Fun Fraction. The percentage of non-overlapping data ranged from 56% through 100%. Limitations, suggestions for future research, and educational implications are discussed.

Keywords

cognitive strategy instruction fractions students with mathematics learning disabilities virtual manipulatives word problem solving

The Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) includes specific standards that address problem solving, which is important for all students to master. Yet, middle school students with mathematics learning disabilities (MLD) often struggle with solving word problems (Montague, Enders, & Dietz, 2011; Montague, Krawec, Enders, & Dietz, 2014). Such difficulties stem from the demands of word problem solving, including the ability to read and understand situations represented with various schema (e.g., compare, change, combine problem types for addition and subtraction), translate the schema to mathematical equations, or perform a series of computations (Tolar et al., 2012). Schema is a framework or structure for representing information and is an important aspect of word problem solving (DiMaggio, 1997). Students can become more proficient problem solvers if they are able to model or represent the structure of the problem in a diagram or graphic organizer (see Jitendra, Star, Dupuis, & Rodriguez, 2013; van Garderen, Scheuermann, & Jackson, 2013). Consider a word problem about doubling a baking recipe. Students must understand such concepts as multiplicative reasoning, multiplication of fractions, and the relationship among fractions (Empson & Levi, 2011; Hunt & Empson, 2015; National Council of Teachers of Mathematics [NCTM], 2010). Thus, word problems with different schema can be challenging for students with MLD to solve because of the need to accurately interpret and represent the structure of the problem.

In addition to difficulties with word problem solving, findings have shown that students with MLD demonstrate difficulties associated with conceptual (i.e., comprehension of ideas associated with a mathematical topic and the interrelationships of these ideas) and procedural (i.e., performing a sequence of steps and operations) understanding for solving word problems with fractions (Hecht & Vagi, 2010; Mazzocco & Devlin, 2008; Newton, Willard, & Teufel, 2014). For students with MLD, improving their knowledge of fraction concepts is an area that requires intensive instructional support (Dougherty, Bryant, Bryant, Darrough, & Pfannenstiel, 2015). Coupled with word problem solving, increased instruction to improve conceptual and procedural understanding on fraction concepts and skills, which are typically challenging for students with MLD, is critical because of their importance as algebra-readiness mathematical ideas.

Fraction Knowledge of Older Students With Mathematics Difficulties

According to Siegler et al. (2010), “A high percentage of U.S. students lack conceptual understanding of fractions, even after studying fractions for several years; this, in turn, limits students’ ability to solve problems with fractions” (p. 6). For example, results from the National Assessment of Educational Progress (NAEP) indicated that 74% of fourth graders with and without disabilities were unable to solve a word problem involving fractions; whereas, 59% of eighth graders were unable to solve a multi-step problem involving fractions (National Center for Education Statistics [NCES], 2013). In a longitudinal study with 147 sixth-, seventh-, and eighth-grade students with MLD, mathematics difficulties (MD), and typical achievers (TA), Mazzocco and Devlin (2008) found students with MLD performed significantly less well on ranking proportions with fractions and decimals (e.g., ranking smallest to largest decimals [numerically displayed] and fractions [pictorially displayed]) than students with MD and who were TA. Findings also indicated that students with MLD exhibited significantly more problems in identifying fraction and decimal equivalence (e.g., .50 = 5/10) compared with students with MD and who were TA. Mazzocco and Devlin (2008) attributed poor conceptual understanding of rational numbers and weak number sense as a possible explanation for this low performance. An understanding of and the procedural knowledge for fractions are necessary for advanced mathematics, such as algebra, but the persistent deficits of students with MLD necessitate interventions for these students (Siegler et al., 2010).

Interventions for Teaching Word Problem Solving to Students With Mathematics Difficulties

Given the breadth of difficulties students with learning disabilities (LD) including MLD exhibit for word problem solving, empirically supported intensive interventions are necessary. Research has focused on several instructional approaches for teaching word problem solving to middle school students with LD. For example, Jitendra, DiPipi, and Perron-Jones (2002) taught students with LD how to translate word problem schema to mathematical equations for solving word problems, which involved multiplication and division calculations. Four middle school students received schema-based instruction; results showed that all four students’ performance substantially improved after they received the intervention.

Other researchers have employed cognitive and metacognitive strategies for solving word problems when teaching students with LD (Montague et al., 2011). Cognitive strategies provide various problem-solving steps such as “Read the problem. Highlight the key words. Solve the problem. Check your work” (Gersten et al., 2009, p. 1210). Metacognitive strategies focus on students’ self-awareness of “what am I doing” and “what have I done” (Montague et al., 2014, p. 470). Montague et al. (2011; Montague et al., 2014) found that middle school students with LD who received a combination of instruction in both cognitive and metacognitive strategies showed a significantly greater rate of growth (effect size = .88) in mathematics problem solving over the course of the school year than students who received typical, “business as usual” classroom instruction.

Yet another approach, the use of computer-assisted instruction (CAI), has been studied as a means of providing interventions to students with MLD because of the ability to easily adapt instructional delivery features (e.g., pacing, difficulty level, practice) for individual student needs (Ok & Bryant, 2015). Research findings have shown that CAI can be used effectively to support students struggling to learn mathematics calculations and word problem solving (Seo & Bryant, 2009). In particular, CAI that incorporates instructional variables (e.g., corrective feedback, systematically sequenced curriculum) was effective in improving students’ mathematics performance (Fede, Pierce, Matthews, & Wells, 2013; Leh & Jitendra, 2013; Seo & Bryant, 2012; Stultz, 2008). For example, Seo and Bryant (2012) taught students with MD how to solve addition and subtraction word problems using cognitive and metacognitive strategies and CAI. Four students with MD in Grades 2 and 3 improved their word problem solving accuracy on computer and paper/pencil-based tests during the intervention phase. Thus, instructional design features, which can be tailored to individual student needs, are a critical factor for the delivery of instruction (Nam & Smith-Jackson, 2007; Seo & Woo, 2010; Weng, Maeda, & Bouck, 2014).

Finally, the use of virtual manipulatives as part of computer applications is another useful means for teaching word problem solving to students with MLD (Shin & Bryant, 2015). Virtual manipulatives are interactive visual representations that allow students to translate the visual models into a problem-solving situation on a computer screen (Sayeski, 2008; Trespalacios, 2010). Researchers have recommended the use of visual representations and manipulatives to remedy students’ difficulty in solving word problems with fractions (Poch, van Garderen, & Scheuermann, 2015; Shin & Bryant, 2015; Siegler et al., 2010). For students with MLD who have difficulties mapping or diagramming the relationships among quantities in word problem structures (van Garderen et al., 2013), technology makes it possible to incorporate visual representations to aid in the solution process (Garofalo & Sharp, 2003). Reneau (2012) investigated the effects of the concrete-representational-abstract sequence with the use of virtual manipulatives on the solution of word problems with fractions. Results showed that five fifth graders with MD and LD made gains during the intervention phase compared with the baseline (percentage of all non-overlapping data = 91%) phase. Thus, the use of virtual manipulatives holds promise for assisting students in conceptualizing and solving word problems with fraction (Trespalacios, 2010).

Despite the evidence of positive effects from cognitive and metacognitive strategies and virtual manipulatives, no computer application incorporates these two components into teaching word problem solving to middle school students with MLD. To address this research need, Fun Fraction (http://funfraction.org) was designed and developed. All example questions in Fun Fraction were obtained from mathematics textbooks and articles focusing on teaching word problems (NCTM, 2010; Taber, 2002; Tsankova & Pjanic, 2009). The current study investigated the effects of a multi-component CAI program, Fun Fraction, on the word problem solving performance of middle school students with MLD. All participating students had Individualized Education Program (IEP) mathematics goals for word problems and fractions. The following research questions guided the study:

Research Question 1: To what extent does the use of the Fun Fraction program improve the accuracy of middle school students with MLD on word problem solving?

Research Question 2: How does students’ accuracy differ by word problem type?

Research Question 3: What are the perspectives of middle school students with MLD on the use of the cognitive and metacognitive strategies, which were incorporated into the Fun Fraction program, for solving word problems?

Research Question 4: What are the perspectives of middle school students with MLD on the use of Fun Fraction program for solving word problems?

Method

Participants

After obtaining University Institutional Review Board approval, parental consent, and student assent, the researchers screened potential participants who had been identified by a special education mathematics teacher in a local school. Five students returned signed parent consent forms and students signed their assent forms. Of the five students, four met the following selection criteria: (a) were enrolled in Grades 6 through 8, (b) were identified by their school or school district as having MLD, (c) had mathematics goals on their IEP (e.g., students will correctly solve word problems with fractions computations), and (d) demonstrated a deficit in the targeted skill by earning a cut score at or below 30% accuracy on a researcher-developed screening test (e.g., Seo & Bryant, 2012). In the middle of the study, one of the four students withdrew from the study due to excessive absences from school, leaving three students (Tiffany, John, and Alec) in Grades 6 through 8 as participants. Table 1 provides demographic, assessment, and screening test results for the three students.

Table 1.

Demographic and Testing Profiles of Participating Students.

Variable	Tiffany	John	Alec
Age (years)	15	15	13
Grade	7	8	6
Gender	Female	Male	Male
Ethnicity	Black	Hispanic/White	Black/White
Home language	English	English	English
WIAT-III (standard score)
Reading Comprehension	83	81	94
Math Problem Solving	72	60	56
Word Reading	77	78	95
Pseudoword Decoding	65	88	82
Numerical Operations	67	72	51
Spelling	63	75	82
Disability	MLD	MLD	MLD
Areas of difficulty	Mathematics	Mathematics	Mathematics
Screening test (percentage)	20	20	30

Note. WIAT-III = Wechsler Individual Achievement Test–Third Edition (Wechsler, 2009); MLD = mathematics learning disabilities.

Setting

Participants attended a private school for students with LD located in a major city in the south-central region of the United States. The school served approximately 40 students in Grades 5 through 12, of whom 87% were White, 4% were Hispanic, 4% were Black, 4% were American Indian/Alaska Native, and 2% were Asian or Pacific Islander. The study was carried out in the school’s conference room during the sixth period from 2:15 p.m. to 2:45 p.m. A Windows-based PC laptop was located in the room. A school wifi Internet connection allowed each student to access the web-based CAI program, Fun Fraction.

Experimental Design

A multiple baseline design, with multiple probes, across participants (Gast, Lloyd, & Ledford, 2014) was used to investigate the effects of Fun Fraction on the performance of middle school students with MLD on solving word problems. It is common to use multiple probe designs to assess mathematics interventions (e.g., Jitendra et al., 2002; Seo & Bryant, 2012). A multiple baseline design, with multiple probes, is recommended when a continuous baseline is unnecessary due to students’ acquisition of targeted skills through the introduction of repeated tests. Thus, intermittent probe data were used to measure student performances during the baseline and intervention phases (Gast et al., 2014). Staggered introduction of the intervention across the three participants controlled for internal validity threats such as maturation and testing. Word problem solving accuracy percentage (correct %) on the progress-monitoring probes was the dependent variable. A functional relation between the progress-monitoring probes and participants’ performance is demonstrated only if their problem-solving performance improves after instruction and if non-instructed participants’ achievement during baseline is stable and near the pre-intervention level (Carr, 2005). Students were randomly assigned to their order of participation to counterbalance a possible order effect.

Measures

Screening and progress-monitoring probes

Researcher-developed paper and pencil tests served as the screening and progress-monitoring probes. The probes measured the percentage of correct word problem solving problems with fractions and multiplication (dependent variable). The fraction content consisted of whole number and proper fraction multiplier and multiplicand. The multiplication factors were greater than 0 and equal to or less than 4.

The content and constructs were the same for both the screening and the progress-monitoring probes. All probes were presented as multiple-choice items and were not timed. There were 10 questions on the screening test and five questions for each progress-monitoring probe. All items were randomly selected from the Modeling, Guided 1 practice, and Guided 2 practice questions in Lessons 1 through 7 in Fun Fraction so that they could be equally distributed across Forms 1 through 13, keeping the forms similar in conceptual content across probes. Further, as proposed by Taber (2002), each progress-monitoring probe contained five word problems that included three word problem types: (a) Combination (e.g., 2/3 of a pie is on each tray. How many pies are on 4 trays?), (b) Partition (e.g., John has 4 pies. He gave 2/3 of them to Jane. How many pies did John give to Jane?), and (c) Multiplicative Comparison (e.g., John has 2/3 of a pie. Jane had 4 times as many pies as John. How many pies did Jane have?). The researcher scored all of the probes after the students completed each progress-monitoring probe (1 = correct answer, 0 = incorrect answer). A doctoral student independently scored the probes in each phase. Sixty-seven percent of the probe items were independently scored for Tiffany during the baseline, 50% for John, and 67% for Alec. Additionally, 50% of the intervention probe items were independently scored for Tiffany, 56% for John, and 56% for Alec. An interrater agreement of 100% was documented across the two phases for the three students.

Cognitive and metacognitive strategy questionnaire

A cognitive strategy questionnaire was administered that included two cognitive strategy-related questions: (a) What did you like best about learning and using the Read, Restate, Represent, and Answer strategies in Fun Fraction? and (b) What was the hardest thing to learn fractions using the Read, Restate, Represent, and Answer strategies in Fun Fraction? There were also two metacognitive strategy-related open-ended questions: (a) What did you like best about the information message (asking you if you have understood the problem and can now move forward) in the Fun Fraction program? and (b) What was hard for you regarding the information message? Screenshots of examples of cognitive and metacognitive strategies were provided in the questionnaire.

Social validity questionnaire

In studies applying single-case research designs, establishing social validity (estimation of the importance and satisfaction regarding a particular intervention) is a common and recommended quality indicator (Horner et al., 2005). In the present study, a social validity questionnaire was administered consisting of four 5-point Likert-type scale (5 = strongly agree to 1 = strongly disagree) questions and one open-ended question. Questions asked students whether the contents, Multiplication Facts, Vocabulary, Lessons, and Review in Fun Fraction helped them to better understand word problems (e.g., “Overall, lessons helped me to better understand word problem solving”).

Moreover, in an intervention study using a computer application, usability is a critical part of social validity as a means of judging the importance of the user experience (Carnine, 1997). In the current study, a usability questionnaire was used that included nine questions rated on a 5-point Likert-type scale (5 = strongly agree to 1 = strongly disagree). The questions were categorized into three aspects of design features (i.e., information, interface, interaction) and each consisted of three questions: (a) information (e.g., Can you easily identify tasks, activities, and content on the website?); (b) interface (e.g., Is the design of the website consistent in terms of colors, font types, and font sizes?); and (c) interaction (e.g., Can you change area models by moving or clicking your mouse?).

Word Problem Solving Instruction

The independent variable was the multi-component CAI, Fun Fraction program. The multi-components included cognitive and metacognitive strategies, virtual manipulatives, and explicit, systematic instruction. Fun Fraction consisted of seven lessons, a 2-min multiplication facts warm-up, and vocabulary words relevant to fractions (e.g., denominator, numerator). The vocabulary words were displayed in a graphic organizer.

All lessons sequentially integrated the following cognitive problem-solving steps: (a) Read (Read the problem carefully), (b) Restate (Click and highlight all important information), (c) Represent (Represent the problem using the area model), and (d) Answer (Write the equation and answer it). In each problem-solving step, the program purposefully guided students to employ metacognitive strategies through self-monitoring statements (I will read the problem. I will reread the problem if I don’t understand it). Students were expected to move to the next step after responding to a self-checking pop-up message in the program (Have I understood the problem and can now move forward?). In the Represent step, students implemented virtual manipulatives in the form of an area model to represent their word problem situation. To represent 2/5 yards of string, students divided 1 unit by 5 pieces and shaded 2/5 yards of string by shifting a slider on the screen. Figure 1 shows screenshots of the word problem solving steps using Fun Fraction.

Figure 1.

Screenshot of word problem solving process using Fun Fraction.

Further, each lesson consisted of explicit, systematic instructional features including Modeling, Guided 1 practice, and Guided 2 practice. Based on the evidence for the use of explicit, systematic instruction with students with MLD (Coyne, Kame’enui, & Carnine, 2011; Gersten et al., 2009), each CAI lesson provided a built-in video tutorial that explicitly demonstrated how to solve word problems using the cognitive and metacognitive strategies. Through the Guided Practices 1 and 2, students received computer-generated immediate feedback. Guided 1 practice activated all Read, Restate, Represent, and Answer steps as buttons; Guided 2 provided the problem-solving sequence as a flow chart image, reducing the amount of scaffolds in guiding the use of cognitive strategies. Additionally, using the built-in devices, students could access the read aloud functions and a multiplication facts table.

Procedures

A screening test was conducted to confirm each student’s eligibility for the study. Students who met the criteria for participation received a computer training session (20 min) with the primary investigator prior to the initiation of the intervention condition. Students worked independently on the computer (Windows-based PC laptop) each day. While the study was in progress, only the primary investigator and a student were present in the room. The role of the primary investigator was to observe and record each student’s treatment fidelity. When a student skipped a step in the procedure or hesitated to proceed in the use of Fun Fraction, the primary investigator prompted him or her to move to the next step.

This study consisted of two phases including the baseline and intervention. Students completed the probes during the baseline phase. No prompting or feedback was provided on the accuracy of each student’s work. The intervention was introduced after each student demonstrated low performance (“contra-therapeutic trend direction”; Gast et al., 2014, p. 258) during the baseline for at least three administrations of the probes. During the intervention, students worked through Modeling, Guided 1, and Guided 2 practice for both the cognitive (Read, Restate, Represent, Answer) and the metacognitive (self-instruction and self-monitoring) strategies for word problem solving. After completing all seven lessons, students reviewed the lessons for 3 days in the Review sessions. The mastery level was 80% correct or above on two out of three review days. This criterion level was decided based on the results of previous single-subject studies on teaching mathematics to students with LD (Maccini & Ruhl, 2000). The study took place for 13 weeks for all three students with the intervention phase occurring approximately twice per week as follows: 20 min for implementing the Fun Fraction program each session (10 sessions for Tiffany, nine sessions for John, nine sessions for Alec) and 10 min for completing progress-monitoring probes.

Treatment Fidelity

The primary investigator observed students’ fidelity of implementation of Fun Fraction and recorded students’ activities using a researcher-developed fidelity checklist. For all lessons, the fidelity checklist consisted of nine items that addressed Multiplication Facts practice, Modeling, Guided 1, and Guided 2 practice (e.g., goes to Modeling and watches the video, goes through all the procedures presented in Read strategy of Guided 1 practice). Regarding review activities, the fidelity checklist consisted of four items that addressed Multiplication Facts practice and three review questions. The primary investigator placed a check mark beside the corresponding box in the fidelity checklist to indicate whether the student followed the instructional procedures. Each student’s word problem solving session (10 sessions for Tiffany, nine sessions for John, and nine sessions for Alec) using Fun Fraction was observed during the intervention phase. Fidelity of implementation was 100% for Tiffany, 94% for John, and 97% for Alec.

Additionally, interobserver reliability of treatment fidelity was examined between the primary investigator and a doctoral student. The doctoral student was instructed for about 20 min in the overall structures of Fun Fraction and trained on how to use the checklists. The primary investigator and the doctoral student observed an individual student while he or she used Fun Fraction and recorded the student’s activities using the fidelity checklist. Interobserver reliability on fidelity of implementation was assessed during the intervention sessions as follows: 30% for Tiffany, 33% for John, and 33% for Alec. A 100% agreement between the primary investigator and a doctoral student was achieved.

Data Analysis

Three types of data analyses were conducted to answer the research questions. The first type of data analysis consisted of a visual inspection of the data to determine the extent to which a functional relation existed between the use of the Fun Fraction program and the percent correct on the word problem solving progress-monitoring probes and to show the strength or magnitude of that relationship (Kratochwill et al., 2013). In describing the functional relation, six features were considered: level, trend, variability, immediacy of effect, overlap, and consistency of the data pattern across similar phases (Kratochwill et al., 2013).

The second type of data analysis was employed to examine the effects of the intervention. Two non-overlap indices were computed. The percentage of non-overlapping data (PND) was obtained by calculating the percent of intervention phase data points that exceeded the single highest baseline phase (Scruggs, Mastropieri, & Castro, 1987). A PND below 50% was considered unreliable, 50% to 70% was questionable, 70% to 90% was fair, and 90% or higher was considered very effective (Scruggs et al., 1987). Additionally, Tau-U was calculated (Parker, Vannest, Davis, & Sauber, 2011). Tau-U is interpreted as “nonoverlap: percentage of the nonoverlap between phases or as trendedness: percent of data showing improvement between phases” (Parker et al., 2011, p. 291). Tau-U tests and controls for trends in the baseline phase; it also tests for differences in baseline and intervention phases (Parker et al., 2011). Tau-U, p values, and confidence intervals (CIs) are computed using a web-based Tau-U calculator (http://www.singlecaseresearch.org/calculators/tau-u).

The third type of data analysis was to identify and compare each student’s accuracy by word problem type. Students’ performances on the three types of word problems were analyzed on the three progress-monitoring probes (Forms 1 through 3) during the baseline phase and eight progress-monitoring probes (Forms 5 through 12) during the intervention phase.

Results

Effects of Fun Fraction on Word Problem Solving Accuracy

Table 2 summarizes data patterns across the baseline and intervention phases. In Figure 2, the accuracy percentage scores are depicted for each student for the baseline and intervention phases.

Table 2.

Summary of Data Patterns.

Participant	Baseline	Intervention	PND	Tau-U
Participant 1 (Tiffany)
M (SD)	13.33 (11.55)	42.00 (22.01)	70	0.70
Range	0–20	0–60
Trend	−10.00	0.61
Participant 2 (John)
M (SD)	30.00 (20.00)	62.22 (35.28)	56	0.56
Range	0–40	0–100
Trend	−2.86	10.33
Participant 3 (Alec)
M (SD)	13.33 (23.09)	82.22 (15.63)	100	1.00**
Range	0–40	60–100
Trend	0	−0.61

Note. PND = percentage of non-overlapping data.

p < .05.

Figure 2.

Accuracy percentage scores across baseline and intervention phases by students.

Tiffany

The level of accuracy for Tiffany was 13.33% during the baseline phase, and the data showed a downward trend (slope = −10.00). After the intervention was implemented, her level of word problem solving accuracy improved by 28.67% with an immediacy effect of 33.34%, and her performance had an upward trend (slope = .61). From the baseline to intervention phase, Tiffany’s data showed a 70% (PND) improvement trend, which was statistically non-significant (Tau-U = .70, CI₉₀ [0.05, 1.35], p = .08). Tiffany failed to reach the mastery level of at least 80% correct on two of the three review days.

John

The level of accuracy for John was 30.00% during the baseline phase, and the data showed a downward trend (slope = −2.86). After the intervention was implemented, his level of word problem solving accuracy improved by 32.22% with an immediacy effect of 0%. As shown in Figure 2, an upward trend on word problem solving performance (slope = 10.33) was particularly noticeable on Day 5 and remained at or above 80% correct. From the baseline to intervention phase, John’s data showed a 56% (PND) improvement trend, which was statistically non-significant (Tau-U = .56, CI₉₀ [−.04, 1.15], p = .12). John reached the mastery level by performing at 100% correct on two consecutive days of the three review days.

Alec

The level of accuracy for Alec was 13.33% during the baseline phase with a downward trend on Day 11. After the intervention was implemented, his level of word problem solving accuracy improved by 68.89% with an immediacy effect of 73.34%. The PND calculation was 100%, which was statistically significant (Tau-U = 1.00, CI₉₀ [.34, 1.66], p = .01). Alec also reached the mastery level by performing at 80% accuracy on two of the three review days.

Accuracy by Word Problem Types

Regarding students’ total accuracy performance by word problem type, during the baseline phase, students’ mean accuracy percentage indicated that Combination problems (M = 0%) were initially the most difficult, followed by Partition (M = 17%), and Multiplicative Comparison (M = 22%) problems. During the intervention phase, students’ mean accuracy percentage scores on the Partition problems (M = 53%) remained lower than the Multiplicative Comparison (61%) problems by 8 percentage points. Students showed the most improvement on Combination problems with an average of 56% correct. Improvements on Partition and Multiplicative Comparison problems were only 36% and 39%, respectively.

Cognitive and Metacognitive Strategy Questionnaire

The results of the open-ended questionnaire showed positive student responses regarding the use of the strategies in solving word problems. In particular, among the four cognitive strategies of Read, Restate, Represent, and Answer, students described the Represent strategy as the most engaging feature. For example, Tiffany and Alec indicated that Represent was the strategy they liked the best because they could change the shading of the virtual manipulatives on the screen, modeling the numerator and denominator of the fractions. By contrast, John considered Represent as the hardest cognitive strategy to learn in Fun Fraction. Specifically, he reported difficulty in shading the rectangular area model and matching the denominator buttons in Fun Fraction. Thus, for John, modeling the fraction quantity apparently proved conceptually challenging.

With regard to the metacognitive strategies, students reported that the strategies were easy to use and that they were comfortable using them. Specifically, students found the self-instructing and self-checking features useful. For example, they could follow task sequences through “I” statements embedded in Fun Fraction for each cognitive strategy and could monitor and check their responses through the information pop-up message shown at the end of each cognitive routine. John said, “It showed me what to do,” and Alec stated, “Yes. I could look back manually.” None of them indicated any difficulty in using metacognitive strategies. Tiffany said, “Nothing that I can think of.”

Social Validity Questionnaire

The social validity ratings showed that students had somewhat positive views of the helpfulness of the intervention for word problem solving (M = 2.67–3.67) in relation to their word problem solving performance. Specifically, the overall mean rating of 3.67 out of 5.00 about the lessons showed a somewhat positive response about using the Fun Fraction program to solve word problems. In particular, Tiffany (rating = 5.00) and Alec (rating = 4.00) showed higher preferences than John (rating = 2.00). On open-ended questions, Tiffany said, “I think this particular design model is very interesting for K–5. Because of the vocab and the design.” However, John said that he might not use Fun Fraction because he disliked the graph.

Regarding the results of the usability questionnaire, students rated the information (M = 4.00) and interaction (M = 4.11) design features higher than the interface feature (M = 3.00). Students expressed positive views on the use of Fun Fraction because the program allowed them to easily identify tasks and content on the website (M = 4.50). Finally, all three students highly agreed that Fun Fraction was engaging, allowing them to manipulate the rectangular area models and receive immediate feedback (M = 4.00).

Discussion

The purpose of this study was to examine the effects of a multi-component CAI, Fun Fraction program, on the word problem solving performance involving fractions and multiplication of middle school students with MLD. Although mixed, overall findings of the three students indicate some improvements in word problem solving abilities. Visual analyses revealed mixed findings where John’s data showed an upward trend after initial poor performance during intervention, and Tiffany and Alec’s data demonstrated an immediacy effect but variability in trend. Other analyses revealed positive findings for Tiffany and Alec (Tau-U = .70 and 1.00, respectively).

Because Fun Fraction is a multi-component CAI program, it is not possible to attribute performance to any one of the components; rather, the different instructional design components likely had varying effects. The cognitive and metacognitive strategies coupled with explicit, systematic instruction might have influenced students’ word problem solving performance because they were embedded in the program; thus, students had to work through the steps of each strategy to advance in the program. The use of a strategy component is supported by findings from other studies (Fede et al., 2013; Leh & Jitendra, 2013; Seo & Bryant, 2012), which demonstrated the effectiveness of using cognitive strategies with CAI as a method for teaching word problem solving.

The virtual manipulatives component in Fun Fraction might have influenced students’ word problem solving performance because they helped students to visually represent the concept of calculating fractions to solve word problems. This finding is similar to previous research on the use of virtual manipulatives as a tool to teach fractions to students with MLD (Reneau, 2012). Additionally, Yuan, Lee, and Wang (2010) proposed that virtual manipulatives can provide students with flexible representations that allow them to examine and test their mathematical ideas.

Finally, the different types of word problems likely affected the student’s outcome on word problem solving performance across the progress-monitoring probes even though all three types of problems were modeled and practiced. The average percentage of improved performance from baseline to intervention for Partition (M = 36%) and Multiplicative Comparison (M = 39%) problems was lower than the average improved performance on Combination (M = 56%) problems. This finding is consistent with previous research on fractions, indicating that the concept of partitioning (Lewis, 2014) and comparing or ranking fractional quantities is challenging for students with MLD (Mazzocco & Devlin, 2008). Research findings support the notion that a student’s understanding of fractions is developed over time through experiences with partitioning (i.e., dividing a whole into equal parts; Lewis, 2014). Similarly, understanding the magnitude of fractions for ranking and comparison purposes is developed throughout the elementary grades. Thus, students who have struggled for years with understanding fractions won’t have the fundamental knowledge that is critical to apply to word problem solving, algebra, and other mathematics topics.

Cognitive and Metacognitive Perspectives

The findings of this study showed students’ positive views about the cognitive and metacognitive strategies in Fun Fraction. In the present study, students liked Represent more than the other three cognitive strategies of Read, Restate, and Answer. Students were able to model the problem situation by manipulating the rectangular area model. The use of visual representations of mathematical ideas, in this case the use of the rectangular area model to show fraction quantities, can promote students’ comprehension of computational procedures for solving word problems (Siegler et al., 2010). Moreover, findings have shown the importance of using visual representations (e.g., schematic diagrams) in helping students with LD solve word problems (van Garderen et al., 2013). Additionally, the students’ positive perspective of using representations in CAI suggests a promising instructional design feature for word problem solving. Importantly, students enjoyed the self-instructing and self-checking metacognitive strategies in Fun Fraction. According to Montague et al. (2014), metacognitive strategies help problem solvers to monitor performance, correct what they have done, and regulate their use of strategies.

Social Validity Perspectives

On the usability questionnaire, students’ average rating regarding the interaction design was 4.11 out of 5, suggesting that they found the dynamic visual representations useful and interesting. This finding is in line with Reimer and Moyer’s (2005) results indicating that the majority of responses from students using virtual manipulatives to learn fractions were positive. As Sayeski (2008) noted, the power of virtual manipulatives lies in their interactivity, whereby students can see the results of their own actions and control the actions of visual representations. The present study demonstrates that the use of virtual manipulatives may be helpful to some degree, but further research in the use of virtual manipulatives as part of CAI interventions is warranted.

Limitations and Future Research

Several methodological limitations are associated with this study. First, the study was conducted with only three middle school students, thereby limiting its external validity. In future research, replication studies using the procedures of the current study should be conducted with a larger sample (Odom et al., 2005). Also, a large-scale study could be useful for better evaluating the effects of the CAI, Fun Fraction program. Second, the intervention was administered in a pullout instructional setting. To address the issue of more authentic settings, studies could be conducted in inclusion settings as supplemental, differentiated instruction or in a tutoring setting as part of a tiered instructional system for students with LD (Hunt, Valentine, Bryant, Pfannenstiel, & Bryant, 2016). Third, the intervention (Fun Fraction) included multiple components. No component analysis was conducted, which limits the ability to pinpoint one component of the program as most effective in promoting mathematics performance on the progress-monitoring probes. To examine the effects of each instructional component of the multi-component intervention, Fun Fraction, future research should conduct a component analysis. Fourth, the progress-monitoring probes were not piloted prior to the study. Future research should address the internal consistency reliability of the probes through test-retest procedures to establish alternative forms. Additional standardized tests, which include word problem solving with fractions and multiplication concepts and skills, should also be included.

Educational Implications

Despite the preliminary nature of this feasibility study of word problem solving through Fun Fraction, the findings point to the promising potential of using interactive computer applications in the classroom. Considering that students with MLD have difficulty understanding word problems and fractions, the cognitive and metacognitive strategies in Fun Fraction could help them by scaffolding the process of problem solving. Furthermore, the instructional design features of virtual manipulatives could help these students actively engage in and monitor their learning by controlling their actions through the computer application. As teachers evaluate software programs to supplement teacher-delivered mathematics instruction, instructional design features that include explicit, systematic procedures and visual representations (virtual manipulatives) should be considered as important variables of a multi-component, computer-mediated intervention.

Footnotes

Associate Editor: Alexandra Trout

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Carnine

(1997). Bridging the research-to-practice gap. Exceptional Children, 63, 513–521. Retrieved from http://www.questia.com/googleScholar.qst?docId=5000464388

Carr

J. E.

(2005). Recommendations for reporting multiple-baseline designs across participants. Behavioral Interventions, 20, 219–224. doi:10.1002/bin.191

Coyne

M. D.

Kame’enui

E. J.

Carnine

D. W.

(2011). Effective teaching strategies that accommodate diverse learners (4th ed.). Boston, MA: Pearson.

DiMaggio

(1997). Culture and cognition. Annual Review of Sociology, 23, 263–287. doi:10.1146/annurev.soc.23.1.263

Dougherty

Bryant

D. P.

Bryant

B. R.

Darrough

R. L.

Pfannenstiel

K. H.

(2015). Developing concepts and generalizations to build algebraic thinking: The reversibility, flexibility, and generalization approach. Intervention in School and Clinic, 50, 273–281. doi:10.1177/1053451214560892

Empson

S. B.

Levi

(2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.

Fede

J. L.

Pierce

M. E.

Matthews

W. J.

Wells

C. S.

(2013). The effects of a computer-assisted, schema-based instruction intervention on word problem-solving skills of low-performing fifth grade students. Journal of Special Education Technology, 28(1), 9–21.

Garofalo

Sharp

B. D.

(2003). Teaching fractions using a simulated sharing activity. Learning & Leading with Technology, 30(7), 36–39.

Gast

D. L.

Lloyd

B. P.

Ledford

J. R.

(2014). Multiple baseline and multiple probe designs. In Gast

D. L.

Ledford

J. R.

(Eds.), Single-case research methodology: Application in special education and behavioral sciences (pp. 252–296). New York, NY: Routledge.

10.

Gersten

Chard

D. J.

Jayanthi

Baker

S. K.

Morphy

Flojo

(2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79, 1202–1242. doi:10.3102/0034654309334431

11.

Hecht

S. A.

Vagi

K. J.

(2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843–859. doi:10.1037/a0019824

12.

Horner

R. H.

Carr

E. G.

Halle

Mcgee

Odom

Wolery

(2005). The use of single-subject research to identify evidence-based practice in special education. Exceptional Children, 71, 165–179. doi:10.1177/001440290507100203

13.

Hunt

J. H.

Empson

S. B.

(2015). Exploratory study of informal strategies for equal sharing problems of students with learning disabilities. Learning Disability Quarterly, 38, 208–220. doi:10.1177/0731948714551418

14.

Hunt

J. H.

Valentine

Bryant

D. P.

Pfannenstiel

K. H.

Bryant

B. R.

(2016). Supplemental mathematics intervention: How and why special educators intensify intervention for students with learning disabilities. Remedial and Special Education, 37, 78–88. doi:10.1177/0741932515597293

15.

Jitendra

A. K.

DiPipi

C. M.

Perron-Jones

(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. Journal of Special Education, 36, 23–38. doi:10.1177/00224669020360010301

16.

Jitendra

A. K.

Star

J. R.

Dupuis

D. N.

Rodriguez

M. C.

(2013). Effectiveness of schema-based instruction for improving seventh-grade students’ proportional reasoning: A randomized experiment. Journal of Research on Educational Effectiveness, 6, 114–136. doi:10.1080/19345747.2012.725804

17.

Kratochwill

T. R.

Hitchcock

Horner

R. H.

Levin

J. R.

Odom

S. L.

Rindskopf

D. M.

Shadish

W. R.

(2013). Single-case intervention research design standards. Remedial and Special Education, 34, 26–38. doi:10.1177/0741932512452794

18.

Leh

Jitendra

A. K.

(2013). A comparison of the effects of teacher-mediated and computer-mediated instruction on the mathematical word problem solving performance of third grade students with mathematics difficulties. Learning Disability Quarterly, 36, 68–79. doi:10.1177/0731948712461447

19.

Lewis

K. E.

(2014). Difference not deficit: Reconceptualizing mathematical learning disabilities. Journal for Research in Mathematics Education, 45, 351–396. doi:10.5951/jresematheduc.45.3.0351

20.

Maccini

Ruhl

K. L.

(2000). Effects of a graduated instructional sequence on the algebraic subtraction of integers by secondary students with learning disabilities. Education & Treatment of Children, 23, 465–489.

21.

Mazzocco

M. M. M.

Devlin

K. T.

(2008). Parts and “holes”: Gaps in rational number sense among children with vs. without mathematical learning disabilities. Developmental Science, 11, 681–691. doi:10.1111/j.1467-7687.2008.00717.x

22.

Montague

Enders

Dietz

(2011). Effects of cognitive strategy instruction on math problem solving of middle school students with learning disabilities. Learning Disability Quarterly, 34, 262–272. doi:10.1177/0731948711421762

23.

Montague

Krawec

Enders

Dietz

(2014). The effects of cognitive strategy instruction on math problem solving of middle-school students of varying ability. Journal of Educational Psychology, 106, 469–481. doi:10.1037/a0035176

24.

Nam

C. S.

Smith-Jackson

T. L.

(2007). Web-based learning environment: A theory-based design process for development and evaluation. Journal of Information Technology Education, 6, 23–43.

25.

National Center for Education Statistics. (2013). NAEP questions tool. Retrieved from http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx?subject=mathematics

26.

National Council of Teachers of Mathematics. (2010). Focus in grade 6. Reston, VA: Author.

27.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. Available from http://www.corestandards.org

28.

Newton

K. J.

Willard

Teufel

(2014). An examination of the ways that students with learning disabilities solve fraction computation problems. The Elementary School Journal, 115, 1–21. doi:10.1086/676949

29.

Odom

S. L.

Brantlinger

Gersten

Horner

R. H.

Thompson

Harris

K. B.

(2005). Research in special education: Scientific methods and evidence-based practices. Exceptional Children, 71, 137–148. doi:10.1177/001440290507100201

30.

M. W.

Bryant

D. P.

(2015). Effects of a strategic intervention with iPad practice on the multiplication fact performance of fifth-grade students with learning disabilities. Learning Disability Quarterly. Advance online publication. doi:10.1177/0731948715598285

31.

Parker

R. I.

Vannest

K. J.

Davis

J. L.

Sauber

S. B.

(2011). Combining non-overlap and trend for single case research: Tau-U. Behavior Therapy, 42, 284–299.

32.

Poch

A. L.

van Garderen

Scheuermann

A. M.

(2015). Students’ understanding of diagrams for solving word problems: A framework for assessing diagram proficiency. TEACHING Exceptional Children, 47, 153–162. doi:10.1177/0040059914558947

33.

Reimer

Moyer

P. S.

(2005). Third-graders learn about fractions using virtual manipulatives: A classroom study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5–25.

34.

Reneau

J. L.

(2012). Using the concrete-representational-abstract sequence to connect manipulatives, problem solving schemas, and equations in word problems with fractions (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 3530320)

35.

Sayeski

K. L.

(2008). Virtual manipulatives as an assistive technology support for students with high-incidence disabilities. Journal of Special Education Technology, 23(1), 47–53.

36.

Scruggs

T. E.

Mastropieri

M. A.

Castro

(1987). The quantitative synthesis of single subject research: Methodology and validation. Remedial and Special Education, 8(2), 24–33. doi:10.1177/074193258700800206

37.

Seo

Y.-J.

Bryant

D. P.

(2009). Analysis of studies of the effects of computer-assisted instruction on the mathematics performance of students with learning disabilities. Computers & Education, 53, 913–928. doi:10.1016/j.compedu.2009.05.002

38.

Seo

Y.-J.

Bryant

D. P.

(2012). Multimedia CAI program for students with mathematics difficulties. Remedial and Special Education, 33, 217–225. doi:10.1177/0741932510383322

39.

Seo

Y.-J.

Woo

(2010). The identification, implementation, and evaluation of critical user interface design features of computer-assisted instruction programs in mathematics for students with learning disabilities. Computers & Education, 55, 363–377. doi:10.1016/j.compedu.2010.02.002

40.

Shin

Bryant

D. P.

(2015). Fraction interventions for students struggling to learn mathematics: A research synthesis. Remedial and Special Education, 36, 374–387. doi:10.1177/0741932515572910

41.

Siegler

Carpenter

Fennell

Geary

Lewis

Okamoto

. . .Wray

(2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE 2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance. Retrieved from https://ies.ed.gov/ncee/wwc/pdf/practice_guides/fractions_pg_093010.pdf

42.

Stultz

S. L.

(2008). The effectiveness of computer-assisted instruction for teaching mathematics to students with specific learning disability (Doctoral dissertation). Regent University, Virginia Beach, VA.

43.

Taber

S. B.

(2002). Go ask Alice about multiplication of fractions. In Litwiller

Bright

(Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 61–71). Reston, VA: National Council of Teachers of Mathematics.

44.

Tolar

T. D.

Fuchs

Cirino

P. T.

Fuchs

Hamlett

C. L.

Fletcher

J. M.

(2012). Predicting development of mathematical word problem solving across the intermediate grades. Journal of Educational Psychology, 104, 1083–1093. doi:10.1037/a0029020

45.

Trespalacios

(2010). The effects of two generative activities on learner comprehension of part-whole meaning of rational numbers using virtual manipulatives. Journal of Computers in Mathematics and Science Teaching, 29, 327–346.

46.

Tsankova

J. K.

Pjanic

(2009). The area model of multiplication of fractions. Mathematics Teaching in the Middle School, 15, 281–285.

47.

van Garderen

Scheuermann

Jackson

(2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Quarterly, 36, 145–160. doi:10.1177/0731948712438558

48.

Wechsler

(2009). Wechsler Individual Achievement Test (3rd ed.). San Antonio, TX: NCS Pearson.

49.

Weng

P.-L.

Maeda

Bouck

E. C.

(2014). Effectiveness of cognitive skills-based computer-assisted instruction for students with disabilities: A synthesis. Remedial and Special Education, 35, 167–180. doi:10.1177/0741932513514858

50.

Yuan

Lee

C.-Y.

Wang

C.-H.

(2010). A comparison study of polyominoes explorations in a physical and virtual manipulative environment. Journal of Computer Assisted Learning, 26, 307–316. doi:10.1111/j.1365-2729.2010.00352.x