The Effect of a Tier 2 Multicomponent Fraction Intervention for Fifth Graders Struggling With Fractions

Fraction Concepts and Skills

Considering that students struggling with mathematics who need Tier 2 interventions did not adequately respond to Tier 1 core instruction, their fraction performances are likely to be below grade level. Thus, to remediate their fraction learning deficits, this intervention covered a broad range of CCSSM content standards for fraction concepts and skills that need to be taught from Grades 3 to 5 (Dougherty et al., 2017). Table S1 (see in the online supplemental materials) shows CCSSM content standards targeted in each lesson. The third-grade content standards include the introduction of fractions and addressing them as a number through presenting the fraction on area models and number lines. Fourth-grade content standards focus on understanding equivalent fractions and comparing fractions using benchmark fractions. Students also learn adding and subtracting fractions with like denominators, multiplying a fraction by a whole number, and the use of decimal notations for fractions. Fifth-grade content standards address using equivalent fractions for adding and subtracting fractions with unlike denominators.

Explicit Instruction

The second intervention component of the fraction intervention was explicit instruction. Over the years, explicit instruction has built an established research base, supporting its effectiveness for improving the mathematics performance of students struggling with mathematics (Doabler et al., 2017; Ennis & Losinski, 2019a). Based on the framework of explicit instruction, each lesson consisted of five sections: Warming Up, Learning To Solve, Practicing Together, Trying It On Your Own (TIOYO), and Wrapping It Up. The Warming Up section was a 5-min lesson opening that consisted of basic skills necessary for the day’s lesson. The Learning To Solve section included interactive modeling of mathematical ideas or problem-solving. The Practicing Together section contained guided practice through pair work. The TIOYO section included four problems for both independent practice and daily progress monitoring. The Wrapping It Up section included summarizing what the students had learned in the lesson.

Multiple Representations

The third intervention component of the fraction intervention was multiple representations. Multiple representations can facilitate students’ conceptual understanding of mathematics by having students represent mathematics concepts and write or think aloud about their representations, using area models (e.g., rectangle models, circle models), set models, length models (e.g., Cuisenaire Rods™, number lines), diagram, symbols, tables, and written and spoken languages (Kara & Incikabi, 2018; Rau et al., 2014). Fractions are one of the mathematics domains in which multiple representations are used extensively (Rau et al., 2014). The use of multiple representations was integrated into each lesson (Dougherty et al., 2016; Kara & Incikabi, 2018). For example, when the students learned about unit fractions, they represented a unit fraction using circle models, Cuisenaire Rods, and number lines, and then compared and made connections between different representations (e.g., a circle model, a number line).

Misconceptions of Students in Learning Fractions

The fourth intervention component of the fraction intervention (Dougherty et al., 2017) was to address students’ common misconceptions in learning fractions within each lesson. Dougherty et al. (2016) defined mathematics misconceptions as “faulty and incorrect ideas resulting from students’ misunderstanding about a mathematical idea or concepts” (p. 92). Misconceptions usually occur due to the application of inappropriate rules, generalization, or insufficient instruction (Dougherty et al., 2016). Before students learn fractions, students are primarily exposed to whole numbers (Dyson et al., 2020). As a result, students misapply principles concerning whole numbers to fractions, which is known as whole number bias (Wang et al., 2019). For example, to understand a fraction as a number, students need to know the role of a numerator and a denominator and how a numerator and a denominator coordinate to yield a fraction, which is one number, even though it contains two whole numbers (Dyson et al., 2020). In addition, mathematics language terms that students used for whole numbers may not be applicable for fractions. For example, when comparing fractions, students often use the terms “bigger fractions” or “larger fractions,” which are not mathematically accurate because they refer to size (Powell et al., 2019). Instead, students need to use the term “greater fractions,” which is a mathematically correct way of referring to quantity for rational numbers (Powell et al., 2019).

The misconceptions addressed in this intervention were chosen by the primary author of the intervention lessons who was a math educator and has been a professor in the field of mathematics education for about 30 years (Dougherty et al., 2017). She chose the misconceptions that students may have related to the lesson objectives for each lesson when developing the intervention. Table S2 (see in the online supplemental materials) includes the misconceptions addressed in each lesson and examples to address the misconceptions from each lesson. In addition, the interventionist addressed the misconceptions through explicit instruction, multiple representations, and progress monitoring. The interventionist provided explicit instruction using multiple representations to address the misconceptions in each lesson. The interventionist identified students’ misconceptions by having them explain their mathematical thinking using multiple representations. In addition, the progress monitoring measure, the TIOYO section, included four multiple choice problems with the response choices for indicative of mathematical misconceptions. The problems could provide information about students’ conceptual understanding and misconceptions.

Setting

After obtaining school permission, this study was conducted at a charter school in the southwestern United States, serving 304 students in Grades K through 5. At the school, 50% of the students were male, 66.1% were Hispanic, 17.4% were White, 10.2% were Black, and 1.3% were Asian. Moreover, 22% of the students had limited English proficiency, 55.2% were eligible for a free or reduced-price lunch, and 9% had individualized education programs. The interventionist provided the intervention within the school in an intervention classroom, which was equipped with whiteboards, desks, and chairs. All sessions took place three times a week, and each session lasted for 54–62 min. Groups 1 and 3 received the intervention in addition to the Tier 1 instruction during regular school hours, and Group 2 received the intervention after school.

Participants

In this article, struggling learners with fractions refer to students who showed low fraction performance determined by scores on a fraction assessment and by their mathematics teacher as having low mathematics achievement. The student participants who were struggling learners in fractions of this study were identified by the following procedure. After obtaining approval from the institutional review board, the researchers screened 22 fifth graders who had submitted parental consent forms and student assent forms declaring their agreement to participate in the screening test for this study. Of the 22 potential participants, the researchers identified six participants based on the following criteria: (a) enrolled in the fifth grade, (b) showed inadequate knowledge of fractions (i.e., scored 50% or lower on fraction pre-assessment tests; Ennis & Losinski, 2019b), (c) identified by their mathematics teacher as learners struggling with mathematics based on teacher observation, and (d) available to participate in the intervention during specific time periods of the school day or after school. The researchers approached the parents of a total of 10 students who met the inclusion criteria and 6 students returned consent forms to participate in the intervention of this study. The researchers assigned two students with the most similar fraction screening test scores to a group, making three groups in total. The six participants’ scores on the screening tests had a mean of 36.04% (SD = 8.42). The other fifth graders (n = 16) who did not participate in the intervention had a mean of 54.96% (SD = 2.86). Of the six participants, one (Zoe) student was receiving special education due to learning disabilities in reading identified through the Response to Intervention process. Two students (Lucas and Zoe) were receiving English Second Language services because of limited English proficiency. Three students (Lucas, Zoe, and Ana) were eligible for a free or reduced-price lunch (see Table S3 in the online supplemental materials for complete participant demographic information).

The Intervention Agent

All intervention sessions were delivered by the first author, a doctoral candidate in the special education department at a major research university in the southwestern United States. She is a certified teacher with 8 years of experience teaching at-risk students and those with disabilities. Three research assistants (RAs) comprising two doctoral students and one master’s student in the special education department conducted observations to measure the fidelity of interventions and assessments and calculate interobserver agreement (IOA). They also completed interrater reliability for the scoring of all outcome measures. The first author trained the observers on the data collection procedures, such as the use of fidelity checklists and how to determine IOA. Before the RAs began to collect data, they achieved an IOA of 100%.

Measures

Procedures

Experimental Design

This study used a SCD, multiple-probe design across groups that is one variation of multiple baseline designs (Kazdin, 2011). In a multiple-probe design approach, researchers collect baseline data intermittently, but consistently, while regularly conducting baseline and intervention phases (Kennedy, 2005). Thus, multiple-probe designs can be helpful for reducing the effects of multiple testing, which otherwise puts the internal validity of a study at risk (Gast et al., 2018). For this study, the researchers intentionally introduced the intervention to the first group of students after collecting three sets of stable baseline data to mitigate the effect of test fatigue on the students. We defined stable baseline data as having relatively flat means, not showing an upward trend, with relatively low variability for the group and the individual students within the group (Swan et al., 2020). A dramatic increase in students’ scores was not expected until the intervention was completed in this study. Thus, we decided to introduce the intervention to a second group when the first group provided three stable intervention data points that showed an increasing trend and when a second group had stable baseline data for the last three baseline data points, while a third group continued at baseline. This process was repeated until all the groups had received the intervention (Gast et al., 2018).

Data Analysis

Graphed data were visually analyzed based on graphing conventions (Ennis & Losinski, 2019b; Kazdin, 2011) including changes in the level, trend, and variability of the graphed data between the baseline and intervention phases. In addition, the researchers calculated means, standard deviation, standard error, and slopes for each phase for each group. The researchers used Tau-U to measure data nonoverlap between baseline and intervention phases and the between-case standardized mean difference (BC-SMD) to interpret the magnitude of change derived from baseline to intervention. First, Tau-U was calculated to gauge an estimate of effects using the following online calculator: http://www.singlecaseresearch.org (Vannest et al., 2016). Tau-U is a percentage of data improved over time that considers phase nonoverlap as well as intervention phase trend after controlling baseline phase trend (Parker et al., 2011). Tau-U is preferable to other ES indexes because (a) it has more statistical power than any other nonoverlap index and (b) it includes baseline levels and trends (Parker et al., 2011).

Second, given concerns regarding the use of Tau-U in isolation, such as its inability to account for magnitude and possible disagreement with visual analysis, BC-SMD was calculated (Matta et al., 2020). The BC-SMD is considered as one of the more robust SCD outcome metrics in SCD research (Valentine et al., 2016). The advantages of using the BC-SMD include the following: (a) magnitude and direction of effect can be assessed and (b) fixed and random effects of baseline, intervention, and trends are considered to estimate the effect size (Matta et al., 2020; Valentine et al., 2016). The BC-SMD are modeled with a hierarchical linear approach (HLA; Matta et al., 2020; Valentine et al., 2016). The restricted maximum likelihood (REML) procedure is applied for the HLA to examine the correlational relationship between the outcome for each individual and the intervention and the data change pattern across cases in the study (Matta et al., 2020; Valentine et al., 2016). Specifically, we used the following online scdhlm app: https://jepusto.shinyapps.io/scdhlm/ (Pustejovsky et al., 2020). We followed the step-by-step explanations provided by Valentine et al. (2016) to use the scdhlm app for calculating BC-SMD of multiple probe designs. For the model specification, we considered visual inspection of data and tried to specify the most parsimonious model possible given the small number of cases of SCD studies (Valentine et al., 2016). After loading the data in the scdhlm app, we used the REML estimation because our model included time trend (Valentine et al., 2016). For phase time trend, we chose no trend for the baseline phase and a linear trend for the intervention phase because the baseline phases were fairly stable and the intervention phases showed an increasing trend (Valentine et al., 2016). We added fixed and random effects for the baseline phases and the intervention phase level because the effect of the baseline and intervention varied across cases (Valentine et al., 2016).

Fraction Probes

Overall, a functional relation was established for each group between the fraction intervention and the fraction probe outcome variable, and all students showed a positive effect (see Figure 1). In the following sections, data are presented for each group and for individual students. The means, standard deviations, slopes, and standard errors are reported for individual students and phases in Table 1. Across all six participants, the BC-SMD was 0.57 (SE = 0.29), and the Tau-U was 0.89 (confidence interval [CI]₉₀ = [.66, 1.00]).

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Figure 1.

Fraction probe scores across the baseline, intervention, and maintenance phases for the three groups of students.

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Table 1.

Outcomes on Fraction Probes.

	Fraction probes (percentage correct)
Student	Baseline			Intervention			Tau-U	Maintenance
	M (SD)	Slope	S E y x	M (SD)	Slope	S E y x		M (SD)
Group 1
Sara	35.00 (8.66)	–7.50	5.00	60.00 (17.32)	4.91	5.48	0.87	90 (0)
Lucas	36.67 (11.55)	0.00	6.67	62.00 (16.19)	4.00	5.12	0.90	85 (0)
Group 2
Mia	40.00 (17.68)	4.94	7.91	87.00 (21.24)	5.09	6.72	0.88	85 (0)
Zoe	42.00 (6.71)	1.22	3.00	67.00 (16.19)	3.88	5.12	0.84	80 (0)
Group 3
David	36.67 (6.83)	1.16	2.79	84.00 (13.29)	4.24	4.20	1.00	95 (0)
Ana	51.67 (6.83)	0.42	2.79	74.50 (16.74)	5.12	5.29	0.83	75 (0)

The baseline mean scores of those in Group 1 were relatively stable with a slightly decreasing trend during the baseline phase and displayed a marked increase in level and trend in the second probe taken during the intervention phase. Both students demonstrated low levels in their percentage of correct answers during the baseline phase (i.e., Sara, M = 35.00; Lucas, M = 36.67). Changes in level, trend, and variability were noted for each student during the intervention phase, with scores maintained 2 weeks after the conclusion of the intervention. The Tau-U score was 0.87 (CI₉₀ = [.22, 1.00]) for Sara and 0.90 (CI₉₀ = [.25, 1.00]) for Lucas. Moreover, both students displayed an exponential increase in level from baseline to intervention. Sara displayed increased variability, as measured by SD and S E y x . Lucas displayed increased variability, as measured by SD, but decreased variability as measured by S E y x . The changes in the trend (i.e., slope) from baseline to intervention were notable for both students: from −7.50 to 4.91 for Sara and from 0.00 to 4.00 for Lucas. Both students displayed increasing slopes that far exceeded the baseline slope.

The baseline mean scores of those in Group 2 displayed relatively higher variability than the other groups with a slightly decreasing trend at the end of the baseline. A marked increase in level and trend was demonstrated on the second probe taken during the intervention phase. Zoe began with a stable baseline, but Mia had higher variability than Zoe during the baseline phase. Changes in level, trend, and variability were noted for each student during the intervention phase, with scores maintained 2 weeks after the conclusion of the intervention. The Tau-U score was 0.88 (CI₉₀ = [.34, 1.00]) for Mia and 0.84 (CI₉₀ = [.30, 1.00]) for Zoe. In addition, both students displayed a significant increase in level from baseline to intervention. Zoe displayed increased variability, as measured by SD and S E y x . Mia displayed increased variability, as measured by SD, but decreased variability as measured by S E y x . Both students displayed increasing slopes that exceeded the baseline slope. The changes in the trend (i.e., slope) from baseline to intervention were notable for both students: from 1.22 to 3.88 for Zoe and from 4.94 to 5.09 for Mia. Both students displayed increasing slopes that exceeded the baseline slope.

The baseline mean scores of those in Group 3 were relatively stable with a slightly decreasing trend at the end of the baseline. Group 3 students demonstrated a marked increase in level and trend across the intervention. Both students began with stable baselines. Changes in level, trend, and variability were noted for each student during the intervention phase, with scores maintained 2 weeks after the conclusion of the intervention. The Tau-U score was 1.00 (CI₉₀ = [.49, 1.00]) for David and 0.83 (CI₉₀ = [.33, 1.00]) for Ana. In addition, both students demonstrated a significant increase in level from baseline to intervention. As both students showed increased levels, both also displayed increased variability during the intervention, as measured by SD and S E y x . Both students displayed increasing slopes that far exceeded the baseline slope. The changes in the trend (i.e., slope) from baseline to intervention were notable for both students: from 1.16 to 4.24 for David and from 0.42 to 5.12 for Ana. Both students displayed increasing slopes that far exceeded the baseline slope. Ana had a lower percentage of correct answers 2 weeks after the conclusion of the intervention but maintained a level well above the baseline.

Generalization Measure

Five of the six students made gains on the generalization measure, easyCBM numbers, operations, and algebra assessment (see Table 2). The student mean score was 67.71 on the pretest (ranging from 56.25 to 81.25; SD = 10.77) and 79.17 on the posttest (ranging from 68.75 to 87.50; SD = 9.41). The student average gain score was 11.46 (ranging from −6.25 to 18.75; SD = 10.77).

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Table 2.

Outcomes on Generalization Measure.

	Generalization (percentage correct)
Student	Pretest	Posttest	Gain
	M (SD)	M (SD)	M (SD)
Group 1
Sara	56.25 (0)	68.75 (0)	12.50 (0)
Lucas	56.25 (0)	75.00 (0)	18.75 (0)
Group 2
Mia	81.25 (0)	87.50 (0)	6.25 (0)
Zoe	75.00 (0)	68.75 (0)	–6.25 (0)
Group 3
David	62.50 (0)	87.50 (0)	25.00 (0)
Ana	75.00 (0)	87.50 (0)	12.50 (0)

Note. Gain = posttest score—pretest score.

Social Validity

Regarding the acceptability of the intervention, the results of the social validity questionnaire indicated that the students had positive perceptions about the intervention and thought that the intervention was an acceptable method for learning fractions (M = 4.45 out of 5.00, SD = 0.78). The questionnaire results by groups were as follows: M = 4.80 for Group 1, M = 4.40 for Group 2, and M = 4.17 for Group 3 (see Table S4 in the online supplemental materials).

The Effects of the Fraction Intervention

Social Validity of the Intervention

The social validity of the intervention was rated by the student participants as high, which was consistent with the findings from previous studies on teaching fractions to fifth graders struggling with mathematics (Ennis & Losinski, 2019b; Flores et al., 2020; Losinski et al., 2019). Because the students’ mathematics teacher did not participate in this study directly, social validity ratings from the teacher were not obtained. However, when the students’ outcome graphs were shared with the mathematics teacher face-to-face, the teacher stated,

When the students learned fractions in the general education class after the completion of the intervention, the students were actively engaged in solving fraction problems and verbally explained the reasoning of solving fraction problems using visuals in front of class. (Personal communication, January 14, 2018)

Limitations and Future Research

This study had several limitations, and hence various suggestions for future research are proposed. First, the researchers conducted this study with only six elementary students, which may limit the external validity of this study. Future researchers are recommended to conduct replication studies using this current study’s procedure but with a larger sample to verify generalization of this study’s findings to larger populations. Second, the researchers measured the maintenance effect of the intervention 2 weeks after the completion of the intervention. Two weeks may not be sufficient to measure students’ long-term gains. Thus, future research should examine whether students can maintain what they learn from interventions after 2 or 3 months to provide better evidence of students’ long-term gains. Finally, when Dougherty et al. (2017) developed the Tier 2 fraction intervention, she used area and length models for all lessons with other representations, but not set models. Thus, future researchers should consider using set models to teach fractions because using set models is helpful for connecting problems to real-life situations that use fractions and for developing easy connections to ratio concepts (Cady et al., 2015).

Implications for Practice

The findings from this study have insightful implications. Students struggling with fractions in Tier 2 who do not show adequate progress within Tier 1 need more explicit and systematic instructional support for learning fractions (Fletcher & Vaughn, 2009; Fuchs, Fuchs, et al., 2016; Jimerson et al., 2016). They also need more instructional support to obtain a sound conceptual understanding of fractions and to become more motivated for learning (Flores et al., 2020; Fuchs, Fuchs, et al., 2016). Teachers are encouraged to consider using the instructional components used in the intervention (i.e., explicit instruction; multiple representations; misconceptions; flexibility, reversibility, and generalizations; self-graphing) to provide more effective support for students struggling with fractions in Tier 2. In addition, during the intervention, the students had to complete multiplication and division fact sheets at the beginning of each session due to their insufficiently developed multiplication and division skills, which are prerequisite skills for learning fractions. In 2009, the What Works Clearinghouse practice guide on mathematics instruction for students struggling with mathematics (Gersten et al., 2009) recommended that interventions at all grade levels should spend approximately 10 min to establish that students could undertake a quick retrieval of basic arithmetic facts. Also, the NMAP (2008) has emphasized fact fluency as one of the critical foundations of algebra. Therefore, when students struggle in learning fractions due to insufficient fact fluency, teachers need to provide practice for such students to improve their relevant fact fluency.