Effects of a Supplemental Intervention Focused in Equivalency Concepts for Students With Varying Abilities

Abstract

Keywords

Tier 2 instruction supplemental instruction mathematics rational numbers

One of the most relentless areas of difficulty in mathematics performance and understanding for all students is rational number concepts and applications (Cawley & Miller, 1989; Hecht, Vagi, & Torgesen, 2006; Mazzocco & Devlin, 2008; National Center for Educational Statistics, 2009). On the 2007 National Assessment of Educational Progress (NAEP), 59% of fourth-grade students could not identify two rational numbers as equivalent or generate additional equivalent rational number relations (National Center for Educational Statistics, 2009). Less than half of eighth-grade students could correctly identify rational numbers given in ascending order or add two fractions with different denominators. If students have difficulties understanding mathematics, we would expect rational numbers to become increasingly more difficult to learn. Indeed, research points to not only quantitative performance differences between students with and without mathematics difficulties but also qualitative differences in the ways students who struggle to learn mathematics understand and approach problems involving rational numbers (Mazzocco & Devlin, 2008).

Despite the differences students experience as they learn rational number equivalency concepts, few studies offer content and instructional guidance to practitioners to help improve understanding of rational number equivalency for students with severe learning difficulties. In the sections that follow, the author addresses (a) content, instructional practice, and mathematics difficulties students who struggle may face as they relate to rational numbers; (b) supplemental approaches to increasing proficiency in rational number concepts for students who struggle with mathematics; and (c) the research objectives of the present study.

Many people view the Common Core State Standards Initiative (CCSS, 2010) as a possible catalyst for improving mathematical proficiency of children and adolescents in the United States. The CCSS offer direct guidance in content and practices regarding rational numbers. In terms of mathematical practices, the CCSS emphasize student reasoning and sense making of problem situations and solutions as well as those posed by other students, student use of visual models, mathematical structures, and relationships, and the ability to abstract concepts through making use of mathematical structure and expressing regularity in repeated reasoning. In terms of content, the CCSS define equivalence as the understanding of two fractions that are the same size or at the same point on a number line. Evidence of a student’s growing understandings of rational number equivalence also includes the ability to recognize and generate simple equivalent fractions (e.g., 1/2 = 2/4, 4/6 = 2/3) and explain why the fractions are equivalent (e.g., by using a visual fraction model).

For students who have difficulty learning mathematics, however, the content and processes involved in learning rational number concepts such as equivalence may be especially challenging. For example, studies (Grobecker, 2000; Hecht et al., 2006; Mazzocco & Devlin, 2008) suggest these students display difficulty understanding beginning equivalency concepts such as understanding relationships involved with unit fractions (e.g., the relationship between 1 and 4 in one-fourth); renaming fractions after they have been repartitioned into equivalent amounts (e.g., renaming one-fourth as two-eighths), and identifying equivalent fraction relations when presented in pictorial and abstract representations. Two studies (Hecht et al., 2006; Mazzocco & Devlin, 2008) report the abilities of students with more severe learning problems as significantly lower than students considered typically progressing in their understanding of mathematics. Moreover, authors report qualitative differences in understanding rational number concepts for these students. “Taking” (e.g., within an area model, shading a number of pieces and then understanding those pieces as “taken away” instead of as a part of a whole) and “halving” (e.g., understanding a partitioned area model as an action instead of a fractional part of a whole), were areas resistant to efforts to improve struggling students’ understanding of equivalency (Lewis, 2007).

Researchers and policy makers outline more complimentary approaches to understanding rational number equivalency in recent literature. The National Mathematics Advisory Panel (NMAP, 2008) and What Works Clearinghouse panel authors (Siegler et al., 2010) recommend using ratio-like situations to build students’ understanding of proportional reasoning and rational number relationships such as equivalency along with approaches outlined in the CCSS such as equal sharing and measurement activities. Ratios can denote part–whole or part–part situations where the quantities are somehow related (Kieran, 1993). Ratios represent the rational number interpretation most relevant to the teaching and learning of equivalency concepts (Behr, Post, & Silver, 1983).

Educators can use ratio-like situations to facilitate understandings of balance, unit relationships (i.e., the relationship between the two numerator and denominator of a single ratio), and patterns evident among equivalency situations (Vanhille & Baroody, 2002). Two ratios that are equivalent possess the same within, or unit, relation. Ratio-like instruction may afford students with learning difficulties access to understanding rational number content as well as a vehicle to grow in their use of mathematical practices outlined in the CCSS (Siegler et al., 2010). Yet, research studies could not be found that utilized a ratio-based approach to rational number equivalency to supplement equivalency instruction occurring in students’ core mathematics instruction.

Response to intervention (RtI) is one supplemental approach used to provide students additional mathematics instruction. The aim of RtI is to raise student performance levels as well as to prevent later identification of learning disability (Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008; L. S. Fuchs et al., 2006). Such approaches utilize universal screening measures to identify students early who fall behind in mathematics performance, provide support services, and monitor progress within the intervention to determine if additional or different supports are needed. Students identified as struggling in the core curriculum (i.e., Tier 1) receive support services in the form of Tier 2, supplemental instructional interventions. Students engage in 20 to 40 min of small group intervention instruction each day, 4 to 5 days a week, for 18 to 23 weeks. Information on the nature of successful early prevention tiered programs is becoming increasingly prevalent. Yet, there is currently little to no research regarding interventions in advanced elementary concepts paired with alternate approaches to building proficiency in rational number in students who struggle to learn.

Research is needed to explore the effects of Tier 2 supplemental instruction in critical content areas such as rational number equivalency for students who demonstrate a need for additional or qualitatively varied instruction. The purpose of this exploratory study was to determine the effects of a supplemental instructional intervention for rational number equivalency emphasizing proportional reasoning combined with elements of effective instruction on performance for students demonstrating a need for additional instruction in equivalency concepts. The intent was to evaluate how the intervention would impact performance on several measures, including those reflective of the core curriculum as well as standard measures more in line with concepts presented during the intervention. A secondary intent was to evaluate whether students with and without mathematics difficulties who received the intervention would display different levels of understanding as demonstrated by the dependent measures. The key research question was as follows:

Research Question: Do students with and without mathematics difficulties receiving a ratio-based intervention demonstrate improved performance on a pretest–posttest and a standard measure of rational number equivalency when compared with similar students receiving core mathematics instruction with no intervention?

It was hypothesized that students with and without mathematics difficulties receiving the ratio-based intervention would demonstrate improved performance on a pretest–posttest and a standard measure of rational number equivalency when compared with similar students receiving core mathematics instruction with no intervention. Past research reflecting the importance of instruction on proportional reasoning combined with mathematical practices served as the basis for the hypothesis (Fuchs et al., 2006; Vanhille & Baroody, 2002).

Method

Participants and Setting

The study included 38 students with and without mathematics difficulties in the third grade from one elementary school. The researcher chose third grade due to state curriculum constraints that set the learning of fraction equivalency concepts to a third grade maximum. All participating students attended the same elementary school in an urban neighborhood in central Florida. A majority of the school population qualified for free and reduced-price lunch. Student characteristics relating to ethnicity, gender, and grade level for the 38 students in the final sample were obtained from the district at the onset of data collection (see Table 1). In the experimental group (EG), 53% of students were males and 47% were females; 53% were White, 21% were African American, and 26% were Hispanic. In the control group (CG), 47% of students were males and 53% were females; 53% were Caucasian, 16% were African American, and 31% were Hispanic. Additional information is listed in Table 1.

Table 1.

Characteristics of Experimental and Control Groups (in Percentages).

	Experimental group	Control group
	n = 19	n = 19
Age
8–0 to 9–4	74	63
9–5 to 10–0	26	37
Gender
Male	53	47
Female	47	53
Ethnicity
White	53	53
African American	21	15
Hispanic	26	32
District Screener	27	31
Key Math-R	18	20
WJ-III	21	23

Note. Key Math–R and WJ-III = mean percentiles. WJ-III = Woodcock–Johnson III.

The researcher used an a priori power analyses to confirm necessary sample sizes using G Power 3 statistical software (Faul, Erdfelder, Lang, & Buchner, 2007) with effect sizes from previous research in fractions for students with mathematics difficulties ranging from 0.50 to 0.75. The power analyses indicated a total sample size of 38 was sufficient to produce a power of .90 for a 2 × 2 between-factors MANOVA with repeated measures, with 19 students in the EG and 19 students in the CG with an alpha level of .05 and noted effect sizes (Faul et al., 2007). MANOVA was used to decrease risk of Type I error.

Measures

The researcher utilized a combination of measures including a pre–posttest measure of fraction equivalency and a standardized measure of transfer performance. Each of the dependent measures is discussed below.

Standardized test

The transfer test consisted of a subtest from the Brigance (1999) Comprehensive Inventory of Basic Skills–Revised. Specifically, researchers used subtest Q6, Converts Fractions and Mixed Numbers, as the standardized measure for this study. The subtest consisted of 16 fraction items measuring fraction equivalency. Four items measured students’ ability to scale up a fraction to produce an equivalent fraction; four items asked students to simplify a fraction to an equivalent fraction. Eight items asked students to convert between mixed numbers and improper fractions. The test did not present these problems in context. Reported reliability and validity ranged from 75% to 85%, respectively (Glascoe, 1999). The test was group administered, and students were allowed 50 min to answer as many problems as possible, yielding a score of total number of problems correct. The problems contained in this measure were reflective of those used in supplemental intervention.

Pre–posttest

The purpose of the ratio-based, Tier 2 instructional intervention was to supplement core instruction (as opposed to replacing it). Thus, measures of performance should also document increases in fraction equivalence as presented in core instruction. This is necessary to ensure increased performance is transferrable. To this end, researchers extracted 20 equivalency items from the district curriculum, Envision Mathematics, Level 3, Chapter 12 (Charles, Caldwell, Crown, & Fennell, 2011). The researcher used items from this chapter to construct a pre–posttest. The researcher examined each lesson within the chapter that taught fraction equivalency and pulled every other problem from the text practice questions to construct the tests. Items included situated problems (e.g., word problems), abstract problems, or problems that require students to judge the correctness of given equivalency statements. Example problems from the pre–post test include items such as (a) A store sells packages that contain 6 pencils and 4 pens. Your friend bought enough packs to get 36 pencils. How many pens did they get?: (b) 5/20 = __/4; and (c) Show 3/4 and 9/12 are the same.

The researcher calculated internal consistency reliability and convergent validity for the pre- and posttests during a pilot study. Results of the reliability analyses generated an alpha coefficient of .71. The researcher measured validity of the pre- and posttests against performance on the Brigance Q6 subtest. Results of the analyses generated a coefficient of .77 and .78, respectively, between the standardized measure and the pre–post test. The researcher obtained evidence of internal consistency and convergent validity because all coefficients were above .70 (Nunnally & Bernstein, 1994).

Procedures

Assessment procedures–District screener

Before the onset of the study, all 93 third-grade students at the elementary school site completed state mathematics testing (the Florida Comprehensive Assessment Test [FCAT]). The state reported internal consistency reliability as an alpha coefficient of .89. The state reported concurrent validity for the screener as an alpha coefficient of .84 between the criterion-referenced portion of the FCAT and the norm-referenced portion (Stanford 9). A score of 1 or 2 confirmed 78 students as needing additional instruction in mathematics and thus eligible for involvement in the study. Identified students were then given parental consent forms for study participation through a routine front-office mailing. Student assent was collected through a signature on the bottom of the consent form. Parents did not receive any follow-up contact from the school or the researcher requesting they return the permission slips. Of the original sample of 78 students, 40 third-grade students who met the selection criteria returned consent forms.

Assessment procedures–Pretest/posttest and overall performance measures

Consenting students then took the pretest and standard measure used in this study; a score of below 50% confirmed students’ need for additional instruction in rational number equivalency concepts. A classroom was set up in the third-grade hallway for purposes of testing; consenting students were excused from their classrooms for two separate 50-min intervals for testing. A member of the research team administered and scored the pretest in one 50-min class period and the standard measure in a separate 50-min class period. The tests were given to students as a group; students were told they were going to complete questions about fractions and to do their best. Researchers gave no other direction. Word problems contained in the pre–post tests were read aloud to students who requested assistance. Testing days were consecutive. Posttest procedures mirrored pretest procedures.

Two students were excluded from the study because they scored above the 50% cutoff on both tests. The 38 students who remained completed three overall performance measures: the calculation subtest from the Woodcock–Johnson III (WJ-III) Test of Achievement along with the Numeration and Mental Computation subtests from the Key Math-R. “Struggling” student performance fell at or under the 25th percentiles on two out of three measures and “typical” student performance fell above the 25th percentile on two out of three measures. The analysis led to identification of 12 students as struggling in mathematics and 26 students as typically achieving. Researchers then matched students on their “student type” and randomly assigned to either a treatment or CG (Borg & Gall, 1989). Matching ensured students were comparable across intervention conditions on relevant characteristics (i.e., student type designation; Gersten et al., 2005). Two members of the research team administered and scored the screening measures, following the standard protocol accompanying each subtest, in the same room as the pretest. Researchers tested each student individually. Combined administration time for all screening measured ranged from 30 to 65 min.

Training

Members of the research team received 3 hr of training on all measures used in the study from the author. For the pre–posttest, members of the research team were given standard directions to read to students before implementing the test and were instructed to give no further directions or help to students during test implementation. Procedures for scoring were included; items were marked as correct (1 point) or incorrect (0 points). The author presented and modeled administration of each measure, including scoring procedures, for the standard pretest measure and the overall performance measures. The author gave additional time for each member to practice implementation of procedures; the author was present during the practice. The research team consisted of two doctoral students in Special Education; each person had completed a Master’s program in Special Education and one member of the research team specialized in assessment.

Instructional procedures–Core instruction

Ten days of core instruction commenced over 2 weeks. Each day of instruction lasted about 45 min. Five third-grade teachers were responsible for delivering core mathematics instruction to all 38 students involved in the study. Core instruction focused on concepts of rational number equivalency and related skills and was based on state mathematics standards. The intent of core instruction was to give students opportunities to learn equivalence concepts through an understanding of how the size of a fractional part related to the number of equal-sized pieces in the whole and then extend the concept to notions of equivalence. Lessons in core instruction included opportunities to partition whole amounts into equal-sized parts (e.g., divide a rectangle into four equal parts). The use of such activities set the stage for students to work with various fraction models and to understand a fraction as a part of a whole. Students then represented various fractional quantities using area, set, and linear models. These experiences were designed to help students see the relationship between one of the equal parts and the whole. Finally, students used area, set, and linear models to represent equivalent fractions as well as identify given fraction representation as equivalent. In these lessons, students began with a representation for a certain fraction (e.g., drawing a circle, partitioning the circle into four parts, and shading three of the four parts). To represent an equivalent fraction, students further partitioned each part (e.g., partitioning each of four parts in half to make eight parts), counted how many parts were shaded (e.g., six), and recorded the equivalence observed with numbers and symbols (e.g., 3/4 is the same as 6/8).

Procedures used by core instruction teachers during the lessons were based on explicit, systematic instruction and included modeling, guided practice, and error correction. During modeling, the teachers demonstrated the steps needed to solve problems or provided explanations of how to perform certain skills (e.g., partitioning a rectangle into four equal pieces). Guided practice consisted of multiple opportunities for students to practice teacher-demonstrated skills and concepts, included partitioning area, set, and linear models of fractions, creating concrete and pictorial representations of various fractions and their equivalents, and writing symbols for identified equivalent fraction representations. Teachers used a series of high, medium, and low prompts during guided practice time with a mastery criterion of 80% of students answer 80% of prompt questions and/or problems correct. Teachers prompted students through the steps of solving each problem during highly supported guided practice; students completed each step as prompted by the teacher until they reach the mastery criterion. The teacher then moved to medium and low levels of guided practice using the same criterion for mastery. When students solved three problems correctly on their own with no prompting from the teacher, students moved to independent practice.

Fidelity observations of core instruction

The research team observed each teacher in the core/control condition for two instructional sessions (20% of all core instruction sessions) to assess the fidelity of mathematics instruction in students’ core classrooms. Two members of the research team trained on an observation instrument used for the core lessons conducted the observations. The forms contained a box for each feature of instruction used as a part of core instruction; raters gave an overall rating of 0 (not used) to 4 (highly used) for each lesson observed. Observers reached a point-by-point interrater reliability of 92% across all observations. Fidelity was rated moderately high by each observer, with all of the lessons observed rated as a 3 or a 4.

Instructional procedures–Supplemental intervention

Twenty sessions of supplemental intervention commenced over 4 weeks with the 19 students in the treatment group. Researchers formed four small groups of four students and one group of three students after the conclusion of pretest and screener administration; groups from based student schedules. Instruction occurred in the same classroom that was designated for pretest and screening measure implementation. Each session lasted 30 min. All sessions were videotaped. The intervention teacher was the researcher.

Procedures used by the supplemental instruction teacher during intervention were based on a three-part instructional sequence containing features of explicit and systematic instruction (e.g., teacher modeling, meaningful practice opportunities, judicious review, concrete and visual representations; Coyne, Kame’enui, & Carnine, 2011); with elements of conceptually based instruction (e.g., purposefully scaffolded task sequencing, active involvement in mathematical tasks, verbalization of strategies; Hiebert, 2003; Kieran, 1993; Siegler et al., 2010). During modeling (i.e., Part One), the teacher used a think aloud to model her problem solving of one to two sample problems. The number of problems thought aloud by the teacher depended on how many students were answering questions presented during the think aloud correctly (i.e., 80% of students answering 80% of questions correctly). As part of the modeling, the teacher talked about the problem situation, produced a corresponding representation and problem solution. Part One lasted about 6 to 8 min on average, although in some sessions Part One lasted up to 10 min. Student-centered practice opportunities (i.e., Part Two) constituted the bulk of the lesson (15–18 min) and consisted of students working on problems given on their own, in pairs, and then as a group. During practice, the teacher displayed a transparency with thinking questions modeled in Part One designed to help students think about the problem. Students solved the problem on their own for 2 min; the teacher told students to keep the questions on the transparency in their heads. Then students shared their solutions with a partner. This lasted another 2 min. As students shared, the teacher looked for a student to present their work to the class. The selected student shared their solution. Students discussed whether they felt the answer was correct. If students could not agree or agreed on an incorrect solution, the teacher utilized a counter argument. If the counterargument did not produce a consensus on the correct answer, the teacher explicitly modeled the problem solution utilizing a think aloud. After students had practiced three to four problems, the teacher used questioning strategies to help students reflect on the reasonableness of their solutions (i.e., Part Three).

Figure 1 shows a contrast of core instruction and supplemental intervention. The supplemental intervention focused on concepts of rational number equivalency and was based on the work of Battista and Borrow (1995). The intent of the intervention was to give students opportunities to learn equivalency concepts through a conceptual understanding of the relationships found within a ratio unit (e.g., the relation between the numerator and the denominator must be maintained) and a procedural understanding of how to use the ratio unit along with mathematical operations to generate equivalent fractional units (e.g., if three boxes make four muffins, how many boxes make 20 muffins). The lessons included opportunities for students to work with quantities iterated, or repeated, several times (e.g., Mauricio ordered five pieces of bacon; Nicosha ordered twice as much; Katy ordered three times as much). The use of such activities set the stage for the use of ratio units. Students were also given a relation (e.g., 1 pancake feeds four children) and examining different quantities of pancakes and children, judged whether the given amount of pancakes was too much, just enough, or too less. These experiences were designed to help students understanding that the relationship between cans and pancakes needed cannot change when additional cans or a number of pancakes are added to a situation. This further established the recognition and importance of the concept of the unit in understanding equivalency. In addition, students worked with missing value situations (e.g., if three boxes make four muffins, how many boxes make 20 muffins; 3 /1 = 6 / 8 = 9 /12 = 12 /16 /15 / 20 or 3 /1 = 15 / 20) using additive and multiplicative strategies to bolster formalized notions of equivalency relations.

Figure 1.

Contrast of supplemental and core instruction.

Intervention fidelity procedures

During implementation of the intervention sessions, a randomly selected sample of 30% of the videotaped sessions was observed for fidelity to the implementation script. A checklist containing the instructional script for modeled problems as well as the procedures for student practice problems was used to evaluate the extent to which the teacher (a) implemented the scripted modeling of problems in each lesson and (b) implemented the procedures outlined for student practice problems (i.e., timing, questioning strategies, counterarguments). The checklist contained a box for each instructional element; raters checked “yes” or “no” for each element of the instructional script observed. Two members of the research team trained on the use of the checklist conducted the fidelity checks. Point by point interrater reliability across instructional elements was 89%. Fidelity was rated moderately high by each observer, with 87% of the instructional elements observed across the six sessions rated as “yes.”

Data Analysis

The researcher utilized a between-factors MANOVA with repeated measures to test the amount of change in the dependent variables as a result of intervention (Tabachnick & Fidell, 1996). The within-subjects factor was time (i.e., before and after intervention). The between-subjects factor was group assignment (i.e., intervention or CG) and student type (i.e., low-achieving or typically achieving). Dependent variables included the pre–post test measure and the transfer test. To investigate the impact of each significant main effect on the individual dependent variables, a Roy–Bargmann step-down analysis was performed. The analysis was used because the dependent variables were correlated. Thus, the sole use of univariate ANOVAs would have increased the risk of Type 1 error. In step down analysis, an a priori decision to prioritize dependent variables was made (i.e., standard measure first priority and pre–posttest second priority).

Experimental Design

Researchers used a pretest–intervention–posttest CG design. Students were randomly assigned to either the EG or CG using matched pairs based on student type designation (Borg & Gall, 1989). Matching ensured students were comparable across intervention conditions on relevant characteristics (i.e., student type designation; Gersten et al., 2005).

Results

A repeated-measures MANOVA was used to evaluate the effectiveness of the intervention on two measures of equivalency. All evaluations of assumptions of normality, linearity, and multicollinearity associated with MANOVA were satisfactory. No univariate or multivariate within cell outliers existed at p = .001 and no data were missing. Box’s Test of Equality of Covariance Matrices was not significant, F(10, 1033.88) = 0.981, p = .459. There were no significant differences in pretest mean scores between the EG and CGs before instruction. A moderate degree of correlation was expected and uncovered (0.54) between the pre–posttest and standard measure pretest scores, because both tests reflect understanding of rational number equivalency. Means for both dependent measures (before and after intervention) can be found in Table 2.

Table 2.

Pretest and Posttest Group Means.

	Mean scores by group
	Experimental group		Control group
	Low achieving	Typically achieving	Low achieving	Typically achieving
	n = 9	n = 10	n = 10	n = 9
Measure
Pretest score (out of 20)	1.25	5.00	2.20	4.12
Posttest score (out of 20)	10.13	13.09	2.10	3.57
Pretest standard measure (out of 16)	1.88	4.18	2.00	4.57
Posttest standard measure (out of 16)	8.50	12.55	2.40	4.64

Multivariate Effects

Table 3 shows multivariate results for significant independent measures by the combined dependent measures. Significant multivariate effects were found for the interaction of group assignment and test time, partial η² = 0.364, group assignment, partial η² = 0.449, student type, partial η² = 0.279, and pre–post testing time, partial η² = 0.378, favoring the EG.

Table 3.

Significant Multivariate Effects.

Variable	p	F	df	Error df
Group Assignment × Test Time	.000	10.597	2	26
Group Assignment	.000	10.234	2	26
Test time	.000	11.362	2	26
Student type	.014	5.032	2	26

Univariate Effects

Table 4 shows significant univariate effects for each dependent variable. The highest priority dependent variable, determined a priori as the standard test, was tested first via factorial repeated measures ANOVA. Significant univariate effects were found for the interaction of group assignment and time of testing, partial η² = 0.45. Students in the experimental condition outperformed students in the control condition. Student type also contributed to increased scores on the transfer test, partial η² = 0.254, indicating a small to moderate relationship between student type and scores on the post-standard test. Students deemed typically achieving (M = 8.12; SD = 3.629; total possible = 16) performed better on the standard measure than students who struggled (M = 4.64; SD = 5.126; total possible = 16) in the experimental and CGs.

Table 4.

Significant Univariate Effects.

Variable	p	F	df	Error df
Standard test
Group Assignment × Test Time	.000	27.818	1	34
Test time	.000	31.549	2	26
Student type	.002	11.605	2	26
Pre–posttest
Group Assignment × Test Time	.004	9.556	3	33

The pre–posttest was tested using a factorial ANCOVA with repeated measures. The standard measure served as the covariate (Tabachnick & Fidell, 1996). The only significant contributor to higher scores on the pre–posttest measure was the interaction of group assignment × pre–post test time, partial η² = 0.225. Students in the EG increased their performance significantly from pretest to posttest (pretest M = 3.42; SD = 3.22; posttest M = 11.84; SD = 3.96; total possible = 20) compared with students in the CG (pretest M = 3.37; SD = 3.303; posttest M = 3.21; SD = 2.84; total possible = 20). No other factors were significant.

Discussion

The purpose of this study was to examine the effects of a Tier 2 supplemental intervention focused on rational number equivalency concepts and applications on the mathematics performance of third-grade students with and without mathematics difficulties. Students who struggle to learn mathematics evidence difficulties understanding beginning equivalency concepts such as relationships involved with unit fractions, renaming fractions after they have been repartitioned into equivalent amounts, and identifying equivalent fraction relations when presented in pictorial and abstract representations. Results of the present study are encouraging in that ratio instruction resulted in an increase in performance on both measures for all students in the EG over and above performance in the CG. It is possible that ratio-like situations may help bolster understandings of balance and unit relationships in rational number equivalency situations (Vanhille & Baroody, 2002) and help students increase their performance on tests of rational number equivalence. Such understandings underscore a conceptual understanding of rational number equivalence relationships. Although the CCSS define equivalence as the understanding of two fractions that are the same size or at the same point on a number line, ratio-based intervention might work to strengthen conceptual understandings of fraction units in students who struggle to understand rational number equivalence through concepts of size alone. Further research is needed to evaluate the validity of the added value of ratio-based intervention on students’ performance in CCSS curriculum.

Although results are preliminary, there is encouraging evidence that the ratio-based intervention not only produced increased levels of performance for students who struggled, but brought their performance levels in line with students who did not struggle with mathematics, which is the intent of Tier 2 interventions. It remains unclear, however, from the results of this study, whether knowledge gained through participation in the intervention may have transferred to or strengthened curriculum taught in the core classroom.

Limitations and Future Research

Several limitations need to be acknowledged in this study. First, the researcher provided all of the instruction. While the instructional sessions were checked for fidelity of implementation by two independent observers, the results of the study provided no evidence of the effects of the instructional sequence implemented by other instructors. Future research evaluating the impact of the ratio-based instructional sequence with classroom teachers or tutors is planned. A second limitation may have been the length of the intervention. The instructional dosage of the current intervention was shorter than what is suggested and implemented by many RtI intervention researchers (Bryant et al., 2008; L. S. Fuchs et al., 2006). Thus, difference results could have been found had the intervention sustained a longer period of time.

Finally, more information is needed on differences in other elements brought with the use of the ratio intervention, such as the fraction interpretation used for instruction and the use of counterarguments to augment student thinking and error correction. Results of the current study suggest the need for different perspective in special education mathematics intervention research to address the question of how these instructional elements might add value to mathematics instruction for struggling students.

Conclusion

Ratio-like instruction may help bolster understandings of balance and unit relationships in rational number equivalency situations (Vanhille & Baroody, 2002) and help students increase their performance on tests of rational number equivalence. The results of the current study reflect potential positive results of adding supplemental instruction that explicates the unit relationship involved in rational numbers while teaching equivalency to students who struggle to learn mathematics. Reports of performance of students struggling to understand rational number equivalency underscore the continued need for intervention work in this foundational area of mathematics.

Footnotes

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.

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