Teaching Fraction Concepts Using the Concrete-Representational-Abstract Sequence

Abstract

Understanding related to fraction concepts is a critical prerequisite for advanced study in mathematics such as algebra. Therefore, it is important that elementary students form conceptual and procedural understanding of fractional numbers, allowing for advancement in mathematics. The concrete-representational-abstract (CRA) instructional sequence of instruction has been shown to be an effective means of teaching conceptual understanding of fractional numbers. The purpose of this study was to compare the effects of CRA with remedial multitiered systems of support (MTSS) Tier 2 instruction for teaching fraction concepts. Thirty-one fifth-grade students participated in two different Tier 2 interventions; one group received typical Tier 2 instruction with their general education teachers and the other received CRA instruction with the researchers. The researchers measured student performance using a pretest and posttest and found significant differences in progress favoring the CRA group. Results and implications are discussed.

Keywords

mathematics elementary prereferral/MTSS concrete-representational-abstract CRA fractions

Fraction knowledge predicts later mathematical achievement (Bailey, Hoard, Nugent, & Geary, 2012). For example, solving algebraic equations requires more than just an understanding of whole number magnitude; it requires fractional understanding regarding the coordination of numerators and denominators and fraction magnitude (Booth & Newton, 2012). Therefore, in preparation for algebra, students must understand what fractions represent and how to use them (Booth, Newton, & Twiss-Garrity, 2014).

The foundation for understanding fractional numbers begins at the elementary level (Common Core State Standards Initiative, 2010). In the primary grades, standards reference halves and equal shares. In third grade, students identify fractions and compare them when given different models. Area models fill equipartitioned shapes to represent fractions. Set models separate groups of objects or pictures to represent fractions. For example, one shows 3/4 with a set of 12. Using the denominator of 3/4, one separates 12 into four sets of 3. Then, using the numerator, one selects three sets of 3; 3/4 of 12 is nine. Length models fill or shade number lines in which the spaces between each whole number are equipartitioned (e.g., number line showing fourths has four equipartitions between 0 and 1, 1 and 2). Fourth-grade standards expand fractional understanding to composition and decomposition into units, begin operations with like denominators, and lay the foundations for procedures related to equivalence and operations. With foundations mastered, mathematics standards in fifth grade and beyond include procedural knowledge related to operations and the use of fractions within prealgebra and algebra (Resnick et al., 2016). Without foundational understanding, students will struggle with complex standards as they advance from grade to grade (Charalambous & Pitta-Pantazi, 2007).

Tiered Intervention

Given the importance of understanding fractional numbers to students’ advancement in mathematics, teachers must identify early signs of failure. Schools currently use instructional models to identify academic deficits and prevent failure, namely, multitiered systems of support (MTSS). The purpose of MTSS is to prevent failure by providing students with successively more intensive academic and behavioral interventions, referred to as tiers (Fletcher & Vaughn, 2009). Within MTSS, all students participate in Tier 1 intervention and receive evidence-based instruction in general education classrooms. Students who do not demonstrate progress move to Tier 2 intervention in which teachers provide differentiated instruction and remediation. Students who do not demonstrate adequate progress within Tier 2 move to Tier 3 that targets specific skill deficits using more intensive methods (Glover & DiPerna, 2007).

Given the MTSS model, students who show signs of struggle with grade-level fractional concepts would move to Tier 2 instruction. Interventions should build their basic conceptual understanding of the coordination between numerators and denominators, fraction magnitude, and other basic concepts so that students can manipulate fractions according to grade-level standards. One intervention for building this conceptual understanding is the concrete-representational-abstract (CRA) sequence, the intervention used in the current study. CRA is a graduated sequence of instruction in which students complete mathematical tasks with the assistance of manipulative objects (concrete) and then pictures or drawings (representational), prior to using just numbers (abstract). The literature has shown the CRA sequence to be effective in teaching fraction concepts (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Jordan, Miller, & Mercer, 1998; Kim, Wang, & Michaels, 2015; Watt & Therrien, 2016).

CRA Research

The CRA fraction research began with Jordan et al. (1998). Using a group design, the researchers taught elementary students with and without learning disabilities in general education classrooms and compared CRA with business as usual instruction. The CRA group used an area fraction model by filling laminated circles with pieces and shading drawings. Based on the results from researcher-made tests, the CRA group improved their performance from a mean of 17.05 to 45.94 out of 50, but the control group only improved from 16.73 to 28.29 out of 50.

Using materials and group sizes similar to Jordan et al. (1998), Butler et al. (2003) used CRA to teach fraction concepts to middle school students with mathematics disabilities. This study differed in that the researchers studied the benefits of concrete instruction and compared CRA with instruction that only included representational and abstract instruction (RA). The researchers taught fraction equivalence, transformation of improper fractions to mixed numbers, and application of those concepts in word problems. Concrete and representational instruction included area models (circular shapes similar to Jordan et al.) and set models (first, a group of objects/pictures is equipartitioned according to a denominator, and then one selects a particular number of those groups based on the numerator). The researchers measured student learning with assessments that included identification of fractions given shaded area models and sets of objects, conversion of a given fraction to an equivalent fraction, and conversion of mixed numbers to an equivalent improper fraction. Students in both groups made similar gains on these assessments; the CRA group improved by 63% and the RA group improved by 62%. However, on fraction identification, students who received CRA instruction improved by 47% and their peers in the RA group only improved by 27%.

Kim et al. (2015) also used the CRA sequence to teach fraction skills, but used a single case design and focused on word problems. The researchers used a set model; however, the materials were different from those used by Butler et al. (2003). The researchers used Computation of Fractions (Witzel & Riccomini, 2009), a fraction curriculum that uses explicit CRA instruction. At the concrete level, cups represented denominators and sticks represented the numerator. At the representational level, students used pictures of cups and sticks, and student drawings. Kim et al. provided one-on-one CRA instruction to three fourth-grade English language learners. The researchers demonstrated a functional relation between CRA and student performance using word problems. Tasks included finding that four out of 12 students is one third of the students and that one-half of 20 students is 10 students. Student performance in baseline was between 0% and 50%; after intervention, scores were 100%.

Watt and Therrien (2016) also implemented CRA with Computation of Fractions (Witzel & Riccomini, 2009). However, these researchers used CRA as a preteaching intervention for sixth graders participating in tiered interventions. Using a group design, the researchers compared small-group CRA instruction with that of a comparison group receiving typical MTSS intervention. Researchers taught comparison, reducing fractions, and addition and subtraction of fractions with like and unlike denominators. On a researcher-generated assessment, the CRA group improved from 9.35 to 13.4 items correct, but their peers decreased from 8.80 to 8.76.

The previous studies showed that the CRA sequence is an effective intervention to teach fraction concepts related to equivalence, computation, and application to word problems. The aforementioned CRA studies also successfully used objects, pictures, and drawings with area and set models. Recent research suggests that the materials and representations of fractions within CRA instruction are effective components of interventions for students at risk for failure; however, other researchers suggest that length models and measurement of fraction magnitude should be included in fraction interventions (Fuchs et al., 2013; Fuchs et al., 2016).

Fraction Research Using Length Models

Length models involve placing fractions on number lines and fraction magnitude is a fraction’s value or amount. The number line provides a more explicit representation of a fraction’s value as compared with 0, 1/2, 1, 2, or any other number. For example, 3/4 is closer to 1 than 1/2, but 3/4 is not as close to 1 as 7/8. Judgment of fraction magnitude is easier with this model because the number line visually displays the distance between a given fraction and other numbers.

Fuchs et al. (2013) compared fraction instructional programs for elementary students identified as at risk for failure. Using a group design, researchers compared an intervention with emphasis on measurement of fractions to a control program with emphasis on whole-part interpretation using area models. The intervention group received small-group instruction for 12 weeks. Instructors used and referenced area and set models, but emphasized length models and measurement. Students used objects, fraction blocks, pictures, and drawings. The control group received 12 weeks of instruction in the same content using similar materials, but with emphasis on the area model. The researchers measured the effects of the programs using assessments with 15 items directly aligned with instructional tasks as well as 18 items from the National Assessment of Educational Performance (NAEP). The NAEP assessment items included both whole-part interpretation and measurement. Students in the intervention group increased their fraction comparison by 5.91 items compared with their peers’ decrease in performance of 1.77 items. Intervention students decreased their percentage of error in identifying fraction magnitude 15%, but their peers only decreased their error by 4%. Out of 18 NAEP items, intervention students increased their performance by 6.02 items and their peers improved by 3.04 items. The researchers further analyzed NAEP performance and found that students’ measurement skills influenced fraction knowledge. Fuchs et al. (2016) compared the intervention used by Fuchs et al. (2013) with a business as usual intervention for elementary students. This second study replicated the results showing that measurement skills positively mediate increases in fraction knowledge.

Given the conclusions of Fuchs et al. (2013; Fuchs et al., 2016), the inclusion of the length model within fraction interventions is promising. Repeated experience with fractions on number lines may provide students with greater understanding of fractions, understanding about how to determine their magnitude or value through the coordination of numerators, and denominators, and understanding comparison or ordering of fractions. Currently, research for CRA fraction instruction is missing the length model. Therefore, the purpose of the current study was to use the CRA sequence to teach students receiving Tier 2 interventions fraction and decimal concepts using area and length models. The research question follows. Is there a difference between CRA instruction including both length and area models and business as usual Tier 2 instruction?

Method

Participants

The participants were 31 fifth-grade students. The criteria for participation was parent permission, student assent, a standard score of at least 85 on the Operations subtest of the KeyMath 3 Diagnostic Assessment (Connolly, 2007), and a pretest score of 65% or less. The pretest comprised items representing fourth-grade standards related to fractions and decimals described in the “Measures” section. The Operations subtest of the KeyMath involves whole numbers, fractions, and decimals, but a standard score of 85 would ensure that students had prerequisite computation skills associated with whole numbers. The intervention group consisted of 17 students who received CRA instruction. The intervention group scored an average 46% correct (SD = 17.52) on the pretest and standard score of 99 (SD = 4.91) or a grade equivalent of 5.3 on the KeyMath. The comparison group consisted of 14 students who received business as usual Tier 2 instruction. They had an average pretest score of 58% correct (SD = 10.12) and operations standard score of 99 (SD = 4.66) on the KeyMath which is a grade equivalent of 5.3. The school did not provide information about participants’ subsidized lunch. None of the participants had identified disabilities. Each group included one student who received English Language Learning (ELL) instruction. Demographic information is located in Table 1.

Table 1.

Student Demographics.

Student characteristics	Intervention (CRA)	Comparison
Gender
Male	n = 9	n = 8
Female	n = 8	n = 6
Cultural background
White	n = 9	n = 7
African American	n = 4	n = 6
Latino/Latina	n = 4	n = 1
KeyMath computation standard scores 91–100	n = 8	n = 4
Grade equivalence 4.1–5.1
KeyMath computation standard scores 101–110	n = 9	n = 10
Grade equivalence 5.2–6.1
Pretest assessment scores 0%–40% correct	n = 7	n = 2
Pretest assessment scores 45%–65% correct	n = 10	n = 12

Note. CRA = concrete-representational-abstract.

Setting and Distinctions Between Groups

The research took place in an elementary school in a small town near a major university in the Southeastern United States. Both groups received Tier 2 instruction for 25 min each day for 4 days per week. The researchers implemented intervention instruction in a classroom that had the same resources as general education classrooms with 25 desks, an interactive white board, document camera, and large dry erase boards on the walls. Students in the intervention group received instruction as a whole group. The ratio of instructor to students was 2:17. At the same time, students in the comparison group remained in their general education homeroom classrooms. The students worked with their general education teachers at small tables in groups of six or less. The ratio of teacher to students was 1:3–6.

Procedures

Due to the requests of school personnel, the researchers and teachers followed existing intervention schedules and kept groups intact. Therefore, the study was a quasi-experimental nonequivalent group pretest and posttest design, using intact groups (Heppner, Wampold, & Kivlighan, 2008). Researchers use this design in schools because groups are previously established based on need, condition, and availability (Heppner et al.). In collaboration with the teachers, the researchers created pretests and posttests to measure fourth-grade standards from the Alabama College and Career Ready Standards that include the Common Core State Standards Initiative (2010). The teachers intended to use the tests to assess readiness for grade-level concepts and provide Tier 2 supports. These tests measured fraction knowledge one would assume mastered in the fourth grade (unit fractions, fraction equivalence, relation between fractions and decimals, and decimal notation). At the beginning of October, the teachers administered pretests to all fifth-grade students to assess their mastery of fourth-grade fraction and decimal concepts. The teachers used these data to determine which students needed Tier 2 intervention. Following the approved Institutional Review Board procedures, by mid-October, the researchers sent home permission forms to all students who scored 65% and below on the pretest, the teachers’ criteria for Tier 2 fraction instruction. Once students returned permission forms, the study began with intact groups of students receiving Tier 2 instruction. The researchers obtained student assent and administered the Operations subtest of the KeyMath 3 Diagnostic Assessment (Connolly, 2007). Starting the first full week of November, the researchers implemented CRA as Tier 2 supplemental instruction. At the same time, the comparison group received business as usual Tier 2 supplemental instruction.

Business as Usual Comparison Group Instruction

The district-prescribed materials for the comparison group were GoMath, a comprehensive K–8 mathematics curriculum adopted for use in all district elementary schools (Dixon, Leiva, Larson, & Adams, 2012). The researchers did not observe instruction or collect fidelity checks. The researchers gathered information about instruction through teacher reports and examination of lesson sheets. Review and practice instruction included two problems, one for demonstration and one for guided practice. Lessons included the area model or the length model, but both models did not appear in the same lesson. For each small group, one teacher modeled a task and guided students in completing another. After guided practice, students independently completed six to 10 problems on their own. Teachers reported that they provided written and verbal feedback.

CRA Intervention Instruction

The researchers implemented the CRA sequence with explicit instruction across all lessons (Hudson & Miller, 2006). All lessons involved an advance organizer (overview), demonstration with physical modeling and thinking aloud, guided practice with back and forth task completion (the instructor completed part of a task and students completed and verbally told the instructor how to complete the next part of the task), independent practice, feedback, and a post organizer (wrap-up). Two researchers were always present for lessons. One researcher at a time demonstrated concepts and led guided practice for the whole group using a document camera or white board. Although one researcher taught a portion of the lesson, the other circulated around the room to ensure students were on the correct part of the lesson or had appropriate materials. The researchers switched roles between tasks (except 1 day per week when one researcher completed the treatment integrity checklist). Both researchers provided feedback to students by circulating throughout the group after independent practice tasks (six to 12 items). If a student made an error on independent items, a researcher provided the student with feedback with another model and guided opportunity as appropriate given that the intervention used explicit instruction. Three or fewer students needed this feedback per lesson. Students progressed to the next lesson if they had 80% accuracy on first-attempt independent tasks. All students progressed without need for lesson repetition. The researchers presented lesson content in the order it appears in the standards and according to the development of conceptual understanding. Table 2 shows an overview of lesson content.

Table 2.

Lesson Overview.

Lesson	Topic introduced	Lesson items
1–3	Concrete DConcrete WP	Model: 2 D area, 2 D length, 1 WP; Guide: 2 D area, 2 D length, 1 WP; IP: 2 D area, 2 D length, 2 WP
4–6	Representational DRepresentational WP	Model: 3 D area, 2 D length, 1 WP; Guide: 3 D area, 2 D length, 2 WP; IP: 3 D area, 2 D length, 2 WP
7–9	Abstract D, WPConcrete EF	Model: 2 D, 1 WP, 3 EF area, 3 EF length; Guide: 3 EF area, 3 EF length; IP: 3 EF area, 3 EF length, 2 D, 1 WP
10–12	Abstract WPRepresentational EF	Model: 1 WP, 3 EF area, 3 EF length; Guide: 3 EF area, 3 EF length; IP: 3 EF area, 3 EF length, 1 WP
13–15	Abstract WP, EFConcrete FD, D, DNL	Model: 1 WP, 3 EF, 3 FD, 2 D; Guide: 3 EF, 3 FD, 3 D; IP: 2 WPs, 3 EF, 3 FD, 3 D
16–18	Abstract WP EFRepresentational FD, D, DNLConcrete FA	Model: 1 WP, 3 EF, 2 FA, 3 FD, 3 DNL; Guide: 1 WP, 2 FA, 3 EF, 3 FD, 3 DNL; IP: 1 WP, 3 EF, 3 FD, 3 DNL, 3 FA
19–20	Abstract WP, EF, DNLRepresentational FA	Model: 1 WP, 3 EF, 2 FA, 3 FD, 3 DNL; Guide: 1 WP, 2 FA, 3 EF, 3 FD, 3 DNL; IP: 1 WP 3 EF, 2 FA, 3 FD, 3 DNL

Note. D = decomposition of fractions; WP = word problem; IP = independent practice; EF = equivalent fractions; FD = fraction to decimal; DNL = decimal on number line; FA = fraction addition.

The materials included sheets divided according to items used for modeling, guided practice, and independent practice. The materials for concrete instruction included plastic fraction tiles, fraction blocks, base 10 blocks, and coins. The materials for representational instruction were pictures and number lines printed on the learning sheets that the students shaded. Materials for abstract instruction were learning sheets with problems or equations with numbers and symbols only. Description of the processes for teaching follow.

Decomposition of fractions to solve word problems (concrete)

A researcher taught students to decompose or break fractions apart and make an addition equation using their unit parts (3/5 = 1/5 + 1/5 + 1/5). First, students made a fraction using blocks. Then, they successively broke the fraction into its unit parts and wrote the corresponding equation. Students used this skill to solve word problems. An example word problem involved three students using four fifths of a pack of paper. One student used two fifths of the pack and two students each used the same amount of the remaining paper. Decomposing four fifths using blocks (4/5 = 1/5 + 1/5 + 1/5 + 1/5) assisted the students in determining that the two students each used one fifth of the pack (4/5 = 2/5 + 1/5 + 1/5).

Decomposition of fractions to solve word problems (representational)

Students decomposed fractions and solved word problems by successively shading parts of shapes (area model) and number lines (length model) as they developed the addition equation. The materials presented an array of equipartitioned shapes or number lines. Given the fraction, students chose the appropriate shape or number line by evaluating the denominator. One unit at a time, the students shaded the shape or number line and made an addition equation. The students solved word problems using the equations made with their drawings and equations.

Decomposition of fractions to solve word problems (abstract)

The researchers taught students to write addition equations corresponding to a given fraction using just numbers (3/4 = 1/4 + 1/4 + 1/4). Students used addition equations involving unit fractions to solve word problems. For example, if two boys ate 3/4 of a candy bar and one boy ate 2/4, the other ate 1/4.

Pictorial representations of equivalent fractions and decimals

The fourth-grade standard associated with this skill specifically states that abstract level procedures are not included, only pictorial. Given that the standard included the word “pictorial,” there was no abstract phase because students cannot demonstrate the standard using only numbers and symbols. Adding fractions with unlike denominators at the abstract level is a fifth-grade standard and this study only addressed fourth-grade standards. This is the only skill area within the CRA intervention in which there was no abstract instruction. Instruction included concrete and representational area models.

Pictorial representations of equivalent fractions and decimals (concrete)

Through explicit instruction, the students learned to show that the fraction 3/10 equals 30/100. At the concrete level, students used squares divided into 10 parts and 100 parts. Using the area model, students filled these shapes with base-10 blocks, tens blocks for tenths and ones blocks for hundredths.

Next, students solved addition problems involving tenths and hundredths (3/10 + 5/100). Students represented each fraction using base-10 blocks. This task involved transforming tenths to hundredths to complete the problem (3/10 became 30/100). The researchers emphasized the need for the units to be the same to complete the problem.

Pictorial representations of equivalent fractions and decimals (representational)

Students shaded squares to show 2/10 = 20/100. Then, students solved addition problems by shading squares. The researchers showed students how this process related to current procedural instruction occurring in their fifth-grade general education classrooms regarding lining up decimal points when adding (0.3 + 0.05); a common error is an answer of 0.08 or 0.8 because students do not recognize that this problem, 0.3 + 0.05 is 3/10 + 5/100.

Equivalent fractions (concrete)

The researchers taught students to complete the following equation with a missing numerator: 3/4 = /12. Students made a multiplication equation showing how to make equivalent fractions: 9/12 = 3/4 × 3/3. At the concrete level, students used fraction blocks placed on shapes (area model) or number lines (length model). The blocks showed how one would make a fraction equal to one used in a multiplication equation (there are three twelfths-blocks for each fourth-block, so fraction of one is 3/3).

Equivalent fractions (representational)

Students shaded equipartitioned shapes (area model) and number lines (length model) of the same size. They used these shaded shapes and number lines to make equivalent fractions (2/6 = 1/3). They made multiplication equations (2/6 = 1/3 × 2/2) by examining the shaded number lines or shapes (for each third, there are two sixths). An example of instruction at the representational level is shown in Figure 1.

Figure 1.

Making equivalent fractions and corresponding multiplication equations at representational level.

Equivalent fractions (abstract)

The researchers provided a partially completed equation (2/6 = /3). Students completed the equation by making a multiplication equation. The students multiplied the fraction with the largest units by a fraction equal to one (2/6 = /3 × 2/2). They found the missing multiplier (2/6 = 1/3 × 2/2) and equivalent fraction (2/6 = 1/3).

Locate decimals on number line (concrete)

The researchers taught students to locate a given decimal on a number line labeled with 0, 1/2, and 1. A number line is a pictorial or representational model. The researchers used the number line to represent magnitude at the concrete, representational, and abstract levels. The researchers placed coins (authentic U.S. currency) next to the number line. The researchers used coins because students were (a) familiar with its notation, and (b) familiar with the use of tenths and hundredths in relation to pennies and dollars. Students located a given decimal by using coins and number lines divided into tenths (dime on each tenth), fourths (quarter on each fourth), and halves (two quarters on each).

Locate decimals on number line (representational)

The researchers substituted pictures of coins printed next to number lines for the actual physical coins. The researchers presented students with a decimal such as 0.15 and a number line labeled with benchmarks of 0, 1/2, and 1. The researchers used an additional sheet of paper with number lines with pictures of coins next to each equal part (picture of a dime at each tenth, picture of a quarter at each fourth, and pictures of two quarters at each half). Given the decimal 0.15, the researchers and students discussed its general magnitude; 0.15 was less than one-half or 0.50 and closer to zero than 0.50. Using sets of number lines with pictures of coins, the researchers taught students to more accurately locate 0.15 using their knowledge of money. The decimal, 0.15 (like 15 cents), is half-way between 0.10 (10 cents) and 0.20 (20 cents). The researchers and students located 0.15 using the sheet with pictures of dimes next to the number line. Then, they moved that sheet close to their learning sheet to mark 0.15 on the number line with benchmarks. Students used this process to locate decimals such as 0.12, 0.76, and 0.24.

Locate decimals on number line (abstract)

The students located given decimals on number lines with benchmark fractions. There were no extra numbers lines with pictures of coins to assist. During abstract instruction, the researchers taught students to think about given decimals and their relation to money and/or fractions. For example, when given the decimal 0.24, researchers taught students to think about the given decimal in relation to a familiar decimal, 0.25 or one fourth. Using this estimate, the given number line would have four equal parts and the mark for 0.24 decimal would be around the first mark from zero.

Measures

A researcher created a pretest and posttest. Each included 20 items corresponding to fourth-grade standards related to fraction and decimal numbers. The pretest and posttest had no specific time limit. The students’ homeroom teachers administered them during a class period and all students completed them within the 50-min period. The test items were as follows: (a) seven items in which students solved word problems using knowledge of unit fractions; (b) two items in which students used pictorial representations of areas models to complete equations of equivalent fractions with denominators of 10 and 100 (3/10 = /100); (c) two items in which students completed addition equations with unlike denominators of 10 and 100 by using pictorial representations (5/10 + 6/100 = /100); (d) three items in which students completed equations showing how decimals were written as fractions (0.76 = /100); (e) three items in which students were given a decimal and they marked its placement on a number line labeled with a 0, 1/2, and 1; and (f) three items in which students completed an equation with a missing numerator (1/4 = /8). With the exception of the number line marking task, each item had one specific answer. To compensate for handwriting, correct answers for the number line could vary 5% in either direction from the correct placement. The researchers calculated this using the percent of absolute error (Siegler, Thompson, & Schneider, 2011). This was the length of the student’s mark minus the length of the accurate mark divided by the total length of the number line. The researcher placed marks that varied 5% in either direction from the correct placement on a template. Researchers used this overlay to score correct answers in the range.

Due to the different fraction and decimal concepts tested, the researchers computed internal consistency estimates. To examine how closely each item correlates to the concepts measured, the researchers employed Cronbach’s alpha. The researchers conducted split half reliability to check the correlation between the two halves of each test. For the pretest, Cronbach’s alpha (internal consistency reliability) was 0.77 and the split half coefficient was 0.86. For the posttest, Cronbach’s alpha was 0.80 and the split half coefficient was 0.82. The researchers asked teachers to assess validity. The criteria for serving as an expert judges was elementary certification and at least 5 years of experience using fourth-grade standards to teach mathematics. The judges were three elementary mathematics teachers with master’s degrees and at least 9 years of experience teaching fourth-grade mathematics standards. The teachers completed a survey that incorporated each problem and the corresponding standard. The teachers reported whether each item assessed its corresponding standard using a Likert-type scale. All teachers strongly agreed that each item assessed the corresponding fourth-grade standard.

Fidelity and Interrater Reliability

A researcher observed intervention lessons 1 time each week (25% of sessions) and completed a checklist of observed behaviors for each phase of CRA. For each checklist, the researcher checked yes or no that teacher behaviors occurred. The following are example items: (a) the teacher models the concept and calculation procedure using appropriate CRA materials; (b) decomposition involves the teacher placing blocks one at a time on number line while writing equation. The checklists indicated 100% treatment fidelity for teacher behaviors across lessons.

The researchers checked all of the pretests and posttests for interrater agreement. A researcher and graduate assistant graded all of the tests and compared their results. Prior to grading, the researcher met with the graduate student to explain how to use the answer key. All but three number line tasks had one answer. The graduate student practiced using the number line overlay to determine whether various scores fell within the correct range. The graduate student demonstrated that she could complete three of these with 100% accuracy before grading. The researchers calculated interrater agreement by adding the number of agreements and dividing that number by the number of agreements and disagreements. The disagreements were items in which the students’ writing was difficult to interpret. For the intervention group, interrater agreement was 99.4% for pretests (338 agreements and two disagreements) and 99.1% for posttests (337 agreements and three disagreements). For the comparison group, interrater agreement was 100% for pretests and 99.3% for posttests (278 agreements and two disagreements).

Analysis

Prior to beginning the intervention, the researchers analyzed the pretests and the standard computation scores of the KeyMath 3 Diagnostic Assessment to check for differences between the comparison and intervention groups. The researchers analyzed group differences using an ANOVA through SPSS. Using a one-way ANOVA, the observed power was 0.63 and the scores showed a significant difference between the groups’ pretest assessments, F(1, 29) = 7.88, p < .01. The intervention group scored lower with an average of 46% and a range of 10% to 65% (SD = 17.52%) correct while the comparison group’s performance averaged 58% with a range from 52% to 64% (SD = 10.12%). There was no statistically significant difference between the students’ standardized computation achievement.

Due to the nonequivalent group design (NEGD), the researchers analyzed the dependent variable (posttest) after controlling for differences (or measurement error) on the pretest. In a NEGD study, randomization of participants does not occur, placing the study at risk for inherent bias between groups. Due to the nonequivalence, there is measurement error in the pretest scores that can be mitigated using a reliability-based adjustment to the pretest scores (Trochim, 2006). A reliability adjustment consists of adjusting the pretest scores for all participants based on the internal consistency reliability coefficient of the instrument. In addition, the researchers utilized an ANCOVA to examine between group differences, while attempting to control for influence from the pretest.

In an effort to be conservative with the main analyses, the researchers completed an ANCOVA using the adjusted pretest scores to reduce the bias between groups to the greatest extent possible. In addition, use of a mixed between–within ANOVA allows for examination of change over time (within analysis) and differences between the groups (between the comparison and the intervention group). This provides evidence of change over time (from pretest to posttest) as well as evidence of differences between groups. Taken as a whole, if researchers find significant results, these tests demonstrate that the intervention significantly affects learning, and that such an increase is significantly different from the comparison group’s performance. Therefore, the pretest scores were adjusted for reliability and these data were analyzed using a mixed between–within ANOVA (Tabachnick & Fidell, 2007; Trochim, 2006). The adjusted average of the pretest scores for the intervention group was 46%. The adjusted scores range from 18% to 61% (SD = 13.59%). The adjusted average of the pretest scores for the comparison group was 58%. The adjusted scores range from 40% to 63% (SD = 7.56%). The means, adjusted means, and standard deviations are presented in Table 3. For this study, the difference in the effects of the treatment on the outcome can be overestimated or underestimated due to nonequivalence of the comparison and treatment groups (or lack of random assignment to groups; Hsu, 2003). Due to this design (NEGD and without random assignment), there is inherent bias in the estimate of the impact of the treatment. This is one of the disadvantages of not having a full experimental study with random assignment (Hsu, 2003). Therefore, this study had a quasi-experimental NEGD (Pallant, 2016). Researchers analyzed the posttests using a one-way ANCOVA, controlling for the bias in pretest scores. Last, the researchers conducted a post hoc power analysis through G*Power (Faul, Erdfelder, Buchner, & Lang, 2009). Analysis through G*Power determined that with a sample size of N = 31 and actual effect size of .67, power for this sample is .95, which was sufficient to find significant results.

Table 3.

Results for Pretests and Posttests.

Ms and SDs	M	SD	Adjusted M	Adjusted SD	ANCOVA M	SD	Effect size	Observed power
Intervention (CRA) group
Pretest	46	17.52	46	13.59	36	13.59
Posttest	89	11.10			89	11.10
							0.67	1.00
Comparison group
Pretest	58	10.12	58	7.56	58	7.56
Posttest	53	18.14			53	18.14

Note. CRA = concrete-representational-abstract.

Results

The results of the ANCOVA indicated a significant change in student performance after intervention, F(1, 28) = 57.28, p < .001, η² = 0.67. To be more specific with the analyses, researchers conducted a mixed between–within ANOVA with the between subjects factor being group (comparison or intervention) and the within subjects factor or dependent variable being the posttest score. There was an interaction between group and the adjusted pretest, Wilks’s Λ = .26, F(1, 29) = 82.98, p = .001, η² =.74. There was a main effect for group, such that there were significant differences between the treatment and comparison group, as indicated above in the original reliability-adjusted ANCOVA. There was also a main effect for time, such that there was significant change from pretest and posttest, Wilks’s Λ = 0.34, F(1, 29) = 55.65, p < .001, η² = 0.66. The means, adjusted means, and standard deviations are presented in Table 3. These results all demonstrate medium effect sizes (Cohen, 1988).

Discussion

The purpose of this study was to compare CRA intervention with MTSS Tier 2 business as usual instruction. The results of this study demonstrate CRA instruction led to greater learning than business as usual instruction. The business as usual group’s performance slightly decreased. This decrease is contrary to the expected change after Tier 2 instruction. Both business as usual and the CRA group had similar prerequisite skills. Both received Tier 2 instruction. The difference in performance may be due to instructional design of the Tier 2 interventions.

The current study’s business as usual comparison group used representational instruction, similar to the comparison RA group in the study conducted by Butler et al. (2003). Butler et al.’s results and current results support the benefit of using concrete instruction because students who received concrete instruction performed better. However, the inclusion of concrete instruction was not the only way instruction differed between groups in the current study. The CRA group received more modeling, guidance, and repeated practice with items. The comparison group had one model and one guided example prior to independent practice. The CRA lesson materials included multiple examples of area models and length models for applicable concrete or representational tasks within modeling and guided practice. These additional chances to observe and practice added to the intensity of the CRA intervention.

Although other fraction interventions including the CRA sequence have results similar to the current study (Butler et al., 2003; Jordan et al., 1998; Kim et al., 2015; Watt & Therrien, 2016), none of the previous CRA research included both area and length models. As recommended by Fuchs et al. (2016), length models may have contributed to the increased performance of the current intervention group, providing students with a greater sense of magnitude. For example, on the posttest, all students in the CRA group marked the decimal, 0.25, as less than one-half, but their peers marked it as more than one-half or at the numeral one. Students in the CRA group experienced repeated practice with the length model, developing a greater sense of magnitude as compared with their peers. The students’ experiences with the number line showed distances between a given fraction or decimal and other numbers in way that was more obvious than the area and set models demonstrated. In addition, instruction included a connection to familiar money concepts at the concrete and representational levels. The length model and connection to background knowledge may have led to greater understanding.

Limitations and Future Research

The current study is limited because the researchers did not randomly assign students to instructional groups. By using intact groups of students, there were differences in pretest scores; the CRA group’s scores were lower. Although researchers accounted for this in the analyses, random assignment or matched pairs may have prevented this difference in the first place. The researchers did not assess treatment integrity for the businesses as usual comparison group. Therefore, the researchers do not know whether comparison instruction followed the materials used or teacher descriptions provided. Another limitation was researcher implementation. This makes the results less generalizable in typical settings. Therefore, future research should include typical implementation by teachers. The study is limited because teacher-to-student ratios differed across groups. The CRA group of 17 students had two researchers and the comparison groups had one teacher per six or fewer students. Research should ensure comparable ratios.

Implications

The researchers provided instruction for 25 min, 4 days per week for 5 weeks to a group of 17 students without splitting them into smaller groups. Similar to Jordan et al. (1998) and Butler et al. (2003), large-group explicit CRA instruction resulted in significant improvement in skills. Large-group CRA instruction provided students with more practice opportunities than the comparison group. This is important since general education teachers may not regularly have the ability to provide small-group Tier 2 instruction, as was available in the current study. The results from the current study support the need for Tier 2 instruction that provides students with multiple opportunities for practice with multiple representations of concepts. The students in the CRA group achieved significantly higher results during the same 25-min, 4-day per week intervention schedule. The improvement of the intervention group was statistically significant with a medium effect size (η² = 0.67). This is also socially significant in that students improved from a percentage that would be considered failing (below 50%) to a percentage that is above average (89%). Furthermore, the intervention group began the study with less skill as demonstrated by lower pretest scores. It was possible to provide explicit instruction with many practice opportunities to a large group of students. Given the time and resources needed for implementation, the results show that Tier 2 explicit CRA interventions may have promise and research should continue to investigate and refine their implementation. It is critical that we seek feasible Tier 2 interventions that prevent further failure. The comparison group did not improve their performance and, within MTSS, may likely move to greater levels of support within Tier 3. However, this increase in support could be unwarranted. Schools must make the best use of limited time and personnel. Effective Tier 2 instruction does more than support student achievement, it saves these resources. Future research is needed to further investigate needs and develop effective and feasible Tier 2 supports in mathematics.

Footnotes

Associate Editor: Margaret King-Sears

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

Bailey

D. H.

Hoard

M. K.

Nugent

Geary

D. C.

(2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113, 447–455. doi:10.1016/j.jecp.2012.06.004

Booth

J. L.

Newton

K. J.

(2012). Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37, 247–253. doi:10.1016/j.cedpsych.2012.07.001

Booth

J. L.

Newton

K. J.

Twiss-Garrity

L. K.

(2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118. doi:10.1016/j.jecp.2013.09.001

Butler

F. M.

Miller

S. P.

Crehan

Babbitt

Pierce

(2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice, 18, 99–111.

Charalambous

C. Y.

Pitta-Pantazi

(2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64, 293–316. doi:10.1007/s10649-006-9036-2

Cohen

J. W.

(1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum.

Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Connolly

A. J.

(2007). KeyMath 3 diagnostic assessment. Minneapolis, MN: Pearson.

Dixon

J. K.

Leiva

M. A.

Larson

Adams

T. L.

(2012). Go math! Grade 4. Orlando, FL: Houghton Mifflin Harcourt.

10.

Faul

Erdfelder

Buchner

Lang

A. G.

(2009). Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41, 1149–1160.

11.

Fletcher

J. M.

Vaughn

(2009). Response to intervention: Preventing and remediating academic difficulties. Child Development Perspectives, 3, 30–37. doi:10.1111/j.1750-8606.2008.00072.x

12.

Fuchs

L. S.

Schumacher

R. F.

Long

Namkung

Hamlett

C. L.

Jordan

N. C.

. . . Changas

(2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105, 683–700. doi:10.1037/a0032446

13.

Fuchs

L. S.

Schumacher

R. F.

Long

Namkung

Malone

A. S.

Wang

. . . Changas

(2016). Effects of intervention to improve at-risk fourth-graders’ understanding, calculations, and word problems with fractions. Elementary School Journal, 116, 625–651. doi:10.1086/686303

14.

Glover

T. A.

DiPerna

J. C.

(2007). Service delivery for response to intervention: Core components and directions for future research. School Psychology Review, 36, 526–540.

15.

Heppner

P. P.

Wampold

B. E.

Kivlighan

D. M.

(2008). Research design in counseling (3rd ed.). Belmont, CA: Cengage Learning.

16.

Hsu

L. M.

(2003). Random sampling, randomization, and equivalence of contrasted groups in psychotherapy outcome research. In Kazdin

A. E.

(Ed.), Methodological issues and strategies in clinical research (pp. 147–161). Washington, DC: American Psychological Association.

17.

Hudson

P. P.

Miller

S. P.

(2006). Designing and implementing mathematics instruction for students with diverse learners. Upper Saddle River, NJ: Pearson.

18.

Jordan

Miller

M. D.

Mercer

C. D.

(1998). The effects of concrete to semiconcrete to abstract instruction in the acquisition and retention of fraction concepts and skills. Learning Disabilities: A Multidisciplinary Journal, 9, 115–122.

19.

Kim

S. A.

Wang

Michaels

C. A.

(2015). Using explicit C-R-A instruction to teach fraction word problem solving to low-performing Asian English learners. Reading and Writing Quarterly, 31, 253–278. doi:10.1080/10573569.2015.1030999

20.

Pallent

(2016). SPSS survival manual, 6th edition: A step-by-step guide to data analysis using IBM SPSS. New York: McGraw Hill.

21.

Resnick

Jordan

N. C.

Hansen

Rajan

Rodrigues

Siegler

R. S.

Fuchs

L. S.

(2016). Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Developmental Psychology, 52, 746–757. doi:10.1037/dev0000102

22.

Siegler

R. S.

Thompson

C. A.

Schneider

(2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273–296.

23.

Tabachnick

B. G.

Fidell

L. S.

(2007). Using multivariate statistics (5th ed.). Boston, MA: Allyn & Bacon.

24.

Trochim

W. M.

(2006). The research methods knowledge base. Retrieved from http://www.socialresearchmethods.net/kb/quasnegd.php

25.

Watt

S. J.

Therrien

W. J.

(2016). Examining a preteaching framework to improve fraction computation outcomes among struggling learners. Preventing School Failure, 60, 311–319. doi:10.1080/1045988X.2016.1147011

26.

Witzel

B. S.

Riccomini

P. J.

(2009). Computation of fractions: Math intervention for elementary and middle grades students. Upper Saddle River, NJ: Pearson.