Concrete-Representational-Abstract–Integrated as a Tier 2 Instruction to Teach Addition

Abstract

Mathematics is crucial to the educational and vocational success of students. The concrete-representational-abstract (CRA) approach is a method to teach students mathematical concepts. The CRA involves instruction with manipulatives, representations, and numbers only in different lessons (i.e., concrete lessons include manipulatives but not pictures or numbers only). Researchers are exploring integrating the phases of CRA, referred to as concrete-representational-abstract–integrated (CRA-I), because it may be a more efficient method of instruction. The CRA-I integrates all phases starting with the first lesson (e.g., Lesson 1 includes manipulatives, pictures, and numbers only) and then fades concrete and representational instruction. The purpose of this article is to provide information about a CRA-I Tier 2 mathematics intervention on additive reasoning for second-grade students in a rural school in the southeastern United States. The researchers describe CRA-I, the lessons implemented, and implications for teacher use.

Keywords

rural special education explicit instruction concrete-representational-abstract (CRA)concrete-representational-abstract–integrated (CRA-I)

Major challenges in rural education involve issues such as poverty, geographical remoteness, curriculum relevancy, and a shortage of qualified personnel (Dexter et al., 2008; Stelmach, 2011). Other challenges include outcomes such as low achievement rates of students who are at risk of or identified with a high incidence disability (Dexter et al., 2008; Stelmach, 2011). Education in rural areas often must address conditions of poverty as well as specific local needs of residents regarding education access, instructional quality, the learning environment, and schools’ capacity to plan and implement education (Stelmach, 2011). More instructional methods should be explored to address student learning within the heterogeneous groups in rural communities (Smit & Humpert, 2012). One promising approach involves multitiered instructional systems.

Multitiered instruction is also a way to allocate resources to meet student needs. It can have broad benefits for students in rural areas because it is a way to employ sound instructional practices; provide alternative learning opportunities for students with cultural, linguistic, or ethnic differences; and separate the need for remediation from a learning disability (Kashi, 2008). In addition, students in remote locations can receive intensive instruction without having to wait for professionals to identify special education eligibility using a multitiered system. The literature on multitiered instructional systems include Response to Intervention (RTI) and Positive Behavior Interventions and Supports (PBIS). Both RTI and PBIS are based on implementing evidence-based instruction, increasing supports based on data, and continually monitoring progress. Levels of support are referred to as tiers. Most schools have three tiers, although some may have more. Tier 1 is a universal evidence-based instruction that all students receive. If students do not show sufficient progress in Tier 1, they receive supplemental instruction known as Tier 2. This tier consists of targeted instruction for students that is focused on a specific skill. Students who do not make adequate progress after Tiers 1 and 2 interventions will receive more intensive instruction, Tier 3.

In the school where this study took place, students were having difficulty using mental strategies in the application of addition. Progress monitoring indicated students lacked conceptual knowledge of addition. Researchers collaborated with the teacher to create a Tier 2 intervention that could be implemented within the classroom setting and also make use of currently available resources.

The Intervention

The purpose of this article is to provide information about this Tier 2 mathematics intervention on additive reasoning for second-grade students in a rural school in the southeastern United States. A veteran teacher with 20 years of experience worked with the authors and implemented an intervention that addressed students’ difficulties with addition. Specifically, students had to recall sums with automaticity and use mental strategies (e.g., use of decomposing numbers of an equation to make it more understandable) to fluently add equations with sums to 20. The teacher needed an intervention that could (a) be incorporated into a 30-min intervention time, (b) address deficits in students’ mathematical understanding that contributed to their struggles, and (c) increase recall of sums with fluency. To ensure conceptual understanding, the intervention needed to teach students additive reasoning before moving to fluency. So, for example, an intervention with a focus on repeated practice would not have been appropriate although much research has confirmed the effectiveness of this strategy (Burns et al., 2010; Codding et al., 2011). An additive reasoning intervention was created and successfully implemented across two second-grade classes. This intervention involved explicit instruction, multiple representations, scaffolding, and repeated practice. The concepts taught included the addition operation as well as number and mathematical knowledge (meaning of equals sign, number magnitude, place value, commutative property, relationship between addition and subtraction). To include explicit instruction (Miller et al., 2011), multiple representations, scaffolding, and repeated practice, the additive reasoning intervention followed the concrete-representational-abstract–integrated (CRA-I) sequence.

Concrete-Representational-Abstract (CRA) Method

The CRA method is a systematic, explicit approach to teaching mathematics using multiple representations (Miller et al., 2011). CRA shows how and why we count, perform operations, and solve problems (Witzel, 2005). It is systematic in that the types of illustrations used to model mathematics operations or concepts are presented in levels or phases. Bouck et al. (2018) stated that implementation of CRA for students as a tiered support could be a potential extension of the CRA literature. Often, CRA is implemented as separate phases: the concrete phase with objects, the representational phase with pictures or drawings, and the abstract phase with numbers only.

Concrete phase

The concrete phase provides students with a visual and hands-on experience. Researchers have found that manipulatives used in the CRA method improve mathematical outcomes for students who are at risk of or have disabilities (Bouck & Park, 2018; Peltier et al., 2020). Teachers can use manipulatives that come in a purchased curriculum, but they may also use readily available materials, for example, plates and paper clips (Miller et al., 2011; Peltier et al., 2020). One example of concrete instruction is having students use counters such as plastic squares grouped on paper plates to create the quantities and relationships of numbers represented in addition or subtraction equations. Students build their conceptual understanding by solving abstract equations (in this case addition or subtraction problems) using objects.

Representational phase

Once students master the operation using objects, they move to the next phase, representational instruction, in which they solve equations using pictures/drawings. Representational instruction includes number lines, pictures, or drawings that share similar features as the objects used in the concrete phase. For example, students might create drawings of 1s (short horizontal lines), 10s (long vertical lines), and 100s (squares) when showing place value, decomposition of numbers, or regrouping. When students create the representation with drawings, they are more actively involved in demonstrating their understanding because they are creating the representation themselves.

Abstract phase

Once students master solving equations at the representational level, they move to the abstract level. Many researchers have included a mnemonic strategy between the representational and abstract phases, with a focus on procedural knowledge (Miller et al., 2011). The abstract, and final, phase is the use of only numbers and symbols in solving problems. This may include a mnemonic strategy to assist students with the procedures associated with the process.

In the past, concrete and representational parts of CRA have been presented systematically, one at a time. However, Strickland (2012) discussed the difficulty that students have in moving from one phase of instruction to the other. Several researchers have shown that integrating the CRA steps (CRA-I) addresses this issue (Morano et al., 2020; Strickland & Maccini, 2013). Transition to the representational phase without objects and the abstract phase without pictures/drawings becomes more accessible because students already have experiences with all illustrations. The CRA-I instructional phases progress in an integrated order: (a) concrete, representational, and abstract representations; (b) representational and abstract representations; and (c) abstract representations only. The first level of CRA-I is solving abstract equations with concrete objects and representational pictures/drawings together. After mastery, the use of objects fades and students solve abstract equations with pictures/drawings which is the second level. After mastery using pictures/drawings, students work with just numbers and symbols and may also use a mnemonic strategy. Working with numbers and symbols is the third level. Table 1 provides a comparison of CRA and CRA-I.

Table 1.

CRA and CRA-I Stages.

Stage	CRA	CRA-I
1	• Solve equations and complete tasks with objects, numbers, and symbols • Demonstrate mastery	• Solve equations and complete tasks with objects, pictures and/or drawings, numbers, and symbols • Demonstrate mastery
2	• Solve equations and complete tasks with pictures and/or drawings, numbers, and symbols • Demonstrate mastery	• Solve equations and complete tasks with pictures and/or drawings, numbers, and symbols • Demonstrate mastery
3	• Solve equations and complete tasks with numbers and symbols • May include mnemonic device to assist with procedures • Demonstrate mastery	• Solve equations and complete tasks with numbers and symbols • May include mnemonic device to assist with procedures • Demonstrate mastery

Note. CRA = concrete-representational-abstract; CRA-I = concrete-representational-abstract–integrated.

To date, CRA-I has been considered a variation of an evidence-based practice; however, Morano et al. (2020) compared CRA and CRA-I instruction and found that both are effective. The difference between the two is the length of intervention. Morano et al. (2020) used a CRA-I involving three fewer lessons. It is possible that CRA-I is more efficient because conceptual and procedural knowledge are taught together as opposed to teaching conceptual understanding of the mathematics content and then procedural understanding (Rittle-Johnson et al., 2015).

Instruction Using CRA-I for Additive Reasoning

Mathematics instruction should focus on mathematical reasoning in the application of operational knowledge. Additive reasoning involves integrating the commutative and complement principles of mathematics to solve problems (Ching, 2016). Additive reasoning problems are one-variable problems in that quantities are put together (part + part = whole), separated (whole − part = part), or compared (larger − smaller = difference; Nunes et al., 2008). Operations of addition and subtraction should be taught together to build understanding of the commutative and complement principles, and problems need to be diverse in that multiple representations of problems are included (Nunes et al., 2008). CRA-I is a method teachers can use to teach additive reasoning.

CRA-I Lessons for Additive Reasoning

Every CRA-I lesson is explicit. Each lesson includes the explicit steps of the advance organizer, modeling, guided practice, independent practice, and postorganizer regardless of phase. The CRA-I lessons begin with an advance organizer in which the teacher gives an overview of the mathematics concept. The next step is modeling, during which the teacher thinks aloud the thought processes in solving the problem, counts, or answers questions aloud about known information. After modeling, the teacher implements guided practice, during which the teacher and students take turns in solving problems. The teacher systematically fades prompts in guided practice to build students’ independence. Once students demonstrate understanding in guided practice, the teacher moves to the next step, independent practice. In this step, students complete problems without assistance. Lessons end with a post-organizer in which the teacher reviews major concepts from the lesson and provides students general feedback about their performance.

The lessons can be separated into four different levels. Level 1 is the use of concrete, representational, and abstract illustrations during instruction. Level 2 is the use of representational and abstract illustrations during instruction. Level 3 is the memorization of a strategy that prepares students for abstract computation and a mnemonic to help students remember the strategy. Level 4 is the use of abstract instruction in which the strategy is applied to equations and word problems using numbers only. Table 2 outlines lesson objects for the addition intervention at each level of instruction.

Table 2.

Addition Intervention Lesson Goals and Objectives.

Lesson	Objectives
1: Students will solve addition equations & word problems; demonstrate knowledge of numbers & symbols using base-10 blocks, number lines.	• Students to add numbers w/ sums 10–20 by manipulating base-10 blocks.
	• Students use counting-on to add numbers w/ sums 10–20 using a number line.
	• Students show meaning of symbols in addition problems (addition & equal signs).
	• Students show meaning of 10s & 1s using base-10 blocks.
	• Students compare the magnitude of numbers using a number line.
2 & 3: Students solve addition & missing addend equations & word problems; demonstrate knowledge of numbers & symbols using base-10 blocks, number lines.	• Students to add numbers with sums 10–20 by manipulating base-10 blocks.
	• Students find missing addends with sums 10–20 by manipulating base-10 blocks.
	• Students use counting-on to add numbers w/ sums 10–20 using a number line.
	• Students show meaning of symbols in addition problems (addition & equal signs).
	• Students show meaning of 10s and 1s using base-10 blocks.
	• Students compare the magnitude of numbers using a number line.
4 & 5: Students solve addition & missing addend equations & word problems; demonstrate knowledge of numbers & symbols using pictures, drawings, & number lines.	• Students to add numbers w/ sums 10–20 by using pictures & making drawings.
	• Students find missing addends w/ sums 10–20 using pictures & making drawings.
	• Students use counting-on to add numbers w/ sums 10–20 using a number line.
	• Students show meaning of 10s and 1s by using pictures & making drawings.
	• Students compare the magnitude of numbers using a number line.
6 & 7: Students use pictures & drawings to transform unknown facts into known facts (doubles + 1 and addition to 10).	• Students use pictures & drawings to transform unknown facts (e.g., 4 + 3) into doubles to plus 1 (e.g., 3 + 3 + 1) and solve.
	• Students use pictures & drawings to transform unknown facts (e.g., 9 + 3) into fact involving 10 (e.g., 10 + 2) and solve.
	• Students show meaning of 10s and 1s by using pictures & making drawings.
8: Students say each step in the FACTS strategy.
9–12: Students solve addition, missing addend, & word problems using the FACTS strategy.	• Students think aloud about their approach to an addition or missing addend equation w/ sums 10–20.
	• Students use the FACTS strategy to solve an addition or missing addend equation w/ sums 10–20.
	• Students solve addition & missing addend word problems w/ sums 10–20.
	• Students think aloud about their approach to addition & missing addend word problems w/ sums 10–20.

Note. FACTS = Focus on the problem; if I do not know the sum from memory then use Another problem, Count on, or make Tallies; and finally Solve.

Lessons 1 Through 3

At the first level of CRA-I, lessons involve solving abstract equations and word problems using objects (base-10 blocks), using pictures (number lines), and drawing on the number lines. The concepts within the lessons were addition, its relation to subtraction, the meaning of the equals sign, the commutative property number magnitude, and place value. The word problems are one-step problems with the required operation being addition or subtraction (missing addend). Figure 1 shows activities in which the teacher explicitly taught the concept of addition, the meaning of the equals sign, place value, and number magnitude. First, students and their teacher represented addends within an abstract problem with base-10 blocks representing 1s. The teacher then pointed to the addition sign and explained that it signified an action to be taken, the combining of two groups. This action would occur on the other side of the equals sign (a line drawn under the vertically written equation). As the figure shows, the sum was composed by putting both addends together into one combined group. The teacher then put addends together twice, each beginning with a different addend to show that the sum did not change (commutative property). The teacher counted the 1s blocks within the sum and brought students’ attention to the fact that there were more than 10. After bringing attention to the fact there were more than 10, the teacher and the students regrouped the blocks by exchanging 10 ones for a 10s block. It is only after this process that the teacher and students wrote the sum in the written equation, with the teacher explicitly showing why the sum was composed of . . . in the 1s place and . . . in the 10s place.

Figure 1.

Concrete-Representational-Abstract–Integrated Level 1: Activities.

Figure 1 also shows this process completed using a number line. Addends were shown on the number line, but the purpose of doing so was to make the counting-on process explicit. We have found, in previous research and in our teaching experience, that students engage in a common error pattern when counting on. Therefore, the teaching methods associated with addition on the number line attempted to ensure that students had appropriate understanding that might circumvent this potential error pattern.

First, the teacher shaded the number line to show the largest addend. When using blocks, students learned that the order of the addends did not change the sum. The teacher explained that using the largest addend (even though it did not appear first in the equation) would make counting-on less difficult and faster. Next, the teacher drew “jumps” on the number line to show the other addend. The teacher and the students shaded and counted aloud as they shaded this area. The systematic counting helped students understand the process of counting on, beginning the first count with the number after the first addend. They stopped shading and counting and arrived at the sum. Using the number line, the teacher taught students how to determine the sum’s proximity to 10 and 20 (number magnitude). Evaluating the number line, they determined whether the sum was closer to 10 or 20.

Finally, these lessons included missing addend equations (beginning in Lesson 2). The meaning of the equals sign (learned in the previous lesson) was a critical component of solving these equations with blocks. As shown in Figure 1, the teacher and students made the given addend and the sum on the other side of the equals sign. They discussed that both sides had to be equal or the same. They systematically added 1s blocks to the left side, under the missing addended spot, counting on until they arrived at the sum. They also discussed another way to write a missing added equation, writing it as a subtraction equation.

Lessons 4 Through 6

At the second level of CRA-I, the teacher and students solved abstract problems and word problems with the same number lines as well as another representation, tallies (long vertical lines represented 10s and short horizontal lines represented 1s). At Lesson 4, concrete objects are no longer used. The teacher used the same methods as in the previous lessons; the only change was the type of representation. Beginning at Lesson 5, the teacher added a new activity. When solving certain problems, the teacher explicitly showed students how to use mental strategies. She taught them how to use another process. For example, in the equation 9 + 8 = _____, the teacher used 8 + 8 + 1. The teacher used 8 + 8 + 1 because it was easier to visualize the sum and the teacher knew that 8 + 1 together are 9, which would be the same as saying 8 + 9. She thought aloud, drew pictures, and wrote the new problem (see Figure 2). The teacher began solving the given problem of 9 + 8. On the other side of the equals line, she made nine tallies. When beginning to make the next addend, she stopped abruptly and brought students’ attention to the fact that she had drawn 10 tallies. She and the students discussed how it is easy to solve addition problems when 10 was the addend. She started a new problem, writing 10 as the addend. The students and teacher counted the remaining tallies that would serve as the addended for the new equation with 10. The teacher and the students quickly identified the sum. They investigated whether the two different equations (9 + 8 and 10 + 7) resulted in the same sum. The teacher used this same explicit process with the drawings and pictures to show how students could solve equations using doubles plus one (see Figure 2).

Figure 2.

Concrete-Representational-Abstract–Integrated Level 2: Use of strategies. (A) Strategy: Using another problem with 10 as the addend. (B) Strategy: Using another problem doubles + 1.

Lesson 7

This lesson involved learning a mnemonic. As in other CRA interventions (Miller et al., 2011), this provided support for their transition to the last level of CRA-I, abstract computation. The mnemonic was FACTS: Focus on the problem; if I do not know the sum from memory, then use Another problem, Count on, or make Tallies; and finally Solve and then check. The first step asked students to focus on the problem. During this step, students thought about the equation (the meaning of the sign and the addends). They asked themselves whether they knew the sum from memory. If not, students could use one of the next three steps—another problem, count-on, or tallies—two of which could be accomplished mentally. The last step directed students to solve the problem and check (see Table 3 for the FACTS mnemonic).

Table 3.

FACTS Strategy Mnemonic.

F	Focus on problem (Look at sign & numbers; ask, “Do I know the answer?” If not, choose strategy):
A	Another problem (doubles + 1, use of 10)
C	Count on (count on to the larger number)
T	Tallies (draw the problem using tallies)
S	Solve the problem and check (use strategy, write the answer, and check your thinking)

Lessons 8 Through 12

The last set of lessons involved solving equations and word problems using the FACTS strategy. The teacher explicitly thought aloud about using FACTS and choosing the more efficient strategy. For example, drawing equations with tallies was the most labor intensive and she encouraged students to be thoughtful about when and how different approaches would be most helpful. The teacher also engaged in games using flashcards. The purpose of these lessons was to increase students’ practice in applying their conceptual understanding and procedural knowledge so that they became more efficient in solving equations.

Discussion

Changes in Student Performance

The teacher implemented the intervention over the course of 2 years with two groups of students. The first implementation was with a group of five students who received Tier 2 intervention. Before, during, and after the intervention, the teacher collected data on students’ additive reasoning, number knowledge, and mathematical knowledge. The items assessed students’ knowledge of number magnitude, the commutative property, use of mental strategies, the relationship between addition and subtraction, and addition (see Figure 3 for an example of the assessment used). Prior to the intervention, students’ accuracy scores were 59%, 17%, 27%, 49%, and 41%. After the intervention, all students demonstrated 100% accuracy across two consecutive assessments. Four weeks after the intervention, their maintenance scores ranged from 89% to 94%. The teacher also gave timed addition assessments to monitor students’ progress toward fluency. Prior to the intervention, students’ fluency ranged from 16 to 24 correctly written digits. These scores were close to the accepted fluency benchmark of 30 (Hosp et al., 2016), but the sums only had one correct digit in most cases (e.g., 7 + 6 = 14). After the intervention, all students wrote 30 correct digits and their sums were correct; 4 weeks after instruction, they maintained their performance. The following year, the teacher implemented the intervention with her entire second-grade class. Compared with another second-grade class whose teacher used the usual instruction and interventions, the fluency gains were greater for the class that received the additive reasoning intervention.

Figure 3.

Addition within 20 conceptual understanding.

Implications for Practice

The components of this CRA-I intervention can be incorporated into any classroom without significant resources. This makes it possible for teachers to implement a Tier 2 intervention even if resources are limited. The CRA-I can address student learning within heterogeneous groups, and it is possible that mathematical concepts within the intervention can be spread throughout Tier 1 addition instruction. For example, teaching the meaning of the equals sign would involve a slight adjustment in instructional procedures so that the teacher can show the sum with objects. After this, students would be primed for missing addend instruction during small-group activities.

The use of objects and drawings associated with CRA-I teaching can be made from available classroom materials. For example, place value representations can be created with paper. Miller et al. (2011) described creation of place value strips. It is also possible that curricular materials used in general education classrooms can provide representations within a CRA-I intervention for additive reasoning, but teachers may not currently use them in the same way as we did to build additive reasoning skills. Teachers could make adjustments to build a deeper understanding of mathematical concepts in daily instructional routines as well as use CRA-I for supplemental Tier 2 instruction. For example, noticing sums’ proximity to 10 and 20 on a number line would be a simple adjustment to instruction. Adding on the number line using the addend with the greatest magnitude and systematically counting on is another simple instructional change.

Last, CRA-I can be easy to implement. The teacher who worked with us found the interventions to be easy to use and effective, as she reported on a survey after each implementation. Students improved in their additive reasoning knowledge even though they had a variety of instructional support needs. It is possible that other teachers could successfully implement CRA-I as a Tier 2 intervention for additive reasoning as well.

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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