The Effect of the Concrete-Representational-Abstract Mathematical Sequence on Addition Skills 0 to 9 for Struggling Learners in Kindergarten

Running head: CONCRETE-REPRESENTATIONAL-ABSTRACT MATH SEQUENCE

The Effect of the Concrete-Representational-Abstract Mathematical Sequence

on Addition Skills 0 to 9 for Struggling Learners in Kindergarten

Single Subject Design

University of Nevada – Las Vegas

December 2014

Janet Van Heck

CONCRETE-REPRESENTATIONAL-ABSTRACT MATH SEQUENCE 2

Abstract

Many students with disabilities struggle in the area of mathematics. This is especially true related

to the development of conceptual knowledge. The concrete-representational-abstract (CRA)

teaching sequence delivered via explicit instruction helps increase both conceptual knowledge

and math performance. A multiple-probe-across-participants design was used to examine the

effects of explicit instruction and the C-R-A sequence to teach kindergarten students how to add

numbers 0 to 9. The three subjects benefitted from the scripted instructional lessons. Their test

scores improved considerably and were able to generalize what they learned on subsequent

practice sheets. Practical implications and suggestions for future research are provided.

Key Words: mathematics disabilities, concrete-representational-abstract teaching sequence,

explicit instruction, manipulative devices.


Many school-aged students with disabilities experience difficulties in the curricular area

of mathematics. Once thought to be quite uncommon, it is now agreed that between 5% and

13.8% of the school population have learning disabilities in mathematics (Barbaresk, Katusic,

Colligan, Weaver & Jacobs, 2005; Mazzocco, 2005). These math disabilities range from mild to

severe (Barbaresk, Katusic, Colligan, Weaver & Jacobs, 2005; Mazzocco, 2005).

One of the most persistent difficulties for students who experience mathematics

disabilities is the development of conceptual knowledge (Hecht, Vagi, & Torgesen, 2007).

Students who demonstrate conceptual knowledge in mathematics have deep understandings

related to the meaning of abstract mathematical symbols. For example, they understand that a

plus sign means to add and a minus sign means to subtract, but they also understand that addition

means combining two groups to get a larger group and/or that subtraction either means beginning

with a large group and taking some away or comparing two quantities to determine the amount

of difference between them (Hudson & Miller, 2006). Unfortunately, many students with

mathematics disabilities learn to perform mathematics operations (i.e., addition, subtraction,

multiplication, division) without understanding the underlying concepts associated with each

operation (Stevens, Harris, Aguirre-Munoz, & Cobbs, 2009). They simply memorize step-by-

step processes to get correct answers or they try to find patterns in problem examples provided in

their texts and the problems they need to solve without understanding why the steps are being

performed or why the pattern exists. When students fail to acquire conceptual understanding

related to the mathematics they are learning, they are less likely to maintain and generalize the

new concepts and skills to real-life activities (Panasuk, 2010; Simon, 2008).

Because of the challenges that students with disabilities display related to conceptualizing

the meaning of mathematics (Das & Janzen, 2004) diverse classroom emphases, adaptations and


sometimes divergent methods are needed (Hough, 2004). The use of evidence-based practices

has the potential to help teachers design effective, efficient, and motivating mathematics lessons.

The purpose of this article is to share information about how teachers can combine the use of the

concrete-representational-abstract teaching sequence and explicit instruction in teaching addition

facts 0 to 9.

The Common Core State Standards include addition and subtraction standards related

conceptual and procedural knowledge for a number of reasons. First, due to the hierarchical

nature of mathematics learning, many students who struggle with math have limited

understanding related to foundational operations (National Mathematics Advisory Panel, 2008).

When students do not master these early skills, they will continue to fail. Also, students who

lack conceptual knowledge tend to rely on memorization to get answers correct. Additionally,

deficits in conceptual knowledge negatively impact a student’s ability to develop and generalize

procedures for mathematics learning. It is this balance between conceptual and procedural

knowledge that causes students to become successful at mathematics (Hudson & Miller, 2006).

While the Common Core State Standards for Mathematics are very useful in identifying

important instructional competencies, the curriculum does not provide information on evidence-

based practices for students who struggle with mathematics. Over the last 20 years, the number

of studies developed to improve math performance in students of all ages has increased and

consistently include such practices as explicit instruction and strategy instruction (Gersten,

Chard, Jayanthi, Baker, Morphy& Flojo, 2009; Miller, Butler & Lee, 1998).

Explicit instruction provides clear models for solving problems, examples for solving

problems, and multiple practice opportunities of the newly learned skill (Components of an


Effective Lesson, 2014). The components of an effective lesson are often used in explicit

instruction (e.g., advanced organizer, demonstrations, guided practice, independent practice and

feedback) (National Mathematics Advisory Panel, 2008).

Strategy instruction focuses on the process of solving problems. This category of

instruction includes teacher-student dialogue and feedback, multi-process instructions, reminders

to use procedures that have been taught, demonstrations, questioning, and systematic

explanations (Montague, 2008). One instructional practice that combines both explicit

instruction and strategy instruction is the concrete-representational-abstract mathematical

sequence.

Concrete-Representational-Abstract Mathematical Sequence

The concrete-representational-abstract (C-R-A) teaching sequence is an evidence-based

instructional practice that is used to support the development of conceptual understanding in

mathematics (Hudson & Miller, 2006; Flores, 2010; Strickland & Maccini, 2013). See the

Literature Review on the Use of the Concrete-Representational-Abstract Sequence in Table 1.

When teachers use the C-R-A sequence, instruction on a new math skill begins at the

concrete level (i.e., manipulative devices are used to represent and solve the type of math

problems being taught). When students reach mastery level (i.e., typically 80% accuracy) at the

concrete level, instruction progresses to the representational level (i.e., pictures of objects or

tallies are used to represent and solve the type of problems being taught). When students reach

mastery level (i.e., typically 80% accuracy) at the representational level, instruction progresses to

the abstract level (i.e., problems are represented and solved using numbers without manipulative

devices or drawings).


Explicit instructional procedures are used at all three levels (i.e., concrete,

representational, abstract) of instruction with a minimum of three lessons at each level. Explicit

instruction involves structured and systematic lessons in which students are guided through the

learning process with clear statements about the purpose and rationale for learning a new skill,

clear explanations and demonstrations of the learning target, and supported practice with

feedback until independent mastery has been achieved (Archer & Hughes, 2011; Hudson, Miller,

and Butler, 2006; Taymans, Swanson, Schwarz, Gregg, Hock, & Gerber, 2009).

A substantial amount of research supports the use of the CRA teaching sequence being

implemented via explicit instructional approaches (Flores, 2010; Strickland & Maccini, 2013).

This approach has been effective for both elementary and secondary students with disabilities

across a variety of math skills (e.g., math facts, place value, regrouping skills, fractions, algebra).

See Table 1.

The purpose of this study is to apply the previous literature on C-R-A instruction to the

new skill of addition facts 0 to 9 for Kindergarten students. The proposed research study will

investigate a new area of C-R-A implementation. Not only may junior high and high school

students benefit from C-R-A instruction, but students as young as Kindergarten will be able to

put this research-based practice to use.

Research question: Will kindergarten students show improvement in their 0 to 9 addition

facts upon completion of the concrete-representational-abstract sequence intervention?


Method

Participants

Seven students who are identified to begin the Response-to-Intervention process will be

screened for mathematic ability. Specifically, this will involve the students completing four

problems in which they described aloud how to solve the problem using manipulative devices,

and four problems where they describe how to solve the problems without the math

manipulatives. The students who earn an average score of 50 or below will be selected for the

study. The students will be tested on their knowledge of addition facts 0 to 9. There will be

three students identified for participation. Students who have had very little exposure to this skill

will be selected for the treatment. This will be done to reduce the chance that previous

instruction would influence participant performance data. Parental permission will be obtained

by having them sign a release form for their child to participate in the study. If students are

absent on a day of instruction, they will start where they left off on the intervention.

Two of the students will be male, and one was female. Two will Hispanic, and one will

be White. Achievement scores on formal assessments will likely from the 1st percentile to the

20th percentile. See Table 2 for information on each participant and their demographic group.

A special education teacher will administer the treatment to each participant. The teacher

will have a master’s degree in special education and at least 5 years of experience in the field.

The teacher will have previous knowledge teaching with the C-R-A sequence.


Setting

The study will take place in an elementary school in a large urban area in the southwest

of the Unites States. There are 620 students enrolled in the school. 61% were Hispanic, 18%

were African-American, 13% were White, and 7% were Asian/Pacific Islander. 85% of the

school population received free or reduced lunches. The lessons and data collections for this

study will be done in the resource room. The students receiving the treatment are the only

students in the classroom at the time of the lessons. Generalization for the study will also take

place in the same setting. However, students will also be monitored in the general education

classroom to evaluate their generalization of the lessons.

The teacher will be seated across from the student at a table. The teacher will collect data

for the baseline condition. She will also teach all of the lessons of the C-R-A sequence to each

student. The experimenter will be the person to score the dependent measure practice sheets and

place the data in an Excel spreadsheet. The teacher will instruct the lessons, and the student will

complete the practice sheets on a one-on-one basis for each of the three students selected for the

study. There will be no non-participants in the room at the time of the instruction or

measurement.

Materials

The materials will consist of scripted lessons, practice sheets, math manipulatives, and

student notebooks (Miller & Mercer, 1992). Fifteen scripted lessons will be used to guide the

lessons. They will include what the teacher will say and do during each of the lesson elements:

advanced organizer, model, guided practice, independent practice, and problem-solving practice.

Ten learning sheets and 5 quizzes will establish student engagement during the lessons. The


problems on the practice sheets are in line with the teacher scripts for each lesson. Manipulative

devices (cubes, rectangles, and base-ten blocks) will be used during the concrete phase, which

comprised the first 5 lessons. Each student will store their completed practice sheets, quizzes,

and progress chart in their folder. The progress charts will track the students’ practice sheet

scores and were used to motivate the students as they progress in their skills. Generalization

stimuli will only consist of progress sheets, quizzes, and the Frog Game. There are no tangible

rewards for participation, but students will gain a sense of accomplishment as their skills

improve and see their steady progress on the progress chart in their folder. The progress charts

will consist of a vertical axis of student performance from 0 to 100, and the horizontal axis will

show the date of the lesson. The score of each quiz and practice sheet will be recorded here.

Measurement Procedures

The instructional routine will be facilitated through the use of the 15 scripted lessons

(Miller & Mercer, 1992). Sessions will take 4 to 5 times per week, depending on work load and

the school schedule. The sessions will take place at 1:15 pm in the afternoon. The maximum

length of each session will be 1 ½ hours. The stimuli will be presented in the order of the C-R-

A, a research-based practice, sequence to be discussed further. All lessons will include

computation problems, and the last three lessons will have word problems. Generalization will

be attained with the use of a Frog Game, where students work the problems mentally, without

using manipulative devices or representative tally marks.

Concrete lessons. The first five lessons will be concrete level lessons. They will

involve the use of base-ten blocks and Practice Sheets 1, 2, 3, 4, and 5 to introduce the concept

of addition facts 0 to 9. The teacher will demonstrate to the student how to use the blocks to add


a certain number of blocks to another amount of blocks. The student counts how many blocks in

total to get the answer.

Additionally, the DRAW strategy (Harris, Miller, & Mercer, 1995) will be introduced.

The students will discover the sign, indicating which mathematical operation to perform. Then,

they will read the problem, reading it aloud for the teacher to hear. The student will answer the

question by counting out the blocks. Finally, they will write the answer (Mercer & Miller,

1992).

Representational lessons. The next three lessons will involve representational level

instruction using drawings or tallies instead of the blocks used in the concrete phase. For lessons

6, 7, 8, 9, and 10, the students draw tally marks on the practice sheets to represent how many

blocks they would count out. Then, they add the next amount of tallies over the next number.

They count together all of the tallies, and this solves the addition problem. The DRAW strategy

will be utilized again as an instructional aid to help students remember what to do (Harris,

Miller, & Mercer, 1995).

Abstract lessons. The last five lessons involved abstract-level instruction. The students

must solve the problems with number symbols only, excluding the use of manipulative blocks or

tally marks. The students will be allowed to look at a learning sheet that has the DRAW strategy

instructions listed on it if they having difficulty remembering what to do. Fluency in adding

addition facts 0 to 9 will be emphasized during this phase (Mercer & Miller, 1992).

Additionally, the students will play a Frog Game, where they have to move their pieces

forward on a board depending on the numbers they add together on a pair of dice. They have to

be able to generalize the skill to the game in order to play correctly. The game and observations


of students in the general education setting will determine the external validity or generalization

of addition facts 0 to 9 taught using explicit instruction and the C-R-A sequence.

Lesson Components

Each of the 15 lessons will include the same five explicit teaching components to teach

the content discussed in the previous section. These components will be advanced organizer,

describe and model, guided practice, independent practice, and problem-solving practice

(National Mathematics, 2008).

During the advanced practice component of each lesson, the teach script will involve

introducing the upcoming lesson, the explicit teaching components used in the upcoming lesson,

handing out the lesson materials, and reviewing what was learned in the previous lesson

(Components of an Effective Lesson, 2014).

During the describe and model component of each lesson, the teacher script will involve

teacher think-alouds related to solving addition facts 0 to 9 using manipulative blocks, drawings,

or the DRAW strategy depending on whether a concrete, representational, or abstract lesson was

being taught. The student will watch the teacher solve three problems while the students write

down the same markings and numbers. The teacher will give corrective feedback if the students

make any errors (Mercer & Miller, 1992).

During the guided practice part of each lesson, the teacher will give verbal support by

asking questions to assist the students as they solve the guided practice problems. The students

will use blocks, tally marks, or the DRAW strategy. A reduction in teacher support will occur

with the second and third guided practice problems. Answers to the last two guided practice

problems will not be discussed (Harris, Miller, & Mercer,1995).


During the independent practice component of each lesson, the student will complete six

problems that are on the practice sheet without assistance from the teacher. No teacher feedback

will be provided as the students solve these problems (Components of Effective Lesson, 2014).

During the problem-solving practice portion of each lesson, the students will solve two

word problems using blocks, tally marks, or the DRAW strategy depending on whether it was a

concrete, representational, or abstract lesson. The teacher will read the first word problem aloud

to the student, and then the student will solve the problem without teacher assistance. The

student and teach will plot the scores on the progress chart in the student notebooks (Mercer &

Miller, 1992). Generalization of the new skill will be measured during this phase.

Experimental Design

The study will employ a multiple probe (Gast, 2010) across students. The multiple probe

design will be replicated one time with two additional students. The design evaluates a

functional relationship by measuring improvement in mathematical performance with each

lesson. The students will be assessed for baseline with a simple test like the practice sheet in

Figure 1. They will be taught and assessed on the concrete, representational, and abstract

methods. Generalization will be assessed by a test where there are asked to compute their

problems in their heads without using the manipulative devices of the concrete phase or the

tallies of the representational phase. They will demonstrate their generalization during the Frog

Game. This generalization exhibits external validity. Internal validity will be established by the

increase in math performance as documented on the progress charts and as seen on the design

graphs in Figure 2. This describes the design for the sole research question. The data will be

interpreted by visual inspection, and the following characteristics will be noted:


Baseline Procedures. Following the procurement of research approval, screening data,

parent permission, and student commitments, the three participants will receive baseline probes

following the parameters the multiple probe design. The probes will be administered in the

setting described previously on a one-on-one basis. Three alternative forms of baseline probes

will be administered in sequence, similar to the probe in Figure 1. The probes will be

administered daily for three days in a row or until level, trend, and slope have been established

(Gast, 2010), and they will not be timed. No assistance will be given to the students as they

complete the baseline probes. This will determine the students’ ability in addition facts 0 to 9

before they have received any instruction in this area. There will be no consequences or rewards

at this point.

Next, the teacher will have a discussion with each student about the importance of

learning addition facts 0 to 9. This discussion will end with the students and teacher agreeing to

do their best to learn and teach the new addition skill, and signing a contract.

C-R-A Intervention Procedures. All three participants will receive the 15 C-R-A

intervention lessons from the teacher following the parameters of the multiple probe design. The

lessons will be administered in the resource room setting, and the students will complete the

practice sheets as the lessons progress through the lesson components. The last ten problems on

the sheets will be used as the intervention probe for the lesson. The probes will be scored with

the student present, and feedback will be provided to the student and the score will be plotted on

the progress chart in each student’s notebook.

The teacher will sit across from the student and present the 15 lessons and document

scores from the practice sheets on the progress charts. As stated in the lesson components


section, students will receive more support in the beginning of each lesson, and gradually have

the support reduced until the student is working independently. The consequence for an

incorrect score will be a lower score documented on their progress chart. Likewise, the

reinforcement for correct answers will be a higher score marked on the progress chart. Each

lesson will be taught using explicit instruction, and instructions will be followed as presented in

Mercer and Miller’s Addition Facts 0 to 9 (1992).

Generalization procedures. Generalization procedures will be used to assess for the

ability to apply addition facts 0 to 9 in a situation, apart from the instructional activities and

solving the problems on the practice sheets and quizzes. This will determine whether they in fact

learned a new skill, and that the intervention was the reason for the students demonstrating the

new skill. The students will demonstrate generalization for three days after the intervention

phase. The students will play the Frog Game. The teacher will measure how many times they

add the numbers on the dice together correctly. These figures will be documented as an accuracy

score. If the student is truly skilled, he or she will be able to win the game, and this will be their

reward for generalizing the skill.

Response definitions and recording procedures. It will be a requirement that students

can read and write numbers and letters, including some words, and that they understand how to

distinguish mathematical symbols (e.g. +, -, /, x). The desired response for this intervention is

for students to improve in mathematical ability in demonstrating addition facts 0 to 9. To see the

interventions and lesson procedures for a student, view Figure 2 in the appendix.

The generalization of this skill would be being able to add any numbers from 0 to 9

without using the concrete (manipulative blocks) or representational (tally marks) phases of the


C-R-A sequence. Examples of this would be applying the skill to advance and win the Frog

Game described in the generalization section. If a student is able to complete a quiz abstractly

(mentally, without using blocks or marks), the student will be demonstrating mastery of the skill.

When a student can add numbers 0 to 9 in the general education setting without being reminded

of the C-R-A sequence, the student is demonstrating generalization at this point.

The recording procedure will be the practice sheets and progress chart documentation for

performance on each lesson following explicit instruction of the C-R-A sequence. Students will

receive a grade from 0 to 100. The responses to be recorded during each experimental condition

are as follows. A score from 0 to 100 will be assigned for each baseline procedure. The only

assessment during instruction is that students are engaged, participating, and exhibiting

understanding. During intervention, the students will be assessed using the practice sheets; the

last 10 questions of the sheet will be used to apply to the data set for intervention.

Results

The baseline, intervention, and generalization probe scores for all three participants are

displayed in Figure 3. Visual analysis of these data shows that all students display immediate

level gains when the intervention lessons start. The baseline probe scores for these participants

range from 40% to 60%, showing that the students already had some prior knowledge of this

skill. The intervention probe scores for the participants ranged from 55% to 100%.

Individual participant means, standard deviations, and percentage point gains are listed

on Table 3. Julie is the only student to who can be measured for baseline mean and standard

deviation. Her baseline mean is a score of 55, with a standard deviation of 15. Her intervention

mean and standard deviation were 92 and 6.32. Her percentage point gain score was 92. Ricky


had a lower intervention mean and a higher standard deviation, with scores of 89 and 12. His

percentage point gain score was 83. Jesse had a slightly higher intervention mean of 90, and a

comparable standard deviation of 10. His percentage point gain score was 90. Ricky showed the

most improvement from baseline to generalization.

All three participants were able to achieve mastery (80 percent or higher) on all 15

lessons. Performance trends were similar for all three participants. The trends for intervention

performance were stable at relatively high performance levels. There was a ceiling level of 100

percent for intervention probes.

C-R-A Performance

Julie. After beginning C-R-A addition facts instruction, Julie will reach criterion on the

first probe of the 15 total sessions. Her range of scores will be 71 through 100. There will be an

immediate change in the level of performance between baseline and C-R-A, and there will be no

overlapping data points between baseline and the C-R-A instruction. Julie’s average

performance will be a score of 92. The data points of the C-R-A intervention demonstrate an

upward path, indicating steady improvement. There is no change in trend. Her learning appears

to be stable, although she does show three drops during instruction, on days 6, 8 and 18. Day 18

was the first day of the abstract lessons, so she does demonstrate some difficulty with applying

the new skill on the abstract level. The median score in of the 15 instructional lessons was 100.

Ricky. After beginning the first of the 15 C-R-A instructional lessons, he reached

criterion immediately. His range of scores was 71 to 100. There will be an immediate change in

level of performance between baseline and C-R-A. Ricky has an average score of 89. Ricky’s

data points show an upward path, but his data points have an unstable path demonstrating that he


struggles with the material considerably. On days, 9, 11, 14, and 17, he had lower scores. Word

problems were introduced on day 9, which would explain why his score would drop. The

representational phase of using tallies to represent numbers was introduced on day 11, and Ricky

had a drop on this day also. On day 17, he struggles with the introduction of the abstract phase,

removing the use of blocks and tallies, and working the problems mentally. There is no change

in trend or any overlaps. The median score of the 15 lessons was 77.

Jesse. After beginning C-R-A addition facts instruction, Jesse reached criterion on the

seventh probe of the 15 total sessions. His range of scores is 75 through 100. He shows an

immediate change in the level of performance between baseline and C-R-A, and there are no

overlapping data points between baseline and the C-R-A instruction. Jesse’s average

performance is a score of 90. He had a low score upon the beginning of the representational

phase, a score of 55. From there until the end of the intervention, the data points demonstrate an

upward path, indicating steady improvement. There is no change in trend. Although he shows

steady improvement, his learning appears to be unstable, with drops in performance on days 11,

13, and 17. The median score in of the 15 instructional lessons was 85.

Evaluation of threats to internal validity. Maturation will not occur because the study

will only take about 4 to 5 weeks for each participant to complete. That will not give them

enough time to simply outgrow the instructional intervention. Multi-treatment interference will

not likely happen because the only mathematic instruction that Kindergarteners receive is

reading, writing, and understanding what each number means. Parents will be requested not to

instruct their children in this skill until after the study is completed. Attrition will probably not

happen because the baseline and intervention together only take about 4 to 5 weeks, so the

likelihood that a student will leave during this short of a period of time is not high.


Procedural Fidelity and Inter-rate Reliability. Fidelity data will be collected for 30%

of all intervention lessons. Two independent observers will doctoral students in the special

education program at University of Nevada, Las Vegas. They will be trained on the C-R-A

sequence and will be trained to make sure the lessons are presented according to standards of

explicit instruction. They will use a checklist to measure to the implementation of instructional

components. These are teacher behaviors that should be followed (advance organizer, describe

and model showing teacher demonstration and student engagement, guided practice going from

high to low support, independent practice, and problem-solving practice). The percentage

agreement that must be obtained for each teacher behavior is 80%. They will collect data every

three days. Based on the formula agreements/ (agreements + disagreements) x 100, the

procedural reliability for fidelity of treatment will be determined. Two independent scorers will

score 100% of the baseline and intervention probes. The inter-scorer reliability will be

calculated. The minimum score for both reliability and fidelity will be 80%.

Social Validity. Social validity will be determined by the results of interviews conducted

before and after the study. The students and teacher will answer questions on the need for the

intervention and recommendations for other students and teachers. The data collected will be

described. The teacher will likely report that that the students’ performance in addition facts 0 to

9 improved. Some students may express that addition facts 0 to 9 was difficult, and they want to

participate in the program. The students will make a commitment to participate and determine to

learn a new way to complete addition problems.

PND. The PND in this study was calculated by identifying the highest baseline probe

among all participants, identifying the number of treatment process from all three participants

that were greater than the highest baseline probe which was Julie and Ricky at 60%, and dividing


the number of treatment probes greater than the highest baseline probe (50%) by the total

number of treatment probes (57) and multiplying by 100 to determine the PND. Thus, the PND

was 93% percent, which represents a large effect size and indicates the interventio n was

effective for these three participants (Gast, 2010).

Discussion

The results of this study reveal that Kindergarten students who struggle in mathematics

benefit from explicit instruction using the C-R-A instructional sequence to learn how to solve

addition facts 0 to 9 problems. These results were obtained following established criteria for

single subject designs (Gast, 2010). One criterion for single case designs is that the intervention

must be systematically manipulated. In this study, the multiple probe design will be used to

stagger the C-R-A intervention. Once intervention begins with the first participant a lesson

mastery criterion of 80 percent across three lessons will be used to determine when the

intervention will begin with other participants.

Additionally, a criterion for single case design is that outcome variables such as baseline

and intervention probes are measured systematically over time by more than one assessor and

that inter-assessor agreement has to be reached for each phase. A minimum percentage

agreement score within the range of 80 percent to 90 percent within each phase of the study is

considered acceptable. For this study, inter-assessor agreement will exceed the 20 percent for

fidelity of treatment and probe scoring.

A third criterion for single subject design is that each phase must have a minimum of

three data points (Gast, 2010). Only three data points were collected during the baseline phase

because stability was reached at this point. Based on meeting the criteria for acceptable multiple


probe designs and the results to be obtained from the visual analysis related to level, variability,

and trend, there appears to be a causal effect between the high student performance and the

intervention lessons.

This study concurs with the findings of previous researchers that investigated the benefits

of the C-R-A sequence (Strickland & Maccini, 2013; Flores, 2010; & Morin & Miller, 1998).

The current study adds to the C-R-A literature that the intervention results in positive student

performance. One research question which could be answered in a replication study is: what are

the effects of C-R-A instruction of students’ maintenance of fluency in computing addition facts

0 to 9?

Other areas for future research could include the importance of providing explicit

instruction designed to help students who struggle with mathematics. One of the most important

practical implications to arise from this study is the importance of giving explicit instruction for

students with learning disabilities to develop conceptual understanding related to addition facts 0

to 9. Another implication from these lessons is the importance of providing memory tools like

mnemonic devices to assist students in remembering the steps needed to solve the problems.

Also, the emphasis on plotting student performance on the progress charts while providing

feedback to the participants about their performance is significant. This is a positive motivator

for the participants.


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Appendix A


Table 2. Participant Demographic Table.

Demographics

Julie Ricky Jesse

Ethnicity Hispanic Hispanic White

Gender Female Male Male

Grade Kindergarten Kindergarten Kindergarten

Age 5.5 5.5 5.8

Mathematics

Achievement

18 on the KTEA 14 on the KTEA 15 on the KTEA

Identified for RTI No Yes Yes


Table 3. Mean Percentage Scores, Standard Deviations, and Percentage Point Gain Scores

Participant Baseline M/SD Intervention M/SD Percentage Point Gain Score

Julie 55/15 92.00/6.32 92.00

Ricky 0/0 89.09/12.21 82.76

Jesse 0/0 90.00/10.00 90.00


Figure 1. Sample Baseline Probe.

Baseline Probe A

0 +1

1 +1

3 +2

2 +1

5 +4

4 +6

1

+6

3

+3

2

+1


Figure 2. Task Analysis Table for C-R-A Instruction for Addition Facts 0 to 9.

Student Name: ________________________________ Grade: __________________

Instructional Task: Addition of numbers 0 through 9 using the Concrete-Representational-Abstract

mathematical sequence.

Student Performance: Scored on a practice sheet on a 0 to 100 grading scale.

KEY: B – Baseline M – Modeled prompt G – Guided I – Independent

Task Analysis Steps:

1 2 3 4 5 6 7 8 9 10 11 12 13

1.Baseline condition: no instruction

Performance on baseline probe.

B

2. Concrete lessons using explicit instruction.

Practice sheet 1.

M

G

I


Performance on practice sheet 2.

M

G

I



M

G

I

5. Representational lessons using explicit

instruction. Performance on practice sheet 4.

M

G

I



M

G

I



M

G

I

8. Abstract lessons using explicit instruction.


M

G

I



M

G

I



M

G

I



I

12. Generalization using the Frog Game.

Students demonstrate skill

I


Students demonstrate skill.

I


Students demonstrate skill.

I


Figure 3. Multiple Probe Across Participants Design

.

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Baseline C-R-A InterventionJulie

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Baseline C-R-A InterventionRicky

0

20

40

60

80

100

120

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

Baseline C-R-A Intervention Generalization

Jesse

Generalization

Generalization