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Running head: ACTION RESEARCH PROSPECTUS

Kayla Easley

Action Research Prospectus

Kennesaw State University: ECE 7543

M.Ed. Program 2016-2017

2ACTION RESEARCH PROSPECTUS

Abstract

This action research explores the impact of the Concrete-Representational-Abstract (CRA)

framework of instruction for English learners. This framework is a scaffolding process beginning

with hands-on learning, transitioning to pictorial representations, and ending with algorithms.

CRA was implemented during a five-week unit of fractions instruction in a third grade

classroom. Students were formatively assessed for engagement every fifteen minutes using a

checklist. Students were also formatively assessed by completing a pre-assessment and post-

assessment. Journaling was implemented biweekly to identify emerging trends and themes in

mathematics. Quantitative data gathered supports that CRA positively impacts conceptual

understanding of English learners, as all students exceeded target growth goals and improved

post-assessment scores significantly. Qualitative data gathered supports that CRA has some

impact on engagement, as most students received consistent engagement marks. Two outlier

students inform future research on engagement of English learners and the affective domain’s

role in learning.

Keywords: Concrete-Representational-Abstract framework (CRA), English learners,

engagement


Introduction

When interviewing for my current position a question was posed, “Can you support

students in a high poverty, English learner (EL) school?” My response was simply, “Yes, I have

the English to Speakers of other Languages (ESOL) endorsement.” In my naïve mind, this

endorsement was sufficient in providing me with the skills necessary to educate ELs. I quickly

learned that my knowledge base was not enough to close achievement gaps between my native

speakers and ELs. I was forced to scour through textbooks and internet articles in hopes of

finding research-based strategies to support these students. It became evident that all subjects,

including mathematics, relied heavily upon reading and vocabulary.

I challenged myself to discover some way to help my ELs succeed, and in doing so I

came across the Concrete-Representational-Abstract (CRA) framework. This framework

incorporates concepts I was taught in my ESOL endorsement classes such as: including realia,

objects and materials from everyday life to aid in teaching a concept and allowing students to

demonstrate understanding through different mediums such as using pictures instead of

algorithms or words. CRA allows students to become active participants in learning by first

learning concepts through hands-on inquiry rather than by learning algorithms. I hypothesize that

CRA will aid my students in understanding the process behind the product of mathematics

concepts. This framework unifies conceptual and procedural knowledge so that students

understand concepts holistically. Through my action research, I hope to discover how CRA

impacts learning for my ELs.

Literature Review


Conceptual knowledge, defined by Miller and Hudson (2007), is the development of a

deep understanding of mathematical concepts by connecting new information to existing

information. These connections result in understanding relationships and patterns among

different pieces of information (Bottge, 2001). Procedural knowledge, defined by Miller and

Hudson (2007), is the ability to solve a mathematical task. A second definition, provided by

Bottge Et al., is the ability to follow a step-by-step procedure to solve a math problem (Agriwal

& Morin, 2016). Research presented by Bottge (2001) indicates that mathematic interventions

for low-achieving students focus upon teaching computational skills and procedures rather than

conceptual knowledge. However, this approach to instruction is flawed; students need to develop

both procedural and conceptual knowledge to become skilled in mathematics (Bottge, 2001).

While there have been many debates about which type of knowledge develops first and is more

important, researchers now understand that an integrated understanding of conceptual and

procedural knowledge is necessary for mathematic proficiency (Agriwal & Morin, 2016).

Hudson and Miller (2007) contend that student-centered practices, as supported by the

National Council of Teachers of Mathematics (NCTM), allow students to focus on a range of

skills by implementing a variety of thinking processes. Students must spend time interacting with

manipulatives, representing concepts in multiple ways, and sharing their knowledge with other

pupils. An evidence-based practice cited by Hudson and Miller (2007) is the use of

manipulatives and pictorial representations, as this strategy helps students develop conceptual

knowledge. This practice can be systematically integrated into explicit instruction through the

CRA framework. This framework ensures that concepts are taught with three-dimensional and

two-dimensional representations, providing different ways for students to understand concepts

(Hudson & Miller, 2007).


The NCTM recommends that students have an opportunity to develop conceptual and

procedural knowledge by engaging in meaningful math instruction (Agriwal & Morin, 2016). A

method of instruction, entitled the Concrete-Representational-Abstract (CRA) framework, helps

learners develop a clear relationship between conceptual and procedural knowledge (Agrawal &

Morin, 2016). The CRA framework was designed for learners who struggle with understanding

mathematical concepts and procedures. These educational deficiencies stem from multiple

sources including poverty and intellectual factors, as well as improper teaching methods. Using

the CRA framework students in individual, small group, and whole group settings in elementary

and secondary school can improve skills in mathematics regardless of skill level or ability

(Mudaly & Naidoo, 2015).

“The CRA sequence of instruction differs from other approaches regarding instructional

methods and activities and teacher involvement. The combination of teacher demonstration,

guidance, and student demonstration of mastery over three lessons differentiates this [CRA] from

other methods (Flores, 2010, p.196).” The CRA framework is a tiered process in which a teacher

guides students through mathematical concepts and the computational process by using

manipulatives, visual representations, and numbers with symbols. The concrete and

representational phases serve as scaffolds, with the overarching goal being that students can

compute at the abstract level. As each phase is encountered, students must be taught to mastery

level before progressing to the next phase (Agrawal & Morin, 2016).

A study conducted by Mudaly and Naidoo (2015) on master mathematics teachers was

designed to determine effective teacher practices. One common practice was the inclusion of

concrete objects to increase procedural knowledge retention. This strategy has roots in Gardner’s

Theory of Multiple Intelligences, as students can retrieve information in various ways including


auditory, tactile, visual, and kinesthetic. A second commonality found was that master teachers

were more focused upon the process of problem solving than the product; the teachers ensured

that the scaffolding was not premature. In some circumstances, the master teachers used visuals

to assist students with limited language proficiency or to clarify concepts being taught. The

researchers concluded that each master teacher in the study was effectively implementing the

CRA framework. The research of Sabainah Akinoso (2015) supports that master teachers should

implement the CRA framework, as students taught using CRA are shown to make higher gains

than those receiving traditional instruction.

The work of Margaret Flores (2010) addresses the impact of CRA on students at risk for

failure. Prior research on students with disabilities recommends the use of visual aids and

explicit instruction with scaffolding when teaching, both of which are embedded in the CRA

framework. Flores concluded that the CRA framework was a successful intervention, as error

patterns were eliminated and most students retained fluency in a six-week follow-up.

Additionally, teachers reported an increase of volunteering, an increase in positive comments

made during instruction, and an increase in willingness to actively participate in instruction.

An equity-based approach to teaching mathematics focuses on the empowerment of the

whole child. Instruction must be adjusted to include multiple identities such as racial, ethnic,

cultural, and linguistic. These identities are utilized as students use background knowledge to

learn and do mathematics. All students must be given an opportunity to learn rich mathematics in

light of background and language (Aguirre, Mayfield-Ingram, & Bernard Martin, 2013, p.9).

Students from different subgroups struggle to meet grade-level expectations in mathematics.

English learners are among those at risk. This group’s presence in U.S. schools has continued to

increase for the past twenty years. Despite the growing population of ELs, schools are often


unequipped to support these students in the development of mathematical proficiency (Doabler,

Clarke, Kosty, Baker, Smolkowski, & Fien, 2016).

English learner is a blanket term using to describe the entire population of ELs in the

United States. There is significant diversity, making matters of instruction more complex. Some

ELs are born in America and are considered conversationally proficient in English. Other ELs

are immigrants who differ in terms of age at immigration, first-language proficiency, English

proficiency upon arrival, and formal education received in the home country. While the abilities

and backgrounds of each EL differ, two phases of second-language acquisition are generally

accepted. Most students take two years to develop conversational skills, also known as basic

interpersonal communications skills (BICS). An assumption made by some educators is that

proficiency in BICS is sufficient for students to have success in academic language, also known

as cognitive academic language proficiency (CALP). However, this is inaccurate. Even with

explicit instruction in vocabulary, students will need five to seven years to develop CALP (Ross,

2014).

The math achievement gap between ELs and English-proficient students appears early in

education and remains stable throughout schooling. In 2009, 43% of fourth-grade ELs scored

below basic in mathematics on standardized tests, the lowest possible score bracket. More

discouraging was percentage of eighth-grade ELs scoring below basic, which increased to 72%

(Valle, Waxman, Diaz, & Padron, 2013). As students who must acquire a new language, ELs

face the double demands of simultaneously learning both English and the language of

mathematics. Is it unclear why ELs have more difficulty with mathematics than their English-

speaking classmates. Early numerical concepts are acquired independent of language, and

studies in neuroscience indicate that mathematical and language processing potentially involve


distinct brain networks. These findings suggest that mathematic tasks with linguistic complexity

mask the mathematical abilities of ELs (Alt, Arimendi, Beal, Nippold, & Pruitt-Lord, 2014).

Emerging evidence cited by Alt, Arimendi, Beal, Nippold and Pruitt-Lord (2014)

suggests that conceptual understanding for ELs is weaker than that of English speaking peers.

Learning a second language requires additional cognitive demands. Cognitive cost is associated

with shifting from one language to another. When solving a problem, a student may translate the

problem into his or her primary language to form a representation, but the cognitive switching is

likely to increase chance for errors. Additionally, literal translation of mathematical terms often

produces errors. Take for example the phrase, “six less than x is 8.” In the mind of an EL, this is

likely to translate to 6-x=8. Despite understanding the meaning of the phrase less than, the syntax

hinders successful completion (Ernst-Slavit & Slavit, 2007).

ELs are often impeded by a lack of familiarity with mathematics vocabulary, including

technical terms and words with multiple meanings. For example, the word foot is used to

describe the accumulation of twelve inches but an EL student could misconstrue this to mean the

body part. Unless explicit vocabulary instruction occurs, little vocabulary knowledge is unlikely

to improve (Alt, Arimendi, Bea, Nippold, & Pruitt-Lord, 2014). To complicate matters more,

vocabulary is broken up into subgroups such as high frequency vocabulary, general vocabulary,

specialized vocabulary, and technical vocabulary (Ernst-Slavit & Slavit, 2007). The assumption

of many educators that mathematics is a universal language is false; Numbers and symbols may

be similar in other languages and cultures but explicit instruction is still necessary to move EL

students beyond shallow conceptions (Leali, Byrd, & Tungmala, 2012).


Most mathematics content is shared through oral language that is dominated by a teacher.

This places critical importance on teachers to develop mathematical language, as this knowledge

base can either promote or inhibit learning. An observed practice in classrooms with ELs is the

oversimplification of mathematical language to communicate concepts. Unfortunately, this

simplification does not grant better access to content or produce better long-term outcomes. This

approach most often simplifies rich mathematical concepts and becomes an obstacle in learning

(Warren & Miller, 2015). As expressed by Alt, Arimendi, Beal, Nippold and Pruitt-Lord (2014),

explicit vocabulary instruction must occur. Rather than watering down curriculum, Warren and

Miller (2015) promote Vygotsky’s Zone of Proximal Development. From this learning

perspective, students can reach their potential capacities for development when given the support

of a more knowledgeable person. Scaffolding provides students support to learn rich

mathematics.

Doabler and his colleagues reviewed research from 2013 to 2016 to determine the

frequency of mathematics intervention studies for English learners. This research revealed only

three cases of studies. Doabler suggests that the lack of research “sheds light on the urgency to

build the knowledge base on effective math instruction for ELs (Doabler, Clarke, Kosty, Baker,

Smolkowski, & Fien, 2016).” Research by Ekwueme, Ekon, and Ezenwa-Nebife (2015) supports

that 75% of learning occurs through activity-oriented instruction. Exemplary mathematics

instruction, therefore, must be centered upon students. Hands-on mathematics instruction guides

students in gaining content knowledge through experience. Through this manipulation of

objects, students can develop into critical thinkers.

Classrooms with high percentages of ELs must provide opportunities for students to

verbalize mathematical understanding and thought processes. Mathematics instruction must


occur explicitly by systematically incorporating visual models to deepen understanding of

concepts and skills (Doabler, Clarke, Kosty, Baker, Smolkowski, & Fien, 2016). Herrell and

Jordan (2014) promote the inclusion of tactile, concrete objects called realia. Through realia,

English learners are given additional access to background knowledge, strengthening academic

vocabulary and success with grappling mathematical concepts.

In a second body of literature, Doabler, Nelson, and Clarke (2016) assert that little rigor

has been applied when conducting research on effective mathematics practices for ELs, resulting

in many students developing mathematics difficulties (MDs). Between the years of 2000 and

2012, Doabler and his colleagues were unsuccessful in yielding any literature about mathematics

interventions with ELs. Despite the lack of research on this topic, evidence-based practices exist

to support ELs in mathematics acquisition. The use of visual models and manipulatives assists

ELs in building connections between fundamental mathematic concepts. Specifically cited

within this strategy is the use of CRA to build conceptual understanding of mathematic concepts.

Teachers can use CRA by beginning instruction with concrete examples and then interweaving

pictorial representations and abstract symbols as students develop conceptual understanding.

While there is a discouraging amount of emphasis placed upon mathematics strategies for

ELs, some research is beginning to surface. A recent study on ELs and mathematic success by

Sorto, Bowler, and Salazar (2016) focuses upon three main facets, one being cultural issues such

as households, parents, and communities. The focus is not placed upon teaching, but rather

relating the world of students to the classroom. A second facet is on the empowerment of

minority students. A third facet is mathematic practices and language. This is especially

pertinent, as native English speakers have difficulty communicating in mathematics discussions

just as ELs do. Differences exist between social language and academic language use. Despite


lingering evidence that ELs struggle with mathematics proficiency, schools with economically

disadvantaged students are outperforming others due to the exposure of rich mathematics and the

implementation of strategies that support learning for linguistically diverse students.

A second body of research presented by (Barwell, 2016) postulates that, while little

emphasis is on evidence-based mathematics strategies for ELs, research on language diversity in

mathematics classrooms is beginning to take form. At present, cognitive research is focusing

upon how language diversity on the individual level influences mathematical thinking and

performance. Discursive research is focusing upon how language diversity impacts participation

in classroom interactions. Finally, socio-political research is focusing upon how language is

more than a communication tool. This research is grounded on the idea that the way language is

used marginalizes certain groups within the population, primarily those who are impoverished or

are diverse in terms of race or language. These three forms of research may provide insight as to

how language impacts mathematical success and why there is a consistent achievement gap

between ELs and native speakers.

Research Question

The purpose of my research is to discover how the CRA method of instruction impacts

learning for English learners. Through observation and the gathering of data, I have identified

gaps in knowledge between native speakers and ELs. To combat these gaps, I plan to integrate

CRA instruction into a mathematics unit on fractions in hopes to enhance learning and increase

retention of concepts for English learners. As evident by the information presented in the body of

the literature review, an inadequate amount of research focuses upon mathematics interventions

and strategies for ELs. While it is well-accepted by educators that language plays an important


role in mathematics, the amount of research on how to nurture the development of mathematical

language and proficiency is disproportionate to the diversity within America. My research

question is posed as follows: How does Concrete-Representational-Abstract instruction impact

learning for English learners?

Participants

The participants of the action research are all third-grade English Learners who are active

participants in English support courses or are in the first year of monitoring. Six students are

male and three are female, bringing the total of students in the study to nine. Of the participants,

one student is currently in tier three of Response to Intervention (RTI). He receives intervention

for both language and mathematics deficiencies.

Setting

The research took place at Chatsworth Elementary School in Chatsworth, Georgia. The

school is a K-6 title 1 establishment with a population of roughly 750 students. Of those students

in attendance, seventy-seven percent of the students are eligible for free or reduced lunch. Sixty-

seven percent of the population is Caucasian, and thirty-one percent of the population is

Hispanic. The remaining two percent is shared between multiracial and Asian students. Twenty-

one percent of the Hispanic population is currently labeled as limited English proficient (“K-12

Public,” 2015).

Methodology

The mathematics unit began with hands-on instruction. The concept was initially taught

using manipulatives, encompassing the concrete phase of the strategy. The same model and


procedures were then used pictorially to represent the process during the representation phase.

The final phase, abstract, required students to practice using mathematical symbols until

automaticity, or mastery of algorithms, occurred. This final phase was reserved for use only if

the concept could not be grasped through prior phases (Mudaly & Naidoo, 2015).

Prior to beginning the mathematics unit on fractions, I administered a pre-assessment.

Data gathered from the assessment was compared to post-assessment data so that growth could

be determined quantitatively. Using measures of central tendency, I created a chart that displays

the mean of pre-test scores for my EL students and the mean of post-test scores for my EL

students. I also included the range for each student by finding the difference between the two

scores. The mean scores served as a quick way for me to see if the average of scores improved

after treatment. I also calculated the standard deviation for both assessments so that I could

lessen the impact that outliers could have on my data. All calculations were completed through

Microsoft Excel. After completing these calculations, I used the REACH growth formula to

determine if students met growth goals.

I gathered additional quantitative data by using a checklist. Using a checklist with the

choices of engaged or disengaged, I selected a choice that indicated the engagement level of each

English learner during the unit. Perceived engagement was recorded in fifteen minute increments

to see if differences emerged as a lesson progressed. The data was coded as a 1 for engaged and

2 for disengaged. After being coded, the data was transferred to a histogram to aid in analyzing

the data. The table allowed me to quickly see the frequency and percentage of students who were

engaged during the unit and to determine how often students were not appropriately engaged.


I gathered qualitative data during the unit by journaling twice per week. Each Wednesday

and Friday, I reflected upon my observations, formative data, and thoughts about the

effectiveness of CRA for my English learners. After completing journaling, I analyzed my

entries for emerging themes by utilizing the website Word Cloud. Recurring themes are

discussed in a subsequent part of this research report.

After analyzing the three data sets, I reflected upon the data and considered what insights

action research has provided me in relation to CRA instruction’s impact on mathematics

instruction for English learners. This reflection takes the form of a narrative.

Data Analysis

Table 1, as shown below, represents student growth as a bar graph. One can conclude

from viewing the data that all students more than doubled in growth.

As shown in Table 2, I recorded the pre-test and post-scores of my English learners. This

table shows the amount of growth, represented in range, for each EL. As indicated in the table,

all students in this study demonstrated growth.

1 2 3 4 5 6 7 8 90

20

40

60

80

100

120

Table 1: Student Growth Bar Graph

Series1Series2

Student Number

Test

Sco

res


The mean pre-assessment score is 21.4. The standard deviation of the pre-assessment

scores is 4.6, indicating a small spread in scores. The mean post-assessment score is 90.2. The

standard deviation of post-assessment scores is 10.2, indicating a small spread in scores. The

mean range in assessment scores is 68.7, supporting that there is significant change in pre-

assessment and post-assessment scores. The standard deviation in range is 11.4, indicating a

small spread in range scores.

After calculating mean averages, standard deviation, and the range I applied the REACH

growth formula ([100-pre] x .5). Table 3, as shown below, reflects REACH growth target data.

Table 3: REACH Growth Data

Student Pre-Test Post-Test Growth Target Actual Growth

1 25 97 37.5 72

2 22 100 39 78

3 22 89 39 67

4 28 92 36 64

Table 2: Comparison Table of Pre-and Post-Assessments ScoresStudent Pre-Test Post-Test Range

1 25 97 722 22 100 783 22 89 674 28 92 645 11 92 816 19 100 817 22 67 458 22 92 709 22 83 61Mean 21.4444 90.2222 68.7778SD 4.6398 10.2442 11.4431


5 11 92 44.5 81

6 19 100 40.5 81

7 22 67 39 45

8 22 92 39 70

9 22 83 39 61

The REACH formula helps to determine the target growth for each student and to indicate if

target growth was achieved. All students exceeded their growth targets, with two students more

than doubling their growth targets.

Student engagement was determined using a checklist in fifteen minute intervals. After

coding engagement with a one and disengagement with a two, the coded data was tallied and

placed in Microsoft Excel. The data was transferred to a histogram to display the frequency of

engagement and disengagement during the unit. During the four week CRA fractions unit, the

mode of each week is coded as a one, indicating overall engagement. Individual discrepancies

that are considered outliers will be discussed in the findings portion of this report in a narrative.

1

1.166666667

1.333333333 1.5

1.666666667

1.833333333More

050

100150200250300350400

Student Engagement Histogram

Frequency

Bin

Freq

uenc

y


Qualitative data was gathered through journaling twice per week using Google Docs as a

platform. Upon completion, the journal entries were uploaded in the Word Clouds platform. I

then reduced the uploads by removing text with a frequency of less than two. I completed a

second reduction by eliminating filler words that muddied up the cloud and student names to

maintain privacy rights. While a preview of the word cloud is provided, a higher quality can be

accessed online http://i.imgur.com/YPmEOzr.png.

Interpretations

When analyzing pre-assessments, it is evident that my English learners had some prior

knowledge of fractions before being exposed to the CRA fractions unit. Some of this prior

knowledge consisted of misconceptions that required corrections. For example, when labeling

fraction models many of the students recognized the shaded parts as the numerator and the

unshaded parts as the denominator. All the students demonstrated an ability to shade fractions,


indicating that little emphasis should be applied to this concept. The small standard deviation in

pre-assessment scores, when coupled with the error patterns and consistency in shading fraction

models, reflects that the students had similar levels of understanding prior to the unit.

All students reached target growth goals for the unit. The standard deviation of the post-

assessment,10.2, is slightly higher than that of the pre-assessment. This increase in the spread of

scores is due primarily to student seven’s post-assessment score, which serves as an outlier.

During multiple times throughout the unit, this student refused to participate. Despite exposure to

a variety of instructional techniques aligned with CRA practices, he was disinterested in the unit

by the second week of instruction. While this student’s engagement fluctuated throughout the

unit, the time spent disengaged had a negative impact on his conceptual knowledge. Although I

provided intensive remedial support, this student received an unsatisfactory post-assessment

score. I hypothesized that student five would receive the lowest post-assessment score, as he is

currently in tier 3 of RTI. However, this hypothesis was incorrect. Effort had a positive impact

on student five’s progress.

After coding student engagement, aggregating the data, and generating a histogram, it

became evident that the CRA unit was successful in engaging students for the duration of the

unit. The mode of each week was a one, revealing that most students were engaged consistently.

Student five received the most disengaged checks throughout the unit. This student struggles

with attention during instruction. I hypothesized that his attention would improve during the

CRA unit. While he was more enthusiastic about mathematics class, his attention span remained

relatively the same. During the second week of instruction, I elected to move student five’s seat.

This seemed to alleviate many of his attention span difficulties, evident by the increase in

engagement checks. I speculated that student five would not meet adequate growth due to his


attention span difficulties. However, this student nearly doubled his expected growth target and

received a satisfactory post-assessment score.

Student seven received the second highest number of disengagement checks. This

behavior is atypical for this student, suggesting that some internal or external stimulus impacted

his engagement in the content. When asked why he refused to work, he did not have an

appropriate answer. I received a shrug and the behavior did not improve. His refusal to

participate encouraged me to pose additional research questions that might help to reveal

underlying causes of disengagement: How does motivation impact how students learn

mathematics? How does the affective domain impact how students learn mathematics?

When analyzing my word cloud, I selected only the words that stood out to me. Some

words, such as EL, give little insight to emerging themes despite the size of the words and were

consequently ignored as potential narrative content. The most prominent word in the cloud is

assessment. Truly, assessment drove my instruction. A formative pre-assessment helped me

determine prior-knowledge including misconceptions, another theme in my cloud, held by

students. I formatively assessed students in terms of engagement and ability through observation

and weekly assessments. I provided a summative assessment so that I could determine the

appropriate next steps for each of my learners. Assessment was a critical factor in the

implementation of my unit.

A second emerging theme is the ability of my students to complete the fractions unit. All

students demonstrated adequate growth during the CRA unit. After realizing that being able

doers of mathematics was a large part of my journaling, I decided to look for additional

emerging themes that enhanced the abilities of my students. An apparent theme connected to


ability is practice. My students were given scaffolded opportunities to practice fractions during

the unit through hands-on learning, pictorial learning, and abstract learning. Another connected

theme is instruction. The students were active participants in rigorous instruction four days per

week. They were expected to complete real-world tasks, Google Classroom practice, and

assessments from the course content.

A third emerging theme is disengagement. This descriptor was used during journaling to

describe the behavior of four students. Two of the students had single, isolated occurrences of

disengagement, warranting little further reflection on my part. These students typically present

no behavioral issues or indicators of disengagement, as evidenced through observation and

assessment data. The remaining two students, however, had multiple incidents of disengagement.

These students, numbers five and seven, are opposite in an academic respect. Student five is in

the third tier of Response to Intervention (RTI). Despite volumes of evidence that suggest the

need for special services testing, this process has been delayed because of his status as an

English learner. Student seven is a strong mathematics student. When given a county benchmark

assessment he scored the highest percentage in the class.

I reflected upon how my English learner who struggles the most could outperform my

English learner who struggles far less. I decided to refer to my engagement scan notes to see if

they could assist. His notes include explanations such as looking around the room, playing with

manipulatives, and requiring constant redirection. Before completing the post-assessment, I

assumed this student would make little gains, as he seemed to be off-task during very critical

parts of the unit. My assumptions about the student’s potential achievement were inaccurate and

unfair. Rather than focusing upon the moments he was engaged and exerted great effort I focused

upon the periods of deficit attention. I postulate that this student was not disengaged, but instead


has difficulty maintaining his attention span, implying that I need to provide more variety during

whole group instruction.

I also referred to my notes to ascertain why student seven was disengaged. Initially my

comments stated that he refused to work. Despite applying consistent classroom management

and behavioral techniques, student seven was unwilling to exert effort. After moving down on

the behavior chart, he would comply by mimicking activities but was not putting forth effort.

Later in the unit, I began to record notes indicating that student seven was confused. This finding

is logical, as he refused to complete prerequisite activities earlier in the unit. In an effort to

remediate student seven’s struggles and prevent future complications, I spoke with the student.

The conversation was fruitless; I received shrugs when asked why he was not trying his best and

what he did not understand. I completed remediation sessions with this student during small

group instruction. Despite this additional support, student seven’s post-assessment demonstrates

a lower growth level than expected. By completing additional research about the affective

domain and mathematics engagement, I hope to discover ways to positively reach student seven

in the future.

Conclusions

The CRA unit was successful in increasing conceptual and procedural knowledge of

fractions. This conclusion is supported by quantitative data collected during the unit. All students

meeting expected growth targets as generated using the REACH formula. Furthermore, all

students exceeded these growth targets. Pre-assessment and post-assessment data also supports

this conclusion, as all students grew by at least 45%. Additionally, all but one student scored

within a passing range of 70%.


Qualitative data gathered during the unit suggests that CRA increases engagement,

although additional action research would need to take place. Of the nine participants, seven of

the students are engaged most the time. The two outlier students are who I hoped to positively

impact regarding engagement. While I saw behavioral improvements in these two students, I

cannot conclude that CRA had a significant impact on student engagement. Therefore, future

research on engagement and motivation for English learners is appropriate.

Reflection and Action Plan

The purpose of my study was to discover how Concrete-Representational-Abstract

instruction impacts learning for English learners. I sought to answer the following research

question: How does Concrete-Representational-Abstract instruction impact learning for English

learners? After gathering quantitative data in the form of pre-assessment scores and post-

assessment scores and qualitative data through engagement scans and biweekly journaling, I

have concluded that CRA has a positive correlation with an increase in conceptual knowledge of

fractions. It appears that CRA has some relation to engagement, although additional action

research needs to be applied to explore this observation fully. Through this research, I learned

the powerful impact that thoughtful, scaffolded instruction has upon the achievement of English

learners. CRA is a powerful framework of instruction that I will implement in future units

beyond fractions.

Future cycles of action research will deviate away from evaluating the impact of CRA,

instead focusing upon how the affective domain and motivation impact student achievement in

mathematics. Particularly, I want this research to focus upon English learners, as this focus

recognizes a diverse group of students as heterogenous regarding educational values and societal


structures. Through future cycles of action research, I hope to discover why two of my nine

participants were seemingly disengaged despite varied instructional settings and strategies. I

hope to discover if disengagement resulted from resistance towards learning, different learning

preferences, attention span, or affective influences. Through this cycle, I can modify my use of

CRA to better match the needs and interests of my English learners.

The CRA framework of mathematics instruction, while often requiring more thoughtful

planning, is a framework that I will continue to follow, particularly because of the impact it had

on my English learners. This framework encourages teachers to implement effective

mathematics teaching practices; students should be exposed to manipulatives and pictorial

representations before relying upon an algorithm to complete a task. The CRA framework

challenged me as an educator to develop conceptual knowledge as fully as possible with the

understanding that procedural knowledge would follow.


References

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