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Running head: SIGNATURE ASSIGNMENT 1

Taking a Literary Approach to Mathematics: An Integrated ELD Model to Improve ELL

Mathematics Performance on Word Problems

Melissa Stencil

National University

SIGNTATURE ASSIGNMENT 2

Abstract

The proposal of the following action research cycle is concentrated around the general

topic of effective mathematics instruction for English Language Learners (ELLs). The primary

research question that the following work will lend towards addressing is how utilizing

mathematical models such as the Three Read Model and the Lesh Model will help K-2 ELL

students better comprehend and solve word problems. Specific sub-questions will also be

addressed including how a focus on systematic vocabulary instruction before and during

problem-solving will assist ELLs’ language acquisition, how presenting word problems through

a literary approach will benefit ELLs, and how translational modes of representation that require

students to express their knowledge in pictures, manipulatives, numbers, and words will assist

the language development and content acquisition of ELL students. Chapter 1 of this research

proposal includes an introduction and background to the topic, as well as the purpose,

significance, assumptions, delimitations, and limitations of the study, and the specific research

questions and definitions pertaining to this study. Chapter 2 comprises the review of literature

and Chapter 3 details the methodology of the proposed study. In sum, this paper seeks to outline

the proposed action research cycle with the hopes of identifying a beneficial integrated ELD

approach to mathematics instruction.


Table of Contents

Chapter 1 ……………………………………………………………………………pgs.4-15

Chapter 2 ……………………………………………………………………………pgs.15-34

Chapter 3 ……………………………………………………………………………pgs.34-46

Chapter 4…………………………………………………………………………….pgs.46-54

References……………………………………………………………………………pgs.55-58


Chapter 1

The number of English Language Learners (ELLs) in American public schools has been

steadily increasing for over a decade, currently calculated at about 5 million students nationwide

(Lynn, 2018). With only 3.8 million students being classified as ELLs in 2000, more than half of

the nation’s states saw increases in this student population between the 2009-10 and 2014-15

school years (Lynn, 2018). More specifically, California (CA) currently hosts about 29% of all

ELL students nationwide, standing as the state with the largest ELL student population (Sanchez,

2017). With particular respects to elementary education, majority of the students classified as

English Language Learners are in lower grade levels, with 67% of ELL students being in

kindergarten or grades 1-5 as of 2015 (Lynn, 2018). The other 33% were in 6th through 12th

grades. Presently and historically, the scores of ELL students in comparison to their reclassified

ELL peers suggests that “being reclassified as English proficient is associated with stronger

academic performance” (Hill, 2012). Coupled with the fact that nearly 60% of ELLs nationwide

come from low-income families with limited educational backgrounds, specific attention has

been paid to this sizeable portion of our nation’s students through federal and state legislation

(Breiseth, 2015).

One example of federal legislation addressing the unique needs of ELLs is the Every

Student Succeeds Act, which was signed in to law by President Obama on December 10, 2015.

This law built on progress made in previous years, specifically reauthorizing the Elementary and

Secondary Education Act, to ensure our nation’s “longstanding commitment to equal opportunity

for all students” (U.S. Department of Education). With regards to ELL education, this law

mandates that states must administer yearly assessments of the English language proficiency of

its ELLs and “develop new accountability systems that include long-term goals and measures of


progress for ELLs” (U.S. Department of Education). At its core, this piece of federal legislation

was drafted by educators and families to focus on the preparation that is needed for all students

to be fully prepared for success in college and careers (U.S. Department of Education).

One increasingly prevalent component of college and career readiness is mathematics and

science content-area achievement. A recent report by the U.S. Congress Joint Economic

Committee stressed the needs of students specifically being prepared for jobs in the STEM field,

stating that the “demand for STEM-capable workers has increased…due to the diffusion of

technology across industries and occupations” (The Johns Hopkins University, 2019). Therefore,

it is crucial to look at the current performance of ELLs in content areas such as mathematics, and

undertake a study focused on effective mathematics instruction for ELL students. Understanding

how to simultaneously develop the English language proficiency and content proficiency of ELL

students during mathematics instruction in the primary grades will shed light on the preparation

that is necessary for ELL students to engage with the increasingly language-based nature of

mathematics across grade levels.

Background

With California’s adoption of the Common Core State Standards in 2010 and

implementation of the standards beginning in the 2014-2015 school year, the “demand for using

more sophisticated language” across content areas surged (Maxwell, 2013). With specific

respects to elementary mathematics instruction, a key shift towards conceptual understanding

was undertaken (Spivey, 2015). This means that practices such as analysis, persuasion, and

comparison, among other typically literacy relegated tasks, have now become feature

components of CCSS math instruction, placing a heightened importance on the understanding

and use of technical vocabulary in content area discourse (Maxwell, 2013). Therefore, the


increasing linguistic demands associated with word problems oblige educators to know how to

teach language and content simultaneously.

Previous and current data reveal that ELL students have underperformed their non-ELL

peers on standardized math assessments across grade levels; but, recent research asserts that

these scores do not necessarily reflect mathematical difficulties (Krick-Morales, 2019). Rather,

these scores are representative of the struggle that ELL students face when they “encounter word

problems in a second language that they have not yet mastered” (Krick-Morales, 2019). Two

instructional strategies have recently gained prominence for their benefits on the language and

conceptual development of students and have many components that could be of specific

assistance to ELL students.

These two models are the Three Read model and the Lesh model. The Three Read model

proposes incorporating literary techniques into the instruction of math, focusing first on

understanding the stories of word problems and clarifying unknown vocabulary, before dealing

with the numbers and operations (Meldrum, 2010). The Lesh model is a problem-solving tool

that asks students to express their knowledge in pictures, manipulatives, numbers, words, and

real-life contexts, placing a particular emphasis on the facilitative role that language plays in

deepening conceptual understanding (Lesh, R. & Doerr, H., 2003). Therefore, a study

specifically looking at the implementation of these two strategies in K-2 classrooms with a high

population of English Language Learners would reveal the benefits that increased attention to

contextual understanding, vocabulary development, and multiple modes of expression have on

the performance of ELL students.

Purpose


The purpose of the proposed qualitative study is to examine the benefits of the two

aforementioned mathematics models for instruction, The Lesh Model and the Three Read Model,

on K-2 ELL students’ mathematics performance in California public schools. Specific focus will

be placed on how three components of these models, including targeted vocabulary instruction,

use of a story stem protocol for problem solving, and the inclusion of translational modes of

representation (pictures, manipulatives, numbers, words, real-life contexts), assist ELLs’ word

problem comprehension and achievement. This will be done through teacher and student

interviews, classroom observations, and data analysis. The research collected through these

methodologies will be used for the proposal of an effective integrated ELD approach to

mathematics in the elementary grades.

Significance

The research about the effects of the two specified mathematics instructional models on

the performance of K-2 ELL students is significant for three reasons: (1) it investigates how to

incorporate The Lesh Model and The Three Read Model into CCSS mathematics curriculum as

an integrated approach to English Language Development; (2) it provides strategies to improve

the language and content acquisition of ELL students to enhance their comprehension of and

problem-solving skills for linguistically rigorous word problems; and (3) it demonstrates the

benefits to teachers of vocabulary instruction and the provision of multiple means of engagement

in mathematical content area instruction.

To begin with, The Lesh Model and The Three Read Model contain many aspects of

instruction, such as a focus on language support and development, the activation of background

knowledge, and an emphasis on contextual support, that have not specifically been looked at in

relation to their potential as effective integrated ELD approaches to mathematics. In addition to


the required designated ELD instruction in California public schools, adopting integrated ELD

approaches across content areas is “the fastest way to bring English learners into full

proficiency” (Neer, 2017). As both of these models of instruction incorporate thinking, talking,

listening, reading, and writing, all ELD domains can be systematically developed with the

regular use of these models during mathematics instruction.

Moreover, with the increasingly verbal nature of mathematics problems in curriculum

and on standardized tests, ELL students need to be provided with strategies that can boost their

language and content acquisition simultaneously to improve their mathematics abilities and

performance. While popular assumption has promoted the notion that math is a “universal

language,” succeeding in mathematics is just as dependent on understanding linguistics as

succeeding in reading and writing (Scholastic, 2019). Therefore, when students are taught

strategies to first focus on understanding the wording and context of math problems, their

performance on math assessments and conceptual understanding of mathematical standards has

been shown to improve to levels on par with their non-ELL peers (Ambrose & Molina, 2010).

Finally, educators need to not only see the benefits of providing vocabulary instruction

and opportunities for multiple means of engagement in their content areas, but also look to

models of tangible strategies they can incorporate into their mathematics instruction to improve

ELL learning outcomes. According to a May 2005 study of California teachers, a substantial

number of teachers reported a lack of knowledge about how to meet the needs of both their ELL

students and their non-ELL students in the same class (Shreve, 2005). However, research has

shown that when teachers receive proper preparation, they are more confident and successful at

instructing their English learners (Shreve, 2005). Therefore, demonstrating the benefits of the


two proposed strategies in this study would further the equipment of general education teachers

with strategies to benefit their students with unique learning needs.

Research Questions

Research Question: How can utilizing problem-solving techniques, with the use of

mathematical models such as the Three Read model and Lesh model, help Kindergarten through

Second Grade ELL students better comprehend and solve word problems?

Related Sub-Questions:

Sub-Q 1: How does presenting word problems through a story stem protocol, with a

focus on systematic vocabulary instruction, benefit ELLs’ ability to generate and solve math

word problems?

Sub-Q 2: How do translational models of representation that ask students to express their

knowledge in pictures, manipulatives, numbers, and words, assist the language development and

content acquisition of ELL students?

Assumptions

Considering the topic and chosen methodologies for conducting this study, there are

several underlying assumptions that propel the research. The first assumption is that the ELL

student population will continue to be a predominant demographic in the mainstream American

classroom. Increasing from 3.8 million students in 2000 to 4.8 million students in 2015 and 5

million students in 2017, the number of English language learners across the United States

continues to rise (Lynn, 2018). A particularly high number of these students reside in California,

claiming upwards of 1.5 million ELL students nationwide (Breiseth, 2015). Because of this


continual increase, their prevalence in American classrooms is worthy of attention from

educators and community members to ensure that they are being educated equitably. With this

upward trend of ELL presence in American classrooms being assumed, it is also assumed that

the success of these ELL students, particularly in content areas such as mathematics, is important

for America’s success in the global economy. A study by the Brookings Institution found that

“workers in STEM fields play a direct role in driving economic growth,” and that STEM workers

earn 26% more than non-STEM workers (The Johns Hopkins University, 2019). This points to

the benefits of developing strong mathematics abilities in ELL students for both our nation’s

advantage and students’ personal upwards mobility, as nearly 60% of ELLs nationwide come

from low-income families (Breiseth, 2015).

Moreover, it is assumed that teachers both need to be and want to be prepared to integrate

language instruction in to their content areas so that they can feel more knowledgeable and

confident about how they are meeting the needs of all of their learners. According to a 2005

study, 43% of California school teachers whose classes are made up of a majority of English

learners, received only one training session in the past five years about how to effectively

instruct these students (Shreve, 2005). Due to a notable lack of teacher preparation about ELL

instruction in mainstream classrooms and voiced concern from California teachers about their

lack of training, it can be assumed that educators have a desire for knowledge about stronger

instructional methods to meet the needs of their ELL students.

Lastly, it is assumed that the qualitative nature of the study will provide for a more

comprehensive look at teacher implementation of the proposed instructional strategies, teacher

outlooks on student participation and performance, and student benefits. Qualitative methods

often allow for the data to have “an enhanced level of detail to it,” allowing greater opportunities


for insights to be derived from it during analysis (Ayres, 2019). Additionally, further depth,

creativity, and authenticity can be gained from participant responses to supplement data collected

and observed, which results in greater accuracy of the study in its predictive ability on

reproduceable outcomes (Ayres, 2019).

Limitations

Due to the scope of this research project, there are several limitations that effect the

potential to generalize the results of this study. To begin with, purposeful sampling will be

utilized as opposed to random sampling of the target population. This will be done for location

purposes as well as to accommodate the shorter timeline of the study. Specifically considering

location, the desired school for selection in this study would be a public school in Southern

California that has a large rural population. Additionally, over 90% of the students qualify for

free and reduced lunches, over 65% of the students are considered to be English Language

Learners, and over 90% of the students are Hispanic, with the primary language spoken by

majority of the ELL students being Spanish. Furthermore, considering the time boundaries on

this study, this would be a short-term study conducted within one school year. While the

limitations of this study prevent it from being widely generalizable to the substantial population

of ELL students nationwide, that come from a multitude of culture and language backgrounds, it

does have the potential for suggestions to be made to this population. However, the results of this

study can be generalizable to primary grade teachers in Southern California who teach classes

with a majority of Spanish-speaking, low-income ELL students.

Delimitations


In terms of delimitations, there are several parameters that I chose to set the boundaries of

my study. To begin with, I chose to focus specifically on the elementary grades K-2, rather than

on all elementary grades K-6. This was chosen to look at the preparation that is needed in the

primary grades to enable students to engage with more rigorous word problems they will

encounter in higher grades. Moreover, this is when students are first encountering word

problems, working towards a reading to learn mindset that will dominate grades 3 and up.

Therefore, specific strategies to build students’ conceptual understandings of word problems in

the primary grades would be applicable for use in higher grades. Secondly, I chose to narrow my

focus of mathematics instruction to word problems for their particular prevalence to ELL

students’ success in mathematics. Word problems traditionally pose difficulties for ELL students

when they are met with problems presented in a language they have not yet mastered and that

might be constructed in culturally dissimilar ways from how they interact with words in their

home cultures. Therefore, looking at strategies designed to improve their word problem attacking

skills is crucial for their language and content acquisition. Finally, the theoretical perspectives

that I chose to adopt to form my foundational beliefs driving this study are Realism and

Pragmatism, supported by the more contemporary theories presented by Stephen Krashen, Jim

Cummins, and Zaretta Hammonds about how ELL students from collectivist cultures learn best.

Taken in synthesis, these theories provide a model for classroom instruction that holds ELL

students to high standards of academic work and performance, while encouraging them to utilize

their cultural, social, and linguistic contexts to problem solve, experiment, and collaborate.

Definitions

The Lesh Model: The Lesh Model is a translational model of representation that

provides spaces for students to demonstrate their understanding of mathematical concepts in five


different modalities: manipulatives, pictures, real-life contexts, verbal symbols and written

symbols (Lesh, R. & Doerr, H., 2003). Other versions of the Lesh Model have four categories

that are generally representative of the original five Richard Lesh proposed. These include math

tools/manipulatives, pictures, numbers, and words (either taken to mean a written explanation of

the solution or a written creation of real-life scenario to represent the given problem). These

different modalities are presented on a graphic organizer where the word problem is centered on

the page and surrounded by each of the five modes of representation. The central premise of this

model is that students show authentic understanding of concepts when they can translate them

from one mean of representation to another. Language is the facilitative tool that assists students

in this translation, being an especially important foundation for students’ ability to document the

problem-solving actions they took with written words (Lesh, R. & Doerr, H., 2003).

Story Stem Protocol: The story stem protocol, also called the “problem stem” protocol,

is a reading of a math word problem that only includes the story or word problem without the

question at the end (SFUSD, 2019). This is done with the purpose of having students focus on

the context and math information before they begin dealing with any questions involved

(SFUSD, 2019). In addition to giving students necessary space to talk through the scenario being

presented, this method for presenting word problems also allows students to generate their own

questions for the given story (SFUSD, 2019).

The Three Read Model: The Three Read Model for mathematics instruction is one way

to perform a close read of a complex math word problem and involves reading the word problem

three times, each with a different purpose. This Model utilizes the story stem protocol described

above for the first read. During the first read, typically both the question and the numbers are left

out of the reading. This is done to ensure student understanding of the text/context (Early Math


Collaborative, 2018). Additionally, this is the read where a focus on systematic vocabulary

instruction can be taken, stopping to clarify unknown or confusing vocabulary words, much as is

done during a close read of a literary text. The problem is read orally, accompanied by

supporting visuals, and students engage in activities to share their understanding of the problem,

including pair shares or even dramatic reenactments of the scenario (SFUSD, 2019). The goal of

the second read is for students to glean information about the numbers involved in the word

problem. Students share their observations about the numerical values and even generate a list of

potential questions that could be asked based on the quantities in their context (Early Math

Collaborative, 2018). Finally, the goal of the third read is to give students the question(s) and

allow students opportunities to collaboratively solve the problem in different ways (Early Math

Collaborative, 2018).

Frontloading Vocabulary Instruction: Frontloading has been widely recognized as an

effective and necessary portion of a lesson to prepare students for the critical thinking and

specific skills they will need for a given lesson (Adams, 2012). Frontloading vocabulary involves

generating a list of high-level or potentially confusing vocabulary words that will be present in a

lesson and then pre-teaching them to students. Pre-teaching the vocabulary words focuses on the

activation of student background knowledge and engagement with them through a range of

visual, kinesthetic, auditory and oral activities (Stowe, 2019).

Chunking Vocabulary Instruction: “Chunking” is a specific way of pre-teaching

vocabulary that Benjamin Woodward presents in opposition to the frontloading method

described above. In the chunking method, vocabulary words are “chunked,” or grouped by word

association, when taught, rather than being taught all at once (Woodward, 2018). This method of

vocabulary instruction tailors to studies about the human brain that reveal limitations of the


human working memory to retain certain amounts of new learned information at once

(Anderson, 2019). Chunking words into thematic or associated categories helps the brain make

authentic connections amongst the words, easing their storage into long-term memory

(Woodward, 2018).

Summary

In review, this study broadly seeks to explore the topic of effective mathematics

instruction for ELL students in the primary grades (K-2). More specifically, this research will

look at the implementation of The Lesh Model and The Three Read Model to document how

their incorporation into mathematics instruction improves ELLs’ strategies to comprehend and

solve linguistically rigorous math word problems. This study will benefit both general education

teachers and ELL students, as it looks to provide a feasible approach to integrated ELD

instruction in mathematics for teachers to utilize in their classrooms. Additionally, it will provide

ELL students with approaches they can make use of to engage independently with word

problems in higher grades when they need greater mastery over literary skills to interpret and

solve word problems. A review of literature will be provided to detail the research that has

already been conducted in this field and make an argument about the added benefits this study

would contribute to this body of literature.

Chapter 2

The following review of literature will provide a synopsis of the challenges faced

uniquely by ELL students when encountering mathematical word problems in English and of the

research that has been done to study instructional techniques aimed at remedying the challenges.

Although existing research discusses both procedural and conceptual mathematics difficulties for


ELL students, this review will specifically focus on comprehension difficulties ELL students

encounter when reading linguistically rigorous word problems that have increased in prevalence

with the CCSS. The specific categorical sections of the review include the following:

Mathematics Difficulties for ELLs, Strategies for Developing Mathematics Vocabulary for

ELLs, Presenting Word Problems through a Literacy Lens, Using ELLs’ Cultures and

Background Knowledge to Aid Word Problem Comprehension, and The Three-Read Method

and Lesh Model. Generally, the research supports the fact that ELLs face difficulties in

mathematics primarily due to the language barriers that inhibit conceptual understanding and

comprehension of increasingly verbal mathematics textbooks, word problems, and assessments.

Findings indicate that improved English proficiency results in improved mathematics scores for

ELLs which points to the need for language instruction to be integrated into mathematics

instruction. Three areas of focus are identified by the literature as ways to embed language

instruction into mathematics instruction. The first area is a prioritization by mathematics teachers

of systematic vocabulary instruction that goes beyond pre-teaching simplistic definitions to

providing multiple ways for authentic student engagement with the vocabulary in meaningful

contexts. The second area is an incorporation of literary techniques, such as close reading and

annotation strategies, into mathematics instruction to develop students into strategic readers of

complex word problems. The last area is a utilization of students’ home languages and cultures

to create personally relevant word problems for mathematics instruction and activate the

necessary background knowledge for students to participate in the modeling needed to solve

word problems. Finally, the review presents two research backed problem solving models, The

Three Read Model and the Lesh Model, which have not been specifically studied in relation to

ELL mathematics performance. Due to the fact that these two models include all three of the


aforementioned areas of focus for integrated ELD instruction in mathematics, the conclusion is

drawn that implementing these two models into mathematics instruction would be a viable way

to integrate ELD instruction into mathematics instruction.

Review of Literature

Mathematics Difficulties for ELLs

Due to increasing linguistic demands associated with math problems, it has become

necessary for students to demonstrate mastery over both language and content in order to exhibit

their proficiency in this subject matter. For English Language Learners, this presents an

additional layer of complexity to solving math word problems when they are still acquiring

fluency in the English language. These difficulties have been highlighted in research that

disaggregates math scores for ELL students and their non-ELL counterparts. “Mathematics

Performance” (2018) is a document that addresses the findings from the National Assessment of

Educational Progress’ (NAEP) student performance assessments in mathematics. NAEP

mathematics assessments have been administered regularly since 1990 and have been analyzed

for grades 4, 8, and 12 in both public and private schools across the nation. Scores range from 0-

500 for grades 4 and 8, and from 0-300 for grade 12. Specifically looking at the elementary

grades, the average mathematics score for 4th-grade ELL students was 217, which was 26 points

lower than the average mathematics score for non-ELL students at 243 (The Condition of

Education, 2018). In grades 8 and 12, the gap widened to about a 40-point differential between

ELL and non-ELL students, revealing an inverse relationship between grade level and

mathematics proficiency (The Condition of Education, 2018).


The relationship between English language proficiency and math scores, as well as

between grade level and math scores, is similarly reported by a study entitled, “Examining the

Relationship Between Math Scores and English Language Proficiency” (Denfield, Nicolae, &

Beate, 2014). This study investigated the capacity of English proficiency to project mathematics

scores, while controlling for gender, socioeconomic status, and grade level among ELLs at a

South Florida elementary school (Denfield, Nicolae, & Beate, 2014). Their analysis revealed that

English proficiency is a statistically significant predictor of mathematics scores, with

mathematics scores increasing simultaneously with English proficiency. From mathematics

scores collected across grades 3-5, English proficiency alone was able to account for 47.9% of

the total variance in mathematics scores (Denfield, Nicolae, & Beate, 2014). Therefore, the study

makes the conclusion that ELLs’ low scores on mathematics assessments might be more

reflective of their inability to “understand the wording of the questions,” rather than of their

“mastery of the mathematical content” (Denfield, Nicolae, & Beate, 2014). This correlates with

the decrease in ELLs’ mathematics proficiency in higher grade levels. As word problems become

longer and more complex to read, “literacy abilities that were functional in the primary grades”

become insufficient to help ELLs decode the more rigorous texts of higher grades (Denfield,

Nicolae, & Beate, 2014).

“Children’s Ways with Words in Science and Mathematics: A Conversation Across

Disciplines” discusses the specific literary challenges that students from culturally and

linguistically diverse backgrounds face in their comprehension of rigorous science and

mathematics texts. Much of the research that has been conducted to understand issues of

educational equity for minority and ELL children focuses on these students’ “ways with words”

(Rosebery, A. & Warren, B., 2000). This is described by Shirley Brice Heath as the differing


ways in which individuals engage in practices such as “argumentation or storytelling in school

and out,” and how “ways of talking and interacting that seem ‘natural’ to members of one

community are experienced as culturally strange by another” (Rosebery, A. & Warren, B., 2000).

Understanding the barriers that ELL students face when it comes to comprehending and

responding to increasingly rigorous word problems needs to take in to account their unique funds

of knowledge, and ways of expressing and communicating this knowledge to others. The ways in

which these students engage with words in their home languages and cultures informs their

understanding, or lack thereof, of the ways in which words are organized in the English

language. With this in mind, Rosebery and Warren conclude that just as children filter input

through their linguistic and cultural lenses, so do teachers; often “misunderstand[ing] children

who say and do things differently from what they expect” (Rosebery, A. & Warren, B., 2000). It

is these interpretive differences which impact the ways in which ELL students are able interact

with mathematical content that is increasingly verbal in nature.

Brenda Krick-Morales also outlines the complexity posed by word problems for ELLs in

that they “require that students read and comprehend the text of the problem, identify the

question that needs to be answered, and finally create and solve a numerical equation” (Krick-

Morales, 2019). Morales specifically underscores the importance for ELLs of not only knowing

the key terminology utilized in mathematical word problems, but also of knowing how to put

together the meaning of the words in their context (Krick-Morales, 2019). This aligns with the

findings of the previous text in that “because words are used differently” in different cultural and

linguistic contexts, students cannot just rely on knowing certain key words. An overreliance on

the specific operations that certain key words signal can lead students away from actually trying

to understand the problems (Krick-Morales, 2019). For example, Krick-Morales looks at a word


problem in which a student is asked to determine how many marbles a boy named Paolo has

knowing that a girl named Maria has 24 marbles which is 8 fewer than Paolo has. If a student is

solely focused on the key words fewer than, they would draw the conclusion that the operation

needed to solve the problem is subtraction, when in actuality it is addition. Therefore, students

need to not only know the meaning of the words, but also be able to “see them in the context of

the whole problem” (Krick-Morales, 2019). The ability to not only decode word meanings but

situate them in a culturally and linguistically different context from their own, has therefore

posed a challenge for ELL students when they encounter math word problems in a language they

have not yet mastered. Literature that proposes instructional techniques to address these

challenges, in light of the findings articulated above, will be evaluated below.

Strategies for Developing Mathematics Vocabulary for ELLs

Kristina Robertson outlines the misleading assumption that ELL students will easily

excel in math because math is a “universal language” (Robertson, 2019). On the contrary,

solving word problems, following directions, and understanding and using mathematical

vocabulary correctly are all skills that require a substantial command over language (Robertson,

2019). For this reason, it is crucial for teachers to make sure that their students understand math

vocabulary and have numerous opportunities to use it across speaking, reading, writing, and

listening modalities (Robertson, 2019). Helping students understand math vocabulary not only

includes teaching math-specific terminology, but articulating the “difference between the

mathematical definition of a word and other definitions of that word” (Robertson, 2019). To do

so requires that teachers encourage their students to explain mathematical concepts to their peers

(offering translations as necessary), provide visual cues, graphic representations, gestures and

manipulatives for students to interact with, and identify key words or phrases to preteach


(Robertson, 2019). Identifying what words or phrases in math word problems might be confusing

for students enables teachers to design and implement necessary scaffolds to guide students

through linguistically and contextually dense problems.

Classroom teacher Nancy Roberts similarly discusses the ambiguities within mathematics

vocabulary and recommends specific strategies to help build ELLs’ understanding of this

vocabulary. Encountering a student whose understanding of the term “base,” based on its

algebraic definition, created confusion with the term “base” being introduced in geometry,

prompted this teacher to ask the question, “Can I afford not to spend time on vocabulary

development?” (Roberts, N. & Truxaw, M., 2013). Roberts explains that mathematics

vocabulary may be more difficult to learn for ELLs than other academic vocabulary for several

reasons: definitions are filled with technical vocabulary, symbols, and diagrams; many

mathematics concepts can be represented in multiple ways; many mathematics words have

multiple meanings; the overlap between mathematics vocabulary and everyday English is

problematic; homonyms and words that sound similar can confuse; and similarity to native

language words can add more confusion (Roberts, N. & Truxaw, M., 2013). Therefore, Roberts

recommends developing a vocabulary list for the unit in order to assess students’ prior

knowledge and preteach new vocabulary, focusing on definitions, pronunciation, and word parts

(Roberts, N. & Truxaw, M., 2013). Roberts specifically highlights word walls and graphic

organizers as interactive ways to help students learn important mathematical vocabulary,

incorporating informal and formal definitions, examples, diagrams, non-examples, and real-life

scenarios (Roberts, N. & Truxaw, M., 2013).

In this same line of thought, Mary Stowe says that frontloading or preteaching vocabulary

to students not only helps them learn the meaning of new words, but “strengthen[s] their


independent skills for constructing meaning from text” (Stowe, 2019). Backing the importance of

developing students’ vocabularies, research has shown that students “must have a working

knowledge of 95% of the vocabulary in a passage in order to comprehend it” (Stowe, 2019).

Similar to Roberts, Stowe says that the first step in teaching vocabulary is to pre-scan the content

to determine difficult vocabulary (Stowe, 2019). She then presents a specific type of graphic

organizer to teach math vocabulary called the Frayer Model. In this model, the targeted

vocabulary word is placed in the center of the organizer and surrounded by definitions,

characteristics, examples and non-examples (Stowe, 2019). These are similar categories to what

Roberts recommended including on a word wall for math vocabulary. However, Stowe contends

that this is only the first of six steps needed to help students effectively learn new vocabulary.

Students should restate the description in their own words, construct a graphic representing the

term, and engage in a range of activities to deepen their knowledge of the term such as

discussing the term with peers, playing games with the term, acting out the term or creating a

song or story with the term, among other options (Stowe, 2019). Differentiating activities to

provide multiple means for engagement with content specific vocabulary is necessary to meet the

needs of a diverse range of learners and “facilitate deeper comprehension of critical content”

(Stowe, 2019).

Benjamin Woodward also explores the effects of preteaching vocabulary on second

language learners, but takes specific aim at whether the frontloading method or chunking method

yielded better results for vocabulary retention. Vocabulary is what “allows language to be

communicated, written, understood, and read,” all components of successfully understanding and

responding to math word problems with the increasing linguistic rigor they are associated with

(Woodward, 2018). Moreover, vocabulary knowledge necessitates that students can not only


recall a definition, but can also determine “how that word fits into the world” (Woodward, 2018).

Woodward conducted a study with two middle school foreign language classes to determine

whether students retained more words from a list of 20 if they were taught all of the words at

once with visuals or if they were taught the words in four chunks that were grouped by word

association (Woodward, 2018). The results of his study revealed stronger vocabulary retention

with students that received the chunking method as opposed to the popular frontloading method.

Woodward asserts that vocabulary knowledge is maintained better when “the brain is able to

make connections and process the vocabulary as something that is more than simply a word”

(Woodward, 2018). Chunking vocabulary words into categorical compartments provides the

necessary setting for the brain to make genuine connections amongst words, better facilitating

their storage in long-term memory (Woodward, 2018). Woodward’s study adds a necessary layer

of specificity to the recommendations made in the previous studies. Helping students learn and

understand the complex content terminology ever more present in math word problems is not just

a matter of preteaching vocabulary, but preteaching vocabulary in a way that allows second

language learners to make essential associations amongst words.

Presenting Word Problems through a Literacy Lens

Learning key content vocabulary is just one component to helping ELLs learn how to

better engage with English word problems. Being able to make sense of that vocabulary in the

context of the word problem requires that ELLs comprehend the story being communicated in

the word problem. This specific task is what Rebecca Ambrose and Marta Molina studied when

they explored whether teaching ELLs with an emphasis on English story problems was

appropriate. Working with primary school teachers of 5-8 year old children, Ambrose and

Molina used the principles of Cognitively Guided Instruction (CGI) as a foundation for their


study, which centers around children’s strategies for solving story problems (Ambrose & Molina,

2010). Key to students’ ability to make sense of story problems is their capacity for “sift[ing]

through information and selecting which components of the story pertain to the problem and

which need to be discarded (Ambrose & Molina, 2010). Ambrose and Molina state that ELLs

possess these “selective attention skills” because of their bilingualism, having “extensive

experience tuning out extraneous information to focus on the important elements” when reading

and listening (Ambrose & Molina, 2010). In classrooms where teachers explicitly guided

students through these metacognitive strategies, ELL students performed slightly higher on an

assessment given in English than in Spanish and only slightly behind their monolingual speaking

peers. Therefore, Ambrose and Molina concluded that when working with ELLs on story

problems, teachers should help children “become aware of their use of selective attention so that

it becomes an asset to their work on story problems” (Ambrose & Molina, 2010). With this type

of instruction, Ambrose and Molina assert that teaching mathematics through story problems is a

worthwhile approach to enhancing the mathematics skills of ELLs (Ambrose & Molina, 2010).

Digging deeper in to the need to instruct ELLs in mathematics with literacy strategies,

such as sifting through relevant and irrelevant information in problems, is Diana Metsisto in her

book “Literacy Strategies for Improving Mathematics Instruction.” Metsisto asserts that a

teacher’s job for mathematics instruction is not “simply a matter of cueing up procedures for

students,” but rather providing training to students on how to read and interpret mathematical

sentences (Metsisto, 2005). The basic strategies that literacy researchers have developed to help

students read to learn need to implemented by mathematics teachers. These include distinct

before reading, during reading, and after reading tasks to communicate the importance to

students of making sense of mathematics texts (Metsisto, 2005). Before reading a word problem,


the teacher should guide students through a preview of the text, noting visuals, labels, and other

print provided (Metsisto, 2005). Additionally, the teacher should activate students’ background

knowledge, discuss pertinent vocabulary and ask questions to set the purpose for the reading

(Metsisto, 2005). During reading, the teacher should check for understanding of the text by

paraphrasing the author’s words, help students use context clues to discern the meaning of

unknown words, and integrate new concepts with existing knowledge (Metsisto, 2005). Finally,

after reading, students should be asked to summarize what has been read, evaluate the

information that has been presented, and apply the ideas in the text to real-world situations

(Metsisto, 2005). All of these facets need to be intentionally prepared for by the teacher and

modeled first before gradually giving more of the responsibility of this “strategic reading” to the

students (Metsisto, 2005). When teachers of mathematics recognize the importance of helping

students learn to read and understand mathematical texts, students can become more

autonomous, self-directed, strategic readers of complex math word problems.

Backing a literary approach to teaching mathematics are Phyllis Whitin and David Whitin

who documented teacher Mirella Rizzo’s experience using books to engage her students in

mathematics. Rizzo anticipated that using math-related literature would allow her students to

grow in their understanding of mathematical content at the same time that they were improving

their reading skills and English language proficiency (Whitin, P. & Whitin, D., 2006). To

introduce her second-grade students to fractional relationships and comparative sets, Rizzo chose

the book How Many Snails? because of its familiar objects and repetitive linguistic pattern.

When Rizzo’s students were unable to generate their own patterns after engaging with her first

book choice, she picked up on her ELLs’ misunderstanding of comparative terms. To provide her

students with additional oral practice using comparative vocabulary, Rizzo chose a wordless


picture book, More, Fewer, Less, to encourage open-ended remarks (Whitin, P. & Whitin, D.,

2006). Her students quickly picked up on the connections between the two books and were able

to do more of the “cognitive work” required to name and describe comparative sets that the

wordless book invited (Whitin, P. & Whitin, D., 2006). Therefore, More, Fewer, Less served as

the “bridge” between How Many Snails? and the students’ own composition process (Whitin, P.

& Whitin, D., 2006). Students were able to create their own illustrations with corresponding

descriptions of attributive sets following the literary pattern of How Many Snails?. This gave the

students an opportunity to reflect further on mathematical relationships and use written means of

communication to express their knowledge to a larger audience at their school’s literacy fair

(Whitin, P. & Whitin, D., 2006). Rizzo’s success with her students highlights the benefits of

using math-related literature, providing students with opportunities to verbalize mathematical

concepts, and having students write regularly to enhance their understanding of mathematics

(Whitin, P. & Whitin, D., 2006).

David Whitin continues discussing the benefits of utilizing literary strategies to teach

mathematical content in his article “Problem in the Elementary Classroom” (Whitin, 2006).

Similarly to how Rizzo selected a wordless book that put the responsibility of observation

making on her students, Whitin says that observations are the start of mathematical

“adventuring” (Whitin, 2006). To encourage students to first make observations, Whitin

advocates for a problem-stem approach to teaching mathematics, in which a problem is discussed

without knowing what the question is (Early Math Collaborative, 2018). By letting students pose

problems based on their observations of numbers and story context, children are encouraged to

look closely, find patterns, offer theories, develop an inquiry mindset, cultivate perseverance and

truly participate in the work of mathematicians (Whitin, 2006). The more children observe, the


more they want to investigate, fueling a process of discovery with students at the forefront

(Whitin, 2006).

Concurring with the need to incorporate strategies that “are typically the province of

language arts teachers” into math lessons is Anthony Rebora (Rebora, 2014). Providing detailed

instruction in close reading, sentence annotation, and writing fluency, as well as ensuring

specific language objectives to accompany the content objectives for math lessons, are all

necessary steps for teachers of mathematics to take to address the linguistic demands that their

ELL students are facing in math (Rebora, 2014). This goes back to Metsisto’s call for teachers to

guide their ELL students through strategic reading strategies to make sense of mathematical text,

applying techniques traditionally taught to students during ELA to their mathematics time.

Rather than reducing the linguistic load placed on ELL students by assigning them “remedial

computation worksheets,” teachers need to provide their students with increased opportunities to

use language, written and oral, to explore mathematic problems and concepts (Rebora, 2014).

Working to support students’ understanding of word meanings needs to happen in the

meaningful contexts of word problems, supported by visual elements and linguistic prompts, to

develop both the content understanding and language acquisition of ELL students (Rebora,

2014). It is a problem-solving, literacy-informed approach to mathematics that brings the

complexity of math word problems into relatable, understandable contexts for ELL students,

boosting both their math performance and English language fluency.

Using ELLs’ Cultures and Background Knowledge to Aid Word Problem

Comprehension

Lynda R. Wiest also concludes that taking a problem-solving approach to mathematics

instruction, in addition to fostering collaboration and communication, drawing on student


background knowledge, and providing students with language support, is beneficial for the

instruction of ELL students (Wiest, 2008). Drawing from the National Council of Teachers of

Mathematics’ (NCTM) five process standards, Wiest says that problem solving is “at the heart of

good mathematics instruction” (Wiest, 2008). This means that students are first asked to discuss

what a story problem is about, without jumping to explanations for how to solve the problem.

Wiest specifically looks at a word problem about chickens and pigs that a fourth-grade teacher

presented to a group of ELL students. Before they jumped into the mathematical tasks of the

problem, students were led through a discussion about characteristics of a barn, pig, and chicken

to immerse them in the scenario and give them opportunities to reflect on the knowledge they

already brought to the problem (Wiest, 2008). By fostering a communal, student-centered

environment where students were encouraged to discuss connections to their primary language

and cultures, this fourth-grade teacher created a classroom climate that valued and utilized the

backgrounds of her ELL students (Wiest, 2008).

Also supporting the use of ELL students’ prior knowledge and cultures to effectively

teach mathematics, the document “ELLs and Mathematics” offers multiple strategies to address

the difficulties that ELLs face in mathematics. This document points out that depending on the

ELL student’s country of origin and previous educational experiences, everything from learning

styles to symbols and mathematical concepts may be distinct (ELLs and Mathematics).

Therefore, several strategies are recommended to draw on students’ prior knowledge and culture,

and thus facilitate deeper understanding of mathematical content. Cooperative learning groups

provide ELL students with necessary interaction amongst peers of varying language and learning

style experiences (ELLs and Mathematics). Personalizing word problems to include students in

real-life scenarios or having students design word problems based on real life scenarios can


increase student motivation and help them make important connections to new content (ELLs

and Mathematics). Additionally, having students create learning logs where they write about

their learning and its application to their lives can also foster students’ connections to their

cultures and backgrounds (ELLs and Mathematics).

Supporting the enrichment of word problems with students’ personal cultures and social

contexts are authors Nonmanut Pongsakdi, Teiha Laine, Koen Veermans, Minna M. Hannula-

Sormunen, and Erno Lehtinen. Noting that word problems have been criticized for their tendency

to encourage students to “apply superficial strategies” to quickly find a solution without deeply

understanding the context of the word problems, these authors designed a Word Program

Enrichment program (Pongsakdi, Laine, Veermans, Hannula-Sormunen, & Lehtinen, 2016). In

this program, teachers were encouraged to use “innovative self-created word problems” with the

goal of improving students’ mathematical modeling and problem-solving skills, as well as to

reprioritize the original focus of word problems to connect the world of mathematics to the world

of everyday life and experiences (Pongsakdi, Laine, Veermans, Hannula-Sormunen, & Lehtinen,

2016). Teachers who adopt this “narrative-oriented approach” provide students with

opportunities to describe situations of the problems in which they see themselves or real-world

experiences and teach them to use this knowledge to deal with the problem (Pongsakdi, Laine,

Veermans, Hannula-Sormunen, & Lehtinen, 2016). Creating word problems independently or

together with students based on interesting, culturally relevant, real-world situations that students

experience directly and indirectly in their everyday life allows them to work through the

modeling and mental representations necessary to accurately determine how to solve the

problems (Pongsakdi, Laine, Veermans, Hannula-Sormunen, & Lehtinen, 2016).


Additionally, Judit Moschkovich also highlights the practice of drawing on ELLs’

personal linguistic and cultural backgrounds as an asset to problem solving. The foundation for

the resources that Moschkovich provides to educators is centered on the premise that ELLs

concurrently develop their mathematic proficiency and their linguistic proficiency to express

mathematical understanding by actively participating in “rigorous mathematical reasoning that is

well scaffolded by instruction” (Moschkovich, 2013). With this premise in mind, teachers need

to deliver instruction that is informed by students’ experiences, language history and educational

backgrounds, viewing these components as resources, rather than as deficits (Moschkovich,

2013). Holding this view as true, teachers can work to make connections between students’

everyday/home languages and academic language and utilize students’ experiences as pathways

to developing academic ways of communicating mathematics knowledge (Moschkovich, 2013).

The Three-Read Method and The Lesh Model

Activating student background knowledge, presenting word problems in literary formats,

and developing the mathematical and linguistic vocabulary of students are all components that

comprise the Three-Read math protocol for word problems (Early Math Collaborative, 2018).

“Exploring the 3-Reads Math Protocol for Word Problems” is an article that documented the

experimentation of this method, also known as the “problem stem” protocol, in four schools in

Chicago across grades K-2. Many of the teachers read Beyond Answers by Mike Flynn which

examines the Mathematical Practice Standards and provides ideas on how to best implement

them in K-2 classrooms (Early Math Collaborative, 2018). Additionally, they used the 3-act

lesson resources authored by Graham Fletcher to better understand how to engage students with

the Three Reads Method (Early Math Collaborative, 2018). This approach involves reading the

situation three times, each with a specific focus. The goal of the first read is to understand the


text/context. As such, it does not include the numbers or the question (Early Math Collaborative,

2018). The goal of the second read is to comprehend the “mathematical structure” of the problem

by introducing students to the numbers of the problem (Early Math Collaborative, 2018).

Students generate a class list of possible questions and share observations about the numerical

quantities, similar to what Whitin advocates for in his problem-solving approach to mathematics

(Early Math Collaborative, 2018). Finally, the goal of the third read is to present students with

the question(s) attached to the problem and allow students opportunities to discuss different ways

of thinking about the steps to the solution (Early Math Collaborative, 2018). By using a method

such as the Three Reads Approach, children can be taught to work first on fully understanding

the context of the problem, a component of word problems that can typically present difficulties

for ELL students according to the literature analyzed above.

Further elaborating on the Three Read strategy for math problems is Adrianne Meldrum

who attended classes on the Three Read strategy at the Idaho Mathematics Conference in Boise,

Idaho. Concurring with Whitin who stated that students participating in a problem-solving

approach to mathematics would develop attitudes about learning such as persistence and an

appreciation for risk taking, Meldrum states that this approach teaches students to “learn to

persevere in problem solving” (Meldrum, 2014). In this version of the Three Read strategy,

Meldrum adds a component to the first read in which students “clarify unknown vocabulary,” an

area of mathematics that can often cause confusion for ELL students (Meldrum, 2014). As

students are discussing the contextual setting of the story problem, they are also working as a

class to discern the meanings of words and phrases that are essential to their understanding of the

problem. Meldrum takes the discussion a step further by bringing literary annotation techniques

in to the process, having her students highlight or underline words and phrases that they think


will be important to the problem and cross out words and phrases that contain inessential

information (Meldrum, 2014). This process of “sifting through information” is what Ambrose

and Molina argued is a skill of ELL students that should be explicitly taught to them as useful for

solving word problems (Ambrose & Molina, 2010). Meldrum iterates that this strategy is

specifically beneficial for ELL students so that they can discuss information or thoughts that

might ultimately get them “stuck” or “distracted” if they cannot verbalize their processing with

their peers (Meldrum, 2010).

Also incorporating the elements essential to effective instruction of ELLs, including a

focus on language development and conceptual understanding, is the Lesh translation model.

This model proposes that elementary mathematical ideas can be represented in five different

modalities: manipulatives, pictures, real-life contexts, verbal symbols, and written symbols

(Lesh, R. & Doerr, H., 2003). These modes of representation are displayed on a diagram for

students to use as a tool when solving word problems, with the word problem being shown in the

center of the diagram and the different modalities each occupying a space around the problem.

This model emphasizes that authentic understanding is demonstrated in a student’s ability to

represent mathematical concepts in multiple distinct ways, as well as their ability to draw

connections amongst these different means of expression (Lesh, R. & Doerr, H., 2003). By

reinterpreting information from one dimension of representation to another, students’ higher

order thinking and critical thinking skills are challenged and deepened (Lesh, R. & Doerr, H.,

2003). Unique to the Lesh model is its emphasis on language; the physical objects, pictures, and

contextual connections all become the objects about which student groups converse (Lesh, R. &

Doerr, H., 2003). Their language facilitates the translations among the different representations,

with student conversations being an important precursor for their ability to record their actions


with written symbols (Lesh, R. & Doerr, H., 2003). The effectiveness of this curriculum was

studied in short and long-term teaching experiments with 4th and 5th grade students. Students in

groups that utilized this instructional model had higher mean scores on overall post and retention

tests than students in groups where traditional instructional texts were used (Lesh, R. & Doerr,

H., 2003). Additionally, interviews conducted during and after instruction with the students

revealed that instruction based on this translation model resulted in higher levels of student

understanding, conceptual development, and verbalization of mathematical concepts (Lesh, R. &

Doerr, H., 2003). By encouraging students’ use of social and academic language to facilitate

their engagement with mathematical content across multiple distinct modalities, components of

math instruction that typically present challenges to ELL students can be addressed for their

benefit.

Conclusion

Taken as a whole, the literature above promotes the notion that ELL students need math

instruction that is also language instruction. Due to the increasing language demands that math

word problems are placing on students, which disproportionately impacts students who do not

have proficiency in the language, it is necessary to teach literary strategies to help ELL students

access mathematical content. The several groupings of literature promote strategies that include

wholistic vocabulary instruction, guided annotation strategies that capitalize on bilingual

students’ natural selective attention skills, the use of a problem stem protocol to foster student

observations and contextual discussions, an integration of real-life scenarios into word problem

instruction, and the incorporation of multiple means of expression to foster deeper, more

authentic conceptual understanding. The Three Read model and the Lesh model are two models

that incorporate these strategies into their instruction by placing language development and


expression at the center of their objectives. Therefore, used in conjunction with each other or on

a frequently rotating basis, these two models could be a viable option to address the differences

in mathematics performance shown between ELL students and non-ELL students. A study

conducted to test and document the impacts that these two models have on K-2 ELL student

performance related to math word problems, would reveal the potential possibility for these two

models to function as effective integrated ELD instruction during mathematics instruction.

Chapter 3

Introduction

The following Methodology section will outline a proposed study to test and document

the impacts that the Lesh Model and the Three Read Model could have on K-2 ELL student

performance related to math word problems. As articulated in Chapter 1 and Chapter 2, ELL

students consistently underperform their non-ELL peers on standardized mathematics

assessments, with the gap in performance widening with each subsequent grade level (The

Condition of Education, 2018). However, research has revealed a correlational increase in ELL

mathematics scores with increasing English language proficiency (Denfield, Nicolae, & Beate,

2014). Therefore, the need to instruct both language and content simultaneously is an area of

research needing further examination in order to advance effective integrated ELD instructional

strategies in the mathematics content area. In order to investigate the research question, “How

can utilizing problem-solving techniques, with the use of mathematical models such as the Three

Read model and Lesh model, help Kindergarten through Second Grade ELL students better

comprehend and solve word problems?”, a qualitative approach and design will be detailed

below.


Participants

The target population of this study are K-2 English Language Learning students in

Southern California. I will specifically be looking at a public elementary school in Vista, CA in

which 65% of the students are classified as English Language Learners, 90% of the students are

considered to come from low-income backgrounds, and over 90% of the students are Hispanic,

with the primary language spoken by most of the ELL students being Spanish. This school site

was purposefully chosen for the sample population as its student demographics largely represent

the characteristics of California’s ELL student population. Around 82.19% of California’s ELL

students speak Spanish as their primary language and nearly 60% of ELLs nationwide come

from low-income families with limited educational backgrounds (Breiseth, 2015). Moreover,

from this school site, a kindergarten class, a 1st grade class, and a 2nd grade class will be chosen

to participate in the study to get a wider range of information about the effectiveness of the two

proposed models for implementation in mathematics curriculum in the primary grades.

Therefore, this purposefully chosen sample population represents a maximum variation sample,

as a variety of characteristics such as background language, socio-economic status, and grade

level can be considered in the data collection and application of the research to the larger

population of California’s ELL students.

Choice of Methodology

The proposed case study will be qualitative in nature to glean a more wholistic view of

the implementation and effects of the proposed instructional strategies on ELL student

performance on math word problems. Qualitative methods allow collection procedures to be

more creative and authentic and give the collected data greater depth and detail, resulting in

higher accuracy of the study to be predictive about reproduceable outcomes (Ayres, 2019).


Triangulation will be the type of qualitative method utilized so that multiple forms of data can be

collected in order to increase the confirmation and validity of the findings (ChrisFlipp, 2014).

Specifically, three means of data collection will be incorporated into this research study,

including focus group interviews, observer as participant observations, and document review.

Focus group interviews will be held with the three K-2 teachers choosing to participate in

the study, as well as with groups of students in their classes. The teachers will be interviewed

before, during and after the proposed study, spanning one semester’s worth of math topic

instructional sequences. Before instruction begins, the teachers will be interviewed to gain their

perspective on the effectiveness of their current math curriculum and the specific difficulties that

they are seeing in their ELL students’ abilities to comprehend and respond to word problems.

They will then be interviewed during and after their instruction with the proposed math models

to track their insights about the effectiveness of incorporating The Lesh Model and the Three

Read Model into their math curriculum across several different topics of study. Furthermore,

groups of students will be interviewed before instruction, during instruction and at the end of

instruction to evaluate their ability to conceptualize math word problems. Questions will be

asked about student attitudes towards word problems, contextualized problem stems, math

vocabulary, and possible solutions to given word problems to determine how confident students

feel conceptualizing word problems and verbalizing their thoughts in group discussions. Focus

group interviews are chosen over individual interviews as they can gather information about

multiple people in one session, can calm the nerves of the participants (specifically of the young

students), and are more representative of the type of collaborative, verbal discussions that

students are expected to participate in during mathematics instruction due to the CCSS

(ChrisFlipp, 2014).


Additionally, I will be conducting observer as participant observations, in which I will

take on more of an observer role than a participant role. I will conduct observations in the

kindergarten, 1st grade, and 2nd grade classrooms before, during, and after instruction begins that

incorporates The Lesh Model and the Three Read Model into instruction. This will enable me to

detail levels of student participation, student understanding, and student performance on math

word problems across several topics of study. This type of observation will also allow me to

have brief interactions with the students and teachers during the math lessons that can be

authentic, rather than deceptive, but will still allow me to maintain a level of objectivity that I

would not have as a complete participant (ChrisFlipp, 2014).

Finally, I will make use of document review as the final medium of data collection. I will

collect students’ work on math word problems prior to them receiving instruction on The Lesh

Model and the Three Read Model, as well as student work using The Lesh Model and the Three

Read Model to solve math word problems. This will allow me to see and evaluate student

thinking in light of the two proposed instructional techniques, revealing any significant changes

in students’ work and ability to respond to complex word problems after being instructed to use

literary strategies, vocabulary knowledge, and multiple means of engagement to comprehend and

solve word problems.

Data Collection/Instrumentation

Several different tools will be used to capture the data gained from the three methodology

choices described in the section above, relating to the focus group interviews, observations and

document review. To collect the data during student and teacher focus group interviews, I will

make use of video or audio tapping, depending both on what is permitted by and feels most

comfortable to the participants. This will allow me to focus primarily on conversing with the


participants, enabling the conversation to be more free flowing and move quicker than if I were

taking notes simultaneously. However, after the conversations have been recorded, I will

transcribe them to have a written record as well. During the observations of math lessons

conducted in the K-2 classrooms, I will take field notes to detail the instruction given by the

teachers and the participation on behalf of the students. The notes will be evaluated in the

beginning of the study, at the half way point of the study, and at the end of the study to track the

evolution of students’ ability to comprehend math word problems and verbalize their

comprehension of math word problems. Finally, document review will be used to collect

samples of student work solving word problems prior to the Lesh Model and Three Read Model

intervention, at the beginning of the intervention, in the middle of the intervention, and at the end

of the intervention. This will allow me to evaluate student thinking prior to receiving the

curriculum intervention and in light of receiving the intervention to reveal any changes to student

comprehension of word problems and their ability to solve the word problems.

Procedure

This study will be conducted in the following steps:

1. The three K-2 teachers participating in this study will be interviewed prior to the

intervention in a focus group to discuss their current mathematical instruction. Open-

ended questions will be asked to evaluate their perspectives on the challenges that their

ELL students face with word problems and how effective they believe their current

instruction is in addressing the needs of their ELL students, specifically in relation to

their performance on word problems.

2. The three K-2 teachers would then receive two 60-minute trainings on The Lesh Model

and the Three Read Model. Teachers would be guided through word problems using each


model, would receive instructions about how to implement them in their specific grade

level, and would be given a plan for implementation for the purposes of the study. This

plan starts with the teachers presenting the models during whole group instruction on a

weekly alternating basis and then incorporating these models into student math rotations

for more independent student completion in collaborative working groups. Word

problems will be generated by the teacher individually or with their students, rather than

being taken from the textbooks, to ensure that these problems connect to real-life

scenarios that are personally relevant to their students, as detailed in the Word Program

Enrichment program generated and tested by Pongsakdi, Laine, Veermans, Hannula-

Sormunen, and Lehtinen (2016).

a. For the instruction of the Three Read Method, teachers would be instructed to

address the following activities during each read, based on a combination of the

steps that the Early Math Collaborative (2018) and Adrianne Meldrum

recommends from her training on the strategy at the Idaho Mathematics

Conference (2014):

i. 1st Read: The teacher reads the story stem of the word problem without the

numbers or the question. The focus should be on facilitating student

discussions about the context of the problem, understanding the story, and

clarifying unknown or high-level vocabulary. The vocabulary for the unit

should be pre-taught with the chunking method, as backed by the research

of Benjamin Woodward (2018).

ii. 2nd Read: The teacher will then read the problem with the numbers

included, but still without the question. The focus will be on guiding


students through annotation techniques, including crossing out irrelevant

parts of the word problem, circling the numbers, and underlining the

question as Metsisto (2005), Rebora (2014) and Ambrose & Molina

(2010) have demonstrated the effectiveness of in their research. The

teacher should help the students generate a class list of observations and

potential questions that could be asked based on their understanding of the

numbers in their context.

iii. 3rd Read: Finally, the teacher will read the word problem with all parts of

the problem involved, including the question. Students will work

collaboratively to solve the problem, discussing multiple ways to solve the

problem.

b. For the Lesh Model, the teachers would be instructed to guide their students

through the following activities in order, as based on Lesh and Doerr’s description

of it and their study of its implementation (2003):

i. The teacher and students would first read the problem, using annotation

techniques instructed during the Three Read Model and discussing the

context and vocabulary pertinent to the problem.

ii. The teacher would then model how to solve the problem using

manipulatives.

iii. Next, the teacher would lead the students though representing their

concrete model with illustrations.

iv. The teacher would then guide the students through expressing their

knowledge of the problem in mathematical equations.


v. Finally, the teacher would have the students verbalize the steps they took

to solve the problem, helping them translate their oral language into

written words, using sentence frames for assistance.

3. Teachers would spend 4 weeks working through the two models with their class during

whole group math instruction for maximal guidance and scaffolding purposes at the

beginning. Week 1 would focus on the Three Read Method, week 2 would focus on The

Lesh Model, week 3 would go back to the Three Read Method, and week 4 would return

to The Lesh Model. Beginning in week 5, the teachers would begin to incorporate these

models into students’ math rotation groups in addition to their rotating whole group focus

on each model weekly. This would become the new pattern of implementation for the

remaining weeks of the semester.

4. During and right after the semester of instruction, teachers and students will be observed

and interviewed in their focus groups to track their attitudes towards and their ability to

conceptualize word problems.

Research Design

This study will begin in the Fall semester of the 2019/2020 school year. A chart of the

proposed timeline is shown below:

Week Activity

Week 1-4 (pre-intervention)

Teachers will participate in a focus group interview, prior to receiving training on new instructional methods.

Students will participate in a focus group interview to determine their initial attitudes towards and abilities to conceptualize math word problems.

Teachers and students will be observed during their math instruction to detail their current math practices.

INTERVENTIO Teachers will receive two 60-minute trainings on the Lesh Model


N and the Three Read Model. The plan for implementation will be reviewed with the teachers. Teachers and students will be observed weekly throughout the remaining portion of the study.

Week 1 Teachers will guide students through the Three Read Method during whole group math instruction.

Week 2 Teachers will guide students through the Lesh Model during whole group math instruction.

Week 3 Teachers will guide students through the Three Read Method during whole group math instruction.

Week 4 Teachers will guide students through the Lesh Model during whole group math instruction.

Week 5- end of the semester

Teachers will continue alternating their whole group math instruction focus between the Three Read Model and the Lesh Model. Beginning in Week 5 and continuing to the end of their semester, teachers will have also have their students participate in the Three Read Model or the Lesh Model once a week during their math rotation groups.

Teachers and students will participate in focus group interviews to track their understanding of the math models and the effects that implementing these math models into curriculum has had on student word problem comprehension and performance.

After instruction Teachers and students will participate in a final focus group interview to evaluate the teachers’ perceived effectiveness of the math models on students’ word problem performance as well as the students’ ability to comprehend and conceptualize word problems.

Data Analysis

As outlined above, notes from the observations and interviews will be analyzed at the

beginning of the study, in the middle of the study, and at the end of the study. Reading through

the transcripts of the focus group interviews from the teachers and the students, as well as

through the field notes from the observations, general themes or patterns that emerge will be

categorized into analysis documents that will be attached below. Table 1 represents the teacher

focus group interviews chart that will be used at the beginning, middle, and end of the study.


Table 2 represents the student focus group interview chart and Table 3 represents the observation

field notes chart, which will also be updated at the beginning, middle, and end of the study.

These charts will allow me to track teacher and student behavior and performance throughout the

study, revealing any shifts or changes that occur as the intervention progresses. In terms of the

document collection, which will also occur at the beginning, middle, and end of the study,

student work on math word problems will be analyzed to detail common patterns, errors, and

strategies that students make/use throughout the progression of the intervention. This will allow

me to see any evidence of literary strategies that students use to comprehend word problems and

evaluate their thinking prior to and after receiving the Lesh Model and the Three Read Model

intervention. A table to record these findings will also be shown below as Table 4.

Table 1: Themes from Teacher Focus Group InterviewsTheme Evidence

Table 2: Themes from Student Focus Group InterviewsTheme Evidence


Table 3: Themes from Observation Field NotesTheme Evidence

Table 4: Patterns/Errors/Strategies from Student WorkPatterns/Errors/Strategies Evidence

Reaction to Networking Tools

Critical to the conduction and completion of field work is the task of exploring the

opportunities made available to the researcher due to networking. Working to develop a

relationship of reciprocal benefits by presenting the research in terms of what it seeks to provide

for students and teachers is an important part of gaining approval to conduct the study (Shenton

& Hayter, 2004). Having approached many teachers at the school that I previously taught at to

complete field work assignments, the value of building trustworthy relationships has always been

shown. Ensuring that all participants are working for the benefit of the teachers and the students

creates more openness and willingness to let a researcher complete observations and interviews

in the classroom. Additionally, positive relationships established with a few teachers opens the


doors to relationships with more teachers, administrators and other stakeholders in a research

project. Therefore, networking is an essential component of the action research process that

begins well before a proposed study and continues during and after the study is completed.

Ethical Considerations

In order to conduct the research study outlined in the preceding sections of Chapter 3,

specific safeguards will be put in place to protect the participants and ensure their respect. Two

such areas of consideration, consent and confidentiality, will be planned for prior to performing

the study. In terms of consent, upon approaching school district officials and site administrators

to obtain permission for the study, K-2 teachers at the chosen school would get to opt in to

participating in the study. Permission slips would then be given to the families of the students in

the participating classes to outline the procedure of the study and the role that their student would

play in the study. After participants have all freely consented to their participation in the study,

measures to protect their confidentiality would be taken. These include changing the names of

the participating K-2 teachers and removing student names from the documents of student work

that are collected. Moreover, video recordings of teacher and student focus group interviews

would be kept on a computer that is password protected to further prevent any confidentiality

breaches. Finally, open and honest relationships with all teachers, students, administrators,

community members, and families will be developed from the onset and maintained throughout

the study in order to respect the dignity of the participants and ensure that their best interests are

at the center of the research study. As the main goal of the study is to benefit the ELL student

population, all necessary measures will be taken to reduce the risk of harm on participants and

amplify the value that this study can contribute to them.

Summary


In sum, the Methodology section above details the proposed qualitative research study to

investigate the effects that the Lesh Model and the Three Read Model have on K-2 ELL

students’ performance on math word problems. Data would collected through three mediums:

focus group interviews, observer as participant observations, and document review. These three

types of data collection would be analyzed according to emerging patterns or themes at several

defined points throughout the study, near the beginning, middle, and end, in order to assess

changes in student word problem performance throughout the intervention. Considerations to

maintain the efficacy of the study would be taken, such as ensuring the free agency of teacher

and student participation in the study and maintaining the confidentiality of participants through

name changes/removal and secure storage of video recordings and collected documents.

Participating teachers, students, faculty and community members are of the utmost importance to

this study as it is designed for their long-term benefit, so appropriate measures would be

followed to protect their dignity. The following chapter will delve deeper into these protective

measures as it first details a negotiations section and then proceeds to discuss reflections about

the implications of the proposed study for practice, teaching, and further research.

Chapter 4

Introduction

As begun in Chapter 3, a discussion on the necessary steps to ensure participant

protection will be further elaborated by detailing a plan for proper negotiations with relevant

stakeholders. To gain entry into the desired site for the research study in Vista, CA, I will contact

the school administrators and appropriate district staff. These parties will be given an

introductory letter that requests their permission to conduct the study at their school site. In this

letter, I will first emphasize my previous experiences teaching preschool at this school as a Teach


for America Corps Member, my desire to return to the community as an elementary school

teacher, and my dedication to providing an equitable education for all students, specifically for

low-income, English Language Learning students. Additionally, this letter will include a

summary of the proposed research project, a copy of the literature review, and a promise to share

the findings of the study with the stakeholders involved. This will be done to ensure the district

officials and school administrators that I have done preliminary research to ground my study and

that I am working in the best interests of their school, with the end goal being to improve

academic outcomes for their students. This approach is based on the reciprocity tactic of gaining

entry to desired field locations developed by Sharp and Howard (Shenton & Hayter, 2004).

If this study is approved at the desired school location, I will provide the formal consent

forms necessary to receive parental permission for their students’ participation in the study. This

consent form will include a brief overview of the research project, the reasons for the study, and

the actions that I will take as a researcher to ensure the least possible disruptions to their

students’ education (Shenton & Hayter, 2004). In addition, these consent forms will be translated

as needed into the appropriate home languages of the families involved to guarantee accurate

understanding of the study that their students will be participating in.

Finally, throughout the study, open and authentic relationships will be maintained with

the district, school administrators, participating K-2 teachers, students, families, and other

involved community members. Conversations with the involved stakeholders can be arranged

throughout the study as requested and desired to provide checkpoints for review and feedback of

the study.

Analysis of Existing Research in Literature


To reiterate, the review of literature in Chapter 2 warrants the proposed research study

and the corresponding negotiations that would need to be undertaken in order to ensure the

conduction of this study. As previously mentioned, mathematics scores for ELL students are

consistently lower than mathematics scores for non-ELL students primarily due to the

increasingly rigorous language demands of word problems that necessitate a sophisticated

understanding of the English language (Krick-Morales, 2019). This presents an additional

challenge for students learning English as a second language, dashing the notion of math as a

“universal language” (Robertson, 2019). Existing bodies of research have studied several aspects

of mathematics instruction that are necessary to address the needs that ELL students present as

learners of language and content simultaneously. These include multi-modal, pre-instruction

vocabulary development, the incorporation of literary strategies into the teaching of mathematics

that guides students to close read math word problems similar to how they would read language

arts texts, the use of story stem formats to foster students’ contextual understanding of word

problems before they jump into the numeric tasks, the integration of personally and culturally

relevant scenarios into math word problems, and the inclusion of multiple means of engagement

in problem-solving strategies. Both the Three Read Model and the Lesh Model encompass these

aspects of instruction into their methods, with language development and expression being the

foundations of these two models. As the procedures for both models include an emphasis on

developing students’ conceptual understanding of word problems and necessitate the use of oral

and written language to work through problems, they inherently work to advance both students’

language and content understanding. Therefore, the proposal for these two models to be used in

conjunction with each other or on a frequently rotating basis, is merited as a significant study to


undertake for the benefits that they present as effective integrated ELD strategies for

mathematics instruction.

Implications for Practice and Teaching

Several benefits can be foreseen based on the results of the proposed study, with the

ultimate desire of the study being to benefit the English Language Learning student population in

specific regards to their mathematics performance. To begin with, it is advantageous to explore

the potential that these two models have to be effective methods of integrated ELD strategies in

the mathematics content area as integrated ELD approaches across content areas are “the fastest

way to bring English learners into full proficiency” (Neer, 2017). Since both of these

instructional models include speaking, listening, reading, and writing into their procedures, all

ELD domains could be developed alongside the development of students’ content knowledge.

Moreover, ELL students could be better equipped with strategies to use when they encounter

word problems in class and on standardized assessments in a second language that they are

currently working on acquiring. Since succeeding is mathematics is now closely intertwined with

linguistic success, students need to be taught strategies to focus on their understanding of the

verbage in word problems in addition to their procedural understanding of the mathematical

operations they are expected to perform. Finally, teachers could be better equipped in how to

provide this type of instruction to their students, gaining crucial development in content area

vocabulary and literary instruction, as well as in the provision of multiple means of engagement

in their content areas. With a substantial number of teachers reporting frustration over their

inability to meet the needs of both their ELL students and their non-ELL students in the same

class, strategies marked by a universal design for learning that benefit all learners, would be

crucial to boosting the preparation and confidence of educators nationwide (Shreve, 2005).


Therefore, demonstrating the benefits of the proposed methods in this study would further the

research on effective integrated ELD strategies for mathematics instruction and would better

prepare teachers and students to face the increasingly linguistic nature of mathematics.

Implications for Further Research and Inquiry

The purpose of this study is to determine a beneficial approach to integrated ELD

mathematics instruction. This study specifically looks at grades K-2 to determine how students

can become prepared for the increasing linguistic and procedural complexity of word problems

in higher grades when they are subsequently assessed on their knowledge. As detailed in the

literature review, student performance on math word problems increases with English

proficiency, but decreases with grade level. Additionally, it has been noted by research that this

decline in performance is due to students’ lack of more sophisticated literacy strategies that

becomes problematic as the verbiage in word problems increases in rigor throughout the grade

levels. Therefore, a long-term study to follow a group of students through the K-6 elementary

school grades with the proposed instructional approach to mathematics would be a desirable next

action research cycle to invest in. This would help determine the adaptations needed to be made

to these instructional approaches in each grade level, and would reveal the usefulness that the

Lesh Model and the Three Read Model could have as approaches to assist ELL students with

word problem comprehension and performance. Studying this group of students in comparison to

a group of students who did not receive the Lesh Model and Three Read Model intervention,

could further the claim that these models provide added benefits to ELL student language and

content development when included in instruction.

Critical Friend


A critical friend is defined by Arthur Costa and Bena Kallick as “an advocate for the

success” of the work that you are engaged in, “pushing you to look through multiple lenses,” and

providing feedback to help you continually focus and refocus on your work through different

perspectives (Costa & Kallick, 1993). The critical friend that I was assigned at the beginning of

this program had prior experience participating in action research, which was one of the most

beneficial “lenses” to have a glimpse into at the start of my research process. Gaining a better

understanding of the expected outcomes and end goals of action research was very beneficial in

narrowing down my research topic to fit the qualifications and scope of an action research

project. Additionally, both my critical friend and I have previous experience working with high

populations of English Language Learning students, so talking through the strengths and needs

that we saw with our students was also crucial in selecting topics that we both saw as presenting

tangible needs for this student demographic. The support of my critical friend was most

impactful in our first course when we were brainstorming and refining topic ideas, as she was

unable to continue with the program as it progressed. I was very thankful to know other people in

the program who were able to continue in the courses and reach out to them for support in

structuring my research moving forward. Therefore, the peers who remained in this course

became critical friends that I could rely on for advice and encouragement, even in the absence of

the critical friend I was initially assigned. Overall, I believe that building the component of a

critical friend into this program was instrumental to the outreach that I continued to have with

other peers in these courses, highlighting the benefits of conducting this research in a community

of likeminded colleagues that are willing and eager to provide constructive feedback.

Summary


Overall, throughout the course of this proposed research study, there are several

components of the Three Read Model and the Lesh Model that I would like to learn about in

relation to their potential as an integrated ELD approach to mathematics instruction. To begin

with, I hope to learn about the effects that these models’ literary elements, such as close reading,

annotation, and vocabulary development, have on ELL students’ comprehension of word

problems. The crossover between literacy techniques and traditional mathematics instruction is

of particular interest to me not only for the effects that they could have on student word problem

performance, but also on student English language development as well. Aside from the literary

techniques that these models incorporate into their instruction, the multiple means of student

engagement with and expression of knowledge that these models invite and necessitate, a

cornerstone of Universal Design for Learning, are also hoped to be evaluated in terms of their

benefits on ELL student performance. The ultimate desire of this research project is for it to be

significant as a means of addressing the needs that ELL students have in content area instruction

alongside their non-ELL peers, providing general education teachers with rigorous curriculum

supplements that can benefit all of their students.

Questions that might arise throughout the research study could include:

1. How do these models need to be specifically adapted for grades K, 1, and 2?

2. How should these models be best incorporated into curriculum in terms of frequency and

usage? In alternation or in conjunction with each other? In whole group instruction, small

group instruction, or both? Daily, weekly, or bi-weekly?

3. What aspects of these models seem to be the most beneficial for ELL student

comprehension of and performance on math word problems? Which aspects do not seem

to have any effect or be beneficial that could be removed or altered?


4. What are the effects of these models on ELL student performance of math word problems

in the long-term?

By carrying out such a study and generating additional questions throughout the research

process, further attention would be brought to the area of effective mathematics instruction for

ELL students. More research is needed to identify and deliver successful integrated ELD

strategies and curriculum approaches for mathematics instruction of ELL students and this study

hopes to be an influential step in this discovery process.

Supporting Documents

Lesh Model Example

Lesh Model Template


Three Reads Protocol


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