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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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).
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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
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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
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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
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(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
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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
SIGNTATURE ASSIGNMENT 23
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
SIGNTATURE ASSIGNMENT 24
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,
SIGNTATURE ASSIGNMENT 25
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
SIGNTATURE ASSIGNMENT 26
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
SIGNTATURE ASSIGNMENT 27
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
SIGNTATURE ASSIGNMENT 28
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
SIGNTATURE ASSIGNMENT 29
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).
SIGNTATURE ASSIGNMENT 30
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
SIGNTATURE ASSIGNMENT 31
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
SIGNTATURE ASSIGNMENT 32
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
SIGNTATURE ASSIGNMENT 33
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
SIGNTATURE ASSIGNMENT 34
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.
SIGNTATURE ASSIGNMENT 35
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).
SIGNTATURE ASSIGNMENT 36
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).
SIGNTATURE ASSIGNMENT 37
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
SIGNTATURE ASSIGNMENT 38
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
SIGNTATURE ASSIGNMENT 39
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
SIGNTATURE ASSIGNMENT 40
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.
SIGNTATURE ASSIGNMENT 41
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
SIGNTATURE ASSIGNMENT 42
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.
SIGNTATURE ASSIGNMENT 43
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
SIGNTATURE ASSIGNMENT 44
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
SIGNTATURE ASSIGNMENT 45
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
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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
SIGNTATURE ASSIGNMENT 47
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
SIGNTATURE ASSIGNMENT 48
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
SIGNTATURE ASSIGNMENT 49
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).
SIGNTATURE ASSIGNMENT 50
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
SIGNTATURE ASSIGNMENT 51
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
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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?
SIGNTATURE ASSIGNMENT 53
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
SIGNTATURE ASSIGNMENT 55
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