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Running head: SUPPORTING ELL STUDENTS IN HS MATH 1

Supporting English Language Learners in High School Mathematics

Lura Ercolano

Seattle Pacific University

Note: Portions of this paper were previously submitted as the Midterm assignment.

SUPPORTING ELL STUDENTS IN HS MATHEMATICS2

Supporting English Language Learners in High School Mathematics

English Language Learners (ELLs) are part of the rich diversity present in today’s

multicultural classrooms. Students with limited English proficiency usually spend all or part of

the school day in the general education classroom where they are expected to meet high state

standards in subjects that are taught in English by teachers who may know their content area, but

who are not specialists in second language acquisition.

It is a fundamental tenet of today’s educational philosophy that all students can be

successful when provided with appropriate support (Friend and Pope, 2005). General education

teachers are expected to differentiate instruction to meet the varied needs of all students, whether

those students are gifted, learning disabled, or learning a second language. Teachers have high

expectations of their students, but often do not have the expertise to determine how to best

differentiate instruction, or the time to work individually with students (Tomlinson, 2001).

Effective teachers need strategies for working with groups of students who share common

learning difficulties (Tomlinson and Strickland, 2005).

This paper will review several articles on adjusting instruction in high school

mathematics classes to support the learning needs of ELL students.

Anticipating language demands for word problems

English Language Learners often struggle in academic areas, or achieve below their

grade level. It is sometimes assumed that mathematics will be relatively accessible to ELL

students because of mathematics’ use of computation, numbers, graphs, tables and symbolic

language structures. Nonetheless, a lack of English language proficiency, and especially English-

reading skills, dramatically hinders the mathematics success of ELL students (Beal, Adams and

Cohen, 2010). For example; Beal, et al. report that more than half of ELL students in Los


Angeles fail Algebra at least once. Beal et al. suggest that ELL students have less “opportunity to

learn” in the math class because they cannot easily understand the materials and explanations;

“Students who must devote substantial cognitive resources to English comprehension will have

less capacity available to devote to math problem-solving operations (Beal et al., p.60).

Beal et al. (2010) examined the performance of 9th grade ELL students in Algebra classes

and found that skill at reading English was strongly correlated to success in mathematics. Many

of the students that Beal et al. studied were coming into the Algebra class with below basic math

skills, but when students’ overall English reading scores were raised, these students were able to

make progress in mathematics. “Students who must devote cognitive resources to understanding

a problem presented in English text perform less well in math than students who are able to read

English well” (Beal et al., p.71). The research project revealed that English conversational ability

was not strongly related to math performance; in this study, reading English was identified as the

essential skill that supported success in mathematics.

Beal et al. (2010) found that the higher reading scores also correlated with increasing

students’ ability to develop a higher self-concept and self-efficacy in math. “Reading proficiency

was a significant predictor of ELL’s math self-concept…but math skill was not.” (Beal et al., p.

69). Pressley and McCormick (2007) state that self-efficacy and higher academic self-concept

are linked to increased academic effort and achievement in general. Pressley and McCormick are

discussing all students, not English Language Learners in particular, but their conclusions

suggest that raising self-efficacy is a significant predictor of future success.

These results indicate the need for ongoing reading support for ELLs, and continued

instruction and practice in reading academic English. The Algebra teacher can take general steps

to support all students’ reading, such as teaching reading strategies and teaching content


vocabulary for mathematics, but Beal et al.’s results are general and do not directly guide the

Algebra teacher in specific ways to modify written material to support the struggling ELL

student.

Many researchers have focused on helping ELL students and struggling readers approach

word problems in mathematics. Orosco (2013) describes a system of “reciprocal learning” in

which a teacher works individually with a student who is working on mathematics word

problems. The teacher provides scaffolding, the word problem vocabulary is simplified, and

students learn to monitor their comprehension. Orosco was reporting on students in elementary

school, but these strategies for word problems could apply at any grade level. For example,

teachers can make word problems “less linguistically complex” by replacing the math term,

“sum” with the everyday term, “total,” or by pre-teaching math vocabulary. Teachers can

provide direct step-by-step strategies for re-reading a word problem with comprehension, and

students can be taught to apply those steps when working collaboratively with a partner.

Linguistic reasoning for higher-order mathematics

Math in high school, however, uses academic language for much more than posing word

problems; higher-order math concepts are built on linguistic reasoning. Lager (2006) describes

how students must organize their knowledge and build their own understanding of abstract

concepts. “Rational constructivists believe that while some mathematical knowledge can be

constructed at some times (Herscovics, 1996), the abstraction of a concept still must be achieved

by the learner” (Lager, p.168).

Herscovics and Linchevski (1994, p.59) describe “the existence of a cognitive gap

between arithmetic and algebra…characterized as the students’ inability to operate

spontaneously with or on the unknown.” As students move from pre-algebra or arithmetic to


algebra, they must move from the least abstract understanding to the most abstract understanding

of mathematics. This is a very difficult transition for many students, yet is a prerequisite for

further advancement in mathematics (Lager, 2006), (Herscovics & Linchevski, 1994).

Lager (2006) used Herscovics’ theoretical learning structure to design a well-ordered

instructional sequence investigating linear patterns. Lager then takes a detailed look at how

specific words, syntax, and language structures impeded learning for some students during that

instructional sequence. Lager studied middle school algebra students (both ELL and non-ELL)

who faced reading challenges and comprehension difficulties while learning about linear

patterns. The study looked in detail at what specific language problems and misconceptions led

to shared patterns of math misunderstanding. Lager’s team analyzed the specific words and

language constructions that tended to cause difficulty.

In this study, students were given a series of diagrams and were expected to identify and

extend a pattern. Lager (2006) explains that ELL students were often making reasonable (but

wrong) guesses based on prior language knowledge. For example, instructions referred to

“Figure number (N)”, which was intended to direct students to imagine the N’th illustration.

Instead, some students interpreted “figure” as a verb that was instructing them to “figure out”

(count) the number of squares. Elsewhere in the problem set, a table had listed “Number of blue

squares (B)”, and some students used that information to infer that “(N)” would therefore be the

number of squares of some different color. And instead of referencing “Figure 6”, some students

invented or guessed at meanings and interpreted the meaning to be a figure with 6 squares, which

was inventive, but also incorrect (Lager, 2006). What is especially interesting about all of these

errors is that they actually demonstrate great creativity, thoughtfulness, and innovation. These

are capable students, despite their academic struggles.


Lager’s detailed analysis reveals ways to improve instruction and increasing clarity for all

students, especially ELLs. “The role of the teacher must evolve. First and foremost, every

mathematics teacher must also be a language teacher. In addition to having profound

mathematical understanding, she must be cognizant of the language she uses in her instruction,

anticipate the language needs of her students, and work with her students to identify any

language misconceptions” (Lager, 2006, p.193).

I am especially interested in this subject, and Lager’s detailed explanations, because this

autumn, I will be teaching mathematics and physics in a high school with a large number of

transitional bilingual students. I will also be co-teaching a newcomer math class with an ESL

specialist. I am eager to learn specific ways to modify my instruction to meet the needs of all

students in the classes I will be teaching.

Some things I learned from reading this report are that many mathematical language

structures have more than one meaning, which can be confusing for all students. Other examples

of this are the way that “less than” can mean all numbers below a certain value, or can mean one

exact number obtained by subtraction.

Students in Lager’s study (2006) were able to self-identify some words as confusing.

Other misunderstandings, however, left students believing they had understood. Following

constructivist theories of learning, this leads to students attempting to build new understandings

on top of the material that was misunderstood. This example points out the need for teachers to

make frequent formative assessments of learning so that misunderstandings can be corrected

promptly.

Lager (2006) points out that ELL students have language strengths that often go

unnoticed. ELL students are already comfortable with the idea that there is more than one way to


express a concept – in English and in Spanish (or another language) – and this can lead to greater

ease than other students with multiple representations of mathematical ideas, and ease in

grasping the concept of a variable. The math teacher can value and build on these strengths.

New State Standards, and Language for Learning

The new Common Core State Standards (CCSS), Next Generation State Standards

(NGSS), and English Language Proficiency (ELP) Standards will change how ELL students are

taught and assessed in Washington State. The standards even tend to redefine the roles of

teachers. Consider this language from the developers of the new ELP standards:

At present, second language development is seen largely as the responsibility of the

ESL/ELD teacher, while content development as that of the subject area teacher. Given

the new standards' explicitness in how language must be used to enact disciplinary

knowledge and skills, such a strict division of labor is no longer viable. Content area

teachers must understand and leverage the language and literacy practices found in

science, mathematics, history/social studies, and the language arts to enhance students'

engagement with rich content and fuel their academic performance. ESL/ELD teachers

must cultivate a deeper knowledge of the disciplinary language that ELL students need,

and help their students to grow in using it. Far greater collaboration and sharing of

expertise are needed among ESL/ELD teachers and content area teachers at the secondary

level. (Understanding Language Initiative, 2012, p.2). (This is also quoted in the ELP

standards; Council of Chief State School Officers, 2014, p.3).

Many states, including Washington State, have adopted the new standards. Figure 1 shows a

sample standard from the new English Language Proficiency (ELP) Standards, specifically

linked to the math practice standard MP3, “Construct viable arguments and critique reasoning of


others,” and the science practice standard SP7, “Engage in argument from evidence” (Council of

Chief State School Officers (CCSSO), 2014, p.204).

Figure 1: Example of ELP Standard aligned to CCSS and NGSS (CCSSO, 2014, p.204).

The Common Core State Standards in Mathematics (National Governors Association

2010), and the Next Generation Science Standards (NGSS, 2013) both state that students will

master explaining, arguing, discussing and writing. Science and math have become disciplinary

practices that students “do” as part of collaborative groups. Active language use is embedded

throughout the Mathematics and Science standards.

Discussing these changing state standards, Valdés, Kibler and Walqui offer, “What is

clear…is that the Standards explicitly include ELLs and clearly frame content learning as


engagement in disciplinary practices – implying an active learning process in which language

plays a key role” (2014, p.10).

Koelsch, Chu, and Bañuelos describe guiding subject-area teachers to provide

pedagogical scaffolding so that ELL students can engage in cooperative discussions in math and

science classes. “Our support of teachers, and their students, is based on Vygotsky’s (1978)

theory that learning occurs first in purposeful social interactions within the zone of proximal

development and is gradually appropriated and internalized” (2014, p.642.). Koelsch et al.

suggest that teachers can support beginning ELLs participation in science discussions through

the use of an “Extended Anticipatory Guide” (EAG) and a system of “Novel Ideas Only (NOI)

(p. 644).

The EAG can be structured for a particular learning segment or topic. The EAG contains

statements about key concepts (including misconceptions and other incorrect statements),

modeled language for discussion, and language for reflecting. The EAG is primarily structured

for dialogue, including statements that provide opportunities for students to agree or disagree,

and to say why. “The success of the EAG hinges on crafting statements that involve key

concepts and invite students to express opinions based on prior, but not prerequisite, knowledge”

(Koelsch et al. , 2014, p. 645).

Novel Ideas Only (NOI) cooperatively builds a concept by having students initially work

in small groups to define a concept or response to a prompt. Then students come together for a

teacher-guided group discussion. Koelsch et al. (2014) give an example of a science class

working together to define the physics term, “force”. Koelsch et al. observe, “The initial

brainstorming provides multiple entry points for students in a more intimate setting, with some


offering a word or phrase, others echoing a response, and all writing it down. The small group

can provide rehearsals of ideas that are then shared with the whole class” (p. 646). Only novel

ideas are shared, so students must listen carefully to how ideas are phrased, in order to determine

whether different wordings present the same or different ideas. Finally, “the teacher modeled the

process of revision to reach a whole class definition: ‘Force is an action of push and pull on an

object to move the object’ ”(p.646).

Similary, Koelsch et al. (2014) suggest that teachers can support discussion in math class

by providing structured “guidance cards.” Guidance cards for interpreting a graph would include

sentence frames such as “The x-axis represents ___”; “When the value of __ is ___, the value of

___ is ____”; and “As the value of __ increases/decreases, the other variable ___” (p.648).

These supports will give all students an entry point for participating in meaningful

discussions in Science and Math classes.

“In contrast to isolated teaching of language in bits and pieces for narrow purposes or

single occasions, focusing on language for learning entails shifting planning and instruction to

center on disciplinary practices and the concepts needed to enact practices. Meanings of key

disciplinary concepts cannot be simply given as definitions but must be co-constructed by

students with peers” (Koelsch, 2014, p.648).

My teacher preparation classes and my student teaching have recommended that high

school math and science courses include student discourse, and guided inquiry units in which

students construct their own knowledge. I appreciate the methods proposed by Koelsch et al. to

make that discourse and inquiry accessible, constructive activities for all students.


Further Research

Valdés, Kibler, and Walqui point out that the ELP standards are based on “consensus-

derived hypothesized progressions” (2014, p.7) and “ELP standards are not based on empirical

evidence (e.g., longitudinal studies) of actual language growth over time by ELLs” (p.7). As

school districts implement instruction aligned to the new standards, there should be opportunities

for researchers to collect data and determine which instructional techniques best meet student

needs.

Ideally, that data should extend well past the standardized tests in high school, and look

at students’ later quality of life, their achievement in higher education and their success in the

work force.

Conclusion

The math teacher can better meet the needs of ELL students by working with English-as-

a-Second-Language departments, by anticipating students’ language needs, by avoiding language

with multiple meanings, by explicitly teaching math content vocabulary and related general

vocabulary, and particularly by identifying and building on ELLs strengths.

Student discourse in science and math class can be made accessible to ELL students

through scaffolded strategies such as the “Extended Anticipatory Guide” and discourse

“Guidance Cards”.

I am excited to implement these strategies in my teaching during the coming year. Above

all, I am eager to welcome my ELL students to my classes, and I look forward to a shared

adventure in building an understanding of mathematics.


References

Beal, C. R., Adams, N. M., & Cohen, P. R. (2010). Reading proficiency and mathematics

problem solving by high school English language learners. Urban Education, 45(1),

58-74. doi: 10.1177/0042085909352143.

Council of Chief State School Officers (CCSSO). (2014). English language proficiency (ELP)

standards. Retrieved from: http://www.k12.wa.us/migrantbilingual/pubdocs/elp/wa-elp-

standardsk12.pdf

Friend, M., & Pope, K. L. (2005). Creating schools in which all students can succeed. Kappa

Delta Pi Record, 41(2), 56-61.

Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra.

Educational studies in mathematics. 27(1), 59-78.

Koelsch, N., Chu, H., & Rodriguez Bañuelos, G. (2014). Language for learning: Supporting

English language learners to meet the challenges of new standards. TESOL

Quarterly, 48(3), 642-650. doi: 10.1002/tesq.181

Lager, C. A. (2006). Types of mathematics-language reading interactions that unnecessarily

hinder algebra learning and assessment. Reading Psychology, 27(2-3), 165-204.

doi: 10.1080/02702710600642475.

http://www.k12.wa.us/migrantbilingual/pubdocs/elp/wa-elp-


National Governors Association (NGA) Center for Best Practices & Council of Chief State

School Officers. (2010). Common Core State Standards for Mathematics (pp. 57-93).

Washington, D.C: Author. Retrieved from: http://www.k12.wa.us/curriculuminstruct/

Next Generation Science Standards (NGSS) Lead States. (2013). Next generation science

standards: For states by states. Washington, D.C.: The National Academies Press.

Retrieved from: http://www.k12.wa.us/science/NGSS.aspx

Orosco, M. J. (2014). Word Problem Strategy for Latino English Language Learners at Risk for

Math Disabilities. Learning Disability Quarterly, 37(1), 45-53.

doi:10.1177/0731948713504206

Pressley, M. & McCormick, C.B. (2007). Child and adolescent development for educators.

New York: The Guilford Press.

Tomlinson, C.A. (2001). How to differentiate instruction in mixed-ability classrooms.

Association for Supervision and Curriculum Development (ASCD).

Tomlinson, C. A., & Strickland, C. A. (2005). Differentiation in practice: A resource guide for

differentiating curriculum, grades 9-12. Association for Supervision and Curriculum

Development (ASCD).

Understanding Language Initiative. (2012). The purpose of English language proficiency


standards, assessments, and instruction in an age of new standards: Policy statement from

the Understanding Language Initiative. Palo Alto, CA: Author. Retrieved from

http://ell.stanford.edu/sites/default/files/ELP_task_force_report_rev.pdf.

Valdés, G., Kibler, A., & Walqui, A. (2014, March). Changes in the expertise of ESL

professionals: Knowledge and action in an era of new standards. Alexandria, VA:

TESOL International Association. Retrieved from:

http://www.literacyinlearningexchange.org/sites/default/files/changes_in_the_expertise_o

f_esl_professionals.pdf

http://www.literacyinlearningexchange.org/sites/default/files/changes_in_the_expertise_o

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