This article was downloaded by: [Northeastern University]On: 11 October 2014, At: 14:23Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Multicultural PerspectivesPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/hmcp20

Acquisition of Mathematical Language: Suggestions andActivities for English Language LearnersMichelle Cirillo a , Katherine Richardson Bruna b & Beth Herbel-Eisenmann ca University of Delaware ,b Iowa State University ,c Michigan State University ,Published online: 18 Mar 2010.

To cite this article: Michelle Cirillo , Katherine Richardson Bruna & Beth Herbel-Eisenmann (2010) Acquisition ofMathematical Language: Suggestions and Activities for English Language Learners, Multicultural Perspectives, 12:1, 34-41,DOI: 10.1080/15210961003641385

To link to this article: http://dx.doi.org/10.1080/15210961003641385

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Multicultural Perspectives, 12(1), 34–41Copyright C© 2010 by the National Association for Multicultural EducationISSN: 1521-0960 print / 1532-7892DOI: 10.1080/15210961003641385

Acquisition of Mathematical Language: Suggestions and Activitiesfor English Language Learners

Michelle CirilloUniversity of Delaware

Katherine Richardson BrunaIowa State University

Beth Herbel-EisenmannMichigan State University

In this article, we describe aspects of mathematicallanguage that could be problematic to English-language learners, provide recommendations forteaching English-language learners, and suggestactivities intended to foster language developmentin mathematics.

Introduction

Public schools in the U.S. now enroll a large numberof students who have been identified by their local schooldistricts as English-language learners (ELLs, Valdes,2001). As the number of ELLs increases each year,mathematics teachers face new challenges. As studentsare encouraged to reflect on their mathematical solutionsand verbalize their explanations, it is imperative thatteachers understand the critical role of language inmathematics education (Ron, 1999).

Despite the fact that mathematical concepts areabstractions that transfer from language to language, thereare many complications when learning mathematics in asecond language. It is a common misconception to believethat if a person is bilingual, he or she automatically knowsthe language of mathematics in both languages (Ron,1999). As noted by Khisty (1995), “We have operatedtoo long with the myth that mathematics teaching

The authors wish to acknowledge that this research was supported,in part, by the National Science Foundation under Grant No. 0347906(Beth Herbel-Eisenmann, PI). Any opinions, findings, and conclusionsor recommendations expressed in this material are those of the author(s)and do not necessarily reflect the views of the National ScienceFoundation.

Correspondence should be sent to Michelle Cirillo, 407 Ewing Hall,University of Delaware, Newark, DE 19716. E-mail: mcirillo@udel.edu

and learning transcends linguistic considerations”(p. 295).

In this article we consider some of the linguisticchallenges faced by ELLs in the mathematics classroom.Based on a review of the research, we first describe aspectsof mathematical language that could be problematic forELLs. Next, we provide recommendations for teachingmathematics to ELLs. Finally, we draw on the high schoolteaching experience of the first author to suggest someactivities that would help secondary mathematics teachersget started with putting some of these recommendationsinto practice. Despite the focus of this article onassisting ELLs in particular, we believe that many of theactivities and suggestions will help all students acquiremathematical language.

Mathematics as a Language

Mathematics itself is sometimes described as beinga language (e.g., Usiskin, 1996). Usiskin (1996) arguedthat because mathematics has a grammar; containsexpressions (2 + 5x), verbs (+, ×, =, ⊥, etc.), andsentences (4x+ 3 < 11); and is a well-constructed syntax,indeed, it is certainly a language. While we realize thatthis metaphor has its limitations (see, e.g., Pimm, 1987),we think it can help to highlight some of the ways inwhich language impacts the teaching and learning ofmathematics for ELLs. One might say that, for an ELL,the mathematics classroom is a domain in which three‘languages’ intersect: The student thinks in his/her firstlanguage (L1), but is required to communicate in English(L2)—using mathematical language and the symbolicsystem. There are many ways in which the intersectionof these three ‘languages’ causes challenges for ELLs.

The Official Journal of the National Association for Multicultural Education

34

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

To explore mathematics as a language and itsimplications for ELLs, we start with the understanding ofmathematics as a written, spoken, and ‘foreign’ language.While the issues addressed here are not always distinctfrom one another, we address them separately in orderto discuss different emphases. Because the L1 of alarge majority of ELLs in the U.S. is Spanish, we useSpanish-language examples.

Writing Mathematics

Perhaps the symbolic nature of mathematics causesus to view mathematics as something special (Usiskin,1996). The symbols of mathematics, just like letters orcharacters in other languages, form the written languageof mathematics. Mathematics has its own symbols (+, =,∦, etc.) and borrows letters from the Latin (x, y, e), Greek(π ,

∑), and Hebrew (ℵ) alphabets. However, cultural

differences found in notation and certain algorithms doexist and can be cause for confusion, even with respect tobasic ideas (Perkins & Flores, 2002). In some countries,for example, students learn that angles (e.g., angle A) arerepresented using the notation A. As a result, studentsworking with the English representation (∠ A) mightconfuse the angle symbol for the less than symbol (<).

Speaking Mathematics

The spoken language of mathematics is also importantfor comprehension of mathematical concepts (Usiskin,1996). The challenges of mathematical conversationsinvolve much more than just vocabulary (Halliday, 1978).There is a certain structure that must be maintained whenone is speaking mathematically. For example, when theteacher makes a statement of the form: if p then q, thelaws of logic are employed. In logical statements, as withmany mathematical statements, order is important, andeach utterance is critical to understanding the concept.Additionally, participants in mathematical conversationsoften refer to diagrams and symbolic statements throughgesture. For these reasons, ELLs may encounter difficultyinternalizing a concept when the teacher is merelyspeaking the language of mathematics (Usiskin, 1996).

Students may also face challenges when they attemptto speak the language of mathematics. Because the lettersof the alphabet, as well as mathematical symbols, arepronounced differently in English and Spanish, studentsmay have difficulty “translating” a mathematical sentenceinto English. Despite the fact that there are no “words”in the mathematical sentence: 4x3 − 3x2 + 8 ≤ 2, ELLsmay find it challenging to read this sentence out loud.When the mathematical sentence, four ex cubed minusthree ex squared plus eight, is less than or equal totwo, is spoken, every element of the sentence translates

differently in Spanish, including the pronunciation ofex as equis. In the case of Spanish, the sentence wouldread: cuatro por equis al cubo menos tres por equis alcuadrado mas ocho es menor o igual a dos.

The issue of mathematics as a spoken language isfurther complicated by the fact that we borrow everydayEnglish terms, such as table, line, and plane, to representsomething mathematical. The mathematical spellings andpronunciations of these words are the same as in everydayEnglish. Even within the language of mathematics,because words like sign and sine sound the same but havedifferent mathematical meanings, they pose a challenge toELLs. In addition, the use of multiple words to representany one concept can be challenging for all students,but especially for ELLs. The concept of addition, forexample, can be implied through a multitude of terms:plus; combine; increased by; more than; sum, and so forth.

Taking the oral and symbolic natures of mathematicsinto account, one can see that challenges related to pro-nunciation exist. Unlike with English or Spanish words,students cannot “sound out” mathematical symbols(Rubenstein & Thompson, 2001). While ELLs may knowhow to articulate mathematical symbols in Spanish, theymay not know how to pronounce those symbols in En-glish. For example, a student might know that, in Spanish,the sign “+” is read “mas,” but she may have a difficulttime finding “plus” written out in her textbook or on theblackboard, impeding her access to the pronunciation ofthe mathematical symbol. Gaining familiarity and easewith the symbolic and oral language of mathematics is acritical objective of mathematics education.

Mathematics as a ‘Foreign’ Language

One reason that mathematics can be thought ofas a ‘foreign language’ (Usiskin, 1996), for manystudents (not just ELLs), is that mathematical languageis learned almost entirely in school, and it is not spokenat home. Some mathematical terms, such as quotient,asymptote, and contrapositive are found only in amathematical context (Thompson & Rubenstein, 2000).Olivares (1996) wrote about three characteristics thatmake communication in mathematics ‘foreign,’ whencompared to everyday communication: First, students arerequired to communicate with abstraction and symbols;second, each element of a mathematical proposition iscritical for understanding the whole proposition; and last,the elements of mathematics propositions are frequentlyso specific that they cannot be rearranged. For example,the term 4x2 has a different meaning from 42xor 42.Each element must be included, and the elements mustbe in a specific order to retain their intended meaning.Recognizing mathematics as a language forces us torethink its teaching (Usiskin, 1996).

Multicultural Perspectives Vol. 12, No. 1

35

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

Understanding the written, oral, and ‘foreign’ demandsof mathematical language helps us identify a teachingagenda for ELLs in the reform-oriented classroom.Moschkovich’s (1999) work has shown that in reform-oriented classrooms, students are not only asked tograpple with the acquisition of a technical vocabularyand problem solving skills, but they are also expected toparticipate, both in writing and verbally, by explainingsolutions, making conjectures, proving conclusions,and presenting arguments. As mathematics classroomsbecome more language dependent, how to best meetthe needs of ELLs will vary depending on their priorexperiences with mathematical discourse in either theirL1 or L2. Teachers, therefore, must be prepared to adaptto their students’ needs.

Recommendations for Teaching ELLs inMathematics Classrooms

In our review, the most prevalent practices advocatedby the authors included: use of cooperative learning(Campbell & Rowan, 1997); use of the native language(de la Cruz, 1998); development of language skills (Ron,1999); use of writing (Garrison & Mora, 1999); use ofvisual representations and manipulatives (de la Cruz,1998); personalization of instruction (Moschkovich,1999); and parental involvement (Lee & Jung, 2004).For the purpose of this article, we focus on the use ofcooperative learning; the development of language skills;and the use of visual representations and manipulatives.We will follow with activities that make use of these threestrategies.

Cooperative Learning

Cooperative learning has become an importantinstructional strategy in reformed mathematics teaching.Campbell and Rowan (1997) advocate for groupingstudents heterogeneously to promote language growthof students who are less proficient. This strategy givesELLs opportunities to use and hear the language ofinstruction in a more context-embedded situation (Faltis,1993). While many students who are learning a secondlanguage are hesitant to speak in front of the whole class,the use of groups or partners affords all students greateropportunities to express their ideas and practice theirEnglish (Garrison & Mora, 1999).

If possible, students should be placed in small groupswhere at least one bilingual student can act as interpreteror language facilitator during the activities (de la Cruz,1998). The language facilitator can be a more proficientELL or the teacher can even encourage English-proficientstudents learning Spanish to practice their secondlanguage skills by acting as a translator. If multiple ELLs

use the same native language in a classroom, the use ofthis language can best be facilitated in small groups (dela Cruz, 1998; Lee & Jung, 2004). Teachers need to beconscious of their individual objectives and understandthat if the goal is for students to learn mathematics, thereare times when the use of the L1 will more effectivelyachieve this goal.

Developing Language Skills

The literature is filled with recommendations forthe development of language skills. One such recom-mendation is to include exposure to etymologies, orword origins, because they help build bridges betweeneveryday language and mathematical language (readThompson & Rubenstein, 2000 for more detail on thisidea). Because words are labels for thoughts, ideas,concepts, and thinking, vocabulary (as well as familiaritywith syntax and discourse patterns) is central to conceptformation, understanding, and articulation (Garrison &Mora, 1999). Thus, it is critical that teachers exposeELLs to mathematized language so that the language ofmathematics and the concepts expressed by it are moreaccessible to students (Ron, 1999).

Use of Visual Representations andManipulatives

One of the basic premises of second-language in-struction is relating new vocabulary to tangible objects(Garrison & Mora, 1999). Visual representations andmathematics manipulatives provide excellent opportuni-ties for students to see and touch while repeatedly hearingand saying the new words (Garrison & Mora, 1999).Manipulatives can be used to demonstrate a new conceptso that new information can be processed, if possible, inboth English and Spanish (de la Cruz, 1998). Teacherscan record the concepts or vocabulary on the board as theywork with the manipulatives. Visual representations andmanipulatives allow students access to the key conceptualideas without being entirely dependent on the language(Lee & Jung, 2004).

Activities for Developing MathematicalLanguage

In this section, we provide examples of activitiesthat can improve students’ familiarity with mathematicallanguage. In these activities, the students developtheir mathematical language skills by working incooperative groups and/or using visual representations or

The Official Journal of the National Association for Multicultural Education

36

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

manipulatives. While these activities are not a panacea,we suggest them as a starting point for assisting ELLs.

Crossword Puzzles

Teachers can create their own crossword puzzles bygoing to the www.puzzlemaker.com webpage. Afterselecting, Criss-Cross Puzzle from the pull-down menu,teachers begin by entering a puzzle title. Teachers (orstudents) can create puzzles simply by typing in theword to be used in the puzzle, followed by a clue; forexample: “vertex another name for the turning point ofa parabola.” Cutting and pasting the created puzzle intoa word processing program allows for modifications andadditional symbolic notation (see Appendix A). In theexperience of the first author, students respond quitefavorably to this activity. The crossword puzzles can beused at the beginning of a unit to teach or remind studentsof vocabulary terms or at the end of a unit or course as areview activity.

I Have . . . Who Has . . . ?

The activity “I have . . . Who has . . . ?” can beplayed as a whole class or as a small group activity.Teachers (or students) create index cards or paper cutoutswith a statement and a question. The activity beginswith a question, such as, “Who has another name foraverage?” The student with the corresponding card reads,“I have the mean,” and then reads the question at thebottom of his/her card: “Who has the formula for thecircumference of a circle?” and so forth (see AppendixB). The game should end when the person who asked thefirst question answers a question. In this activity, studentsare developing language on two levels—learning themathematical language while developing verbal fluency.

Paired Practice

Rubenstein and Thompson (2001) suggested a partnertranscription activity where one partner reads a symbolicexpression or sentence while the other writes what he orshe hears using symbols. For example, one student canread, “sine squared theta plus cosine squared theta equalsone,” while their partner transcribes the equation on paper.Students can also be given the symbolic representationsof expressions such as AB ∼= CD or 4x2 + 2x ≤8 andpractice reading these expressions aloud.

In another variation of the transcription activity,students can practice using mathematical language bydirecting their partners to draw figures (see Figure 1)given to students on index cards.

One student describes the drawing, and the otherstudent draws it from the description. Students could also

F

E

C

A B

D

Figure 1. Figures can be replicated in Paired Practice.

use protractors and compasses to replicate figures, suchas triangles, using only verbal clues from their partners.In this activity, students are given the opportunity to usemathematical language with their peers while interactingwith a visual representation.

Jeopardy

Jeopardy, the popular television game show, can alsobe used to engage students in practicing mathematicallanguage while recalling mathematical facts and formulas.The teacher can make up a Jeopardy game using fivedifferent categories (see Appendix C). A student canhelp keep track of the questions that have been answeredby erasing or covering the monetary values as they arechosen. Teachers may read or show the chosen questionon the overhead projector. Students can play in teams orindividually, depending on the size of the class.

Conclusion

In this article, we described the written, spoken, and‘foreign’ aspects of mathematical language; discussedrecommendations for working with ELLs; and suggestedactivities that make use of these recommendations. Webelieve that mathematics teachers need to explicitly focuson mathematical language in their lessons. The activitiespresented here provide a starting point to integrate explicitteaching of mathematics and language. Making use ofthese strategies will help make mathematics accessible toall students.

References

Campbell, P. B., & Rowan, T. E. (1997). Teacher questions + studentlanguage + diversity = Mathematical power. In J. Trentacosta& M. J. Kenney (Eds.), Multicultural and gender equity in themathematics classroom: The gift of diversity, 1997 Yearbook (pp.60–70). Reston, VA: NCTM.

Multicultural Perspectives Vol. 12, No. 1

37

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

de la Cruz, Y. (1998). Issues in the teaching of math and science toLatinos. In M. L. Gonzalez, A. Huerta-Macias, & J. V. Tinajero(Eds.), Educating Latino students: A guide to successful practice(pp. 161–176). Lancaster, PA: Technomic Publishing Company,Inc.

Faltis, C. (1993). Joinfostering: Adapting teaching strategiesfor the multilingual classroom. New York: MacmillanPublishing Co.

Garrison, L., & Mora, J. K. (1999). Adapting mathematics instructionfor English Language Learners: The language-concept connection.In L. Ortiz-Franco, N. G. Hernandez, & Y. De la Cruz (Eds.),Changing the faces of mathematics: Perspectives on Latinos (pp.35–47). Reston, VA: NCTM.

Halliday, M. A. K. (1978). Language as social semiotic: The socialinterpretation of language and meaning. London: University ParkPress.

Khisty, L. L. (1995). Making inequality: Issues of language andmeanings in mathematics teaching with Hispanic students. In W.G. Secada, E. Fennema, & L. B. Adajian (Eds.), New directionsfor equity in mathematics education (pp. 279–297). New York:Cambridge University Press.

Lee, H. J., & Jung, W. S. (2004). Limited-English-Proficient (LEP)students: Mathematical understanding. Mathematics Teaching inthe Middle School, 9(5), 269–272.

Moschkovich, J. N. (1999). Understanding the needs of Latino studentsin reform-oriented mathematics classrooms. In L. Ortiz-Franco,N. G. Hernandez, & Y. De la Cruz (Eds.), Changing the faces of

mathematics: Perspectives on Latinos (pp. 5–12). Reston, VA:NCTM.

Olivares, R. A. (1996). Communication in mathematics for studentswith limited English proficiency. In P. C. Elliott & M. J. Kenney(Eds.), Communication in mathematics, K–12 and beyond (pp.219–230). Reston, VA: NCTM.

Perkins, I., & Flores, A. (2002). Mathematical notations and proceduresof recent immigrant students. Mathematics Teaching in the MiddleSchool, 7(6), 346–351.

Pimm, D. (1987). Speaking mathematically: Communicationin mathematics classrooms. NY: Routledge & KeganPaul.

Ron, P. (1999). Spanish-English language issues in the mathematicsclassroom. In L. Ortiz-Franco, N. G. Hernandez, & Y. de laCruz (Eds.), Changing the faces of mathematics: Perspectives onLatinos (pp. 23–34). Reston, VA: NCTM.

Rubenstein, R. N., & Thompson, D. R. (2001). Learning mathe-matical symbolism: Challenges and instructional strategies. TheMathematics Teacher, 94(4), 265–271.

Thompson, D. R., & Rubenstein, R. N. (2000). Learning mathemat-ics vocabulary: Potential pitfalls and instructional strategies.Mathematics Teacher, 93(7), 568–574.

Usiskin, Z. (1996). Mathematics as a language. In P. C. Elliott & M. J.Kenney (Eds.), Communication in mathematics, K–12 and beyond(pp. 231–243). Reston, VA: NCTM.

Valdes, G. (2001). Learning and not learning English. New York:Teachers College Press.

The Official Journal of the National Association for Multicultural Education

38

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

Appendix A

Across

3. We use the equation x = −b2a

to find the axis of .5. In the equation y = mx + b, the m represents the .8. If a circle is centered at (0, 0), we say it is centered at the .

Down

1. In 7 down, we would take the square root of 25 to obtain the2. If we graphed the equation y = x2 – 5x + 6, the graph would be a .4. Another name for the turning point of a parabola.6. If we were to graph the equation y = 3x – 2, the graph would be a .7. If we graphed the equation (x − 4)2 + (y +2)2 = 25, the graph would be a .

Multicultural Perspectives Vol. 12, No. 1

39

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

Appendix B

I have a permutation. I have a rotation.

Who has a stretch or shrink in the plane? Who has the formula for circumference of a circle?

I have a dilation. I have pi times diameter.

Who has a number that never ends or repeats? Who has the type of triangle that you would apply thePythagorean Theorem to?

I have an irrational number. I have a right triangle.

Who has the formula for the axis of symmetry? Who has the formula for area of a trapezoid?

I have x = −b over 2a I have (1/2) height times (base 1 plus base 2).

Who has another name for the turning point of aparabola?

Who has the name for the distance around the outside ofa polygon?

I have the vertex. I have perimeter.

Who has the relationship between the slopes ofperpendicular lines?

Who has the most famous Pythagorean Triple?

I have negative reciprocals. I have a 3, 4, 5 right triangle.

Who has the transformation that “flips” a figure over aline?

Who has the slope—intercept form of a line?

I have reflection. I have y = mx + b.

Who has the transformation that turns a figure? Who has the equation of a circle centered at the origin?

The Official Journal of the National Association for Multicultural Education

40

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014

Appendix C

Circles Transformations Quadratics Proof Hodge Podge100 100 100 100 100200 200 200 200 200300 300 300 300 300400 400 400 400 400500 500 500 500 500

Circles Transformations Quadratics Proofs Hodge PodgeThe name of the

chord that passesthrough the centerof the circle

The type of trans-formation that“stretches orshrinks”

Another name for theturning point of aparabola

What intersectinglines form

7% of 100 is equal tothis number

The measure of thethis angle is halfthe measure of thecentral angle

The type of transfor-mation that “turns”a figure in the plane

The equation thatfinds the axis ofsymmetry

What perpendicularlines form

A relation is a functionif it passes this test.

The name of theline segment thatintersects the circleat only one point

The type of transfor-mation that “flips”a figure over a linesegment

The formula for thesum of the roots

The method used toprove that 2 trian-gles are similar

A relation is one-to-oneif it passes this test

The name of theline segment thatintersects the circleat exactly 2 points

This is a transforma-tion that preservesdistance.

The quadraticformula

The type of sides thatare in proportionif 2 triangles aresimilar

The name of therelation: 3x2 − 3y2

= 12

The angle formed bya diameter and atangent line

A word to describethe “order” ofthe vertices of apolygon

The description ofthe roots whenthe discriminant isequal to zero

The product of themeans is equal tothis

The name of therelationship between3 + 2i and 3 – 2i

Multicultural Perspectives Vol. 12, No. 1

41

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

14:

23 1

1 O

ctob

er 2

014