Mathematics: A Second LanguageAuthor(s): Bradley R. Jones, Peggy F. Hopper, Dana Pomykal Franz, Libby Knott and ThomasA. EvittsSource: The Mathematics Teacher, Vol. 102, No. 4, Data Analysis and Probability (NOVEMBER2008), pp. 307-312Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20876351 .

Accessed: 22/03/2013 23:06

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions

CONNECTING Bradley R. Jones, Peggy F. Hopper, and Dana Pomykal Franz

Mathematics:

A Second Language

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are

triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

? Galileo Galilei (1564-1642), Opere U Saggiatore, p. 171

For many mathematically challenged stu

dents, the thought of working a mathematics

problem?processing numbers and more numbers?evokes fear, a tightening of the stomach, and overpowering feelings of anxiety. This feeling, commonly described as "math anxiety," is more

formally defined as "a feeling of intense frustration or helplessness about one's ability to do math" (Pla tonic Realms 2007). Such anxiety is just one of the factors contributing to many students' struggle to learn mathematics. U.S. students as a whole wrestle

with mathematics, but a closer analysis across sub

jects and grade levels reveals an unfortunate link between socioeconomics and school achievement,

due in part to academic challenges relating to pov erty (Bower 2001). However, students also struggle with language barriers resulting from the mismatch of academic language and their home language or the fact that they are primary speakers of languages other than English (Hacker 2007).

No matter the psychological or socioeconomic

reasons, poor mathematical ability has serious

consequences, and as educators we must address the question of why so many students are failing. One solution may be to create a classroom where fears can be left at the door and immersion in the

language of mathematics can occur. This article describes how mathematics can be defined as a second language and the instructional methods that result from this perspective.

DEFINING MATHEMATICS AS A SECOND LANGUAGE Greece?home to such great mathematicians as

Pythagoras, Plato, and Euclid?stood as the math ematical center of the ancient world. Although Greece later fell to the Romans, Greek culture and mathematical language characteristics did not. As

Horace, a Roman poet, wrote, "Captured Greece held its ferocious conqueror captive and introduced the arts to rustic Latium" (Varner 2006, p. 282). The

most powerful empire could not suppress the math ematical language of the Greeks. Therefore, our first task is to consider mathematics as a language.

Students can clearly comprehend that math ematics is a powerful tool, but many have difficulty comprehending mathematics as a language. Adams

(2003, p. 786) notes that Wakefield identified several commonalities between mathematics and language:

Abstractions (verbal or written symbols representing ideas or images) are used to communicate.

Symbols and rules are uniform and consistent.

Expressions are linear and serial.

This department consists of articles that bring research insights and findings to an audience of teachers and other mathematics educators. Articles must make

explicit connections between research and teaching practice. Our conception of

research is a broad one; it includes research on student learning, on teacher think

ing, on language in the mathematics classroom, on policy and practice in math

ematics education, on technology in the classroom, on international comparative work, and more. The articles in this department focus on important ideas and

include vivid writing that makes research findings come to life for teachers. Our

goal is to publish articles that are appropriate for reflection discussions at depart-. ment meetings or any other gathering of high school mathematics teachers. For

further information, contact the editors.

Libby Knott, knott@mso.umt.edu

University of Montana, Missoula, MT 59812

Thomas A. Evitts, taevit@ship.edu

Shippensburg University, Shippensburg, PA 17257

Vol. 102, No. 4 November 2008 | MATHEMATICS TEACHER 307

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions

Table 1

Adaptation of Gunderson's English Proficiency Scale to Mathematics

Proficiency Level Criteria

Zero-level mathematics Cannot answer yes/no questions

Unable to identify and name any objects

Understands no mathematics

Often appears withdrawn and afraid

Very limited mathematics

Responds to simple questions with mostly yes or no or one-word responses

Speaks in one- to two-word phrases

Attempts no extended conversation

Seldom, if ever, initiates conversation

Limited mathematics Responds easily to simple questions

Produces simple sentences

Has difficulty elaborating when asked

Occasionally initiates conversation

Limited fluency in mathematics

Speaks with ease

Initiates conversation

Makes errors in more syntactically complex utterances

Freely and easily switches code

Understanding increases with practice. Success requires memorization of symbols and rules.

Translations and interpretations are required for novice learners.

Meaning is influenced by symbol order. Communication requires encoding and decoding. Intuition, insightfulness, and "speaking with

thinking" accompany fluency. Experiences from childhood supply the founda tion for future development.

According to Adams, "Mathematics is a language that people use to communicate, to solve problems, to engage in recreation, and to create works of art and mechanical tools" (2003, p. 786). Mathematics

apparently shares many of the characteristics that define English as a language, and although histori

cally mathematics was rarely considered a lan

guage, current views enable us to consider it as such. Today, unless children grow up in homes in

which parents speak and model fluent mathemat

ics, it can be legitimately viewed as a "second lan

guage" (Adams 2003; Wakefield 1999, 2000). In today's increasingly diverse world, teaching

language is a timely issue. Therefore, if mathemat ics is viewed as a second language, why not teach it as one? The methodologies of teaching English as a second language (ESL) may be applied to the

methodologies of teaching mathematics as a second

language, or MSL (Curtin 2005). Ruddell notes:

ESL is used to designate classes that are immersion

classes, and students in these classes generally have

various primary languages and varying language lev els of English literacy and oral fluency.... Transition

programs are programs intended to bridge ESL class rooms to regular classes, and a method often practiced

is one of Sheltered Instruction (SI). (2005, p. 193)

Sheltered instruction is viewed by language educa tors as an effective approach to language instruc tion for ESL students in transitional classes and includes "rich language interaction, focus on student's prior knowledge, integrated/collaborative learning, repetition of ideas and concepts, allow ance for students to choose which language to use at any given moment and a low-risk environment for second-language use" (Ruddell 2005, p. 203).

If a student evaluates his or her own mathemati cal language ability and compares it with a class

mate's, he or she may feel a kinship with a student in the traditional ESL classroom. A student may even be able to place himself or herself in Gunder son's graduated scale of English oral-language proficiency (see table 1), modified by replacing the word English with mathematics (Ruddell 2005, pp. 194-95). Thus, a student can rate himself or herself

308 MATHEMATICS TEACHER | Vol. 102, No. 4 November 2008

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions

as a Mathematics Language Learner (MLL), a desig nation similar to English Language Learner (ELL).

Once a specific mathematical language level is identified, level-appropriate instruction can

be introduced. Thus, teaching mathematics as a second language through a variety of ESL and sheltered instruction strategies instead of more

traditional methods may result in higher student

comprehension and achievement.

DECODING MATHEMATICS AS A SECOND LANGUAGE Dissecting a passage of text in a language other than one's native language is a daunting task and

requires a strategy. When dissecting mathematical

language, readers are faced with the same chal

lenges, whether the mathematics is in the form of an equation or in the form of a word problem. To aid students in accomplishing such tasks, teachers can apply the framework for problem solving cre

ated by George Polya. Polya's (1962) four steps are

(1) understanding the problem, (2) devising a plan, (3) carrying out the plan, and (4) looking back. The first and last steps seem particularly relevant to our

claim that mathematics is a second language and, therefore, should be presented to students through the use of a second language.

The problem that follows, solving for a system of linear equations, offers a perspective on how each step of the solution relates to mathematics as a

language. The problem also illustrates aspects of the Process Standards as defined in NCTM's Principles and Standards for School Mathematics (2000). These

Standards?specifically Problem Solving, Com

munication, and Representation?are intertwined

throughout the process. The problem follows:

Solve this system of equations:

X + 4Y + Z = 285 (1) 27+2Z = 90.88 (2) Y+3Z= 75.68 (3)

From here, students can dissect the problem while

simultaneously following a step-by-step analysis according to Polya.

STEP 1: UNDERSTANDING THE PROBLEM This beginning step can be related to a novice for

eign language learner who has just been handed a

paragraph in the newly studied language. Two steps are critical in understanding the passage: knowing the vocabulary and understanding the structure of the language. Consequently, the first step in reading any passage is understanding how to get started.

Many mathematical words take on meanings differ ent from their everyday English meanings. A student's

ability to translate these words into mathematical

language is critical for success. After the student has

applied the appropriate definitions, the next step is to reread the sentence and notice any words, sounds, or

subject-verb structures that provide any contextual clues. This process is similar to decoding a foreign language, and "math learners benefit when equations are considered from a subject and verb approach" (Wakefield 1999, p. 6). The subject is the aspect of the mathematical equation a reader can understand and connect to past knowledge or work in mathemat

ics, and the verb indicates the direction in which the reader must go to solve the problem successfully.

An additional challenge for the mathematics reader at this step is to understand the unique struc tures of mathematical language. Like English, Span ish, and French, mathematics can be read from left to right; like old Egyptian, it can be read from right to left; like Japanese and Chinese, it can be read

vertically. This uniqueness often requires the math ematics reader to sweep visually from right to left as well as, possibly, up and down, diagonally, and left to right (Barton, Heidema, and Jordan 2002) to understand the "text." In this example, students are asked to read three separate equations from left to

right and align the variables and numbers from top to bottom for clarification. This process requires the student to maintain continuous visual sweeps from left to right and from top to bottom:

Reading a three-variable equation from left to right as well as reading a column of variables from top to bottom cannot be done simultaneously.

To meet the NCTM Representation Standard, students must "select, apply, and translate among

mathematical representations to solve problems" (NCTM 2000, p. 67). Some students maybe able to

comprehend this problem if they think of X as a

video game console, 7 as video game controllers, and Z as actual video games. Still others may connect best if they think of X as a prom dress, Y as pairs of

shoes, and Z as pieces of jewelry. Translating this mathematics problem into the language each student

speaks may be the cornerstone to success. Compre

hension is achieved when students are able to immerse themselves in the material and create con nections to their own lives, a goal that every math ematics teacher sets for his or her students. Edward

Thorndike, frequently referred to as the father of educational psychology, defines intelligence as an individual's ability to make these connections (Ber liner 1993, p. 17). However, educators have yet to create the ideal environment in which these math

LX + 47+lZ = 285 0X + 27+ 2Z= 90.88 0X + 17+ 3Z= 75.68

(1) (2) (3)

Vol. 102, No. 4 November 2008 | MATHEMATICS TEACHER 309

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions

ematical connections can take place. The purpose of this article is to help create this ideal environment.

STEP 2: DEVISING A PLAN To understand any passage, whether in mathemati cal or any other language, the reader must be able to translate the text. Translation occurs when a reader is able to define the words and successfully decode the relevant information. Adams found that "a stu dent's ability to recognize and employ the formal definition is key to understanding and applying concepts when reading mathematical text" (2003, p. 787). In the system of equations presented earlier, the student must grasp the concept of elimination as

well as feel comfortable working with two different

equations simultaneously. Knowing that elimination is key to solving the problem, the student must then be able to discern which variable is most suitable for removal. Students should realize that selecting equa tions 2 and 3 and variables 7 and Z for initial elimi nation will be the most direct method.

STEP 3: CARRYING OUT THE PLAN Once a student has quickly reviewed the problem and dealt with any language challenges, he or she must "sum up" all the information and discern

meaning?that is, discern the problem's solution. The student must now carry out the devised plan. In choosing equations (2) and (3), variables 7 and Z become obvious choices. Of those two, 7 becomes a clear choice, because eliminating Z might intro duce fractions into the problem.

0Z + 27+ 2Z = 90.88

2(0X + 17+ 3Z) = (75.68)2

0X+27+2Z= 90.88 0Z +27+6Z= 151.36

From this step, the student is asked to subtract

downward, thus forming one equation with one

variable and removing the other variable completely:

-4Z = -60.48

At this point, the student will divide both sides of the equation by -4 or multiply both sides by -1/4.

Arriving at the conclusion that Z = 15.12, the stu dent will then substitute the value for Z into equa tion (2) and conclude that 7= 30.32. Ag ain, the student is asked to substitute values for 7 and Z into

equation (1) and will realize that X = 148.60. The final solution set will be of the following form:

X= 148.60 7=30.32 Z= 15.12

STEP 4: LOOKING BACK The student solving the problem presented here must take a moment to look back and ask, "What do those numbers mean?" Teachers hope that students

will ask themselves this question, but if they do not, teachers must lead them to do so. Looking back on a solved problem "lets students engage in discus sions about the problem-solving process to further enhance their reasoning skills and abilities to

explain and justify solutions" (Adams 2003, p. 791). This fourth and final step hinges on another

NCTM Process Standard?Communication, defined

by NCTM as students' ability to "communicate their mathematical thinking coherently and clearly to peers, teachers, and others" (NCTM 2000, p. 60). In this example, after students make the substitu tions for each variable, it is important to direct them to ask themselves whether their answers

make sense. Students show a complete understand

ing of mathematical concepts when they know whether their answers seem logical or not.

For the stated problem, both positive and nega tive answers would be acceptable. However, if the

problem had been designed around video games or articles of clothing, would a negative answer

have made sense? Can a pair of shoes really cost

-$40.67? If a student truly comprehends a prob lem, he or she should be able to communicate these ideas with a listener of any mathematical skill level.

Writing is arguably the best way for a student to reflect on a completed task; it "supports learning because it requires students to organize, clarify and reflect on their ideas?all useful processes for

making sense of mathematics" (Burns 2004, p. 30). Writing down all the steps, ideas, and solution

strategies applied to a problem allows students to

step back and reflect on their own processes as well as develop future methods most appropriate to their

unique learning styles. Applying Polya's four steps for solving problems

is one effective way of teaching mathematics as a

second language. The steps are logical and may be familiar to mathematics educators. Other effective

ways can be drawn from strategies often used in

second-language instruction.

INSTRUCTIONAL METHODS FOR THE LANGUAGE OF MATHEMATICS Students may benefit from sheltered instruction

strategies that result from a teaching perspective of mathematics as a second language. Sheltered instruc tion incorporates three key concepts: small-group collaborative learning; connecting; and immersion.

Small groups are ideal for creating Mathematics as a Second Language (MLS) classrooms. Working through the mathematics as a language concept as a teacher may be important to classroom success, but

310 MATHEMATICS TEACHER | Vol. 102, No. 4 November 2008

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions

fostering students' process of learning mathematics as a language through sheltered instruction meth

odologies may further affect students' abilities to

experiment and acquire mathematical knowledge in depth. Allowing students to meet in small groups enables them to speak with one another quietly and formulate their own conclusions. Working with

peers allows them to communicate in their first

language and also removes the pressure of speaking in front of the entire class. Sheltered instruction calls for a mixture of group members of all speaking levels (remember Gunderson's scale) and provides an outlet for students to collaborate. Once the small

groups adjourn and the class comes back together as a whole, students have a chance to share their

group's ideas through the language of mathematics. The process of connecting with the language was

previously demonstrated through Polya's four steps for problem solving. For students to comprehend a mathematical topic, they must be able to make a

connection to previously learned materials or ideas; and to provide students with the chance to make a connection, teachers need a basic understanding of the different cultures and homes each student comes from. Many students come from homes in which neither the mother nor the father has a math ematics-related job. Others may come from different cultures in which no great emphasis is placed on the study of mathematics. Hence, it is important for the teacher to relate the student's world to the vast

world of mathematics. Students need to understand that mathematics is relevant in every area of society, not just the mathematics classroom or even school itself. Although most mathematics teachers would

acknowledge the concept of connection, thinking of it as immersing the student in the "culture of math ematics" sheds new light on the idea.

Inviting community members who represent a

variety of professions to explain how they use math ematics in their jobs could demonstrate to students the various applications of mathematics. Tracing the

history of mathematics from the Egyptians to the

Babylonians, the Indians, the Greeks, the Chinese, and today's mathematicians would be a vivid way of relating each culture to the only language that has survived from the beginning of time. Educa tors should note that the goal of any connection should be to "develop opportunities for students to

strengthen their understanding of mathematics ter

minology and concepts" (Adams 2003, p. 789). Immersion has been identified by language educa

tors as one of the best ways for students to learn a

foreign language. Although mathematical immersion in the classroom should not mean "sink or swim," the importance of being able to speak mathemat ics can still be reinforced. Teachers may find that

through immersion, students in mathematics class

rooms develop a deeper understanding of mathemat ics concepts. An MSL classroom, like the traditional ESL classroom, might display illustrations with the

corresponding words underneath. Students could be

required to have dictionaries that help them under stand and translate words. Teachers could speak in the formal language, but, to further understanding, they would explain in the students' first language. And parents may be encouraged to immerse their children in mathematics at home.

These strategies attempt to aid students in

acquiring a language through continuous immer sion rather than periodic instruction (Wakefield 1999, p. 4). If immersion is an excellent strategy for learning such languages as English, Spanish, French, and German, then it seems only logical to teach the language of mathematics the same way.

CONCLUSION The intent of this article was to answer the ques tion, How should we teach mathematics to our

students with an emphasis on comprehension? Thinking about mathematics through the lens of ESL instructional methods may provide insight into weaknesses present in current traditional

approaches to teaching mathematics. The non

traditional idea of using approaches more com mon to language instruction offers suggestions for

pedagogical reform that may have the potential for

promising results. More accurately, the specific question implied here is this: Is there room in math ematics reform for teachers to use language-specific methods for teaching mathematics such as sheltered instruction and reading strategies? (Franz and Hop per 2007). By treating the mathematics classroom as a classroom for language acquisition, many read

ing strategies find a home.

BIBLIOGRAPHY Adams, Thomasenia L. "Reading Mathematics: More

Than Words Can Say." The Reading Teacher 56, no. 8 (2003): 786-95.

Barton, Mary L., Clare Heidema, and Deborah Jordan.

"Teaching Reading in Mathematics and Science." Educational Leadership 60 (2002): 24-31.

Berliner, David C. "The 100-Year Journey of Edu cational Psychology from Interest, to Disdain, to

Respect for Practice." In Exploring Applied Psychol ogy: Origins and Critical Analysis Master Lectures in

Psychology, edited by Thomas K. Fagan and Gary R. VandenBos. Washington, DC: American Psycho

logical Association, 1993.

Bower, B. "Math Fears Subtract from Memory, Learn

ing: Learning Anxiety in Poor Math Students." Sci ence News (June 2001).

Burns, M. "Writing in Math." Educational Leadership

(October 2004): 30-33.

Vol. 102, No. 4 November 2008 | MATHEMATICS TEACHER 311

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions

Curtin, Ellen. "Teaching Practices for ESL Students."

Multicultural Education 12, no. 3 (Spring 2005): 22-28.

Hacker, Holly. "TAKS Math Scores Slump in Ninth Grade." Dallas Morning News. June 30, 2007.

Franz, Dana P., and Peggy F. Hopper. "Is There

Room in Math Reform for Preservice Teachers to

Use Reading Strategies? National Implications." National Forum of Teacher Educational Journal 17, no. 3 (March 2007): 1-9.

Furman University. Mathematical Quotations Server.

"Mathematical quotations?G." 2008. math.furman

.edu/ ~ mwoodard/mquot.html.

Platonic Realms. "Platonic Realms MiniTexts: Coping with Math Anxiety." 2007. www.mathacademy.

com/pr/minitext/anxie ty/index. asp. National Council of Teachers of Mathematics

(NCTM). Principles and Standards for School Math ematics. Reston, VA: NCTM, 2000.

Polya, George. Mathematical Discovery: On Under

standing, Learning, and Teaching Problem Solving. New York: John Wiley and Sons, 1962.

Ruddell, Martha R. Teaching Content Reading and

Writing. Phoenix: John Wiley and Sons, 2005.

Varner, E. R. "Reading Replications: Roman Rheto

ric and Greek Quotations." Art History 29, no. 2

(2006): 280-303.

Wakefield, Dara V. "Se Habla Mathematics? Consid eration of Math as a Foreign Language." Classroom

Teacher 52 (1999): 2-11. -. "Math as a Second Language." The Educational

Forum 64, no. 3 (Spring 2000): 272-79. oo

PBRADLEY R. JONESf jones.brad.rd> gmail.com., a graduate student in sec

ondary education at Mississippi State

University, teaches mathematics at Starkville High School in Starkville,

Mississippi. PEGGY F. HOPPER, pfh7@ msstate.edu, is an assistant professor at Mississippi State University in sec

ondary English and language arts ed ucation. Her areas of research include international studies; content area

reading, specifically reading in math

ematics; and teacher quality. DANA POMYKAL

FRANZ, df76@colled.msstate.edu, is an assistant

professor of secondary mathematics education at Mississippi State University. Her current re

search focuses on specific reading strategies for USe in high SChOOl mathematics. Photographs by Bradley R. Jones and Russ Houston; all rights reserved

312 MATHEMATICS TEACHER | Vol. 102, No. 4 November 20C

NCTM Membership Works. Is It On...and Receive Rewards. w As aifWCTM member, you already know that NCTM

membership ojfcrs support, guidance, resources, and insights thaflflflu can't get anywhere else. Help spread the word!

Participate in the TCftf Membership Referral Program and, to Wmk you, we'll send you rewards. The more members you recruit the greater the gifts* you'll receive. Plus, each referral* enters your name in a grand prize drawing for a free trip to NCTM's 2010 Annual Meeting in San Diego.

Get started today? everything you need is online.

www.nctm.org/membership

See Web site for complete details.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

(800) 235-7566 | WWW.NaM.ORG

This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:06:08 PMAll use subject to JSTOR Terms and Conditions