TESOL 2005
San Antonio, TX
April 1, 2005
The Language of Math
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How to use this presentation
Important language features of math problems and
math instruction
Activities to reinforce the language of math
Kathryn Heinze
Assoc. Professor
Center for Second Language Teaching and
Learning
Hamline University A 1790
1536 Hewitt Ave.
St. Paul, MN 55104 kheinze@hamline.edu
Copies of this handout can be downloaded from http://kathrynheinze.efoliomn2.com
The Language of Math
Vocabulary
(1) Mathematics includes words that are specific to mathematics such as divisor, denominator,
and quotient. These words are new to most students. (2) But math also includes common
everyday vocabulary that takes on a different meaning in math. Examples of these words
include: square, rational, column, and table. (3) Math also combines math vocabulary in phrases
that can take on a brand new meaning: the least common multiple and a quarter of the apples.
Words for Addition
add
and
plus
sum
combine
increased by
Subtraction
subtract from
take away
minus
Words for Multiplication
multiplied by
times
Division
divided by
into
Words for Answers
total
gets you
results in
 It’s not enough for students to learn lists of words. They must learn what they mean in a
particular mathematical expression. Examples: (1) 3 multiplied by 10 is vastly different than 3
increased by 10, (2) divided by and divided into will give entirely different results.
 When students learn mathematical symbols such as , , , >, <, and they must learn to relate
them to (1) mathematical processes or operations, and (2) translate them into everyday concepts.
 If students have had math education in their home countries, they may find differences in
symbolic use confusing.
300.000
1,73
Take special note of vocabulary that has both common and technical meanings. This can be
particularly challenging for ELLs and needs to be drawn attention to.
column
product
quarter
place
table
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Syntax (Word Order)
The language of mathematics, like all technical or specialized vocabulary, has special syntactic
structures and special styles of presentation.
Here are some examples of structures that are frequently used in math and can be difficult to
master by ELLs.
greater than/less than
n times as much (as)
as in
as in
as . . .as
-er than
as in
as in
Numbers used as nouns
(Rather than adjectives)
as in
Prepositions
Passive voice
as in
as in
all numbers greater than 4
Choua earns six times as much as I do. Choua earns
$40,000. How much do I earn?
The tennis ball is as big as the plastic ball.
Fernando is three years older than Frank. Frank is
25. How old is Fernando?
Twenty is five times a certain number. What is the
number?
eight divided by four and eight divided into four
when 15 is added to a number, the result is 21. Find
the number.
 One of the principal characteristics of the syntax used in mathematical expressions is the lack
of one-to-one correspondence between mathematical symbols and the words that they
represent. For example, if the expression eight divided by two is written word for word in the
order in which it is written, the resulting mathematical expression could be 8  2 =
As a result, ELLs tend to duplicate the surface word order of mathematical statements. Example:
“The number a is five less than the number b” is often interpreted by students as:
a=5–b
instead of
a=b5
 Logical connectors are another language device used in mathematics: if . . .then, if and only if,
because, that is, for example, such that, but, consequently, either . . . or. Students must know
what situation is signaled, cause and effect or reason and result, similarity, contradiction,
chronological or logical sequence. To use these connectors, they also need to know where
logical connectors occur: in the beginning, middle, or end of a clause. Some connectors can only
be used in certain positions; others’ meaning changes if the position is changed. Elementary
students may, for the most part, be limited to having to puzzle out if. But they do show evidence
of having difficulty understanding the hypothetical situations signaled by this connector in
problems like those below:
If Mohammed can type one page in 20 minutes, how much time will it take him to type
two pages?
However, they have much less difficulty when this problem is expressed as:
Mohammed types one page in 20 minutes. How much time does it take him to type two
pages? OR How much time does he (Mohammed) need to type two pages?
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Ellipsis and Cohesion
Language, all languages, are full of redundancies. But because repetition in a language you know
can become annoying, writers of even math problems use several linguistic features to avoid
repetition. Two ways that redundancy is avoided include (1) the use of ellipsis (leaving words
out), and (2) the use of pronouns and articles.
In ellipsis, words may be left out of a phrase because native speakers can very easily figure out
what the missing words are.
All numbers greater than four = All numbers (that are) greater than four
Maria earns six times as much as Peter ( ).
 Check to see if students can supply missing words or phrases. If they can’t, it may indicate that
they are having problems inferring the deleted information.
In pronoun reference, pronouns such as she, he, it, her, him, they, them, their, theirs, hers, his,
its, etc. refer to nouns that have been mentioned before.
Rachel had 17 toy cars. She gave 11 of them away. How many toy cars does she have
now?
 Other pronouns that can be even harder to identify include words such as the one(s), that, these,
those, this.
Spread your thumb and first finger as far apart as you can. Do this in the air. Don’t use
your other hand to help. Trace them on the board.
 If students can’t figure out what nouns or ideas pronouns refer back to they miss out on
understanding how the sentences relate to each other; they don’t understand the cohesion of the
text, or how it hangs together. This can be a serious stumbling block to their understanding of a
text, whether it is in math or another subject area. Students who claim to know all the words, but
can’t understand the message or meaning may have trouble identifying which nouns the pronouns
refer back to .
The use of articles and determiners (the, a, an, this, that, these, those) can also be troublesome.
Note the following problems:
When 15 is added to a number, the result is 21. Find the number.
Twenty is five times a certain number. What is that number?
One number is ten times another number
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ELLs need to be explicitly taught the meaning of articles/determiners. Words such as the,
this, that, these, those usually indicate that it is a number that has been mentioned before.
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Other Language Considerations
To be proficient readers, ELLs need to know how sentences, groups of sentences, paragraphs, or
whole texts function together to convey a message. Understanding vocabulary alone is not
enough. Students have to know what bank of prior knowledge about the world to tap into
because the language of math is: (1) very dense and highly packed with concepts, (2) requires upand-down as well as left-to-right eye movements, (3) requires a slower reading rate and multiple
readings, (4) uses numerous symbolic devices such as mathematical symbols, charts, and graphs,
and (5) contains a great deal of technical language that conveys precise meanings.
 Students have to apply mathematic concepts, procedures, and applications with language. So
students with a weak math background and second language skills are doubly challenged.
 Students need to need to recognize which previously learned math concepts, procedures, and
applications must be applied to the text they are reading. They must also know when everyday
background knowledge can be applied to what they are learning.
Adapted from Richards-Amato & Snow (1992), Cocking & Mestre (1998), and Carasquillo & Rodriguez (1996).
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Problems from the MN Basic Standards Test for Math
Kim drove an average of 50 miles per day for the first 4 days of her work week. What would her
daily average be for her five-day work week if she drove 35 miles on the fifth day? (if clause)
In a football game, a team gained 4 yards, then another 12 yards, was penalized and lost 15 yards
and gained 7 yards. What was the net gain of their three successive plays?
(vocab: net, successive, gained, penalized)
A machine does a job in ¾ of an hour. It takes a person 6 ½ hours to do the same job by hand.
How much less time is required to do the job by machine? (articles)
A house plan is drawn to the scale of ¼ inch = 4 feet. What is the length of the house if it
measures 3 ½ inches on the floor plan? (synonymous phrases)
Jim spent 1 hour and 30 minutes raking the yard in the morning and 1 hour and 45 minutes in the
afternoon. How long did he rake altogether? (ellipsis)
An item was regularly priced at $63. The item went on sale at 1/3 off the regular price. How
much was saved?
A pen costs $11.39. Sarah bought 3 pens. Not including tax, how much did Sarah pay?
At 7:00 a.m., Jose’s thermometer registered 5 F. By 5:00 p.m., the temperature had dropped
18F. What was the temperature at 5:00 p.m.?
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The Word Problem Procedure
Word Problem Procedure (WPP)
1.Choose a partner or partners. Write your names above.
2.Choose a problem. Write the problem in the space below.
3.One student read the problem out loud. Discuss the vocabulary and circle words you don't understand.
Write the words below.
4.Use a dictionary for help. Ask your partner or teacher for help.
5.What does the problem ask you to find? Write this below:
6.What should you do to solve the problem? Add? Subtract? Multiply? Divide? Write this below.
7.Solve the problem below.
8.Check your answer below.
9.Explain your answer to your partner(s). Write your explanation below.
10.Explain your answer to the class.
11.Write a similar problem on the back of this page.
The various steps of the WPP make it possible for students to practice academic English, apply
mathematical rules, and use learning strategies and tactics in the course of the activity. It also provides
opportunities for teachers to be explicit and direct in their instruction and to pinpoint specific strategies and
tactics that they wish their students to practice. Students use metacognitive strategies such as selective
attention in focusing on unknown words (Step 3) and attending to what the problems ask them to find
(Step 5), advance preparation in reading the problem to each other (Step 3), and self-management in
explaining the problems (Step 9) or writing similar problems (Step 11). They use cognitive strategies such
as resourcing in using their dictionaries (Step 4), and elaboration and deduction in solving the problems
(Steps 7 and 8). They use social-affective strategies such as cooperative learning and questioning for
clarification in working their way through the various steps. The use of this format thus creates
opportunities to introduce and practice learning strategies while solving word problems. In the context of
this study, it also makes it possible to analyze student performance in reference to their ability to follow
individual steps and sequences of steps, and to learn and apply learning tactics and strategies.
Discussion of an Example of the WWP
To read a more detailed analysis of how students dealt with each stage of this procedure, go to
http://www.ncela.gwu.edu/pubs/symposia/third/spanos.htm. In my presentation, I will only
discuss some of the steps in this procedure.
Word problem: Sam's truck weighs 4,725 pounds. The truck can carry 7,500 pounds. What is the total
weight of the truck and a full load? (addition of whole numbers)
Step 2 (Choose a problem and write it). Simply copying the word problems from the review sheet was
difficult for many of the students. Their work was marked by poor handwriting, spelling errors, sentence
fragments, run-ons, and mechanical errors involving capitalization and punctuation.
Step 3 (Read the problem out loud. Find difficult words). It was difficult to convince students that they
should take turns reading the problem to each other. Even after the instructor selected two students to
model the practice for the whole class, most pairs still refused to read aloud. However, they did enjoy
circling or underlining words they did not understand and writing them in the space provided. Some of the
items selected from the word problems were the following:
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weighs, carry, load
Step 5 (Write the question). Students were not initially aware that the word problems ended with a
question. Once they were, Step 5 became a routine matter of finding the question and copying it. As with
Step 2, students made copying errors such as omitting question marks and other punctuation, or writing
puzzling statements. For example:
What is carry?
Steps 6 and 7 (Find the operation and solve the problem). There were numerous cases of incorrect
selection and application of mathematical operations, incorrect representations of the problems, and
computational errors. For example:
4,725 x 7,500 = 31,500
Steps 9 and 10 (Explain orally and in writing). It was difficult for students to give a coherent verbal or
written English explanation of their work. A typical response was to simply write the numerical answer
from Step 7 along with a statement such as: "This is the answer to the problem." Some of the more
interesting written explanations follow:
“I assembly together and I get this count it is 12,225 I think it is the car it goes two trip first one carry 4725
and second carry 7,500 after that I use add. ”
“First we saw the problem and then we thought about the problem. If the truck weight 4,725 pounds and
can carry 7,500 pounds all we have to do is add.”
Step 11 (Write a similar problem). The initial response to this step was to simply copy other problems
from the review sheet or any other materials or texts at hand. (Perhaps they believed that all word problems
were similar—hard!). Or, perhaps not understanding what “similar” meant, they were groping for the most
convenient written response. Whatever the case, upon noticing their difficulties during the first WPP
session, the instructor spent the next class period teaching the students how to write their own problems.
They were instructed to change numbers, names, and situations which appeared in sample word problems.
But they were not allowed to change the mathematical operation. This exercise resulted in problems which
were similar but not exactly the same. When satisfied that enough students grasped the distinction, the
instructor had them attempt Step 11. Most students were still unable to take the step without coaching by
the instructor, but the following are some of the more interesting results:
“Delmis wieght 145 pound, and Ali wiegh 150. What is the when we together 145 is Delmis and
Ali is 150?”
“Sam's truck weight 2,354 galon. The truck can carre 1,453 galon. What is the total weight of the
truck and full load?”
“My brother weighs 155 pounds and he can carry 125 how mus dos he weighs together.”
Excerpt from: Spanos, G.(1993). ESL Math and Science for High School Students: Two Case Studies.
http://ncela.gwu.edu/ncbepubs/symposia/third/spanos.ht
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Math and ELLs:
A Print Bibliography
Andrini, B. (1998) Cooperative Learning and Mathematics. Kagan.
Chock full of ideas for using cooperative learning activities to teach math. Written for native and
non-native speakers alike. Contains copiable black-line masters.
Bamberger, H, & Hughes, P. (1995) Super Graphs, Venns & Glyphs: Hundreds of Great Data
Collecting Activities to Build Real-Life Math Skills. Scholastic.
Good resource if you are running out of ideas for connecting ‘real life’ to data collection.
Not just for ELLs, but most appropriate for K-12. However, ideas can be adapted for older
students.
Bresser, R. (2003). Helping English-language learners develop computational fluency. Teaching
Children Mathematics, February, 294-299.
Especially valuable for its list of strategies to help ELLs.
Carasquillo, A. L. & Rodriguez, V. (1996) Language minority students in the mainstream classroom.
Multilingual Matters.
Contains a good chapter on integrating language and mathematics. Good bibliography for further
reading on the subject.
Chamot, A. & O’Malley, J. M.. (1994) The CALLA Handbook: Implementing the Cognitive Academic
Language Learning Approach. Longman.
A great resource book that helps ESL and mainstream teachers look at what students need to know
in content areas and provides an entire chapter on mathematics.
Clement, L. L. (2004). A model for understanding, using, and connecting representations. Teaching
Children Mathematics, September, 97-102.
Provides several examples of how native speakers of English can misinterpret math language.
Cocking, R. R. & Mestre, J. P. (1998) Linguistic and Cultural Influences on Learning Mathematics.
Lawrence Erlbaum.
Has excellent chapters on the cultural and linguistic problems that may affect ELLs. May be more
comprehensible to ESL teachers, but reading it is worth the effort for teachers who really want
answers to what is difficult in math for ELLs
Dale, T. C. & Cuevas, G. J. (1992) Integrating mathematics and language learning. In RichardAmato, P. A. & Snow, M. A. (Eds.) The Multicultural Classroom: Readings for Content-Area
Teachers. Addison-Wesley.
An excellent chapter that outlines the language difficulties that ELLs may have with math
instruction
DeBolt, V. (1998). Write! Mathematics, 4-9: Multiple Intelligences & Cooperative Learning Writing
Activities. Kagan.
Lots of good ideas for getting students to use language to talk (or write) about
important concepts and processes in math. Activities can be adapted for older students.
Elliott, P.C. (1996). Communication in Mathematics, K-12 and Beyond. Reston, VA: NCTM.
This yearbook has many chapters concerning the importance of communication in teaching and
learning mathematics. Two chapters specifically are important for ESL teaching and the language
of mathematics: “Communication in Mathematics for Students with Limited English Proficiency”
by Rafael A. Olivares and “Mathematics as a Language” by Zalman Usiskin.
McCallum, R. & Whitlow, R. (1994) Linking Mathematics and Language: Practical Classroom
Activities. Pippin.
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Great collection of activities that can be used to introduce or reinforce math concepts.
Murray, M. (2004). Teaching Mathematics Vocabulary in Context. Heinemann.
Wonderful resource that offers a range of strategies that highlight the important role of
language in math.
NCTM, (2000). Principles and Standards for School Mathematics. Reston VA: NCTM.
NCTM. (1999). Changing the Faces of Mathematics: Perspectives on Latinos. Reston, VA: NCTM.
A nice collection of articles how culture, previous schooling, and language affect the learning of
mathematics. Two other books in this series address multi-cultural issues and the math education
of Native American Indians.
Stephens, A. C. (2003). Another look at word problems. Mathematics Teacher, 96 (1), 63-66.
Argues for the value of having students create their own word problems as a way to make certain
that they understand math concepts.
Whitin, D. J. & Wilde. S. (1992) Read Any Good Math Lately?: Children’s Books for Mathematical
Learning, K-6. Heinemann.
Great resource for elementary teachers who want ideas for contextualizing math instruction.
Texts for Students
Chamot, A. &. O’Malley, J. M. (1988). Mathematics Book A: Learning Strategies for Problem
Solving. Addison Wesley.
Helps students figure out how to solve problems. Highly recommended by a middle school
bilingual teacher who has successfully used it, but it isn’t necessarily the magic bullet. Limited
to basic math.
An On-line Bibliography
The following articles about teaching language and mathematics can by accessed by using the
URLs below or by going to: http://ncela.gwu.edu. Click on “Resources About,” then enter
“Mathematics” in the site search box. Or you can enter the URLs of the articles below that
interest you, skim them, and then decide which ones may be of interest to you.
Alecio, R. Lara., Parker, R., Aviles, C ., Mason, S. and Irby, B.J. (1998).Un Estudio de
Evaluación Educativa Manipulativos en el Aprendizaje de las Matemáticas con
Estudiantes Hispanos Adquiriendo Inglés Académico como Segunda Lengua.
http://brj.asu.edu/v22234/articles/art7.html
Black, C. and Huerta, M.G. (1994). Teacher Insights: The Use of Language Functions in
Mathematical Group Games. http://www.ncela.gwu.edu/pubs/nabe/brj/v18/18_34_black.pdf
Buchanan, K. and Helman, M (1993). Reforming Mathematics Instruction for ESL Literacy
Students. http://www.ncela.gwu.edu/pubs/pigs/pig15.htm
Crawford, K. (1990). Language and Technology in Classroom Settings for Students from
Non-Technological Cultures. http://www.enc.org/topics/equity/articles/documents/0,1946,ACQ111330-1330,00.shtm
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Dalton, S. and Sison, J. (1995). Enacting Instructional Conversation with Spanish-Speaking
Students in Middle School Mathematics. http://www.ncela.gwu.edu/pubs/ncrcdsll/rr12/index.htm
Henderson, R. W. and Landesman, E.M. (1992). Mathematics and Middle School Students of
Mexican Descent: The Effects of Thematically Integrated Instruction.
http://www.ncela.gwu.edu/pubs/ncrcdsll/rr5.htm
Khisty, L.L. (1993). A Naturalistic Look at Language Factors in Mathematics Teaching in
Bilingual Classrooms. http://www.ncela.gwu.edu/pubs/symposia/third/khisty.htm
Leon, R. (1994). The Effects of the Presence of Extraneous Information In Mathematical Word
Problems on the Performance of Hispanic Learning Disabled Students.
http://www.ncela.gwu.edu/pubs/nysabe/vol9/mathprob.htm
Mather, J.R.C. and Chiodo, J. (1994). A Mathematical Problem: How Do We Teach
Mathematics to LEP Elementary Students?
http://www.ncela.gwu.edu/pubs/jeilms/vol13/math13.htm
Padron, Y. N. and. Waxman, H. C. (1993). Teaching and Learning Risks Associated with Limited
Cognitive Mastery in Science and Mathematics for LEP Students.
http://www.ncela.gwu.edu/pubs/symposia/third/padron.htm
Raborn, D. (1995). Mathematics for Students with Learning Disabilities from
Language-Minority Backgrounds: Recommendations for Teaching.
http://www.ncela.gwu.edu/pubs/nysabe/vol10/nysabe104.htm
Radford, L., Netten, J. and Duquette, G. (1997). Developing Target Second Language Skills
Through Problem-Solving Activities in Mathematics. (PDF only)
http://www.ncela.gwu.edu/pubs/nysabe/vol12/nysabe125.pdf
Reyhner J. and Davison, D.M. (1993). Improving Mathematics and Science Instruction for LEP
Middle and High School Students Through Language Activities
http://www.ncela.gwu.edu/pubs/symposia/third/reyhner.htm
Secada, W. G. (1998). School Mathematics for Language Enriched Pupils.
http://www.ncela.gwu.edu/pubs/florida/workforce98/secada/
Secada, W., Carey, D., and Schucher. R. (1989). Innovative Strategies for Teaching Mathematics
to Limited English Proficient Students. http://www.ncela.gwu.edu/pubs/classics/pig/10math.htm
Spanos, G. (1993). ESL Math and Science for High School Students: Two Case Studies.
http://www.ncela.gwu.edu/pubs/symposia/third/spanos.htm
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