I Understanding and Supporting Children’s Mathematical Vocabulary

Understanding and
Supporting Children’s
Mathematical Vocabulary
magine a teacher running her hands across her desk as she tells her students, “A plane is a perfectly
flat surface.” The students listen quietly, but one of them is thinking, “I thought a plane was something that flies.”
As this example illustrates, the language of
mathematics can sometimes be confusing. Many
mathematical words are unusual, some terms have
different meanings in everyday usage than in technical mathematics, and some are used in mathematics in more than one way. However, terms,
phrases, and symbols are essential in communicating mathematical ideas; and becoming fluent with
them is vital for children’s mathematical learning.
In everyday English, new vocabulary can be
assimilated into daily conversations; students’
opportunities to use the language of mathematics,
however, often are limited. Teachers must present
classroom activities in which children read, write,
draw, and explain mathematical ideas. Our goal is
for mathematical language to become fluid, natural, and, as Principles and Standards for School
Mathematics states, a “powerful tool . . . to foster
the learning of mathematics” (NCTM 2000,
p. 128).
As adults, we are comfortable with the varied
meanings and implied understandings in words and
phrases. In this article, we step back and consider
how these words come across to children. Our
work is built on the writings of Earp and Tanner
(1980), MacGregor (1990), Shuard and Rothery
(1984), Usiskin (1996), Thompson and Rubenstein
(2000), and Whitin and Whitin (2000).
To be more aware of, and sensitive to, issues
of mathematical language acquisition and to be
more creative and persistent in finding ways to
support children’s learning, teachers must first
understand children’s difficulties in making
sense of mathematical language. This article
begins by sharing difficulties children sometimes
face in learning mathematical language and continues with strategies teachers may use to help
prevent or overcome those difficulties.
Language Challenges
Table 1 outlines some common difficulties in
learning mathematical vocabulary and lists one or
more examples of each. As you study the table,
we invite you to place the words below in one or
more rows of the table and to think of additional
words for the grade level with which you are most
familiar. These terms were selected from a word
list of a commercial textbook series.
• Grade 1: square, foot, odd
• Grade 2: cube, cent, face, sum
• Grade 3: scale, factor/multiple, dividend, polygon, round
• Grade 4: pictograph, isosceles, slide
• Grade 5: radius/diameter, similar
Rheta N. Rubenstein and
Denisse R. Thompson
Rheta Rubenstein, [email protected], teaches at the University of Michigan—Dearborn. She is interested in making mathematics accessible to all learners. Denisse Thompson,
[email protected], teaches at the University of South Florida. Her interests
include curriculum development, the use of literature in teaching mathematics, and assessment issues.
Copyright © 2002 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Vocabulary difficulties and examples
Category of Difficulty
Some words are shared by mathematics
and everyday English, but they have
different meanings in the two contexts.
Right angle versus right answer
Right angle versus right hand
Reflection as flipping over a line versus reflection as thinking about
Foot as 12 inches versus the foot on a leg
Some mathematical words are shared
with English and have comparable
meanings, but the mathematical meaning
is more precise.
Difference as the answer to a subtraction problem versus difference as a
general comparison
Even as divisible by 2 versus even as smooth
Some mathematical terms are found
only in mathematical contexts.
Quotient, decimal, denominator, quadrilateral, parallelogram, isosceles
Some words have more than one
mathematical meaning.
Round as a circle versus to round a number to the tenths place
Square as a shape versus square as a number times itself
Second as a measure of time versus second as a location in a set of
ordered items
A side of a rectangle is a line segment versus a side of a prism is a
Some words shared with other
disciplines have different technical
meanings in the two disciplines.
Divide in mathematics means to separate into parts, but the Continental
Divide is a geographical term referring to a ridge that separates
eastward- and westward-flowing waters.
Variable in mathematics is a letter that represents possible numerical
values, but variable clouds in science are a weather condition.
Some mathematical terms are homonyms Sum versus some, arc versus ark, pi versus pie, graphed versus graft
with everyday English words.
Some mathematical words are related,
but students may confuse their distinct
Factor and multiple, hundreds and hundredths, numerator and
A single English word may translate into
Spanish or another language in two
different ways.
In Spanish, the table at which we eat is a mesa, but a mathematical table
is a tabla (Olivares 1996).
English spelling and usage have many
Four has a u, but forty does not.
Fraction denominators, such as sixth, fifth, fourth, and third, are like
ordinal numbers, but rather than second, the next fraction is half.
Some mathematical concepts are
verbalized in more than one way.
Skip count by threes versus tell the multiples of 3
One-quarter versus one-fourth
Students may adopt an informal term as
if it is a mathematical term.
Diamond for rhombus
Corner for vertex
Our placement of the words in the table can be
found at the end of the article.
Teachers might use a variety of approaches to
enhance their support of children’s mathematical
language learning and their own awareness of
these difficulties with language. A major premise
of all strategies is to connect new terms or phrases
to ideas children already know. Children should
first do activities to build concepts, then express
their understanding informally, and finally, when
ideas solidify, learn the formal language. When
formal words are introduced, they should be care108
fully spoken, written, spelled, illustrated, and used
to ensure that students know exactly what is said
and meant. When students use terminology incorrectly, teachers might restate the sentence appropriately so that children hear the correct usage.
Following are a variety of strategies that support
children’s language acquisition. These strategies are
meant to serve as examples, not an exhaustive list.
We encourage readers to identify and share other
strategies they have found to be successful by submitting articles like this one or writing letters to this
journal’s “Reader’s Exchange.”
Many language-learning strategies can be built
directly on language arts ideas. For example, reading teachers often keep a “word wall” of new
terms. New words are posted on a bulletin board,
along with definitions, pictures, and uses, most
provided by the students themselves. Similarly, a
“mathematics word wall,” as shown in figure 1,
has words, related pictures, definitions, and other
images to help make each new word meaningful.
When mathematics definitions need to be differentiated from everyday or science meanings or
when commonly confused word pairs (see table 1,
rows 1, 2, 5, 6, 7) need to be clarified, students can
be invited to discuss the differences, then write
sentences or draw pictures contrasting the two
meanings. For example, a student might write,
“The difference between my two pens is that one is
blue and one is red. The difference between 12 and
5 is 7 because 12 – 5 = 7.”
A more general strategy is to have students
write journal entries, stories, cartoons, bumper
stickers, skits, raps, songs, or poetry about the topics and terminology they are learning in mathematics. For example, consider two rap verses for the
distributive property:
3 times the quantity
2 plus 5
is 3 times 2
plus 3 times 5.
It doesn’t matter
what numbers you use,
just follow the pattern
with whatever you choose.
Another language arts strategy is to invite students to invent their own terminology. As one
example, David Whitin (1995) reported an activity
in which students explored the rectangles that
could be created with a given number of square
tiles. The students noticed that some numbers of
tiles made only long, narrow rectangles while others made both dense and narrow rectangles. They
decided to call the groups “sidewalk” and “patio”
numbers. Later, the teacher introduced the corresponding formal vocabulary, prime and composite
numbers. Although opportunities to have students
invent language may seem scarce, these occasions
can be planned. Simply withhold the formal terminology. Let students use materials to explore ideas,
suggest their own terms, and explain their rationales (see Whitin and Whitin [1997] for more
examples). Of course, formal terms must be introduced eventually, and students must be able to
Using Language Arts
Section of a “mathematics word wall”
translate between informal and standard words, but
by inventing, they realize that terms come from
people thinking about new ideas. This realization
aligns with an important goal we have for students:
to be thinkers and creators in the world of the
Writing about mathematics is another opportunity to understand mathematical terminology. Not
only do students use mathematical vocabulary in
writing, but also their output gives teachers the
opportunity to assess students’ understanding of the
terms they use. For example, one of the authors was
planning geometry lessons for use in a fifth-grade
classroom. After giving a textbook-generated test at
the end of a recent geometry unit, the classroom
teacher thought that students had learned very little.
We began the planned lessons by having students
complete the open-ended task in figure 2. As the
work indicates, the student knew some of the geometry ideas studied and needed help with others.
Both the teacher and the author were reminded that
such tasks allow students to use mathematical
vocabulary to share details about their understanding. Open-ended writing prompts also give teachers
information about students’ misconceptions that
can inform future instruction. The student’s mistaken notion that parallelograms could not have
right angles opened a discussion about how some
shapes, such as rectangles, can be special cases in a
broader category, such as parallelograms.
Student’s work on an open-ended geometry task
Student’s work describing likes and
dislikes of polygons
Using Literature
Trade books offer a wealth of resources for mathematics learning in general and language development in particular. For example, a good book for
helping children distinguish between sets and elements, as well as singulars and plurals, is Only
One (Harshman 1993). The book uses contrasting
phrases, such as “There may be 3 musicians, But
there is only one trio” or “There may be 50,000
bees, But there is only one hive.” When youngsters are first exploring odd and even numbers,
they can read Even Steven and Odd Todd
(Christaldi 1996), in which Steven likes things
only in even amounts and Todd likes things only
in odd amounts. The many examples provided in
the book help youngsters figure out the distinction for themselves. When children learn about
customary units of measure, knowing the origins
of the words foot or ruler enriches their meanings.
How Big Is a Foot? (Myllar 1991) is a good
source for these historical connections. Geometry
terms can be explored in Pluckrose’s (1995) colorful Math Counts: Shapes for lower elementary
school students or Crosbie’s (1993) Architecture
Shapes for older students, which includes wonderful connections to construction. Marilyn
Burns’s The Greedy Triangle (1994) helps students learn geometrical terms and shapes. Circle
vocabulary is playfully explored in Sir Cumference and the First Round Table: A Math Adventure (Neuschwander 1997) and Sir Cumference
and the Dragon of Pi (Neuschwander 1999).
After reading The Greedy Triangle to the same
fifth-grade class mentioned above, we asked students to write about a shape, as shown in the work
in figure 3. This kind of activity allows students
to write freely about how they perceive shapes
and contrasts with more traditional activities of
simply labeling figures. Again, as in the earlier
discussion, the writing suggests questions that the
teacher might want to ask the student. For example, does the student understand that a rhombus
(“rhombum”) must have four congruent sides?
What does the student mean when he says that a
triangle looks “plain”?
Using Word Origins
The origins of words may also help older students
make connections. When we share with children the
“words behind the words,” they connect terms that
sound “foreign” to words they already know. For
example, parallel comes from “alongside” (para-),
as in a paraprofessional who works “alongside professionals.” Percent means literally “for each hundred,” from which we derive “divided by 100” or
Word origins also can alleviate some mix-ups
with word pairs that are commonly confused. For
example, numerator and denominator are difficult
for many children to distinguish. Using the roots
helps. The root nom means “to name.” The denominator names the fraction, for example, fifths, as a
nominator names a person to run for office. A numerator tells the number of parts of interest. Consequently, three-fifths means that something is divided
into five equal parts and we are referring to three of
those parts.
Another confused pair is radius and diameter.
The radius, like a ray of sunlight or a beam from a
radio tower, radiates out from a point. In fact, the letters in italics all stem from one root, ray. The diameter is a measure (meter) of the line segment through
the center of a circle. Dia- means “across or through”
and appears as well in diagonal, a line segment that
goes through an angle (gon).
Factors and multiples both relate to multiplication
and are difficult for students to distinguish between.
Mnemonic devices can be created from the etymologies: “Just as factories make products, so do factors
make products” and “Every number has a multitude
of multiples.”
Opportunities for integrating mathematical language learning with other subjects abound in selfcontained elementary school classrooms, in which
all subjects are taught by one teacher, or in schools
where teachers of special subjects are in close communication. For example, when students are studying pendulums in science and observe the movement
of a weight on a string, they can see that when the
pendant on the pendulum comes to rest, it is perpendicular to the ground. The italicized letters come
from a root meaning “to hang.” In geography, students can recognize that the equator is equally distant from the two poles. As teachers become familiar
with these simple etymological connections, they can
help students identify them. For more ideas about
using word origins, see Rubenstein (2000). For a dictionary of word origins, see Schwartzman (1994).
Using Games
Card games can be a fun way to help students recognize equivalent verbal, symbolic, story, or picture
representations of the same values. Figure 4 shows
playing cards with eight sets of equivalent values.
Adapting similar sets of cards for such games as concentration, old maid, rummy, or go fish is easy. Figure 4 includes a variety of mathematical strands,
such as number, geometry, money, time, and measurement, but any one of these strands could be
expanded for its own special deck of cards. For a
richer learning experience, students can be invited to
create their own sets of equivalent cards for new
terms and symbols they have learned.
A set of cards using terminology, symbols, and
pictures can be presented for the “I have . . . Who has
. . . ?” game format. For example, “I have an isosceles triangle. Who has a figure with all sides congruent
and all right angles?” “I have a square. Who has an
equilateral triangle?” The cards may have pictures,
terms, definitions, symbols, or combinations of these.
Another game is “step forward and take a
bow.” In this whole-class activity, several students stand in front of the room showing large
cards that are labeled, for example, with numbers. Then a clue is given, such as “If you have a
prime number greater than 30, step forward and
take a bow” or “If you have a factor of 24, step
forward and take a bow.” This game offers direct
assessment of students’ understanding. Those
who are not standing can participate by showing
whether or not they agree with the students who
have stepped forward and by discussing their
ideas. In the lower grades, using number cards,
students can step forward if they have “the sum
of 5 + 6,” “an even number,” or “twice an odd
number.” Similarly, the game can be played with
geometric shapes (Florida Department of Education 1994).
Because one of the few places students have to “talk
mathematics” is in our classrooms, we as teachers
must give attention to mathematical language learning. In this article, we have tried to identify some of
the difficulties in learning mathematical language
and some strategies to deal with these difficulties.
We hope we have increased teachers’ sensitivity to
issues in language learning and inspired ideas for
creating strategies to help children become fluent in
mathematical communication.
Burns, Marilyn. The Greedy Triangle. New York: Scholastic,
Christaldi, Kathryn. Even Steven and Odd Todd. New York:
Scholastic, 1996.
Crosbie, Michael J. Architecture Shapes. New York: John Wiley &
Sons, 1993.
Sample sets for card games
Each row represents a set of equivalent values.
a fair share when 3
pies are shared equally among 4 people
3 sets of 4
the number of outfits
that can be made from
4 tops and 3 pairs of
a quarter after 6
15 minutes past 6
the area of a rectangle
that measures 3 units
by 4 units
40/100 of a dollar
$0.50 – $0.10
10 minus 4
four less than ten
4 from 10
5 plus 1
20 centimeters
two-tenths of a meter
two decimeters
1/5 of a meter
line segment in a
polygon joining two
vertices that are not
next to one another
line segment going
through two angles of
a polygon
line segment joining
two points on a circle
and passing through
the center of the circle
length equal to two
radii of a circle
Earp, N. Wesley, and Fred W. Tanner. “Mathematics and Language.” Arithmetic Teacher 28 (December 1980): 32–34.
Florida Department of Education. Opening the Gate. Tallahassee,
Fla.: Florida Department of Education, 1994.
Harshman, Marc. Only One. New York: Cobblehill Books, 1993.
MacGregor, Mollie. “Reading and Writing in Mathematics.” In
Language in Mathematics, edited by Jennie Bickmore-Brand,
pp. 100–108. Portsmouth, N.H.: Heinemann, 1990.
Myllar, Rolf. How Big Is a Foot? New York: Young Yearling,
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.:
NCTM, 2000.
Neuschwander, Cindy. Sir Cumference and the Dragon of Pi: A
Math Adventure. Watertown, Mass.: Charlesbridge, 1999.
———. Sir Cumference and the First Round Table: A Math
Adventure. Watertown, Mass.: Charlesbridge, 1997.
Olivares, Rafael A. “Communication in Mathematics for Students
with Limited English Proficiency.” In Communication in
Mathematics, K–12 and Beyond, 1996 Yearbook of the
National Council of Teachers of Mathematics (NCTM), edited
by Portia C. Elliott and Margaret J. Kenney, pp. 219–30.
Reston, Va.: NCTM, 1996.
Pluckrose, Henry. Math Counts: Shapes. Chicago: Children’s
Press, 1995.
Rubenstein, Rheta N. “Word Origins: Building Communication
Connections.” Mathematics Teaching in the Middle School 5
(April 2000): 493–98.
Schwartzman, Steven. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Wash-
ington, D.C.: Mathematical Association of America, 1994.
Shuard, Hilary, and Andrew Rothery. Children Reading Mathematics. London: John Murray Publishers, 1984.
Thompson, Denisse R., and Rheta N. Rubenstein. “Learning
Mathematics Vocabulary: Potential Pitfalls and Instructional
Strategies.” Mathematics Teacher 93 (October 2000):
Usiskin, Zalman. “Mathematics as a Language.” In Communication in Mathematics, K–12 and Beyond, 1996 Yearbook of the
National Council of Teachers of Mathematics (NCTM), edited
by Portia C. Elliott and Margaret J. Kenney, pp. 231–43.
Reston, Va.: NCTM, 1996.
Whitin, David. Talk presented at Umbrella Language Conference,
Windsor, Ont., 1995.
Whitin, Phyllis, and David Whitin. “Ice Numbers and Beyond:
Language Lessons for the Mathematics Classroom.” Language Arts (February 1997): 108–15.
———. Math Is Language Too: Talking and Writing in the Mathematics Classroom. Urbana, Ill.: National Council of Teachers of English; Reston, Va.: NCTM, 2000.
Authors’ answers to word placement task for table
1: (Answers may vary.) Grade 1: square (rows 1, 2,
and 4); foot (row 1); odd (row 1). Grade 2: cube (1,
2, 4); cent (6); face (1); sum (6). Grade 3: scale (1,
5); factor/multiple (7); dividend (1, 5); polygon
(3); round (4, 8). Grade 4: pictograph (3); isosceles
(3); slide (1). Grade 5: radius/diameter (3, 7), similar (2). ▲