artículo clement (sept 2004, p 97)

8/12/2019 Artículo Clement (sept 2004, p 97)

http://slidepdf.com/reader/full/articulo-clement-sept-2004-p-97 1/6

Teaching Children Mathematics / September 2004 97

Consider the following scenario:

Teacher. Can you solve this problem? (Gives

student paper with 4 – 1/8 written at the top)

Student. (Writes 3/8 as answer ) Three-eighths. I

subtracted one from four, and then kept the denom-

inator, eight, the same.

Teacher. Suppose you had four large brownies

and you ate one-eighth of one brownie. How many

brownies would you have left?

Student. (Pauses, then draws four rectangles,

cuts one of the rectangles into eight pieces, and

shades one of the pieces) Three and seven-eighths.

(Writes 3 7/8)

Now consider another scenario:

Teacher. I will tell you a number, and you write

it. Please write for me the number one-half.

Student. (Writes 1 1/2)

Teacher. Please write for me the number one

and one-half.

Student. (Writes 1 1/2) It is the same thing.

Teacher. (Writes 1/2 and points to it ) What

would you call this?Student. Half.

I have observed both of these scenarios, and I

suspect that most teachers have seen something

similar. The first scenario captures the fact that

real-life, or relevant, situations may be more com-

prehensible to students than are written symbols.

The second scenario exemplifies the fact that the

connections children make between language and

written symbols may differ from the connections

adults make. This article describes five representa-

tions: relevant situations, pictures, manipulatives,

spoken language, and written symbols. It also usesthese two scenarios to explain how understanding

the connections among representations is useful for

making sense of students’ responses to tasks, and

discusses an instructional sequence to help stu-

dents make connections among representations.

The Representations ModelLesh, Post, and Behr (1987) identify five ways to

represent a mathematical idea and emphasize

their interconnectedness. Principles and Stan-

dards for School Mathematics (NCTM 2000)

states, “Representations should be treated as

essential elements in supporting students’ under-

standing of mathematical concepts and relation-

ships; in communicating mathematical approaches,

arguments, and understandings to one’s self and

to others; in recognizing connections among

related mathematical concepts; and in applying

mathematics to realistic problem situations

through modeling” (p. 67). The five representa-

tions presented in this article are a subset of those

that a teacher or students might draw on when

examining mathematical ideas.

PicturesAlthough the representation “Pictures” can refer to

the pictures of mathematical ideas that teachers

Lisa Clement, [email protected], is an assistant professor at San Diego

State University. She is interested in classroom discourse and using children’s

mathematical thinking to support teachers’ practice.

By Lisa L. Clement

A Model for Understanding,Using, and Connecting

Representations

Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All r ights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.



draw or that are in textbooks (such as rectangles

that are cut into four equal pieces with one piece

shaded), I tend to think of pictures as those that

children draw themselves. When teachers ask chil-

dren to draw pictures, they can learn more about

what a child understands about a particular math-

ematical idea, and they can use the different pic-

tures that children create to provoke a discussion

about mathematical ideas. For example, when

teachers ask children to draw one-fourth of a

brownie, they can determine which childrenunderstand that every one-fourth piece should

have the same area as the others, or that they

should use the entire brownie when cutting it into

fourths. If some children use equal parts and oth-

ers do not, the teacher can lead a discussion about

these differences. Constructing their own pictures

can be a powerful learning experience for students

because they must consider several aspects of

mathematical ideas (such as equal parts) that are

often assumed when pictures are pre-drawn for

students.

Manipulatives

Children might use Unifix cubes, base-ten blocks,

or fraction circles to solve a problem. Manipula-

tives are objects that students can touch, move, and,

often, stack. Because children can physically

manipulate these objects, when used appropriately

they give children opportunities to compare relative

sizes of objects that represent mathematical ideas

such as fractions or place value. They also allow

children to identify patterns, put together represen-

tations of numbers in multiple ways, and so on.

Spoken languageTwo uses of “spoken language” exist. One is a way

for students to report their answers, and another is a

way for students to express their reasoning. Tradi-

tionally, teachers often use the spoken language of

mathematics but rarely give students the opportunity

to grapple with it. When students have opportunities

to express their mathematical reasoning aloud, how-

ever, they may be able to make more explicit the

knowledge that had been implicit for them.

98 Teaching Children Mathematics / September 2004

P h o t o g r a p h

b y H e a t h e r A n d e r s o n ; a l l r i g h t s r e s e r v e d



Written symbols“Written symbols” refers to both mathematical

symbols and the written words that are associated

with them; for example, 1/4, 2/8, 0.25, 25%, and

one-fourth, two-eighths, twenty-five hundredths,and twenty-five percent. Written symbols tend to be

more abstract for students than are the other repre-

sentations. I usually introduce symbols after stu-

dents have had opportunities to make connections

among the other representations so that they have

different ways to connect the symbols to mathe-

matical ideas, thereby increasing the likelihood

that the symbols will be comprehensible to them.

Relevant situationsThe fraction 1/4 can be represented in a variety of

relevant situations; for example, “Juan ate 1/4 of the pepperoni pizza” or “Alicia and her three best

friends shared a large chocolate bar fairly. How

much of the candy bar did each person get?” A rel-

evant situation can be any context that involves

appropriate mathematical ideas and holds interest

for children. It is often, but not necessarily, con-

nected to a real-life situation. For example, one

teacher created a story about leprechauns that,

although not a real-life situation, held great interest

for her students; therefore, this context is consid-

ered a relevant situation.

Drawing on Connectionsbetween Representationsto Understand Children’sThinking

When teachers focus their instruction on the repre-

sentations that may be the most familiar to chil-

dren—relevant situations, manipulatives, pictures,

and spoken language—they may provide children

with greater access to and enjoyment of mathemat-

ics. Teachers can help students make connections

from these representations to written symbols, butonly after students have begun to connect the for-

mer representations to one another. For most chil-

dren, the meaning of the symbols is more abstract,

and therefore more difficult to comprehend, than

are the other representations.

For example, when teachers verbally prompt

students to write the mathematical symbol for one-

half, children will often incorrectly write 1 1/2, as

in the second scenario at the beginning of this arti-

cle. Attending to connections between relevant sit-

uations, spoken language, and written symbols

helped me consider one possible explanation of

this occurrence: Children use half, not one-half, in

their everyday spoken language for sharing things

equally. For example, one child may ask another,

“Can I have half your brownie?” Therefore, thatmany students write 1 1/2 when the teacher asks

them to write one-half is not surprising. They write

the symbol “1” to represent the one in one-half,

then write the symbol “1/2” to represent half, a

familiar term to them.

Knowledge of the representations is also helpful

for reflecting on which of them would be most

meaningful to students when mathematical ideas

are first introduced. For example, when asked to

solve the problem 4 – 1/8, many students will

answer incorrectly with responses such as 3/8 (sub-

tracting 1 from 4 and leaving the denominator thesame, as in the first scenario)

or 3/4 (subtracting 1 from 4,

then subtracting 4 from 8).

The students who make these

errors tend to think of the

numerator and denominator as

separate numbers to be oper-

ated on, instead of treating the

fraction—in this case, 1/8—as

a single number. In my experi-

ence, however, the same stu-

dents are much more success-

ful when asked to solve a

problem that corresponds to 4

– 1/8 but is set in a relevant

context. One example of this

is “Maria has four big brown-

ies. She eats one-eighth of one

brownie and saves the rest for later. How much does

she have now?” Setting a problem in a relevant con-

text helps students develop an image that they can

use in deciding how to solve the problem. In this

example, even students who are still unsure of the

final answer almost always know that three brown-

ies and some part of another brownie remain. Theproblem also invites students to draw a picture (or

use manipulatives, if available) to represent their

thinking. The symbols alone rarely suggest to stu-

dents that they should draw a picture because, at

least initially, the symbols tend to have little mean-

ing for most students.

I use a picture (see fig. 1) to remember not only

the representations but also the importance of the

connections between them. The picture provides a

quick visual image of the representations model

that guides my teaching.


Children usehalf , not

one-half , in

their everydayspoken language



The Model in Action:A Decimal LessonI recently had the opportunity to tutor two fifth-

grade students to help them compare decimals. I

knew from my previous work with students that

comparing decimals is often challenging. Many

students compare decimals by ignoring the decimal

point, and treat the numbers as whole numbers. For

example, students who use the whole-number

approach would compare 0.73 and 0.8 and say

incorrectly that 0.73 is greater, because 73 is

greater than 8. Other students could get correctanswers by adding zeroes to the decimal number

with fewer places until both decimals had the same

number of places to the right of the decimal point.

For example, a student who uses this approach

would add a 0 to the right of 0.8 to get 0.80. Then,

when comparing the decimals, the student could

say that 0.73 is less than 0.80. Although this

approach helps students get correct answers, stu-

dents still might not understand that decimals less

than one are parts of wholes. They might still think

of decimals as whole numbers, only now that they

have made both decimals the same number of

places, they can get correct answers.

In a preassessment, the students I was tutoring

compared pairs of decimals in written, symbolic

form (for example, 0.73 versus 0.8); neither stu-dent could successfully compare them. I wanted to

begin instruction by using a real-world context

(relevant situation) that I thought might be famil-

iar to them, instead of beginning with the written

symbols. I also wanted them to use spoken lan-

guage and base-ten blocks—flats, longs, and sin-

gles—to support the context so that they would

have multiple access points for thinking about the

relative size of decimals. Therefore, I began the

lesson by describing the community in which I

live; it has many apartment buildings and very lit-

tle land for things such as gardens. I then intro-duced the flat (a manipulative) as a plot of com-

munity land that is cut into small pieces so that

neighbors have a place to garden, if they so

choose. I asked the students to identify the number

of pieces into which the garden was cut (100). I

also asked them to identify each small piece (one-

hundredth of the garden) and row (one-tenth of the

garden) and why it was so named. I orally asked

the students to show me, with the manipulative

pieces, one-tenth, three-tenths, and thirty-five hun-

dredths of the garden. Later, I selected manipula-

tive pieces—two small pieces, five longs, and two

longs with seven small pieces—and asked the stu-

dents to orally tell me what fractional part of the

garden I had selected. I began with spoken lan-

guage and had them respond with manipulatives;

then I presented manipulative pieces and had them

respond with spoken language. At this point, my

instruction was focused on helping the students

think about decimal fractions by using a relevant

situation, manipulatives, spoken language, and the

connections among all three.

I next attempted to support the students’ think-

ing about multiple ways to name the same section

of garden. I was not only thinking about connec-tions between a situation, manipulative, and spo-

ken language, but also wanting the students to

understand that connections can be made within

the same representation; in this case, multiple ways

exist to name the same part of the land, such as

three-tenths and thirty-hundredths. Of course, this

naming requires more than the students’ ability to

vocalize the correct words. The students must be

able to view and think about the same parcel of

land using different-sized pieces, in this case,

tenths and hundredths.


Figure 1

The five representations



After working on similar problems, the students

could build a fraction of the garden when the frac-

tion was provided verbally, name the parts of the

garden in two ways, and explain why both ways

were appropriate. I asked the students to write thesymbols for the fractions with which they were

working. At this point in the lesson, they had built

with manipulatives seventy-three hundredths of

one garden and eight-tenths of another garden. One

of the students already knew how to write fractions

in decimal form, so he was able to represent the

same fraction in written symbols in multiple ways,

such as 73/100, 7/10, 3/100, 0.73, 0.7, and 0.03. He

would name the fraction 73/100 seventy-three hun-

dredths, but the same fraction, when written 0.73,

as point seventy-three, instead of also seventy-three

hundredths. Many students use this language. Theymay do so because fractions written in decimal

form do not indicate the existence of a denomina-

tor, for example, with a symbol such as “/” or “÷”;

in the example of 0.73, the written symbol does not

include the denominator 100. The denominator

helps students recognize that decimals are tenths,

hundredths, thousandths, and so on. I suggested

that my student use the same language for 0.73 that

he used for 73/100, because the language might

help him remember that decimals are fractions

rather than whole numbers.

My lesson design started with a context that was

relevant to the student and that was further sup-

ported by the use of manipulatives and spoken lan-

guage by both the teacher and the student. The

introduction of written symbols late in the lesson

was purposeful and based on the preassessment,

which indicated that the students had not yet made

appropriate meaning of the symbols. I decided to

delay the introduction of the symbols until the stu-

dents had made appropriate connections within and

among the representations of relevant situations,

manipulatives, and spoken language. In a post-

assessment, these students were able to appropri-

ately compare several pairs of decimals (all lessthan one) in symbolic form and explain which pair

was greater with an explanation that went beyond

adding a zero.

SummaryThe representations model provides a lens for mak-

ing sense of students’responses to tasks and a guide

for planning lessons. Teachers should first consider

whether all or most of the representations should be

a component of a set of lessons. The model is also

helpful for selecting an order in which to introduce

the representations. Teachers should begin with

those that are most meaningful for students. When

analyzing their students’ work, teachers may want

to consider the following questions:

• What representations did I use when I posed the

task?

• What representations did students use to under-

take the task?

• Were any translations among representations

particularly challenging or helpful for students?

• How did the representations and connections

among them support the students’ mathematical

reasoning and flexibility in their thinking?

When considering an important mathematical ideathat they plan to teach, teachers

should consider the following

questions:

• Which representation or

representations will be most

meaningful for my stu-

dents?

• In what order does it make

the most sense to introduce

(or invite the use of) the dif-

ferent representations?

• Which representations or

translations will promote

more powerful mathemati-

cal thinking for my students?

Looking back on my first few years of teaching, I

now realize that I spent much class time talking,

instead of allowing my students to discuss mathe-

matical ideas, and using written mathematical

symbols without relevant situations, manipulatives,

or pictures to support students’ understanding of

the symbols or language. The representations

model helped me recognize gaps in the way I rep-resented mathematical ideas to students and pro-

vided structure for planning lessons. I now reflect

on the most appropriate order for introducing rep-

resentations that will best support my students’

understanding. Teachers with whom I have shared

the model have had similar experiences. Increasing

the number of connections among representations

that students can use to grapple with mathematical

concepts may increase the number of students who

report good experiences with mathematics and

increase their mathematical understanding.


Therepresentations

model provides alens for making

sense of students’responses to tasks



ReferencesLesh, Richard, Tom Post, and Merlyn Behr. “Rep-

resentations and Translations among Represen-tations in Mathematics Learning and ProblemSolving.” In Problems of Representation in theTeaching and Learning of Mathematics, editedby Claude Janvier, pp. 33−40. Hillsdale, N.J.:Lawrence Erlbaum Associates, 1987.

National Council of Teachers of Mathematics(NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

The author would like to thank Bonnie Schap-

pelle, Vicki Jacobs, and Randy Philipp for

their comments on earlier drafts of this arti-

cle. Preparation of this article was supported by a grant from the National Science Founda-

tion (NSF) (REC-9979902). The views

expressed are those of the author and do not

necessarily reflect the views of NSF. ▲


Editor’s Note

Looking for additional resources? NCTM offers a variety of resources related to

representations, such as the following:

Lubinski, Cheryl, and Albert D. Otto. “Meaningful Mathematical Representa-

tions and Early Algebraic Reasoning.” Teaching Children Mathematics 9

(October 2002): 76–80.

National Council of Teachers of Mathematics (NCTM). The Roles of Represen-

tation in School Mathematics. 2001 Yearbook of the National Council of

Teachers of Mathematics. Reston, Va.: NCTM, 2001.

Perry, Jill A., and Sandra L. Atkins. “It’s Not Just Notation: Valuing Young Chil-

dren’s Representations.” Teaching Children Mathematics 9 (December

2002): 196–201.

Visit www.nctm.org for more information on all of NCTM’s resources, includ-

ing professional development offerings and publications available in the online

catalog. —Ed.

artículo clement (sept 2004, p 97)

Documents