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518 Teaching Children Mathematics / May 2008

Darrell Earnest, dearnest@berkeley.edu, is a graduate student in cognition and development

at the University of California—Berkeley in Berkeley, CA 94720, and formerly a researcher and

curriculum developer at TERC in Cambridge, Massachusetts. He is interested in the develop-

ment and support of mathematical reasoning in elementary and middle school. Aadina A.

Balti, aadinab@gmail.com, is a fourth-grade teacher in the Boston Public Schools. She holds

master’s degrees in child development from Tufts University and in special education from the

University of Massachusettes—Boston, and is particularly interested in supporting students in

special education.

By Darrell Earnest and Aadina A. Balti

Incorporating algebra into the elementary grades

has become a focus for teachers, principals, and

administrators around the country. Algebra is

commonly regarded as a gateway to future opportu-

nity (e.g., Moses and Cobb 2001), and elementary

mathematics standards at both the state and national

levels now reﬂect this effort to provide students

with opportunities to learn critical concepts before

middle and high school (California Department of

Education 1997; NCTM 2000). However, imple-

menting algebra standards at the elementary level is

challenging—how do mathematics educators effec-

tively and meaningfully incorporate algebraic ideas

into K–5 curriculum? When elementary teachers

are unfamiliar with early algebra, lessons designed

and labeled as algebraic may become arithmetic

exercises; the algebra then remains hidden from

both the teacher and students in the implementa-

tion. The result is that the algebra standard is only

superﬁcially addressed.

From 2003 to 2006, we were members of a

collaborative team of teachers and researchers on

a project to develop an algebraic lesson sequence

in grades 3–5. The goal of the larger project was

to support students’ algebraic reasoning through

functions and multiple representations of those

functions, including verbal descriptions, tables,

graphs, and letter notation (Schliemann, Carraher,

and Brizuela 2006; Carraher et al. 2006). In the

context of this project, we regularly discussed how

our lessons were algebraic. These discussions were

informed by the Algebra Standard (NCTM 2000),

which emphasizes “relationships among quantities,

including functions, ways of representing math-

ematical relationships, and the analysis of change”

Findings from a Research-Practice Collaboration

Instructional Strategies for Teaching

Algebra in Elementary School:

George Clerk/iStockphoto.com

Copyright © 2008 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.

Teaching Children Mathematics / May 2008 519

(p. 37). The vision is that elementary school pro-

vides the conceptual foundation for formal alge-

bra in middle and high school. Yet what effective

instructional strategies make algebra accessible and

meaningful to elementary students? How can we

foster algebraic reasoning in the classroom?

This article examines our experience with teach-

ing the Dinner Tables problem in third grade, a lesson

commonly used in upper elementary grades for its

algebraic potential. The goal was to engage students

in recognizing, extending, and predicting the use of

patterns. We wanted to ensure that the lesson did not

turn into an arithmetic exercise. Using the context of

this lesson, we describe how we began to recognize

the recurring role of particular instructional strate-

gies throughout the entire lesson sequence. When

these strategies were used, the students began to

extend and predict using mathematical patterns,

instead of completing pattern and function activities

by rote without considering the algebra involved. As

described, these strategies were the use of (1) unex-

ecuted number expressions, (2) large numbers, and

(3) the representational context.

Three Instructional Strategies

for Early Algebra

The Dinner Tables problem was presented as fol-

lows: Students’ tasks included ﬁnding the maximum

number of people that can be seated at as many as

seven tables in both a function table and in a draw-

ing. Students were then asked to construct a rule or

formula that uses the number of tables to determine

the number of people. The purpose of this question

was to have students consider the input-output rela-

tionship of the function. For example, a total of four

people can sit at one table; two tables will accom-

modate six people; three tables can accommodate

eight people; and so on (see ﬁg. 1).

This lesson was designed to be algebraic. How-

ever, implementation can impact how algebraic

a lesson really is. For example, the lesson could

have easily turned into a problem of “adding on by

twos” when students completed a function table,

potentially rendering the lesson an arithmetic task

instead of an algebraic one. Three strategies helped

keep the algebra in the foreground of the lesson.

Using unexecuted

number expressions

An unexecuted number expression refers to a non-

computed sequence that allows for the consider-

ation of the numbers and operations within it. For

example, in the case of three tables, a total of 3 + 3 +

2, or 3 × 2 + 2, people can be seated. Whereas these

expressions are equal to one value, 8, the numbers

within the expressions became a springboard for

discussion in our classroom and a tool to extend and

predict. Students said that three tables would accom-

modate a total of eight people, which was repeated

back to students using an unexecuted expression:

Teacher: Yeah. We can just count them. There

are three on top [the head of the table], three on

the bottom [the foot of the table]. And how many

are on the sides?

Richard: Two.

Teacher: So three and three and two is going to

give us?

Students: Eight.

Students found eight as the total value, but “three and

three and two” drew attention to the numbers within

the expression and how these numbers map onto the

drawing in ﬁgure 2. After asking a similar question

for four tables, the teacher moved to ten tables.

Instead of needing to continue from ten people and

add on by twos, students began to generalize the

idea by making use of the patterns, moving from

3 + 3 + 2 and 4 + 4 + 2 to 10 + 10 + 2, building off

the original problem context that a certain number

An activity sheet helped students explore the relationship between a

function table and a drawing that depicted the maximum number of

seats at the restaurant’s tables.

Dinner

Tables Show How Number of

People

1 4

2

3

4

Figure 1

520 Teaching Children Mathematics / May 2008

of people sit at the head and foot of the table with

two more on the sides.

Teacher: There are ten tables. How many people

will there be? Ashley, do you have a guess?

Ashley: Twenty-two.

Teacher: You think twenty-two.

Richard: Oh, that’s what I guessed!

Teacher: Can you tell me where that number

came from, Ashley?

Ashley: There will be ten on the top, then ten on

the bottom, then two on the sides.

As Ashley says that “there will be ten on the top,

then ten on the bottom, then two on the sides,” the

teacher writes this on the whiteboard both as a ﬁnal

value—22—and also as an unexecuted expres-

sion—10 + 10 + 2 (see ﬁg. 3), setting the stage for

students to extend the pattern further to predict for

100 tables and then for any number of tables.

Making the unexecuted strand of numbers

explicit provides a way to record information from a

problem and encourages students to discern emerg-

ing patterns. (Kaput and Blanton [2005] provide

an example in teacher professional development.)

Evaluating the parts of the unexecuted expression is

a powerful, strategic tool for instruction in early alge-

bra. In the case of 3 + 3 + 2, what do these numbers

represent? In this context, what justiﬁes a particular

operation? When considering 3 + 3 + 2 and then 4 +

4 + 2, what patterns begin to emerge? In the class-

room, an expression like 3 + 3 + 2 not only retains a

connection to the original problem context but also

allows students to extend patterns to 4 + 4 + 2 or

5 + 5 + 2 to eventually predict the output for any

number of tables—an important goal of the abstrac-

tion involved in early algebra. Considering only the

ﬁnal value hides important pieces of mathematics.

Using large numbers

Extending the Dinner Tables problem to include large

numbers of tables, such as 100 or 1,000, provides a

compelling reason for students to consider the rela-

tionship between the input (number of tables) and

the output (number of people) and make it abstract—

an essential piece of the algebra. One common way

to organize this information is in the form of a func-

tion table. However, a common complaint about this

particular representation is that students often move

down columns and rarely move across rows, thereby

seldom considering how a given input relates to the

corresponding output. Students frequently ﬁnd the

difference from output to output as a way to ﬁll in

missing values in the table, a task that can be more

arithmetic than algebraic, as the function table in

ﬁgure 4 shows.

Schliemann, Carraher, and Brizuela (2001)

provide a way to promote algebraic reasoning with

function tables through the use of large numbers.

In a function table, they make use of a break in the

Analyzing the drawings led to generaliz-

ing the numeric expression.

Figure 2

4 people

6 people

8 people

10 people

A break in the number sequence on the func-

tion table allowed students to predict values,

which demonstrated algebraic thinking.

Figure 3

Photograph by Janice Gordon; all rights reserved

Teaching Children Mathematics / May 2008 521

sequence of inputs, as illustrated in ﬁgure 3 when

moving to ten tables. It was possible for students

to draw ten dinner tables and count the seats or to

keep adding on by twos until reaching the output

for ten tables. Drawing 100 dinner tables shows this

strategy to be inefﬁcient and time consuming, a fact

students quickly grasped. After introducing a large

number to have students move from input to output,

the teacher had Shayne explain why he and other

students said there would be a total of 202 people.

Teacher: Can someone explain? Shayne, do you

want to quickly explain how you came up with

that number?

Shayne: Because 100 times 2 equals 200, and then

200 plus 2, and [pause] 200 plus 2 equals 202.

Shayne successfully provided a number expression

to solve for 100 that did not rely on the response for

99 tables, as some responses for 10 tables relied on

the result for 9 tables, and 8 before that, and so on.

Rather, he made a generalization to consider how to

move from the input to the output.

Teacher: So, he’s saying he multiplied 100 times

2, which is 200, and then he added 2. Why did he

do 100 times 2? Who can explain that? Ashley?

Ashley: Because there are 100 on the top and 100

on the bottom.

By incorporating large numbers into the lessons,

we found that students sought out new patterns

and generalizations in order to answer questions

in an algebraic way. Large numbers dissuaded the

use of the arithmetic adding on strategy and at the

same time promoted algebraic reasoning about the

input-output relationship. We frequently used the

strategy of large numbers to consider the functional

relationship of input to output.

Using representational context

The term representational context refers to the

interactions and discourse constructed in a class-

room around a particular representation (Ball 1993,

p. 157). Representational contexts serve as a way

to ground students’ developing mathematical ideas

and reasoning. The use of a representational context

is prevalent throughout Principles and Standards

(NCTM 2000) and in much reform curricula.

In the case of the Dinner Tables problem, third-

grade students were able to use an algebraic expres-

sion, n × 2 + 2, to represent the problem. Does this

necessarily mean they were making connections

about this expression related to aspects of the origi-

nal drawing? What does it mean when students use

letter notation, and what is it they are representing?

After students brought up this notation in the ﬁnal

class discussion, the teacher asked them what the

parts of the expression meant.

Teacher: She has n times two plus two. What

does that stand for, Sean?

Sean: Any number!

Teacher: But any number of what?

There was a pause in the classroom as students

thought about the question. Most students had

picked up on the idea that letters could be used

to stand for any number, but now they were being

asked to apply this rule to the original context.

What is the signiﬁcance of n in the Dinner Tables

problem? After a long pause, one student suggested

that n stands for the number of people; others said

that n stands for the number of tables. After some

consideration, students were able to generate a

correct response, but the question that connected

the algebraic expression to the original problem

context was not trivial.

We found that having a rich problem context

was not enough in and of itself to promote algebraic

thinking. Rather, strategic and purposeful use of

the representational context was necessary to push

students to reason algebraically and make connec-

tions across representations. By asking what the

expression n × 2 + 2 stands for, the teacher helped

make the link between the notation and the original

problem context. We found this type of question

The same task can be more arithmetic than algebraic if students

move down columns instead of across rows.

Figure 4

Photograph by Janice Gordon; all rights reserved

522 Teaching Children Mathematics / May 2008

essential to providing a motivation for students to

use letter notation at all.

In order to recognize when letter notation pro-

vides a useful way to represent a situation, connec-

tions such as this one across representations—in

this case, drawing tables and writing a symbolic

expression—are crucial. After establishing that n

stands for any number of tables, the teacher then

asked about the “times two,” which Alizé—who

had struggled with this idea during individual

work—now answered with conﬁdence.

Teacher: Why am I multiplying it by two? If I

know the number of tables, why do I need to

multiply it by two? Alizé?

Alizé: You have to multiply it by two because

when you put the tables together, the top and the

bottom have the same number of chairs.

Alizé’s explanation provides additional meaning to

the expression n × 2 + 2. Applying the rule to the

representational context is a way to help students

reach a deeper and more ﬂexible understanding of

the algebraic expression, one essential reason to

develop algebraic reasoning with young students.

Conclusion

Incorporating algebra into an already packed

mathematics curriculum can be an elusive task,

and educators still have much to learn about how

to facilitate this implementation. Most adults

learned arithmetic and algebra as separate strands

of mathematics, with algebra kept a secret until

grade 8 or 9. Therefore, we have little personal

experience as learners to relate to our practice. Our

collaboration as a researcher and a teacher, respec-

tively, helped us see how algebra can be success-

fully incorporated into the elementary classroom.

Using the instructional strategies discussed in this

article helped make the implementation of early

algebra less elusive by allowing students to see

patterns, make generalizations, and move across

representations—core goals of algebraic reasoning

in elementary school.

The Dinner Tables problem in particular is

rich with opportunities to generalize and make

abstractions. Nevertheless, these strategies can

apply to any mathematical domain and in any

classroom in which teachers are striving to push

kids to think algebraically. Although neither

author currently works with the project from

which these examples came, we each continue to

use these strategies in classrooms.

We encourage all elementary teachers to incor-

porate algebra into their mathematics classrooms.

Keep in mind that learning how to do this is a

process, one that is not always transparent. Col-

laboration with colleagues provides the opportunity

to share the effectiveness of instructional strategies,

compare stories of student learning, and ultimately

learn from one another. The process takes time,

but effective instruction in early algebra ultimately

creates powerful young mathematical thinkers and

provides an essential conceptual foundation for

later work in higher mathematics.

References

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ing Representational Contexts in Teaching Fractions.”

In Rational Numbers: An Integration of Research,

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and Thomas A. Romberg, pp. 158–96. Hillsdale, NJ:

Lawrence Erlbaum Associates, 1993.

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Kindergarten through Grade Twelve. Sacramento,

CA: CDE Press, 1997.

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———. Bringing Out the Algebraic Character of Arith-

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The project was supported by the National Science

Foundation as part of NSF-ROLE grant #0310171,

“Algebra in Early Mathematics,” awarded to D.

Carraher and A. Schliemann.