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410 Teaching Children Mathematics / April 2007

A

grade 4 mathematics homework question

instructed students, “Use the empty number

line to solve the following,” causing nine-

year-old Emily to respond, “No one should tell me

what strategy to use. I should be allowed to make

up my own mind!” Emily found it easier to solve

some two-digit addition and subtraction computa-

tions by splitting them into tens and ones rather

than using a strategy that could be recorded on a

number line.

Recently, a new mathematics syllabus for

K–6 students was introduced in New South Wales,

Australia (Board of Studies 2002). This document

reﬂected an international trend in its emphasis on

the development of mental strategies in the early

elementary years (e.g., NCTM 2000); it also for-

mally delayed instruction of traditionally taught

algorithms for the four operations and introduced

some new instructional “tools.” One such tool was

the empty, or blank, number line. This article is the

result of my daughter Emily’s and my decision to

explore for ourselves the origins and potential ben-

eﬁts of using an empty number line.

What Is an Empty

Number Line?

The empty number line—a number line that is

presented with no numbers or markers—is a visual

representation for recording and sharing students’

thinking strategies during mental computation

(NSW Department of Education and Training

2002). Starting with an empty number line, students

mark only the numbers they need for their calcula-

tion. For example, ﬁgure 1 shows Emily’s strategy

for solving the problem 53 – 26 as she recorded it

on an empty number line.

Why Use an Empty

Number Line?

The ﬁrst recorded use of the empty number line

that we could ﬁnd occurred in the Netherlands in

the 1970s. According to Gravemeijer (1994), the

empty number line was developed as a “new” tool

to help overcome difﬁculties associated with the

common “procedure only” use of base-ten materi-

als regularly used when modeling the standard writ-

ten algorithms. Early experiments with the empty

number line were not as successful as hoped, pos-

sibly because it was introduced in a measurement

context and its similarity to a standard ruler, with

its rigid calibrations, made students feel uncertain

when approximating the position of numbers on a

line with no given calibrations. However, Treffers

(1991) and Beishuizen (2001) found that students

could successfully use the empty number line to

record and make sense of a variety of solution

Janette Bobis, j.bobis@edfac.usyd.edu.au, is a mathematics educator at the

University of Sydney, Sydney, Australia.

By Janette Bobis

The Empty

Number Line:

A Useful Tool or Just

Another Procedure?

Copyright © 2007 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 / April 2007 411

strategies for two-digit addition and subtraction.

The empty number line is now widely accepted in

the Netherlands as an important didactical tool for

working with numbers up to 100 and beyond (Van

den Heuvel-Panhuizen 2001).

Advantages of using the empty number line, as

outlined by Gravemeijer (1994), include—

• the need for a linear representation of number;

• the close alignment of young children’s intuitive

mental strategies with the empty number line;

and

• its potential to foster the development of more

sophisticated strategies in children.

Clearly, base-ten materials, such as Dienes blocks,

are best for dealing with quantities and actual

problems, but linear representations, such as the

empty number line, are best for modeling distance

or measurement. The strong links between the

empty number line and young children’s intuitive

mental strategies are evident in the way that chil-

dren, when attempting to solve number problems

up to 100, naturally tend to focus ﬁrst on counting

strategies—counting all, counting on, or counting

down. Students who are more proﬁcient mental

calculators use a combination of counting strategies

(usually in chunks of 10) with partitioning strate-

gies. In partitioning, children “take apart” numbers

in ﬂexible ways to make them more convenient

to calculate mentally. These strategies typically

approximate the jumps on a number line. Note that

to solve the problem 53 – 26 (ﬁg. 1), Emily used a

combination of counting back in chunks of 10 and

then partitioning the 6 into 2 lots of 3 to bridge the

decade more easily.

The empty number line’s potential to support

the development of more sophisticated strategies is

the result of children’s using the line to record their

computation strategies, thus revealing to others (and

to themselves) their thinking processes. Hence, we

see not only the level of thinking but also the errors

in thinking that might occur. Instructional decisions

can then be made to assist the development of more

efﬁcient strategies. Given this last point, it seems

logical that such a visual recording of children’s

thinking strategies can provide a stimulus for class-

room discussion and sharing of mental strategies.

Students can explain their strategies by “showing”

them to others. This fact makes the empty number

line a very powerful tool for enhancing communi-

cation in the classroom.

Many of the beneﬁts of using an empty num-

ber line are supported by Emily’s reﬂections on

its use. Currently in grade 4, she was introduced

to the empty number line at the age of 8, when

she was in grade 3. She now considers the empty

number line “easier to learn and remember than

the pencil-and-paper method” because “I get to

record the strategy I’m thinking, not what I think

the teacher wants.” Furthermore, she explains, “If

you make a mistake, it’s easier to ﬁnd it.” From

Emily’s perspective, the empty number line is

“easier” to use because—

• she understands how it works (it represents her

thinking);

Photograph of Emily Bobis by Janette Bobis; all rights reserved

Emily recorded “jumps” on a number line to solve the problem

53 – 26.

Figure 1

412 Teaching Children Mathematics / April 2007

• it keeps a record of each step in her thinking;

• it allows her to track errors; and

• it enables her to think about what to do next

when the computation is too demanding to do

solely “in her head.”

Introducing the

Empty Number Line

Before children are introduced to the empty num-

ber line, they should already be familiar with a

linear representation of number—a number line

with numbers. Buys (2001) recommends that the

empty number line be introduced through strings

of beads that alternate in color every 10 beads.

Although Emily’s teachers did not use this particu-

lar approach, they used another type of number line

that supported the linear representation and assisted

Emily’s and her classmates’ understanding of an

empty number line. Figure 2a shows a number line

that displays only the numbers 0 and 100. Emily

was asked to move a clip along this number line

to a designated number determined by a teacher or

another child. After she positioned the clip on the

number line at the point she estimated to be the des-

ignated number, Emily “ﬂipped” the number line to

reveal a more complete number line (see ﬁg. 2b).

She could then check how close her estimate was to

the designated number.

Before children use the empty number line to

record more complex mental strategies involving

two-digit addition and subtraction, certain counting

skills and knowledge are required. Two essential

strategies that children must understand and use

effectively before they can use the more sophisti-

cated empty number line include (1) counting by

tens, both on and off the decade; and (2) jumping

across tens (or bridging tens).

Counting by tens, both on and off the decade,

allows a student to start with any number and count

forward or backward in multiples of 10. When ﬁrst

introduced to this skill, children can use manipula-

tives, such as strings of 10 beads or bundles of 10

popsicle sticks, to model the counting process and

record their counting on either a hundreds chart or

as jumps on an empty number line.

Bridging tens requires that children be able to

ﬂexibly partition numbers. For example, to solve

the problem 8 + 5, the ﬁrst number remains as a

whole and the second number, the 5, is partitioned

(for example, into 2 and 3) and added in parts. The

process is easier if a part of the 5 (the 2) is added

to the 8 to “make 10” before the ﬁnal part (the 3)

is added; hence, 8 + 2 = 10 and then 10 + 3 = 13.

This same strategy can then be applied when bridg-

ing tens in higher decades (e.g., 38 + 5 = (38 + 2) +

3 = 40 + 3 = 43).

After learning the strategies for counting by

tens and bridging tens, students can be intro-

duced to the jump strategy (or sequential strat-

egy) for two-digit addition and subtraction. The

fundamental characteristic of this strategy is

that one number is treated as a whole while the

second number is added or subtracted in man-

A predrawn number line, indicating multiples of 10, is used to solve

the problem 27 + 24.

Figure 3

A number line indicating only the numbers 0 and 100 (a) and the

reverse side of this number line (b), which includes decade numbers

added by the teacher

Figure 2

2a.

2b.

Photographs by Janette Bobis; all rights reserved

Teaching Children Mathematics / April 2007 413

ageable chunks of tens and ones In calculating a

problem such as 66 – 29, it is often more efﬁcient

to apply a compensation strategy—for example,

by subtracting 30 and adding 1 (66 – 29 = (66 –

30) + 1).

For instructional purposes, the children

should record only the relevant numbers on the

empty number line. As their strategies become

more sophisticated, the number of jumps should

decrease. Emily recalls some confusing experi-

ences with the empty number line during grade 3,

when she was provided with a predrawn number

line that started at zero and had all the multiples

of 10 marked (see ﬁg. 3). Still a relative novice at

using the empty number line, Emily thought that

she needed to start at the ﬁrst number marked on

the line—zero. The result was the meaningless,

“procedure-like” execution of unnecessary jumps.

Hence, while a proven powerful tool for develop-

ing mental computation in young children, the

empty number line and associated jump strategy

must be introduced thoughtfully and with well-

chosen examples. Inappropriate use could result in

the execution of a less sophisticated strategy or in

learning a meaningless procedure. Recent work by

Van den Heuvel-Panhuizen (forthcoming) refers

to similar concerns about the introduction and

appropriate application of the empty number line

with children in the Netherlands.

Another basic strategy for mental computation

of two-digit addition and subtraction is referred to

as the split strategy. This strategy involves “split-

ting” the tens and ones and addressing these sepa-

rately. Emily considered the split strategy a better

(“easier”) method to use for two homework prob-

lems—42 + 26 and 56 + 32—because “the digits

[in the ones place] don’t add up past ten.” Hence,

to calculate 42 + 26, Emily performed two addition

operations—40 + 20 = 60 and 2 + 6 = 8—and then

combined the tens and ones again to get 68. No

written recording was necessary because the num-

bers were simple enough for her to do completely

“in her head.”

Conclusion

We teachers need to be aware that children will

vary their strategy use according to the numbers

involved. The rigid application of just one tool

or one procedure will severely limit their ability

to apply mental strategies in ﬂexible and ﬂuent

ways. Hence, children must be given oppor-

tunities to develop a variety of strategies and

representational tools on their road to ﬂuency.

Eventually, children also need to make decisions

about what strategy to use and be accountable for

these decisions.

The empty number line was introduced into

our curriculum to help children move away from

meaningless manipulations of algorithms using

only pencil and paper, but we must be careful that

we do not unintentionally adopt a similar “pro-

cedural only” approach toward mental strategies.

The development of mental computation strategies

could be prone to the same difﬁculties we face in

teaching the standard pencil-and-paper algorithm

unless we learn to listen to our greatest critics—our

students—and let them make up their own minds

about the best strategy to use.

References

Beishuizen, Meindert. “Different Approaches to Mas-

tering Mental Calculation Processes.” In Principles

and Practices in Arithmetic Teaching, edited by Ju-

lie Anghileri, pp. 119–30. London: Open University

Press, 2001.

Board of Studies, New South Wales. Mathematics K–6.

Sydney: Board of Studies, 2002.

Buys, Kees. “Progressive Mathematization: Sketch of

a Learning Strand.” In Principles and Practices in

Arithmetic Teaching, edited by Julie Anghileri, pp.

107–18. London: Open University Press, 2001.

Gravemeijer, Keeno. “Educational Development and

Educational Research in Mathematics Education.”

Journal for Research in Mathematics Education 25

(1994): 443–71.

National Council of Teachers of Mathematics (NCTM).

Principles and Standards for School Mathematics.

Reston, VA: NCTM, 2000.

New South Wales (NSW) Department of Education and

Training. Developing Efﬁcient Numeracy Strategies

Stage 2. Sydney: NSW Department of Education and

Training, 2002.

Treffers, Adri. “Didactical Background of a Mathematics

Program for Primary Education.” In Realistic Math-

ematics Education in Primary School, edited by L.

Streeﬂand, pp. 21–57. Utrecht: Centrum voor Didac-

tiek vanWiskunde en Natuurwetenschappen, 1991.

Van den Heuvel-Panhuizen, Marja, ed. Children Learn

Mathematics: A Learning-Teaching Trajectory with

Intermediate Attainment Targets for Calculation with

Whole Numbers in Primary School. Utrecht: Freuden-

thal Institute, Utrecht University/SLO, 2001.

——. “Learning from ‘Didactikids’—An Impetus for

Revisiting the Empty Number Line.” Mathematics

Education Research Journal (forthcoming).

The author wishes to acknowledge the assistance

of her daughter, Emily Bobis, in the preparation

of this article. Emily is a student at Beacon Hill

Elementary School in Sydney, Australia. s