Young Children 89

®

2, 3, 4

Bradley S. Witzel, PhD, associate professor of education at Winthrop University, Rock

Hill, South Carolina, has taught general and special education to students who excelled

Response to

Intervention in Math (Corwin Press) and the Institute of Education Sciences’ practice

guide Assisting Students Struggling with Mathematics.

Christine J. Ferguson, PhD, professor of Early Childhood Education at Winthrop Uni-

Deborah V. Mink,

Strategies for Teaching Mathematics (Shell Educa-

Number Sense

Understandingmathematics is important

not only for select careers

but also in daily life. From

paying bills to predicting

expenditures to calculat-

ing mileage, we all do math

most every day. Students

in the United States lack

the skills to compete in an

international marketplace,

especially in ! elds requiring

competence in mathematics

(Gonzales et al. 2008). While

there is evidence that US

children’s performance in

mathematics has improved

since the passage of the No

Child Left Behind Act of 2001

(Gonzales et al. 2008), others

believe that we are still losing ground

in the global math environment

(Pearse & Walton 2011). Further prog-

ress is needed in the early childhood

classroom to prepare children for a

global job market (National Mathemat-

ics Advisory Panel [NMAP] 2008).

Improved performance is expected,

but there are implications for children

in the early childhood years (pre-K

to third grade), especially those with

learning dif! culties and disabilities,

who typically perform below their

peers (Lembke & Foegen 2009). By

being well-prepared in math in pre-K

and the early grades, children will be

more likely to succeed later.

Altering the poor pattern

Between 5 and 8 percent of students

in kindergarten through grade 12

evidence a mathematics-based dis-

ability (Geary 2004). Geary, Hoard,

and Hamson (1999) found that, com-

pared to their classmates, ! rst grade

children with math dif! culties often

struggle with counting knowledge,

number naming and writing, and

memory retrieval. Geary and his col-

leagues argue that these early errors

in basic math may affect their future

mathematics learning. Gersten and

Chard (1999) label the problems listed

by Geary and his colleagues as dif-

! culty with number sense. The NMAP

de! nes number sense as “an ability

to immediately identify the numerical

value associated with small quantities,

90 Young Children

a facility with basic computing skills,

and a pro! ciency in approximating

the magnitudes of small numbers of

objects and simple numerical opera-

tions” (2008, 27).

More advanced forms of number

sense involve understanding of place

value, composition and decomposi-

tion of number, and the concept of

basic arithmetic operations (NMAP

2008). For example, a young child with

more advanced number sense may

be able to turn simple addition facts

into a more complex idea by break-

ing down or decomposing numbers.

Instead of asking her to regurgitate the

answers to basic number facts, ask a

question like: “How many ways can

you make the number 5?” She will be

able to decompose the number 5 into

number sentences and answer: “There

are six ways to make the number 5

using addition: 5 + 0, 4 + 1, 3 + 2, 2 + 3,

1 + 4, and 0 + 5.”

Number sense development in

young children has been linked to

future math achievement in a man-

ner similar to the way phonological

awareness (i.e., children’s awareness

and use of sounds within a language

to make meaning) has been linked to

reading achievement (e.g., Kosanov-

ich, Weinstein, & Goldman 2009). That

is, they may be indicators of future

achievement. Even though some chil-

dren may memorize the basic facts

and recite them, if they are unable to

use those facts when they move to

larger numbers, confusion may lead

to dif! culties with subsequent math-

ematics skills (Witzel 2003).

It is dif! cult for children to under-

stand the concept of regrouping

(borrowing and carrying) in addition

and subtraction if they learn how to

regroup using step-by-step procedures;

they must also understand why they

are performing these steps. For exam-

ple, in subtraction with grouping, such

as 24 – 9, children learn to cross out

the number in the tens place and add

the 1 to the number in the ones place

to make (10 + 14) – 9, or 10 + (14 – 9).

Then children subtract the numbers in

the ones place (14 – 9 = 5) and ! nally

subtract the numbers in the tens place

(10 – 0).

Children sometimes do this without

understanding why or how this works.

They just follow the procedure that

they are asked to do. If you tell second-

graders that the problems on a page

in their math book all require regroup-

ing, they will follow the procedure

described above and get most, if not

all, of the answers correct. However,

if you tell them you don’t know if all

the problems require regrouping, most

children will use the same method

without thinking about whether they

must regroup and get some problems

wrong. This overgeneralization of a

single problem-solving strategy gets

more dif! cult as math becomes more

complex and children have to subtract

three- and four-digit numbers.

However, if children understand the

procedures and the concept behind

them, they are more likely to suc-

cessfully progress to more complex

concepts. Children acquire conceptual

knowledge by taking pieces of informa-

tion they have already gained from

experience and connecting them with

things they have learned into existing

mental structures (Reys et al. 2009). In

the regrouping example, if students use

concrete objects to subtract by taking

away the larger number of objects from

the smaller, they will ! nd that they

don’t have enough objects. In order to

perform the take-away strategy, they

will have to trade a ten for ten ones.

They now are able to take away the

correct number of objects. To progress

in mathematics, students must possess

both the knowledge of procedures and

the concepts behind those procedures

(Mink & Fraser 2005).

The importance of number sense

cannot be overemphasized. In a study

of 454 children, Jordan and colleagues

(2007) found that knowledge of num-

ber sense was a reliable and powerful

predictor of mathematics achievement

in both the ! rst and third grades. In an

analysis of assessments and interven-

tions used with young children who

struggle in mathematics, Gersten, Jor-

dan, and Flojo (2005) found that the

dif! culties children experience change

over time. Additionally, the research-

ers found that the most valid means

of predicting mathematics dif! culties

in young children involves some of

the basic principles of number sense.

In a recent analysis of number sense

assessments, Gersten and colleagues

(2011) identify ! ve components of

developing numerical pro! ciency:

magnitude comparisons, strategic

counting, retrieval of basic arithmetic

facts, word problems, and numerical

recognition.

Number sense development in young children has

been linked to future math achievement in a man-

ner similar to the way phonological awareness

has been linked to reading achievement.

The researchers found that the most valid means

of predicting mathematics difficulties in young

children involves some of the basic principles of

number sense.

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Magnitude comparisons: comparing

the size of two numbers as greater or

less than. (Is 21 greater than or less

than 16?)

Strategic counting: counting up or

down from a given number. (Count

three numbers up from 8.)

Retrieval of basic arithmetic facts:

accuracy and ! uency of single-digit

computation (i.e., fact families).

(What is 6 + 3? What is 3 + 6?)

Word problems: using language to

explain mathematics. (There are four

groups of three erasers each. How

many erasers total?)

Numerical recognition: connecting an

abstract Arabic symbol (a numeral)

with its number equivalent. (Match

the numeral 5 to the set of objects.)

Three strategies to help young chil-

dren who struggle in math develop

number sense are: (1) using concrete

experiences to develop number sense

and numeration, (2) teaching skills

to pro# ciency, and (3) incorporating

language experiences. To explain each

method, we use examples within the

context of advanced forms of number

sense as stated above.

Use concrete experiences

Concrete representations, such as

colored plastic counters or base 10

blocks, are not used often enough to

help children with math dif# culties

develop an initial sense of count-

ing and number (Witzel, Smith, &

Brownell 2001). Young children often

develop counting in a rote fashion

in isolation from the actual number

of objects involved (Smith 2012). In

other words, to some children, count-

ing aloud is no different than singing

a song: they don’t understand the

meaning behind the counting. Once

children understand the counting

principle, they will be able to count

the number of objects in a given set.

For example, a teacher can stack

one group of eight counters next to

another stack of four counters. This

concrete example helps the child

compare the larger or taller stack to

the smaller or shorter stack. The child

can then make a visual comparison to

see which stack is larger.

Vary the concrete experiences

There is no single concrete object

that is the most effective in helping

children develop number sense (Kamii

& Housman 2000). We, as educators, do

not want children to begin to associate

92 Young Children

mathematics with merely the manipu-

lation of a given set of objects. It is

important to provide several different

types of objects, whether they are

commercially produced for the early

childhood classroom or from natural

objects or items found in the home.

For example, plastic bears, counting

chips, linking cubes, base 10 blocks,

craft sticks, beans, and shells can all

be used to teach number, numerals,

and patterns successfully.

Many times, teachers can use !n-

gers to connect counting principles

to other objects. For example, hold

up three !ngers and say “three.” Then

say “three” and hold up a different set

of three !ngers to show conservation

of quantity. After the child develops

a consistent and successful pattern

of !nger counting to a given number,

show the child different representa-

tions of objects.

Transitioning from using !ngers to

using concrete objects when teach-

ing number sense gets children out

of their seats to see that math exists

beyond the classroom. Show chil-

dren that math exists at home, in the

school hallway, and on the way to

and from school. As children walk

through the halls, have

them count the pieces of

hanging artwork or the

#oor tiles. Take them out-

side to show how counting

and number relate to their

everyday lives. Count cars,

trees, bugs, and cracks in

the sidewalk. Ask them

to count the number of

stoplights they pass on

their way to school. Once

their use of the counting

principle becomes more

consistent, they will better

understand that counting

is connected to quantity.

For example, after a child

counts three trees, hold

up three !ngers and say,

“There is a total of three

trees.” Invite parents and

other stakeholders to

partner with you in helping children

expand their knowledge (Ferguson,

Steele, & Witzel 2008).

Adjust textbook work

accordingly

Most current math textbooks for

the primary grades try

to address the number-

to-numeral connection.

A textbook page or work

sheet may show a group

of objects on one side of

a page and on the other

side a list of numerals

for the child to choose

from. While this is a valid

means of assessing a

child’s understanding of

the number-to-numeral

connection, there are

two important strate-

gies to consider with

children who have math

dif!culties. One strategy

is to alter the format of

questions. For example,

students who have dif-

!culty matching number

to numeral may !nd suc-

cess by cutting out pictures of objects

and gluing them next to—or even on

top of—the connected numeral.

Another strategy is to ask the chil-

dren to draw shapes such as circles

that tie to the numerals rather than

simply drawing connecting lines on

publisher-illustrated work sheets. It

also helps to use concrete objects to

teach the connection before turning

to textbook lessons with pictures of

objects (Witzel, Riccomini, & Herlong,

forthcoming). Instead of asking kin-

dergartners to connect lines between

pictures of animals and their represen-

tative numeral, have the children con-

nect numerals with groups of real-life

objects, such as three toy snakes with

a note card showing the numeral 3.

Encouraging physical interaction

with number-to-numeral connections

(as with the three toy snakes) sets up

an important learning sequence that

starts with the concrete and builds

to the abstract (Riccomini, Witzel,

& Riccomini 2011). This learning

process, however, may be lengthy

Young Children 93

or brief. While states continue to set

early learning standards for math by

grade or age, actual rates of devel-

opment remain individual to each

child. Children move from skill to skill

depending on many factors, includ-

ing personal interest and explicit

home connections to math principles.

Teaching should focus on if children

have gained pro!ciency in a math-

ematics topic rather than when.

In response to children’s individual

needs, we educators can control what

and how we teach. Children who have

dif!culty learning math often require

the teacher to model mathematics

principles while explaining what is

happening and why. We often encour-

age children to perform new and

relatively dif!cult math skills at the

expense of accuracy. For example, a

young child who has not mastered

counting to 10 may skip several num-

bers when being asked to count to

20. It is important to comment on

what was correct and encourage the

child’s effort. It is equally important,

however, to arrange lessons for maxi-

mum success and provide appropriate

feedback within each step in the learn-

ing sequence. Letting children repeat

errors may allow them to practice

math incorrectly. For example, saying

ve-teen might make sense to the child

when learning to count, but it requires

correction so that he can communi-

cate the number accurately.

While explicit instruction is neces-

sary for struggling learners (NMAP

2008), children also need time to

explore and to become problem solv-

ers (Latterell 2003). Children need

opportunities to practice and general-

ize their skills in new situations and in

different contexts so that their educa-

tion consists of more than introduc-

tion and exposure.

Make language connections

The language of math is one of

the most important aspects of math

education. Many mathematics edu-

cators call mathematics a second

language. Since children with math

dif!culties often struggle in reading

as well (Fletcher 2005), it is impor-

tant to teach them math language and

the meaning of mathematical terms

and symbols. There are many liter-

ary genres that incorporate number

sense and other math principles

(Huber & Lenhoff 2006; Mink, Fergu-

son, & Long 2008). Use !ction and

non!ction books, maps, calendars,

architectural sketches, and newspa-

per graphs and statistics to connect

math to children’s lives and interests

(Ferguson 2001). For example, chil-

dren delight in singing Raf!’s “Five

Little Frogs” or “Five Little Monkeys

Jumping on the Bed” while counting

on their !ngers.

Aside from publisher-created liter-

ary connections, teachers and chil-

dren can combine math language with

conversational language. For example,

there is evidence to suggest that

some children learn to count as they

learn to talk (Ball 2003). However, it

is important that they understand the

quantity represented by the numbers

they are saying.

English language interpretations

of numerals do not always connect

counting to number. Consider that

many countries use place value in

their language, such as ten and one,

ten and two, and ten and three, while

in English, we say eleven, twelve, and

thirteen. The place value organiza-

tion of our language is often awkward

and sometimes missing altogether. In

“Alternatives for Solving a Math Prob-

lem,” the photos depict two ways to

examine the problem “28 subtracted

from 52,” or “52 minus 28.” The !rst

photo shows a commonly used

approach. The second photo shows

an interpretation by a child who

understands the place value of the

numbers. Both means may be used to

reach an accurate answer; however,

the second photo uses place value lan-

guage to help children establish the

reasoning behind computation, and

therefore provides a means to back up

or reverse their problem-solving steps

in order to explain the thinking pro-

cess of their computation.

When talking with children about

math concepts in and out of school,

include frequently used mathematical

key words that aid in solving word

Teaching should

focus on if children

have gained profi-

ciency in a math-

ematics topic rather

than when.

Alternatives for Solving

a Math Problem

These two pictures show different

place value interpretation, while the

second shows direct place value

94 Young Children

problems. For example, “I have ! ve

! ngers and two more. How many

! ngers do I have now?” The problem

statement implies addition. Make

explicit that and means addition here.

Another example: “We have three

groups of two teddy bears. How many

teddy bears total?” Show how group-

ings in a problem—as well as the word

of—mean multiplication.

As part of the language instruc-

tion, encourage children to verbalize

their mathematical reasoning. Child-

initiated think-alouds are important

not only for assessing children’s

math development, but for guarding

against impulsive problem-solving

approaches. That is, many children

who have learning dif! culties should

verbally explain their reasoning when

solving math problems. Thus, rather

than a child merely saying “Four times

three is twelve,” she might say, “Four

rows, each with three items, is twelve

total” or “Four groups of three is the

same as twelve.”

Conclusion

There is a great need for improved

mathematics teaching and learning in

the United States. We can start early

helping young children who show

signs of dif! culty learning math by

teaching them to make connections

from numeral to number and to under-

stand the value of numbers. Concrete

experiences, teaching to pro! ciency,

and connecting language to math are

three ways to help children improve

their number sense. However, it is

important to develop and try more

techniques. Acclaimed researchers

and national research groups (NMAP

and National Center for Education

Statistics, for example) have recently

turned their eyes toward improving

children’s number sense; thus more

teaching suggestions continue to

emerge. Meanwhile, educators must

continue to adjust the curriculum and

improve practice to help all children

succeed and to value mathematics.

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Copyright © 2012 by the National Association for the Edu-cation of Young Children. See Permissions and Reprints online at .

Concrete experiences,

teaching to proficiency,

and connecting lan-

guage to mathematics

are three ways to help

children improve their

number sense.