GUIDING CHILDREN'S LEARNING OF MATHEMATICS

128 Part 2 Mathematical Concepts, Skills, and Problem Solving

Figure 8.8 Ice cream sundae a. Model with colored paper
combinations ch ch ch ch hf hf hf hf s s s s

v ch s cc v ch s cc v ch s cc

b. Sketch of sundaes with toppings s
ch hf

v ch s cc v ch s cc v ch s cc

c. Diagram of sundaes
ch ch ch ch hf hf hf hf s s s s
v ch s cc v ch s cc v ch s cc

d. Cross diagram of possible sundaes
v ch s cc

ch
hf
s

ACTIVITY 8.7 Targets (Reasoning, Communication)

Level: Grades 2–4 • After the students have played several games, ask them
Setting: Learning center what the highest possible score is and what the lowest
Objective: Students determine the possible scores from throwing possible score is.

three dart balls at a target. • Ask whether the score could be 4, 9, 14, 15, 18, or 20 if
Materials: A Velcro target with Velcro balls. Label the target with all three balls stuck on the target. Ask for their thinking
behind their answers. Is there a pattern that helps them
point values, such as 9 for the center, 5 for the middle ring, and 3 determine which scores are possible and which are impos-
for the outside ring. sible? Ask them to list all the possible combinations using
their score sheet.
• Have students play the target game and keep score on a
score sheet with three columns showing the number of Game Center, Middle, Outer, Total
points for each round of three balls. 9 points 5 points 3 points

9 #1 1 1 1 17

5 #2 1 0 2 15

3 #3 0 1 2 11

#4 0 3 0 15

Extension

• Change the values of the targets and the number of balls.
Ask students to think about the high, low, possible, and
impossible answers.

Chapter 8 Developing Problem-Solving Strategies 129

three balls. Identifying the high and low scores gives Starting with 47 ϩ 10, 323 ϩ 100, or 4,567 ϩ 1,000,
boundaries to the possibilities, and a table helps to the pattern of adding an easy number is established
organize the information and show patterns. Pascal’s before working on 47 ϩ 9, 323 ϩ 99, or 4,567 ϩ 999.
triangle has many applications in mathematics and Learning the principle of compensation by adding
is a good subject for an investigation. In Activity 8.8 and then subtracting with simple numbers encour-
it is used to find several patterns. ages mental computation strategies for addition and
subtraction in many situations.
Solve a Simpler Problem, or Break the Problem
into Parts. Some problems are overwhelming be- Work Backward. Working backward is helpful
cause they appear complex or contain numbers that when students know the solution or answer and are
are large. Breaking a complex problem into smaller finding its components. Some teachers introduce
and simpler parts is an important problem-solving working backward in “known-wanted” problems.
strategy. Sometimes, substituting smaller numbers Students begin with the solution and think about the
in a problem helps students understand what is information that would give that result.
going on in the problem. Solving a simpler problem
gives students a place to start. • Edmundo has seven pets that are dogs and cats.
Five are dogs. How many cats does he have?
Many realistic problems are solved in parts. When
measuring the area of an irregularly shaped room for Children can model or draw seven pets, identify five
carpet, students can measure the room in parts and as dogs, and find that the missing part is two cats. The
add the parts together (Figure 8.9a). The surface area number sentence is a subtraction problem that can
of a cereal box (rectangular prism) is found by add- be written either in addition or subtraction form.
ing the areas of each face (Figure 8.9b).
D D DD
Many mental computation strategies are based
on making a more difficult combination into an DD
easier computation. For example, a teacher wants
students to work on mental computation strategies Edmundo has 7 pets. 5 are dogs. The rest are cats.
for adding 9’s. If 4,567 ϩ 999 is too hard as a first ex-
ample, the teacher could start with 47 ϩ 9. Students 7 pets ϭ 5 dogs ϩ ____ cats
can find that answer by using a number line and 5 dogs ϩ ____ cats ϭ 7 pets
counting forward to 56. They can also see that add- 7 pets Ϫ 5 dogs ϭ ____ cats
ing 47 ϩ 10 is easier to compute mentally, but the
sum has to be corrected by subtracting 1 to get 56.

(a) (b)

Figure 8.9 Breaking area into smaller parts: (a) area of a room to be carpeted, (b) surface area of a cereal box

130 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 8.8 Pascal’s Triangle (Reasoning, Communication)

Level: Grades 4–6 • Ask students to conjecture what numbers are in the next
Setting: Whole group row and to explain their thinking.
Objective: Students explore the patterns in Pascal’s triangle.
Materials: Copies of Pascal’s triangle • Reveal the next row, and have students compare the num-
bers to the conjecture.
• Tell students that they are going to work with Pascal’s tri-
angle. (Pascal was a French mathematician who invented • Ask students what they notice about the order of each
the first computer, but it could only add and subtract row. (Answer: The numbers in each row are palindromic—
numbers.) they are the same front to back and back to front.)

• Display the 1 • Ask students about the number of numerals in each row.
first few rows 11 (Answer: Some are odd, others are even.)
of Pascal’s 121
triangle, one 1331 • Ask students if they see a pattern in the sums of each
row at a time. 14641 row. (Answer: The sum of each row is double the sum
After three or 1 5 10 10 5 1 of the previous row. The sum of each row is 2 raised to a
four rows, ask 1 6 15 20 15 6 1 power: 20 ϭ 1; 21 ϭ 2; 22 ϭ 4; and so on.)
the students 1 7 21 35 35 21 7 1
what pattern
they see.

A similar process of working backward from the Single cupcakes ϩ 120 ϩ 120 ϭ 240 cupcakes
known is used in a more complex problem. packaged cupcakes ϭ total:

• Chen had $20.00 when he went to the grocery In these examples students work backward to
store. After buying a chicken for $2.09, celery for find the elements that are part of a known total or
$0.79, milk for $1.39, and a loaf of bread, he re- answer. Rush Hour is a challenging spatial puzzle
ceived $13.96 in change. Estimate how much the that begins with toy cars and trucks in a gridlock.
bread cost. Students rearrange toy cars and trucks to undo the
gridlock and free the red car stuck in traffic. The dif-
Knowing that the total has to be $20, students can ficulty of the puzzles increases with more vehicles
find several ways to express their thinking. in the traffic jam.

$14 ϭ $20 Ϫ $2 Ϫ $1 Ϫ $1 Ϫ bread A number puzzle also illustrates working forward
$2 ϩ $1 ϩ $1 ϩ bread ϩ $14 ϭ $20 and backward.
$20 Ϫ $14 ϭ $2 ϩ $1 ϩ $1 ϩ bread
• Pick a number, triple it, add 3, double the result,
All the number sentences involve starting with $20 subtract 6, divide by 3. If Sue’s answer was 12,
and backing out the amounts that are known until what was her beginning number?
only $2 is left. Another example of solving a similar
problem and working backward is separating the Try several numbers, and see if you find a pat-
proceeds of a bake sale into two parts. tern. Start with 12 and go backward, reversing each
step.
• The sixth-grade class at Bayview School sold
cupcakes at a carnival and collected $50.00. Single Undo step 5 Multiply by 3 12 ϫ 3 is 36
cupcakes cost 25 cents, and packages of three Undo step 4
cost 50 cents. The sale of single cupcakes was $30. Undo step 3 Add 6 36 ϩ 6 is 42
How many cupcakes did the sixth-graders sell? Undo step 2
Undo step 1 Divide by 2 42 divided by 2 is 21

Subtract 3 21 Ϫ 3 is 18

Total sales ϭ sales of singles $50 ϭ $30 ϩ $20 Divide by 3 18 divided by 3 is 6 Sue started
ϩ sales of packages: with 6.

Each dollar for single 30 ϫ 4 ϭ 120 cupcakes This number puzzle can be made simpler with
cupcakes buys four cupcakes: fewer steps or more complex with larger numbers.
Students can create their own puzzles: “How would
Each dollar for packages 20 ϫ 6 ϭ 120 cupcakes
of three buys six cupcakes:

Chapter 8 Developing Problem-Solving Strategies 131

you make 7 into 99?” or “Make 99 into 7.” Many teach- Being flexible is less a strategy and more a mind-
ers maintain a file of number puzzles for warm-ups, set of seeing alternative possibilities. In the number
learning centers, and sponge activities for the odd puzzles in Figure 8.6, students rearrange the num-
minutes in the school day. bers several times to get the sums and the order of
numbers to work. When one answer is correct, stu-
E XERCISE dents resist changing it even when it is necessary to
solve the entire puzzle.
Sudoku is a number puzzle that has become popu-
lar. The rule for solving the puzzle is simple: Fill Children may be less set in their ways of think-
each row, column, and nine-square section with the ing and therefore may have less difficulty changing
numbers 1 through 9. However, the combinations their point of view compared with adults, who can
are not simple. Look for an easy Sudoku puzzle in become fixed in their thinking. Both children and
the newspaper, a book, or on the Internet to work adults need to learn to ask themselves, “Is there an-
with a friend or classmate. Which strategies did you other way?” In Figure 8.10 squares are drawn on a
grid. The first square has an area of 1 unit. The sec-
use? ••• ond square has an area of 4 units. The challenge is
to draw other squares with areas of 2 square units,
3 square units, 4 square units, 5 square units, 6
square units, 7 square units, 8 square units, and 9
square units. To be successful in this task, students
must change their point of view and recognize that
squares can be drawn at different orientations and
that some squares may not have solutions.

Break Set, or Change Point of View. Creative 1 12
thinking is highly prized in today’s changing world. 34
Persistence is an important attribute of good prob-
lem solvers. Problem solvers also need to under- Figure 8.10 Grid for area puzzle
stand when they have met a dead end. At a dead
end they have to change their strategy, or how they E XERCISE
are thinking about the problem.
How did you have to change your point of view to
Being able to break old perceptions and see draw the squares in Figure 8.10? How many were
new possibilities has led to many technological and
practical inventions. The inventor of Post-it Notes possible? •••
was working on formulating a new glue and found
an adhesive that did not work very well. Instead of
throwing his failure out, he thought how it might be
useful for temporary cohesion.

Put nine dots on your paper in a 3
ϫ 3 grid. Connect all nine dots with
four straight lines without lifting your
pencil from the paper. After several
attempts, you may agree with others
that the problem cannot be solved.
However, when lines extend beyond the visual box
created by the nine dots, the solution is not difficult.
Having permission to try something “outside the
box” opens up new possibilities. Past experience
can be helpful or limiting.

132 Part 2 Mathematical Concepts, Skills, and Problem Solving

Implementing a and state standards. Because a problem-based class-
Problem-Solving Curriculum room may look messier or seem noisier than a tradi-
tional classroom, teachers should be proactive with
Flexibility of thinking is essential in problem solv- principals and parents by explaining the importance
ing. In the past teachers might present only one way of problem solving. Parent information sessions or-
to approach a problem or one way to think about it. ganized by teachers at the first of the year can al-
Students who had alternative ideas were frustrated leviate tension about problem-based mathematics
and discouraged. Most problems can be solved in curriculum. Parents often want to know that their
a variety of ways; learning to use problem-solving children will learn basic computational skills, and
strategies encourages students to try several ap- teachers can reassure them that computation is an
proaches. Even when teachers introduce strategies important goal. Students need reasons for learning
by themselves, students soon learn that the strate- the facts and applying them in interesting problems.
gies are more powerful when they are used together. Because the process and answer are both impor-
As children mature, complex problem situations in- tant, assessment of problem solving focuses on
volve a broader range of mathematics topics and whether students understand a problem, can devise
concepts. Solving a variety of problems in different a strategy, and can come to a reasonable solution
ways develops many skills and attitudes that support that they can explain. Assessment suggestions for
algebraic thinking. problem solving are found in Chapter 7.

Many teaching/thinking lessons and informal ac- Newsletters to parents suggest games and activi-
tivities that link problem solving to algebra are found ties for learning facts and for developing mathemati-
in the Navigations Through Algebra series (Cuevas cal thinking. At workshops or parents’ mathematics
& Yeatts, 2001; Greenes et al., 2001), in Teaching Chil- nights, students can teach their parents how they
dren Mathematics, on the NCTM website (http://www are solving problems with strategies. Parents expe-
.nctm.org), and in supplemental materials from educa- rience problem-solving activities that model how
tional publishers. Good problems are found in many students learn mathematics in new ways. Marilyn
resources, including textbooks and supplemental Burns (1994) produced a video for teachers and par-
materials. Puzzles and games provide many oppor- ents titled What Are You Teaching My Child? which
tunities for problem solving. Classroom situations describes why problem solving is essential for all
over the school year also offer many problems that students and how it is used in real life (the video
students can work together to solve, as discussed in is available at http://www.mathsolutions.com/mb/content/
Chapter 2. publications). The videotape also shows the modern
elementary classroom and why it looks different
Teachers who promote thinking and problem from the classroom that many parents remember.
solving for their students find support in national

Chapter 8 Developing Problem-Solving Strategies 133

Take-Home Activities

Dear Parents,

We have been working with pattern blocks and solving number problems
with them. Your student has a zipper bag with 20 pattern blocks in it. The
small green triangles are worth 1 point, the blue parallelograms are worth
2 points, the red trapezoids are worth 3 points, and the yellow hexagons are
worth 6 points. Using the pattern blocks, you and your child can solve the
following problems:

• Make a turtle with the pattern blocks. Write a number sentence showing
the total value of the blocks you used.

• Make a model of something using pattern blocks that has a value of 26
points. Trace around the picture and write the number sentence you used.

• Make a pattern with the pattern blocks using red and blue blocks. Read
the pattern.

• Show four combinations of blocks with a total value of 18 points.

After solving these problems, see if you can make up other problems using
the pattern blocks.

Sincerely,

134 Part 2 Mathematical Concepts, Skills, and Problem Solving

Take-Home Activities

Dear Parents,

How do students spend their time? We are investigating this question and ask-
ing students to keep a log of their activities one day this week. Using a daily
schedule, students mark which activity is most important in each half hour.
One copy of the schedule is for midnight to noon, and another is for activi-
ties from noon until midnight. After collecting this information, we will make
circle graphs and compare the amount of time students spend on different
activities.

Name:

Activity School/Study Play Sleep Eat/Bath Other

12:00
12:30
1:00
1:30
2:00
2:30
3:00
3:30
4:00
4:30
5:00
5:30
6:00
6:30
7:00
7:30
8:00
8:30
9:00
9:30
10:00
10:30
11:00
11:30

Thank you for your help.

135

Summary Andrews, A. G., and Trafton, P. (2002). Little kids—pow-
erful problem solvers. Westport, CT: Heinemann.
Problem solving has been the focus of instruction in el-
ementary education for several decades. Students act Burns, M. (1994). What are you teaching my child?
as mathematicians as they discover and refine concepts Sausalito, CA: Math Solutions Inc. (video). Available at
and procedures needed to solve a variety of problems. http://www.mathsolutions.com/mb/content/publications
Problem solving also involves process skills of reason-
ing, connecting, communicating, and representing math- Egan, L. (1999). 101 brain-boosting math problems. Jef-
ematical ideas. Creative and critical thinking skills are ferson City, MO: Scholastic Teaching Resources.
important attributes for working in the technologically
demanding world of the twenty-first century. Findell, C. (Ed.). (2000). Teaching with student math
notes (v. 3). Reston, VA: National Council of Teachers of
Learning processes and strategies for solving prob- Mathematics.
lems begins in elementary school and continues through
secondary education. The problem-solving process Greenes, C., & Findell, C. (1999). Groundworks: Alge-
suggested by George Polya leads elementary students braic thinking series (grades 1–7). Chicago: Creative
through four steps: understand, plan, carry out the plan, Publications.
and check to see if the solution is appropriate and sen-
sible. Development of problem-solving strategies, such Kopp, J., with Davila, D. (2000). Math on the menu: Real-
as finding and using patterns, guessing and checking, life problem solving for grades 3–5. Berkeley, CA: UC
breaking a problem into smaller parts, or making a table Berkeley Lawrence Hall of Science.
or graph, provides students with tools that apply to differ-
ent problems, often in conjunction with each other. NCTM. (2001). Navigations through algebra. Reston, VA:
National Council of Teachers of Mathematics.
Teachers have several responsibilities for building
an environment that encourages flexibility of thinking. O’Connell, S. (2000). Introduction to problem solving:
They need to choose worthwhile and interesting math- Strategies for the elementary math classroom. Westport,
ematical tasks that interest children. Many classroom CT: Heinemann.
and interdisciplinary situations are good problem-solv-
ing tasks. Games, puzzles, and informal activities also O’Connell, S. (2005). Now I get it: Strategies for building
develop problem-solving skills and attitudes. When confident and competent mathematicians. Westport, CT:
teachers recognize how algebra is embedded in many Heinemann.
problem-solving situations, they can help students grow
in their understanding and skill in algebra. Shiotsu, V. (2000). Math games. Lincolnwood, IL: Lowell
House. Available from [email protected]

Trafton, P., and Thiessen, D. (1999). Learning through
problems: Number sense and computational strategies.
Westport, CT: Heinemann.

Study Questions and Activities Children’s Bookshelf

1. Which of the 11 problem-solving strategies have you Anno, M. (1995). Anno’s magic seeds. New York:
used? Which strategies do you think are most impor- Philomel. (Grades 3–5)
tant? Why?
Bayerf, J. (1984). My name is Alice. New York: Dial
2. What is your understanding of teaching via problem Books. (Grades 1–3)
solving? What do you need to do to become more
skilled at this approach? Burns, M. (1999). How many legs, how many tails? Jef-
ferson City, MO: Scholastic. (Grades 1–3)
3. Find five problems for students at a grade level of
your interest in resource books, in teacher’s manu- Ernst, L. (1983). Sam Johnson and the blue ribbon quilt.
als, and on the Internet. Solve them and analyze New York: Lothrop, Lee & Shepard. (Grades 1–3)
your thinking. Which strategies did you use in solv-
ing the problems? Share your problems with fellow Hutchins, P. (1986). The doorbell rang. New York: Green-
students to build a file of classroom problems. willow Books. (Grades 3–5)

4. How do you interpret the following statement: “A Pinczes, E. (1993). One hundred hungry ants. Boston:
good problem solver knows what to do when he or Houghton Mifflin. (Grades 2–4)
she doesn’t know what to do.”
Scieszka, J., & Smith, L. (1995). The math curse. New
Teacher’s Resources York: Viking. (Grades 3–6)

Algebraic thinking math project. (1999). Alexandria, VA: Singer, M. (1985). A clue in code. New York: Clarion.
PBS Mathline Videotape Series. (Grades 4–6)

Weiss, M. (1977). Solomon Grundy, born on Monday.
New York: Thomas Y. Crowell. (Grades 4–6)

136

For Further Reading Rowan, T. E., & Robles, J. (1998). Using questions to
help children build mathematical power. Teaching Chil-
Civil, M., & Khan, L. (2001). Mathematics instruction dren Mathematics 4(9), 504–509.
developed from a garden theme. Teaching Children
Mathematics 7(7), 400–405. Teacher questions have a major impact on class-
room discourse and reasoning. Examples of questions
Making a garden motivates students to confront are given as models, with three vignettes of classroom
many mathematics and interdisciplinary problems and questioning practice.
issues.
Schneider, S., & Thompson, C. (2000). Incredible
Contreras, Jose (Ed.). (2006). Posing and solving prob- equations: Develop incredible number sense. Teaching
lems. Focus issue of Teaching Children Mathematics, Children Mathematics 7(3), 146–147.
12(3).
Children create extended equations and develop
This focus issue contains five feature articles about number sense as they solve them.
ways to engage students with problems and to develop
their thinking. Silbey, R. (1999). What is in the daily news? Teaching
Children Mathematics 5(7), 190–194.
Evered, L., & Gningue, S. (2001). Developing mathemati-
cal thinking using codes and ciphers. Teaching Children A newspaper report about the blooming of cherry
Mathematics 8(1), 8–15. trees in Washington, D.C., stimulates student inquiries
and problem solving.
Codes and ciphers demand reasoning and persever-
ance for problem solving. Silver, E., & Cai, J. (2005). Assessing students’ math-
ematical problem solving. Teaching Children Mathemat-
Methany, D. (2001). Consumer investigations: What is ics 12(3), 129–135.
the “best” chip? Teaching Children Mathematics 7(7),
418–420. Problem posing is presented as an important aspect
of learning to solve problems. Students understand the
Nutritional data and taste preferences are consid- process better when they are actively engaged in find-
ered in a classroom research project to find the best ing problems.
chip.
Yarema, C., Adams, R., & Cagle, R. (2000). A teacher’s
O’Donnell, B. (2006). On becoming a better problem- “try”angles. Teaching Children Mathematics 6(5),
solving teacher. Teaching Children Mathematics 12(7), 299–303.
346–351.
Problems and patterns provide background for num-
O’Donnell presents a classroom example of how ber sentences and equations.
teachers can expand their implementation of problem
solving. Young, E., & Marroquin, C. (2006). Posing problems
from children’s literature. Teaching Children Mathemat-
Outhred, L., & Sardelich, S. (2005). A problem is some- ics 12(7), 362–366.
thing you don’t want to have. Teaching Children Math-
ematics 12(3), 146–154. Young and Marroquin make suggestions and pro-
vide resources for teachers to develop children’s books
Outhred and Sardelich conduct classroom action as the source of interesting mathematical problems.
research of primary students engaged in a problem-
solving activity and show how the students improved
their skills.

CHAPTER 9

Developing
Concepts
of Number

earning about numbers, numerals, and number systems is a
major focus of elementary mathematics. Children’s number
sense and knowledge of number begin through match-
ing, comparing, sorting, ordering, and counting sets
of objects. Rote and rational counting are important

milestones in the development of number. Number is
represented in stories, songs, and rhymes and with concrete objects and
numerals. As students count beyond 9, they encounter the base-10 numera-
tion system used for larger numbers. They also explore number patterns in
the base-10 number system through 100 using a variety of experiences and
activities in the classroom.

In this chapter you will read about:
1 Basic thinking-learning skills for concept development in

mathematics
2 Characteristics of the base-10 numeration system
3 Different types of numbers and their uses
4 Activities for developing number concepts through manipulatives,

books, songs, and discussion
5 Rote and rational counting skills and problems some children have

with counting numbers
6 Assessment of children’s number conservation, or number

constancy

137

138 Part 2 Mathematical Concepts, Skills, and Problem Solving

Numbers and counting probably emerged from Pre-K–2 Expectations:
In prekindergarten through grade 2 all students should:
practical needs to record and remember informa- • sort, classify, and order objects by size, number, and other

tion about commerce and livestock. Today, num- properties;
• recognize, describe, and extend patterns such as se-
bers are used in many ways, but the basic utilitarian
quences of sounds and shapes or simple numeric patterns
nature is the same. Accountants and bankers track and translate from one representation to another.

and manage millions of dollars around the globe; Primary Thinking-Learning Skills

social and natural scientists research population Thinking-learning skills appear to be innate in hu-
mans. Brain research shows that infants actively
trends and topics in medicine, physics, or chemis- make sense of their world. Thinking-learning skills
provide a foundation for all cognitive learning, in-
try using computers; computer programmers and cluding development of numbers and number oper-
ations. Three skills are discussed in this chapter, and
analysts manipulate numbers and symbols to de- patterning is developed more fully in Chapter 16:

velop new languages and applications. In daily life • Matching and discriminating, comparing and
contrasting
families track income and expenses. Counting and
• Classifying, sorting, and grouping
number development start early in life. Children use
• Ordering, sequence, and seriation
numbers naturally as they play games, sing songs,
Matching and Discriminating,
read picture books, and solve problems. The NCTM Comparing and Contrasting

standards for algebra include basic thinking skills of When children match and compare, they find simi-
lar attributes. Discriminating and contrasting involve
classifying, sequencing, and patterning. Such skills identifying dissimilar attributes. Infants respond dif-
ferently to familiar or unfamiliar faces, sounds, and
are also fundamental for the development of num- smells. They gaze at pictures resembling faces and
avoid pictures showing scrambled face parts. Rec-
ber concepts. ognizing similarities and differences can often oc-
cur simultaneously, such as when children match
NCTM Standards for Number the mare with the colt, the cow with the calf, and the
and Operations bear with the cub; children see the relationship be-
tween the adult and young animals at the same time
Instructional programs from prekindergarten through grade they recognize the differences. Matching begins with
2 should enable all students to: the relationship between two objects. Relationships
Understand numbers, ways of representing numbers, rela- can be physical (large red triangle to large red tri-
angle), show related purpose (pencil with ballpoint
tionships among numbers, and number systems pen, key with lock, or shoe with sock), or connect
Understand meanings of operations and how they relate by meaning (a cow with the word cow).

to one another Learning activities based on similarities and dif-
Compute fluently and make reasonable estimates ferences extend from early childhood through high
Pre-K–2 Expectations: school and were found by Marzano (2003) to be
In prekindergarten through grade 2 all students should: the most effective teaching strategy for increasing
• count with understanding and recognize “how many” in student achievement. In the primary grades many
games are based on finding similarities and differ-
sets of objects; ences. Lotto games have many variations; children
• use multiple models to develop initial understandings of

place value and the base-ten number system;
• develop understanding of the relative position and mag-

nitude of whole numbers and of ordinal and cardinal
numbers and their connections;
• develop a sense of whole numbers and represent and use
them in flexible ways, including relating, composing, and
decomposing numbers;
• connect number words and numerals to the quanti-
ties they represent, using various physical models and
representations.

NCTM Standard for Algebra

Instructional programs from prekindergarten through grade
2 should enable all students to:
Understand patterns, relations, and functions
Represent and analyze mathematical situations and struc-

tures using algebraic symbols
Use mathematical models to represent and understand

quantitative relationships
Analyze change in various contexts

Chapter 9 Developing Concepts of Number 139

might match a red ap-

ple with a green apple;

a calf with a cow; three

oranges with three ap-

ples; or three oranges

with the numeral 3. A

teacher can place sev-

eral objects in the feel-

ing box (Figure 9.1) and

suggest matching and

discriminating tasks

for students. Later, chil-

dren can bring objects

for the feeling box and

create similar tasks for

classmates.

Children develop Figure 9.3 What is different about each clown?

Feeling box matching and dis- are almost identical in Figure 9.3, but each one has
criminating skills and a distinguishing characteristic for children to find.
Other search puzzles may show hidden objects or
comparing and con- ask which picture is different.

trasting skills through a variety of experiences and Card games, such as Go Fish and Crazy Eights
and Uno, match and discriminate by number, color,
activities. or suit. In the game Set, each card has four attributes:
number, color, shape, and shading. Students make
Matching Tasks Discriminating Tasks “sets” of three cards (Figure 9.4) that match on each
Find two keys. attribute or differ on the attributes. Children match
Find two keys—one objects to shape outlines in Activity 9.1. In Activity
Match two balls of the heavy and one light. 9.2, children explore bolts, nuts, and washers and
same size. Find two balls—one find which ones match.
Find two pencils with large and one small.
the same length. Find two pencils—one Matching and discriminating tasks are found
long and one short. in other subject areas. In language and literature,
metaphors and similes express comparisons and
A teacher can adapt an activity from Sesame Street contrasts.
with four objects so that “one of these things is not
like the others” (Figure 9.2). Drawings of five clowns

Figure 9.2 One of these things is not like the others. Figure 9.4 Set game cards

140 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 9.1 Matching Objects to Pictures (Reasoning)

Level: Grades: K–1 Variations
Setting: Learning center
Objective: Students match objects to pictures or outlines. • Students recognize many things by shape or logo: stop
Materials: Various sign, McDonald’s golden arches, Toyota icon, and many
others. Make a collection of shapes and icons on cards or
• Sketch pictures or outlines of common objects on a large in a book to which students can add new examples.
sheet of heavy tagboard. The pictures may vary from
detailed to outline-type illustrations. • Make a mask from a file folder with a small hole cut in it.
Put a picture inside the file folder so that only part of the
• From a collection of common objects in a box, students picture can be seen. Ask students to identify the pictured
match the real objects with their pictures and outlines. object from the part they can see.

ACTIVITY 9.2 Nuts and Bolts (Reasoning)

Level: Grades K–2
Setting: Learning center
Objective: Students match, classify, and seriate objects.
Materials: Nuts, bolts, and washers of various sizes

• Collect an assortment of nuts, bolts, and washers of
various sizes. Put different numbers and sizes of nuts,
bolts, or washers in different boxes so that each box has
combinations that match or do not match.

• Ask students to match, classify, and line up the objects in
the box.

Simile Examples Metaphor Examples • Compare and contrast Lewis and Clark’s explora-
tions of the Pacific Ocean to the first explorations
The sky was like an The sky was an ocean of space.
ocean of stars. of stars.
The road was like a The road was a silver • What is alike and different about the stories The
silver ribbon. ribbon. Little Red Hen and The Grasshopper and the Ant?

Compare and contrast tasks encourage student • What is alike and different about Charlotte’s Web
reasoning at all grades and subjects: as a book and as a movie?

Chapter 9 Developing Concepts of Number 141

• Compare and contrast the monthly rainfall in has attached a label to a group of furry, four-legged
Portland, Maine, and Portland, Oregon. animals. When an adult replies, “Spot is a dog. He
barks,” or “Spot is bigger than cats,” the child begins
• Compare and contrast the expected and ob- to form two classes of animals that are small, furry,
served outcomes of rolling two dice. and four-legged. A broader classification system is
developed as children learn about a variety of other
• Describe the relationship between dog years and animals with different attributes. Elephants have
people years. four legs but are large and have a trunk. Gradually
children develop a schema, or understanding, of the
• Describe how measuring the length of the foot- attributes that define specific animals and describe
ball field in centimeter cubes and with a trundle complex relationships between different animals.
wheel would be similar and different. They are able to group animals in many ways: size,
type of skin covering, what they eat, where they live,
• Show one-half in six different ways with objects, sounds they make, and many other ways to orga-
pictures, and symbols. nize animals by their characteristics. This ability to
create flexible and complex classification systems is
Classifying critical for conceptual learning.

Classification, also called sorting or grouping or Consistent classification develops over time and
categorizing, extends matching two objects that with experience. When the child lines up a red shoe,
are similar to matching groups of objects that share a green shoe, a green car, a police car, a police of-
common characteristics, or attributes. Classification ficer, and a firefighter, the thinking behind the sort-
is an important skill in all subject areas. In science ing may not be obvious (Figure 9.5). By listening, a
children sort objects that sink or float and objects that teacher may hear the child “chain” the objects rather
are living or nonliving. Groupings of tree leaves and than classify: “Red shoe goes with a green shoe, a
animals are based on similarities and differences. green shoe goes with a green car, a green car goes
In reading children find words that rhyme, have the with a police car.” Attribute blocks emphasize sim-
same initial consonants, or have long vowel versus ple classifications by color, shape, size, thickness,
short vowel sounds. They use classes of letters called or texture or two-way classification. Real-life objects
consonants and vowels that have different uses and are typically more complex in the ways they can be
sounds and find that classification systems can have organized because they have many attributes.
exceptions. Identifying needs and characteristics of
people around the world is a core concept in so- 123
cial studies. People living in different places have
unique languages, art, and music, but they all ex- Figure 9.5 Chaining objects
hibit these cultural characteristics in some way. All
peoples have a staple food that is cooked, mashed, Children may sort buttons by size (big, medium,
and eaten with other foods, whether that staple is small), texture (rough, smooth), color, number of
rice, wheat, corn, or taro root. The commonalities holes, and material, as well as not-button. They also
help children understand the similarities and differ- see that buttons have more than one attribute. The
ences among cultures. big red button is big and red and belongs to two
classes at the same time. The double-classifica-
MULTICULTUR ALCONNECTION tion board in Figure 9.6 challenges students to find
two characteristics at the same time. Sorting boxes
Different cultures have a food staple made of a grain (e.g., contain collections of objects, such as plastic fish-
wheat, rice, quinoa, or corn) or a root vegetable (e.g., potato ing worms, washers of various sizes, assorted keys,
or taro). Ask parents of students from different cultures small plastic toys, buttons, plastic jar lids, and other
to demonstrate how they cook the staple into a bread or common items. These collections provide many
porridge. Students can develop a Venn diagram of how the
different foods are alike and different.

As students become more sophisticated in classi-
fications, simple classes give way to complex classi-
fications. Calling “kitty” demonstrates that the child

142 Part 2 Mathematical Concepts, Skills, and Problem Solving

E XERCISE

How would you describe a chair? Which attributes
are essential for a chair? Why is a sofa or stool
not a chair? What attributes keep them from
being a chair? Make a chart or diagram show-
ing the relationships between different types

of seating. •••

Figure 9.6 Double classification board Ordering, Sequence, and Seriation

experiences for sorting objects by different attri- Another basic thinking-learning skill involves find-
butes: color, shape, material, letters on them, and so ing and using the orderly arrangements of objects,
on. Activity 9.3 demonstrates students’ classification events, and ideas. Order has a beginning, a middle,
skills with collections of objects. and an end, but placement within the order can be
arbitrary. When threading beads on string, children
may put a red bead first, a yellow bead second, and
a green bead last, but they could easily rearrange
the beads. Juan could be at the first, middle, or end
of the lunch line. In sequence, order has meaning.
Days of the weeks have fixed sequence, but the class-
room schedule may vary from day to day depending
on special events. Calendars mark the passage of
days, months, and years, and time lines show the

ACTIVITY 9.3 Sorting Boxes (Problem Solving and Reasoning)

Level: Grades Pre-K–2 pants?” “Are there enough washers for all the bolts?”
“Do you have enough garages for all the trucks?” “Which
Setting: Learning center shells do you like best? Why?” “I see three big dinosaurs.”
“I see two blue cars and three green trucks.”
Objective: Students classify objects based on common
characteristics. Variation

Materials: Collections of objects, such as toy cars, toy animals, With older children the same skills can be developed with
bolts, pencils and pens, coins, canceled stamps, socks, shells, other materials or examples: names of the 25 largest cities
beans, rocks in the United States, rainfall in the 25 largest cities in the
United States, rivers and their lengths, cards with zoo
• A math box is a collection of objects that children can ma- animals, examples of different kinds of rocks.
nipulate and organize in various ways. Watch as students
work with these collections.

• Ask questions or make statements to model language
and extend thinking. “Are there enough shirts for all the

Courtesy of Steve Tipps
Courtesy of Steve Tipps
Courtesy of Steve Tipps

Chapter 9 Developing Concepts of Number 143

sequence of historical events, geological periods, Beginning Number Concepts
or presidents. The sequence of events in a story
provides structure for the plot. Drawings of plant Experiences with matching, discriminating, classify-
growth show the sequence in development. Seria- ing, sequencing, and seriating are important skills
tion is an arrangement based on gradual changes needed for development of number concepts. Ob-
of an attribute and is often used in measurement. jects can be classified by size, color, or shape, but
For example: they can also be sorted by the number of objects:
five apples, five cars, five pencils. “Five” or the idea
• Children line up from shortest to tallest. of “fiveness” is the common characteristic. Number
is an abstract concept rather than a physical char-
• Red paint tiles are arranged from lightest to acteristic; it cannot be touched, but it can be repre-
darkest. sented by the objects.

• Bolts, nuts, and washers go from the largest to the Order, sequence, and seriation also play a role in
smallest in diameter. number concept. Counting is a sequence of words
related to increasing number: 1, 2, 3, 4, 5. . . . Chil-
• Each stack of blocks has one more block in it dren learning to count may say numbers out of order
than the last one. or have limited connection between a number word
and its value. An adult asks, “How old are you?” The
Comparative vocabulary develops with seriation: toddler responds with three fingers and says “two.”
good, better, best; big, medium, small; lightest, light, With more counting experience, children recognize
heavy, heaviest; lightest, light, dark, darkest. Activ- number in many forms: objects in a set, spoken
ity 9.4 describes comparison of length with drinking words, and written symbols.
straws.
Beginning number concepts are a major focus in
Games follow a sequence of turns and rules; most early childhood because they are the foundation for
board games are based on roll, move, and take the
consequences, as in Activity 9.5.

ACTIVITY 9.4 Drawing Straws (Reasoning)

Level: Grades K–3
Setting: Learning center
Objective: Students compare and seriate objects by length.
Materials: Five or six straws or dowels cut into different lengths

• Make a format board by tracing the straws or dowels in a
row from the shortest to the longest on poster board.

• Put the straws or dowels into a box. Have students draw
straws or dowel pieces and place them on the board.

Variations
• Have students arrange straws from smallest to largest

without the board.
• Make a board with only a few lengths of straws drawn

on it. Children take turns picking out straws. They can
choose straws that match the lengths, or they can place
shorter straws or longer lengths in the correct positions
between those drawn.
• Make a board on which the lengths are marked at
random.

144 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 9.5 Follow the Rules (Problem Solving)

Level: Grades 1–3 • The arrows represent rules for changing the shapes in the
Setting: Small groups rectangles:
Objective: Students develop a sequence of attribute blocks that → Change one attribute (a big red rough square to a
small red rough square)
follow a change rule. ⇒ Change two attributes (a big red rough square to a
Materials: Attribute materials or a set of objects with at least three small red rough circle)

attributes, transparency attribute blocks • Display a pattern on the overhead using transparency at-
• Make a game board such as the one shown. tribute blocks. Each block differs in one attribute from the
previous attribute block.
START
• Ask students which blocks they can find to continue the
STOP sequence.

• Distribute a set of attribute blocks to each group. Most
sets of 32 attribute blocks have four shapes in four colors
of two sizes.

• Have children put a shape in the Start box and follow the
rules to the Stop box.

• Ask them to start with a different block and show the
same sequence rule—one attribute changes.

Variation

Older students can make a sequence that has a “two at-
tributes change” rule.

understanding the Hindu-Arabic numeration system across the Mediterranean into Europe. The Hindu-
and number operations. Although numbers up to 10, Arabic system gradually replaced Roman numerals
20, or even 100 may seem simple to adults, number and the abacus for trade and commerce in Europe.
concepts and number sense are important cognitive For a brief time the two systems coexisted, but even-
goals for young children. Teachers and other adults tually the algorists, who computed with the new
encourage numerical thinking through the activities system, won out over the abacists. By the 16th cen-
and conversations about number. tury the Hindu-Arabic system was predominant, as
people recognized its advantages for computation.
The Hindu-Arabic numeration system is a
base-10 system that developed in Asia and the Middle Five characteristics of the Hindu-Arabic system
East. As early as A.D. 600, a Hindu numeration system make possible compact notation and efficient com-
in India was based on place value. The forerunners putational processes:
of numeric symbols used today appeared about
A.D. 700. Persian scholars translated science and 1. Ten is the base number. During early counting,
mathematics ideas from Greece, India, and else- people undoubtedly used fingers to keep track
where into their language. The Arabic association of the count. After all fingers were used, they
with the system came from translation and trans- needed a supplemental means for keeping track.
mission to other parts of the world. The Book of al- A grouping based on 10, the number of fingers
Khowarazmi on Hindu Number explained the use of available, was natural.
Hindu numerals. The word algorithm comes from
the author’s name, al-Khowarazmi; algorithms are 2. Ten symbols—0, 1, 2, 3, 4, 5, 6, 7, 8, 9—are the
the step-by-step procedures used to compute with numerals in the system.
numbers (Johnson, 1999). Increased trade between
Asia and Europe and the Moorish conquest of Spain 3. The number value of a numeral is determined by
further spread the Hindu-Arabic numeration system the counting value and the position: The nu-
meral 2 has different values in different positions
of 2, 20, and 200. Place value is based on pow-

Chapter 9 Developing Concepts of Number 145

ers of 10 and is called a decimal system. The Number Types and Their Uses
starting position is called the units or ones place.
Positions to the left of the ones place increase by Numbers have three main uses: to name or desig-
powers of 10. nate; to identify where objects and events are in
sequence; and to enumerate, or count, sets (Fig-
105 104 103 102 101 100 ure 9.7). Identification numbers or numbers on foot-
100,000 10,000 1,000 100 10 1 ball jerseys and hotel rooms are nominal because
they are used to identify or name, although they
Place value to the right of the ones place de- may code other information. Single digits on foot-
creases by powers of 10, so that negative powers ball uniforms may indicate players in the backfield.
of 10 represent fractional values: A room number of 1523 may be the 23rd room on
the 15th floor or the 23rd room on the 5th floor of
100 10Ϫ1 10Ϫ2 10Ϫ3 10Ϫ4 10Ϫ5 the first tower. Automobile license tags are identify-
ing numbers that may also code information such as
1.0 0.1 0.01 0.001 0.0001 0.00001 county of residence.

A decimal point signifies that the numbers to the Ordinal numbers designate location in a
left are whole numbers and that the numbers to sequence:
the right are decimal fractions. From any place
in the system, the next position to the left is 10 • Evan is first in line; Kelly is last.
times greater and the next position to the right • Tuesday is the seventh day of school.
is one-tenth as large. This characteristic makes • Toni ate the first and fourth pieces of cheese in
it possible to represent whole numbers of any
size as well as decimal fractions with the system. the package.
The metric system for measurement and the U.S.
monetary system are also decimal systems. Cardinal numbers are counting numbers be-
cause they tell how many objects are in a set. Number
4. Having zero distinguishes the Hindu-Arabic
system from many other numeration systems (a) Numeral
and allows compact representation of large
numbers. A zero symbol in early Mayan writings 6
also has been documented. The term zero is
derived from the Hindu word for “empty.” Zero is (b) Ordinal numbers
the number associated to a set with no objects,
called the empty set. Zero is also a placeholder First Second Third
in the place-value system. In the numeral 302, 0 (c) Cardinal numbers
holds the place between 3 and 2 and indicates
no tens. Finally, 0 represents multiplication by
the base number 10 so that 300 is 3 ϫ 10 ϫ 10.

5. Computation with the Hindu-Arabic system is
relatively simple as a result of algorithms devel-
oped for addition, subtraction, multiplication,
and division. Algorithms are step-by-step cal-
culation procedures that are easy to record. Al-
though most people learn a single algorithm for
each operation, several algorithms are discussed
in Chapter 12.

Children begin learning about the Hindu-Arabic OJ OJ OJ OJ
system by counting objects and recording numerals.
Throughout the elementary years they learn about 123 4
the place-value system and computational strategies Figure 9.7 Three uses of numbers
that the system makes possible.

146 Part 2 Mathematical Concepts, Skills, and Problem Solving

describes an abstract property of every set from 1 include many counting and number ideas. “Five
to millions. Little Monkeys” is a counting-down favorite that ap-
pears in song and story versions. Each verse has
• Ishmael has 1 dog. Tandra has 2 cats. Nguyen has one fewer monkey jumping on the bed, illustrating
3 goldfish. counting down or backwards.

• We counted 23 cars, 8 pickup trucks, and 12 vans Many counting and number books invite count-
and sport utility vehicles in the parking lot. ing aloud. Some involve puzzles and problems for
students to solve through counting. Eve Merriam’s
• The distance to the moon ranges between text and Bernie Karlin’s illustrations in 12 Ways to
356,000 kilometers and 406,000 kilometers. Get to 11 build a subtle message about number con-
stancy. On each two-page spread the number 11 is
• The movie Pirates of the Caribbean earned more represented in different situations and different ar-
than $600 million at the box office. rangements. As a follow-up, children could make
boxes for the numbers 11 through 19 and fill them
Written symbols for numbers are numerals; 19 is with different objects each day for several weeks.
the numeral for a counted set of 19 objects. Activity 9.6 describes an activity using Anno’s Count-
ing Book.
Children use nominal, ordinal, and cardinal
numbers from an early age. They learn addresses Children’s literature promotes understanding of
and telephone numbers that are identified with mathematical concepts in situations and stories that
particular people. They know what first and second relate to children. Number books, counting books,
mean when they run races from one end of the play- and number concept books have many advantages
ground to the other. By singing counting songs and for teachers and children (National Council of
reading counting books and through many other Teachers of Mathematics, 1994, p. 171):
experiences, children construct a comprehensive
understanding of numbers, their many uses, and the • Children’s literature furnishes a meaningful
words and symbols that represent them. context for mathematics.

Counting and Early • Children’s literature celebrates mathemat-
Number Concepts ics as a language.

The growth of children’s understanding of num- • Children’s literature integrates mathematics
ber is so subtle that it may escape notice. Children into current themes of study.
younger than 2 years of age have acquired some idea
of “more,” usually in connection with food—more • Children’s literature supports the art of
cookies or a bigger glass of juice. Counting words are problem posing.
heard in the talk of 2- and 3-year-old children. When
young children hold up three fingers to show age, Counting and number concept books are listed
they are performing a taught behavior rather than at the end of this chapter in the Children’s Book-
understanding what “three” means. They repeat a shelf section. Bibliographies found on the Internet,
verbal chain of 1, 2, 3, 4, 5 that receives approval such as http://www.geocities.com/heartland/estates/4967/
and attention from adults. Some children count up math.html, contain a wide variety of mathematical
to 10, 20, or 100 using memorized verbal chains, al- books on counting, number, and other mathemati-
though the meaning behind the words may not have cal concepts.
developed.
Rote counting is a memorized list of number
Early number work involves oral language. Num- words. The verbal chain of 1, 2, 3, 4, . . . provides
ber rhymes, songs, and finger-play activities stimu- prior knowledge for number concepts. Rational
late children’s enjoyment of number language and counting, or meaningful counting, begins when
support the connection of number words with num- children connect number words to objects, such as
ber ideas. Through rhythm, rhyme, and action chil- apples, blocks, toy cars, or fingers. Three becomes
dren associate counting words with their meanings. a meaningful label for the number of red trucks or
Books of finger plays and poems for young children bears or mice or candles on the cake. Counting is a
complex cognitive task requiring five counting prin-

Chapter 9 Developing Concepts of Number 147

ACTIVITY 9.6 Counting with Anno

Level: Grades K–2 • Show the book through the first time without interrup-
tion. Have the students look at each picture.
Setting: Whole group
• The second time through, ask students what they notice
Objective: Students relate number to counting. about each page. Model counting, and have students
count with you or by themselves.
Materials: Anno’s Counting Book, by Mitsumasa Anno (New York:
HarperCollins Children’s Books, 1977), Unifix cubes • Place the book and Unifix cubes at a reading station. Ask
students to match a number of cubes with the pictures.
Anno’s Counting Book engages children in counting from
0 to 12 in a series of pages that depict the development • Assign different numbers to groups of two or three, and
of a community over seasonal changes. This wordless let students make posters using stickers.
book invites children to develop understanding of the
numbers 0 through 12. An almost blank page shows • Have individuals or small groups write stories about the
a winter snow scene that introduces zero. Each page posters or list what they find on a page.
displays more objects to count. As the spring thaw sets
in, children see two trees, two rabbits, two children, two Extension
trucks, two men, two logs, two chimneys.
• Use the book to introduce multiples and skip counting.
Each page shows progressively more complex illustra-
tions that require a more intense search for numbers. On
each page a set of blocks illustrates the number featured
in the illustration.

ciples, which were identified by Rachel Gelman and four, five,” then ask questions such as “What is the
C. R. Gallistel (1978, pp. 131–135): last number you counted?” or “How many cars did
you count?” A commercial number chart, or one
1. The abstraction principle states that any collec- made by the children, demonstrates the “one more”
tion of real or imagined objects can be counted. idea between counting numbers, as shown in Fig-
ure 9.8. Counting sets aloud connects the sequence
2. The stable-order principle means that counting
numbers are arranged in a sequence that does Counting
not change.
1
3. The one-to-one principle requires ticking off the
items in a set so that one and only one number is 2
used for each item counted.
3
4. The order-irrelevance principle states that the or-
der in which items are counted is irrelevant. The 4
number stays the same regardless of the order.
5
5. The cardinal principle gives special significance
to the last number counted because it is not only 6
associated with the last item but also represents
the total number of items in the set. The cardinal 7
number tells how many are in the set.
8
Through counting objects and interactions with
adults and peers, children begin to understand the 9
pattern of numbers and their names: 1-more-than-1 is
two, 1-more-than-2 or 2-more-than-1 is three, and so 10
on. Numbers are developed progressively with con-
crete materials: 1 through 5, followed by 1 through Figure 9.8 Counting chart
10, then 1 through 20. Stress the last or cardinal num-
ber of a set as objects are counted: “One, two, three,

148 Part 2 Mathematical Concepts, Skills, and Problem Solving

of numbers and the objects, as shown in Activity 9.7. ever, children have had experiences with “all gone”
Activities with manipulative materials demonstrate and “no more” that can be used to introduce zero.
numbers created in different ways so that children In Activity 9.10 sorting objects into color groups and
become flexible in their ideas about number (see story situations illustrates zero, or the empty set.
Activities 9.8 and 9.9).
Number Constancy
Listening to children count shows whether
students have learned the sequence of numbers. By using Piaget’s conservation of number task,
By counting objects, children coordinate number teachers can assess a child’s level of number under-
words with objects one to one. Conservation of standing and skill as preconserver, transitional, or
number, or constancy of number, indicates an un- conserver (Piaget, 1952). Adults who are not familiar
derstanding that the number property of a set re- with this task are usually surprised at children’s an-
mains constant even when objects are rearranged, swers to what appears to be a simple task. However,
spread out, separated into subsets, or crowded to- the development of number constancy, or conserva-
gether. Without number constancy children may tion, is a critical cognitive milestone. Without num-
count two sets of six objects but believe that one set ber constancy or number permanence, work with
is larger because it looks longer or more dense or larger numbers, place value, operations, symbols,
more spread out. and number sense is difficult or impossible. Most
students develop number understanding at about
Zero requires special attention because children 5 or 6 years of age because they have sufficient
may not have encountered it as a number word. How-

ACTIVITY 9.7 Counting Cars (Representation)

Level: Grades K–2 • Ask students to line up the cars so that only one car is in
Setting: Learning center one square. Start with the largest number of cars on the
Objective: Students sort and graph objects. bottom line, and place the smallest number on the top
Materials: Plastic cars or other small objects in assorted colors, line. Fill in the middle lines so that the sets of cars are ar-
ranged from largest to smallest.
paper grid with car-size squares
• Ask students to count the number of cars in each line.
• Place plastic cars (or animals or colored cubes) on the Add the numeral to the chart.
table so that each color of cars is a different number, 1–9,
depending on the numbers being emphasized. • Read Counting Jennie, by Helena Clare Pittman (Minne-
apolis, MN: Carolrhoda Books, 1994). Have students take
• Ask students to sort the cars by color. a counting walk and record what they count.

• Have a piece of paper marked with squares large enough
to hold the cars.

Chapter 9 Developing Concepts of Number 149

ACTIVITY 9.8 Unifix Cube Combinations (Representation)

Level: Grades K–2 about the number 7. Repeat the lesson with other num-
bers and objects until students understand that a number
Setting: Learning center or small group can be made with many combinations.

Objective: Students demonstrate that the same number can be
made in different ways.

Materials: Unifix cubes

• Create a Unifix tower of seven cubes of the same color.

• Ask students to make other towers the same height using
two colors of cubes. They will make tower combinations
of two colors for 1 ϩ 6, 2 ϩ 5, 3 ϩ 4, 4 ϩ 3, 5 ϩ 2, and
6 ϩ 1. Some students will notice that 1 ϩ 6 and 6 ϩ 1 are
similar but reversed.

• Ask them to make towers of 5 or 7 with three colors.
Have students discuss all combinations to see if they have
found them all without duplication. Ask if any of the
towers have the same numbers made with the colors in
different orders.

• Ask how many cubes are in your tower and how many are
in each of their towers. Ask them to tell what they learned

ACTIVITY 9.9 Eight (Representation)

Level: Grades K–1 • Ask: “How are these sets the same?” (Answer: Each one
Setting: Small groups has eight in it.)
Objective: Students discover cardinal property of number for sets.
Materials: Objects for sorting and counting, paper plates • Ask: “How can we make each of the sets have seven?
nine?”
• Arrange two or three sets of eight objects, each set on a
separate paper plate. • Ask: “What was done to keep all the sets equal in
number?’’
• Ask: “How are these sets different?” Discuss the dif-
ferences the children see, such as size, color, or type of • Read Ten Black Dots, by Donald Crews (New York:
objects. Greenwillow, 1986), and have students make pages for a
number book using black dots.

ACTIVITY 9.10 No More Flowers (Representation)

Level: Grades 1–2 • Repeat the story two more times and remove two
flowers. Ask how many are left in the garden. Students
Setting: Small groups respond with “none,” “not any,” “they are all gone,” and
so on.
Objective: Students recognize the word zero and the numeral 0 as
symbols for the empty set. • Show several empty containers (boxes, bags, jars). Ask
students to describe their contents. Some will say they are
Materials: Felt objects and flannel board, or magnetic objects on a empty. They may or may not know zero. If they do not,
magnetic board introduce the word zero and the numeral 0 for a set that
contains no objects.
• Put three felt objects on a flannel board.

• Tell a story about the objects, such as: “These flowers
were growing in the garden. How many do I have?”

• Remove one of the objects and ask: “When I went to my
garden on Saturday, I picked one of the flowers. How
many were left?”

150 Part 2 Mathematical Concepts, Skills, and Problem Solving

mental maturity and have had appropriate experi- terview are the same for each concept (Copeland,
ences with objects and language. Teachers and par- 1984; Piaget, 1952; Wadsworth, 1984):
ents of young children may be tempted to “teach”
children the right answer to the conservation task. Step 1. Establish that the two sets or quantities are
The ability to conserve is not learned directly but is equivalent.
constructed through interactions with objects and Step 2. Transform one of the sets or objects.
people as well as through mental maturation. Within Step 3. Ask whether the two sets or quantities are
a short time a student may move from preconserver still equivalent.
to transitional to conserver through his or her own Step 4. Probe for the child’s reasoning.
reasoning and understanding process. Step 5. Determine the level of reasoning on the task.

Piaget developed assessment interviews on Activity 9.11 describes how to conduct an as-
many concepts in mathematics: number, length, sessment on number constancy, or conservation.
area, time, mass, and liquid volume. Although ma- Whether the child thinks the sets of checkers are
terials change for each concept, the steps of the in-

ACTIVITY 9.11 Number Conservation (Assessment Activity and Reasoning)

Level: Grades K–2 (can also be used diagnostically with older Step 3. Ask whether the two sets or quantities are still
children experiencing problems with number concepts) equivalent. “Look at the checkers now. Are there more
checkers here (pointing to the top row), more checkers
Setting: Individual here (pointing to the bottom row), or is the number of
Objective: Students demonstrate number conservation. checkers the same?” The student may respond “more
Materials: Red and black checkers, 5–10 of each color red,” “more black,” or “the same.” Sometimes the stu-
dent is confused or does not answer.
An interview for conservation, or constancy, of num-
ber determines whether a child can hold the abstract Step 4. Probe for the child’s reasoning. This is the most
concept of number even when objects are rearranged. important part of the interview because it reveals how
This protocol is for individual assessment. The interviewer the child is thinking. “How did you know that the number
should refrain from teaching at this time. The purpose is of black checkers was more (or that the number of red
developmental assessment. checkers was more or that the number of red and black
checkers was the same)?”
Step 1. Establish that the two sets or quantities are equiv-
alent. Put seven red checkers and seven black checkers in Step 5. Determine the level of reasoning on the task.
parallel rows so that the one-to-one correspondence be- Based on the child’s answer, determine the level of
tween the rows is obvious. Say: “I have some red checkers conservation.
and some black checkers. Do I have the same number of
red checkers and black checkers?” Preconserver: Child does not yet recognize equivalence
of the sets. Some may not even understand what the
If the child does not recognize that the number of question is or may not have the vocabulary to under-
checkers is the same in both rows, the interview is over. stand the situation.
The child does not conserve number yet or does not
understand the task. If the child responds that the num- Transitional: Child recognizes equivalence in parallel
ber of red and black checkers is the same, the interview lines but is confused when the sets are rearranged.
continues. These children look intently at the checkers and think
out loud: “I thought they were the same, but now they
Step 2. Transform one of the sets or objects. The teacher look different.” “They aren’t the same anymore, but
changes the arrangement of one of the rows by push- they used to be.” “They look different, but they are the
ing the checkers closer together, spreading them out, or same.” Wavering between different and same is typical
clustering them as a group. The rearrangement is done in of the transitional learner.
full view of the child.
Conserver: Child recognizes, often immediately, that
the two sets were equivalent and are still equivalent
regardless of the arrangement. When asked to explain
why the two are the same, their reasoning may vary,
but students who conserve number are absolutely sure
about their answer.

Chapter 9 Developing Concepts of Number 151

the same number or different, the teacher asks for 55 4
their reasoning. Some children recount the sets of
checkers or rearrange them. Others reply that “none 3 1
were added, none were taken away”; their reasoning 8
is more abstract. Transitional students are puzzled (a)
by the task because they are not sure whether the 5
number is the same or different. Some children offer 3
no reason for their answer.
(b)
Children move through the concrete operational
stage during the elementary years as they develop Figure 9.9 (a) Unifix cubes and number boats; (b) Unifix
constancy of length, area, volume, elapsed time, cubes and number indicators.
mass, and other concepts. Piaget’s structured inter-
view is a good model for assessment of any mathe- structured material, show the numeral with the cor-
matics content: student engagement with a specific rect number of peg holes. Students fill up the puzzle
assessment task, followed by questions to discover for each number and then put the numbers together
the child’s thinking. in correct numerical order, as shown in Figure 9.10.

Teachers who focus on student thinking design Children also develop mental images of num-
concrete experiences and avoid abstractions and bers. After working with dice, dominoes, and cards,
symbolizations too early. Throughout elementary children develop a strong visual image for number
school, children need objects as aids in their think- that they recognize without counting the dots or
ing because abstract thinking begins only around pips. This ability is called subitizing and shows a
ages 11 to 13 and develops slowly into adulthood. growing sophistication with number. Students soon
Manipulatives in mathematics are objects to think call out the numbers while playing board games
about. Also, adults often use concrete materials and and connect the number with the action. A number
models to help children learn about new ideas and line is a spatial, or graphic, arrangement of counting
solve problems. numbers (Figure 9.11). The number line introduces

Linking Number and Numeral 1234 5

Many experiences with manipulative and real ob- Figure 9.10 Interlocking number puzzles with pegs
jects, games, computer software, books, songs, and
conversation develop number concepts and skills. 0 1 2 3 4 5 6 7 8 9 10
Children who enter preschool and kindergarten with (a)
meager informal number and counting experiences 20 21 22 23 24 25 26 27 28 29 30
or limited English language proficiency need ex- (b)
tended time and varied experiences. Unfortunately, Figure 9.11 Number lines
children who need additional oral language and con-
crete experiences are sometimes rushed into sym-
bolic experiences without sufficient background.

As children gain proficiency with verbal and
physical representations of number, they also en-
counter the written symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
matched with sets. Unifix number boats and num-
ber indicators, shown in Figure 9.9, structure con-
necting number to numeral because the number
of blocks inside the boat matches the numeral.
Unifix number indicators fit on any tower of cubes,
so children must determine which indicator goes
with which tower. Peg number puzzles, another

152 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 9.12 Plate Puzzles and Cup Puzzles (Representation)

Level: Grades K–2 • Write a numeral on an inverted paper cup. Use golf tees
Setting: Learning center to punch the corresponding number of holes in the bot-
Objective: Students connect sets with numerals. tom of the cup. Children fit the correct number of tees
Materials: Paper plates, scissors, paper cups into the holes in the cup.

• Make puzzles out of paper plates. Cut each paper plate so • Have children make their own cup and plate puzzles.
that it has a distinct configuration.
1 23 34 5
• On one piece of the plate, write a numeral; on the other
piece, draw a picture or put stickers corresponding to the
numeral.

ACTIVITY 9.13 Matching Numeral and Set Cards

Level: Grades K–2 • If students need more clues, the cards can match by color
as well as by number. Additional sets should not have the
Setting: Learning center color cues.

Objective: Students match sets with numerals. Variations

Materials: Pocket chart and sets of cards for matching • Reverse the order of the sets, or mix up the order of the
sets.
A pocket chart can be used in many ways. If made
with library pockets on a poster board strip, it makes • Add a set of number words to match to the sets and the
a learning center that can sit on the chalk tray. At first, numerals.
the matches made are simple, but the variations suggest
many ways of extending the use of the pocket chart. • Add a set of addition cards (5 ϩ 1) and subtraction cards
(4 Ϫ 3) to match the numbers.
• Put cards that have pictures of sets on them in the
pockets for 0–9 or 1–10. Pictures or stickers can be used. • The pocket chart can be used later as a place-value chart
Children can also be given the task of making additional for large numbers.
sets of cards.

• Put a set of picture cards in the pockets. Ask students to
match with other picture cards or with numeral cards.

✽ ❃❃ ▲▲ ❒❒ ❤❤ ✈✈ ☎☎ ✉ ✉ ✫✫✫
▲ ❒❒ ❤❤ ✈✈ ☎☎ ✉ ✉ ✫✫✫
❤ ✈✈ ☎☎ ✉✉ ✫✫✫
☎ ✉✉

123456 78 9

Chapter 9 Developing Concepts of Number 153

the idea of counting up by walking forward on the Writing Numerals
number line. Students can also count down by walk-
ing backward or turning around. Counting up with Writing names and numerals is a major achieve-
objects or on the number line connects counting to ment that gives children more independence in
addition and subtraction. recording their mathematical ideas. Number work
does not have to be delayed until children can write.
Games and puzzles are informal activities that Numeral cards allow children to label sets and se-
are introduced and placed in learning centers as quence numbers. Children usually have developed
exploratory and independent activities. Activity 9.12 many number concepts and can recognize many
shows simple puzzles to match sets with numerals. numerals before they can write them. They may also
In Activity 9.13 picture cards are matched with ap- type or use a computer mouse to select numerals
propriate numeral cards. Activity 9.14 is a concentra- without controlling a pencil.
tion game for matching numerals and sets.
When handwriting instruction begins, correct
E XERCISE form is modeled by the teacher and in the materi-
als used. Students trace over dotted or lightly drawn
Play a board game, card game, or number puzzle numerals to build their proficiency. Children who
game with primary students and observe them. write numerals correctly and neatly during writing
Which mathematics concepts and skills are they practice sessions may be less careful at other times.
learning and practicing? Are all the students equally A chart showing the formation of numerals can be
proficient with the skill? What could you do to help posted on the wall, or a writing strip of letters and
the student be more successful with the game and numerals can be placed at the top of each desk. Dif-
ferent schools adopt different handwriting systems,
skill? •••

ACTIVITY 9.14 Card Games for Numbers and Numerals

Level: Grades K–4 both cards. An alternative is to turn over two cards and
Setting: Learning center the highest sum takes all the cards. Ask students to sug-
Objective: Students match numerals and sets. gest what should be done if the two cards have the same
Materials: Sets of playing cards or index cards with stickers on sum.

them 7

Concentration

• Make a set of 10 index cards with 1 to 10 stickers on
them and another 10 cards with the numerals 1 to 10 on
them. Arrange the cards face down in a 4 ϫ 5 array.

• A child turns over two cards at a time.

• When a match is made between number and numeral,
the child keeps the two cards. If there is not a match, the
cards are returned to their places face down, and the next
child turns over a pair.

Variations

• Match two sets of numeral cards.

• Pick a target number, and turn over combinations of cards
with that sum.

• Play battle with the cards. Each child has half a deck, and
they turn over the cards simultaneously. High count wins

154 Part 2 Mathematical Concepts, Skills, and Problem Solving

12 • A child may make omission errors when
one or more items are skipped.
D’Nealian numeral forms
• A child may make a double-count error
such as the D’Nealian number forms shown in Fig- by counting one or more items more than
ure 9.12. You will probably need to practice the ad- once.
opted handwriting systems so that you are a good
model for children. • A child may use idiosyncratic counting
sequences such as “1, 2, 4, 6, 10.”
Writing opens up many new opportunities for
children to make their own books and displays. On a If children have difficulty with counting and
page, “one” and “1” are written, and the child draws make coordination, omission, or double-counting
or pastes examples of one object. On the “two” page, errors, they can count objects on a paper or a mag-
two objects are drawn or pasted. Posting children’s netic board with a line marked down the center. As
work or making books for the class library extends children count each object, they move it from the
interest as children go back to these products again. left half to the right half. This action reduces the op-
Booklets are also good home projects, from simple portunity to begin or end the count improperly, to
counting projects to special number topics such skip one or more objects, or to double-count some.
as odd and even numbers, multiples, and number After counting proficiency is achieved, a child may
patterns. not need to move the objects.

Misconceptions and Problems Reversal errors in writing numerals, as with 6
with Counting and Numbers and 9 or 5 and 2, are common among children. Eye-
hand and fine muscle coordination are needed be-
Despite well-planned developmental activities, some fore children can write with accuracy and comfort.
children experience problems with number. Some If children count accurately, written reversals should
children have developmental delays or other special not cause concern. If the problem persists into the
needs that interfere with language, motor control, or second or third grade, the teacher may refer the
cognitive development. Children with limited vision student for visual or motor coordination problems.
or certain physical disabilities may need larger ma- Adapting material for students with visual or motor
terials that are easier to see and handle. In analyzing impairments might include providing larger materi-
children’s counting behaviors, Gelman and Gallistel als or using materials with rough surfaces.
(1978, pp. 106–108) observed and identified a num-
ber of common errors: Introducing Ordinal Numbers

• A child may make a coordination error Ordinal numbers describe relative position in a se-
when the count is not started until after the quence or line. Ordinal numbers occur in many situ-
first item has been touched, which results ations that students and teachers can discuss and
in an undercount, or when the count label. Children can stand at the first, sixth, or tenth
continues after the final item has been place from zero on the number line. When children
touched, which results in an overcount. line up for recess or dismissal, they notice who is
first, last, second, fifth, or middle. The bases in base-
ball are also first, second, third, and home (or last)
base. Discussing sequence in stories, in days of the
week or month, or in each day’s events uses ordinal
numbers. Asking questions calls attention to ordinal
concepts:

• Who was the third animal that Chicken Little
talked to?

• Who is second in line today?

• What happens during the second week of this
month?

Chapter 9 Developing Concepts of Number 155

When children see the ordinal usage occur in com- A hundreds board or chart is a visual way for chil-
mon classroom situations and hear adults use the dren to investigate skip counting (see Activity 9.15).
various terms correctly, most have little trouble with
this use of numbers. One-to-One and Other
Correspondences
Other Number Skills
Children use one-to-one correspondence as the ba-
From their first encounters with numbers and count- sis for relating one object to one counting number.
ing, students learn some important number skills They also need experience with other correspon-
that extend their understanding of how numbers dences and relations: one-to-many, many-to-one,
work. Skip counting and concepts of odd and even and many-to-many. One-to-many correspondences
are introduced in primary grades. and many-to-one correspondences are used every
day in many reversible situations; some examples
Skip Counting are shown in Table 9.1.

First experiences in counting are based on one- A good problem-solving strategy for many-to-
to-one correspondence between objects and the many correspondences is a rate table. If three pen-
natural numbers 1, 2, 3. Skip counting is performed cils cost 25 cents, how many pencils will 75 cents
with objects that occur in groups of 2, 3, 4, 5, or oth- buy? The ratio between pencils and cents is 3:25
ers. Skip counting encourages faster and flexible which is extended as far as it is needed to answer
counting and is connected to multiplication and the question.
division.
Pencils 3 6 9 12 15 18 21 24 27
• How many eyes are in our room? Count by two. Cost 25¢ 50¢ 75¢ ? ?

• How many chair legs are in the room? How can E XERCISE
we count them quickly?
Look at a state map. Using the scale on the map,
• We have 15 dimes and 7 nickels. What is a quick estimate the distance between two places in your
way to count our money?
state. •••

ACTIVITY 9.15 Patterns on the Hundreds Chart

Level: Grades 2–5 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Setting: Small groups 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
Objective: Students find patterns on the hundreds chart. 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
Materials: Hundreds chart on paper or transparency (see Black- 61 62 63 64 65 66 67 68 69 70
Line Master 9.1) 71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
• Ask students to color numbers as they skip-count by twos 91 92 93 94 95 96 97 98 99 100
on the chart. Ask students to describe the design made by
multiples of 2.

• Have students in each group color the hundreds chart
for multiples of 3, 4, 5, and so on, and watch for designs
made by the patterns. After students have completed indi-
vidual number pages, make a book in which they describe
the patterns found.

• Pose questions such as, “What skip-counting pattern
begins at the top left and moves downward toward the
bottom right corner?” “Name a skip-counting pattern that
begins at the top right and moves diagonally downward
to the left.”

156 Part 2 Mathematical Concepts, Skills, and Problem Solving

TABLE 9.1 • Some Common One-to-Many, Many-to-One, and Many-to-Many Correspondences

One-to-Many Many-to-One Many-to-Many

Place value 1 ten ϭ 10 ones 10 ones ϭ 1 ten
Value 1 nickel ϭ 5 pennies 5 pennies ϭ 1 nickel
Linear measure 1 foot ϭ 12 inches 12 inches ϭ 1 foot
1 meter ϭ 100 centimeters 100 centimeters ϭ 1 meter
Time/distance 1 hour for 55 miles 55 miles in 1 hour
Shopping 1 dozen has 12 bagels 12 bagels in 1 dozen
Time 1 day has 24 hours 24 hours in 1 day
Maps 1 inch stands for 25 miles
Fuel economy
Food 15 gallons goes 350 miles
Tires 3 cans for $1.45
4 tires cost $125.00

Odd and Even Numbers • Beginning at 2, every other number is an even
number; beginning at 1, every other number is
Counting by twos prompts children’s thinking about odd.
odd and even numbers. Counting eyes, or ears, or
feet, or twins helps students recognize naturally oc- • The column on the left side of the chart contains
curring situations of even numbers. Even numbers all odd numbers ending in 1. Each alternate col-
are a set of numbers divisible by 2 with no remain- umn ends in 3, 5, 7, and 9.
der. Odd numbers are the set of numbers not divis-
ible by 2 evenly and they cannot be organized into • The second column from the left contains all
pairs. Skip counting by 2 is a way to identify even even numbers that end in 2.
numbers. In Activity 9.16 children use plastic disks
to determine which numbers are even and which • Each alternate column across the chart contains
are odd. Objects that can be arranged in pairs rep- even numbers ending in 4, 6, 8, and 0.
resent even numbers. Knowledge of even and odd
numbers establishes a pattern that is also used for • Even numbers are all multiples of 2.
finding other patterns and relationships.
Similar conclusions can be made about 3, 4, 5,
The number line and hundreds chart are used and other numbers as students explore patterns.
for further investigations with even and odd num- Intermediate-grade students can study even and
bers. Children make a variety of observations about odd numbers at a more abstract level. They should
the number chart: develop and justify these generalizations during
their investigations.

Chapter 9 Developing Concepts of Number 157

ACTIVITY 9.16 Even and Odd (Reasoning)

Level: Grades 1 and 2 • Ask students to find other numbers that do not make two
Setting: Student pairs equal rows. Put these numbers on the board, or mark
Objective: Students recognize odd and even numbers by pairing them with a different color on the hundreds chart.

objects. • Say: “Numbers that make two equal rows have a special
Materials: Counting disks or other objects name.” If students do not volunteer “even,” supply it.
• Ask students to arrange eight disks in two rows. Ask if
• Say: “Numbers that cannot make two equal rows also
they can put the same number in each row. have a name.” If students do not know “odd,” supply it.

(a) • After every even number is an odd number, which can be
expressed as 2n ϩ 1.

Students may be asked whether zero is an even num-
ber and to explain their thinking.

• Ask students to find other numbers of disks to make two
equal rows. Put their numbers on a chart, or mark them
on a hundreds chart.

• Ask students to arrange seven disks in two rows. Ask if
the two rows are equal.

(b)

Research for the Classroom •
Jones and colleagues (1992) reported that first-grade
Various research findings have determined that young students engaged in a tutoring program became more
children learn to count and use numbers in purposeful flexible in their numerical strategies and were successful
ways at an early age. Caulfield (2000) suggests that the at tasks that were more complex than usually expected
human brain is “born to count” as a natural way of orga- of them. The tutoring program involved working in pairs
nizing the physical world. Suggate and colleagues (1997) with a teacher on inquiry and exploratory mathematical
studied mathematical knowledge and strategies of 4- and tasks. The mentor-teachers also changed their practices
5-year-olds over a year. Most students had a high level and beliefs about what students could learn as a result of
of numerical proficiency at the beginning of the year and participation in the program.
improved over the year. However, a few students showed In summary, young children have many numerical skills
little improvement over the year. Wright (1994) also stud- and can learn new strategies if teaching is interactive, rel-
ied the growth of forty-one 5- and 6-year-olds for a year. evant, and encourages thinking about number rather than
However, his findings emphasize the mismatch between memorizing processes without understanding.
the children’s numerical competence and the demands of
the curriculum.

158

Summary Praxis
The Praxis II test includes items on early mathemat-
Children use numbers in many ways: to label, to order, ics. Try the following three items to check your un-
to count, and to solve problems in their lives. Young chil- derstanding of the concepts and teaching methods
dren classify and sequence objects based on physical in items similar to those found on the Praxis test.
qualities. Through a variety of manipulative and lan- 4. When Mrs. Rodriquez greets children by counting
guage experiences, children learn about number as an them first, second, third, fourth, . . . , which use of
abstract property of sets that can be represented with number is she demonstrating?
objects, in pictures, and with symbols, called numerals. a. Cardinal
Interactions with numbers in books, songs, poems, puz- b. Nominal
zles, games, and objects contribute to the development of c. Counting
number and numerals. Rote counting, rational counting, d. Ordinal
and number conservation are important achievements 5. Mr. Kinski asks students to classify classroom ob-
in learning about numbers. In rote counting children can jects into three groups. What mathematical concepts
recite number words, whereas rational counting demon- and skills are used?
strates the relationship between number words and ob- a. Recognizing similarities and differences
jects. Understanding that number is constant regardless b. Counting
of the arrangement of objects is called number conser- c. Relating numbers to numerals
vation or constancy and is a developmental milestone d. Recognizing two-dimensional and three-
about age 6 for children.
dimensional shapes
When children encounter numbers in meaningful 6. Miss Shalabi asks students to extend the pattern
settings and situations, they begin to think about num-
bers in terms of number patterns, reasonableness, and made with the following attribute blocks. Benito
estimation. Questions such as what number is after 99, places a square then a triangle at the end of the pat-
what number is between 43 and 45, or which numbers tern. What response would be the best way to help
are odd and even serve as the foundation for number Benito?
sense with larger numbers, computation, and algebraic
thinking. Recognizing patterns in numbers up to 100 sets a. No, that is wrong. Put a circle then the square
the stage for understanding the Hindu-Arabic, or base- and triangle.
10, numeration system. Hindu-Arabic numeration is a
place-value system based on the number 10 that allows b. Read the pattern with me. What pattern do you
numbers of any size to be represented in an economical hear?
way. Developed over centuries, the Hindu-Arabic system
provides efficient algorithms for computation. c. How many shapes are in the pattern?
d. Let me show you how to finish the pattern.
Study Questions and Activities Answers: 4d, 5a, 6b.

1. Number sense is a major goal of instruction in num- Teacher’s Resources
bers. How aware of numbers are you in everyday
life? Do you notice how much groceries cost? how Kamii, C., & Housman, L. (2000). Young children rein-
much you tip a server? how long you wait on line? vent arithmetic: Implications of Piaget’s theory (2nd ed.).
how much you save on an item on sale? Williamston, VT: Teachers College Press.
Shaw, J. (2005). Mathematics for Young Children. Little
2. Ask several young children, ages 4, 5, or 6, to count Rock, AR: Southern Early Childhood Association.
aloud for you to 20. Then ask them to count a set of Wheatley, G., & Reynolds, A. (1999). Coming to know
12 pennies or other small objects. Record what each number: A mathematics activity resource for elementary
child does. Did you find any differences in their school teachers. Tallahassee, FL: Mathematics Learning.
rote and rational counting abilities? Did any of them Whitin, D., & Wilde, S. (1995). It’s the story that counts:
demonstrate the common counting errors listed by More children’s books for mathematical learning, K–6.
Gelman and Gallistel? What procedures would you Portsmouth, NH: Heinemann.
recommend for helping a child overcome counting
errors?

3. Read a counting book with a group of young chil-
dren. Observe their reactions to the pictures, numer-
als, and story. What did you learn about children’s
understanding of numbers and number concepts?

Wright, R., Martland, J., and Stafford, A. (2000). Early 159
numeracy. Thousand Oaks, CA: Corwin Press.
Pittman, Helena Clare. (1994). Counting Jennie. Minne-
Children’s Bookshelf apolis: Carolrhoda Books. (Grades K–2)

Anno, Mitsumasa. (1977). Anno’s counting book. New Schmandt-Besserat, D. (1999). The history of counting.
York: Harper-Collins. (Grades PS–3) New York: William Morrow. (Grades 4–6)

Bang, Mary. (1983). Ten, nine, eight. New York: Green- Zaslavsky, Claudia. (1989). Zero: Is it something? Is it
willow. (Grades PS–1) nothing? New York: Franklin Watts. (Grades 1–4)

Blumenthal, Nancy. (1989). Count-asaurus. New York: For Further Reading
Macmillan. (Grades PS–3)
Fuson, K., Grandau, L., & Sugiyama, P. (2001). Achiev-
Coats, Lucy. (2000). Nell’s numberless world. London: able numerical understanding for all young children.
Dorling Kindersley. (Grades 2–4) Teaching Children Mathematics 7(9), 522–526.

Crews, Donald. (1986). Ten black dots (rev. ed.). New Fuson and colleagues describe young children’s
York: Greenwillow. (Grades PS–3) developmental understanding of number and methods
to enhance understanding.
Dee, Ruby. (1988). Two ways to count to ten. New York: Huinker, D. (2002). Calculators as learning tools for
Henry Holt. (Grades K–3) young children’s explorations of number. Teaching Chil-
dren Mathematics 8(6), 316–322.
Ernst, Lisa Campbell. (1986). Up to ten and down again.
New York: Mulberry. (Grades K–2) Young children with calculators make dynamic
discoveries about numbers, counting, and number
Feelings, Muriel. (1971). Moja means one: A Swahili relationships.
counting book. New York: Dial. (Grades PS–3) Kline, Kate. (1998). Kindergarten is more than counting.
Teaching Children Mathematics 5(2), 84–87.
Freeman, Don. (1987). Count your way through Russia.
Minneapolis: Carolrhoda. (Grades 1–3) The ten-frame is a visual image for representing
numbers and counting ideas.
Hoban, Tana. (1985). 1, 2, 3. New York: Greenwillow. Pepper, K., & Hunting, R. (1998). Preschoolers’ counting
(Grade PS) and sharing. Journal of Research in Mathematics Educa-
tion 29(2), 164–183.
Hoban, T. (1998). More, fewer, less. New York: Greenwil-
low. (Grades K–2) Research report on early counting strategies used in
sharing situations.
Johnson, S. (1998). City by numbers. New York: Viking. Reed, K. (2000). How many spots does a cheetah have?
(Grades 2–6) Teaching Children Mathematics 6(6), 346–349.

Merriam, Eve. (1992). 12 ways to make 11. New York: Children explore the number of spots on a cheetah
Simon & Schuster. (Grades 1–3) and invent counting and estimation strategies.

Morozumi, Atsuko. (1990). One gorilla. New York: Farrar
Straus & Giroux. (Grades K–2)



CHAPTER 10

Extending
Number
Concepts and
Number Systems

s children master basic number concepts and can count
and write numbers through 20, understanding larger
numbers using the base-10 place-value system be-
comes the focus of instruction. Representing larger

numbers with the Hindu-Arabic numeration system is a
foundation for computation and number sense. Reasonable-
ness and estimation are essential for development of number sense—the
ability to think with and about numbers. In this chapter activities develop
understanding of place value and emphasize numbers sense.

In this chapter you will read about:
1 The essential role of number sense and number awareness in

elementary mathematics
2 Representing numbers in many forms
3 Using the base-10 numeration system to represent larger numbers
4 Activities and materials for teaching exchanging and regrouping

in base 10
5 Activities and materials for learning about larger numbers
6 Activities and materials for number sense, rounding, and estimation
7 Extending understanding of number through patterns, operation

rules for odd and even numbers, prime and composite numbers,
and integers

161

162 Part 2 Mathematical Concepts, Skills, and Problem Solving

NCTM Standards and two pets? three pets? How could we find out how
Expectations many pets we really have in our classroom?

Understand numbers, ways of representing numbers, rela- • The newspaper reports that the population of
tionships among numbers, and number systems. our state is growing by 30,000 each year. If 1 out
of 3 individuals is of school age, how many new
In prekindergarten through grade 2 all students should: schools do we need to build each year?
• Count with understanding and recognize “how many” in
Children with number sense see how numbers
sets of objects; are represented and operated on in various ways, al-
• Use multiple models to develop initial understandings of lowing them to use number flexibly in computation
and problem solving.
place value and the base-10 number system;
• Develop understanding of the relative position and mag- • Half a gallon of ice cream is 12, 0.5, or 50% of a gal-
lon or 2 quarts or 4 pints or 8 cups.
nitude of whole numbers and of ordinal and cardinal
numbers and their connections; • If an $80 coat is reduced by 25% on sale, one-
• Develop a sense of whole numbers and represent and use fourth of $80 is $20. The coat costs $60 plus tax.
them in flexible ways, including relating, composing, and
decomposing numbers; • One meter is slightly longer than 1 yard; running
• Connect number words and numerals to the quanti- a 100-meter race should take slightly longer than
ties they represent, using various physical models and running a 100-yard race.
representations.
In grades 3–5 all students should: • The product of 3.8 and 9.1 is approximately 4 ϫ 9
• Understand the place-value structure of the base-10 ϭ 36.
number system and be able to represent and compare
whole numbers and decimals; At the core of number sense is flexible under-
• Recognize equivalent representations for the same num- standing of numbers and how they can be repre-
ber and generate them by decomposing and composing sented in various ways. Rather than memorizing
numbers; rules, students are asked to develop numbers as
• Explore numbers less than 0 by extending the number line they solve problems, estimate, and draw reasonable
and through familiar applications. conclusions from numerical information.

Number Sense Every Day Teachers who engage children in mathematical
conversation encourage children to think about
Number sense should be an everyday event in class- numbers and their meaning. Ms. Chen wants to
rooms. Teachers stimulate children’s thinking with know whether her second-grade students recognize
and about numbers by posing questions based on different representations for 27 using a hundreds
daily occurrences. chart and base-10 materials.

• Last week, the estimation jar held 300 cotton Amani: Twenty-seven comes after 26 and before 28.
balls. This week it is full of golf balls. Do you think
the jar holds more than 300 golf balls or fewer Guillermo: Twenty-seven is three more than 24
than 300 golf balls? Why do you think so? and three smaller than 30.

• Our class has 22 students, and 20 are eating in the Lisette: Twenty-seven is two groups of ten and
cafeteria. If lunch costs 85 cents, will 20 lunches seven ones.
cost more or less than $20?
Yolanda: Twenty-seven is 10 more than 17 and 10
• Sada put 200 color tiles in the bag. When groups less than 37 on the hundreds chart.
each picked out five samples of 20, they reported
these results: Ian: I can count to 27 by threes: 3, 6, 9, 12, 15, 18, 21,
24, 27, but counting by fours is 28.
Red Blue Yellow Green
Jermaine: Twenty-seven is an odd number be-
Group 1 43% 24% 17% 16% cause you skip it when you count by twos.
Group 2 35% 28% 20% 17%
Group 3 39% 25% 21% 15% When making 27 with base-10 materials, some dis-
play 27 with two orange rods and seven white cubes;
What is your best estimate of the number of tiles
of each color in the bag?

• If everybody in our classroom had one pet, how
many pets would we have? What if everybody had

Chapter 10 Extending Number Concepts and Number Systems 163

10 ϩ 10 ϩ 7 MISCONCEPTION beads) are useful for rep-
resenting values to 1,000
10 ϩ 17 English counting words (see Photo 10.1).
above ten (eleven, twelve,
thirteen through nineteen) Place-value materi-
hide the place-value als can be either propor-
meaning. Eleven means tional or nonproportional.

5ϫ5ϩ2 10 ϩ 1, twelve is 10 ϩ 2, Proportional materials
30 Ϫ 3 and thirteen is 10 ϩ 3. show value with the size
Connecting the values of the materials. When the
28 Ϫ 1 from 11 to 19 to their manipulative piece repre-
place-value meaning is
3ϫ9
9ϫ3 important. Some teachers senting one unit is deter-
have the students use mined, the tens piece is
Figure 10.1 Several ways of representing 27 “count ten, ten plus one, 10 times larger, and the
ten plus two,” and so on hundreds piece is 10 times
others line up 27 loose cubes, 2 bean sticks and 7 as an alternative counting the tens place. Some ma-
beans, or 9 groups of 3 (Figure 10.1). language up to 19. Above

By observing and listening to children, the 20 the place-value system terials allow students to
teacher learns whether they understand many nu- and counting language create their own base-10
meral expressions for 27. From student comments correspond: 21 means 2 materials. A tower of 10
and demonstrations, Ms. Chen believes that students tens plus 1. Unifix cubes represents
have developed a flexible understanding of number
and its many representations, so she is ready to in- 10. If one tongue depres-
troduce the place-value system in a more compre-
hensive way. Extending their understanding of num- sor or popsicle stick is the unit, a bundle of 10 repre-
ber and place value to hundreds and thousands is
critical for understanding number operations with sents tens, and 10 bundles are 100. A paper clip is 1,
larger numbers.
a chain of 10 paper clips would be the tens, and ten

chains hooked together show 100. Base-10 blocks,

Cuisenaire rods, and bean sticks are proportional

materials with the tens unit already joined together.

Nonproportional materials can also represent

place values, although the one-to-ten relationship is

Understanding the Base-10 not shown in the size of materials. Money is the most
Numeration System
familiar nonproportional material. Dimes are not 10
When students are learning to count, they often think
of each number as having its own unique name. times as large as pennies, and dollar coins or bills
Numbers 0–9 do have unique names, but learn-
ing a different name for all the numbers from 10 to are not 10 times as large as dimes. Nonproportional-
999,999 would be impractical and impossible to re-
member. Instead the place-value system allows the ity makes money exchange confusing for younger
same 10 symbols to have different values depending
on the position in a numeral: 6 could have a value 1 2 3 4 5 6 7 8 9 10
of 6 in the ones position, 60 in the tens position, or 11 12 13 14 15 16 17 18 19 20
600 in the hundreds position. The hundreds chart 21 22 23 24 25 26 27 28 29 30
(Figure 10.2) illustrates for children the pattern of 31 32 33 34 35 36 37 38 39 40
tens and ones that structures the base-10 numera- 41 42 43 44 45 46 47 48 49 50
tion system. 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
Counting by tens and the hundreds chart intro- 71 72 73 74 75 76 77 78 79 80
duce the pattern of tens in the number system. Base- 81 82 83 84 85 86 87 88 89 90
10 materials represent numbers concretely. Com- 91 92 93 94 95 96 97 98 99 100
mercial and teacher-made materials (bean sticks,
bundles of stir sticks, graph paper, or strings of Figure 10.2 Hundreds chart (see Black-Line Master 10.1)

164 Part 2 Mathematical Concepts, Skills, and Problem Solving

Image not available due to copyright restrictions Image not available due to copyright restrictions

Courtesy of Didax Educational Resources, Peabody, MA
Courtesy of Scott Resources, San Francisco
c d
Photo 10.1 Base-10 materials: (c) Unifix cubes; (d) chip trading materials

children. They have to remember the relationship numeration system is also nonproportional because
rather than seeing the relationship. Chip trading numerals are evaluated by their position.
materials and an abacus are also nonproportional
materials. Value of a number is indicated by color Both proportional materials and nonpropor-
and position rather than by size. The Hindu-Arabic tional materials are useful, but proportional materi-
als emphasize the relative value of the places and

ACTIVITY 10.1 Trains and Cars (Representation)

Level: Grades K–2 • Let the students tell what they have in their bowl. Have
each pair compare the number of cubes in their trains
Setting: Pairs with those in other trains.

Objective: Students recognize 10 as an organizer for counting • Ask two groups to combine their trains and extra cars to
numbers larger than 10. see how many trains they have together.

Materials: Unifix cubes • Count trains by tens, and ask students if any group has
10 trains. Emphasize that 10 trains is 1 hundred.
• Give pairs of children a bowl of Unifix cubes. Have each
student count 10 single cubes and put them together to
make a train.

• Ask them how many trains they can make with the cubes
in their bowl. Ask how many trains and how many extra
cars.

Chapter 10 Extending Number Concepts and Number Systems 165

are recommended for initial instruction in the base- and activities illustrate exchanges with any place-
10 system. Children enjoy making a set of base-10 value manipulatives. Activity 10.3 is an important in-
materials by bundling stirring sticks to show groups formal introduction to creating numbers up to 100 or
of 10, then bundling 10 groups of 10 into a big bundle 1,000 by accumulating and trading up with propor-
of 100. Working with a variety of materials shows tional materials of beans and sticks. Trading down
children that place value is a characteristic of the begins with the 10 tens sticks, and children remove
number system rather than a particular manipula- beans and sticks as they roll the die. This activity
tive. Activity 10.1 introduces place value with Uni- also serves as an informal assessment of children’s
fix cubes and trains of 10. Activity 10.2 introduces understanding of place value. Nonproportional ma-
a train-car work mat with a tens column and a ones terials, such as poker chips or color tiles, can be
column. Additional task cards guide children during used to play the same game. Beans spray-painted
independent learning activities. yellow, blue, green, and red are inexpensive substi-
tutes for commercial materials. Game mats can be
Place-value materials and activities continue made of file folders or tagboard.
through grade 6 as students represent larger num-
bers and model problems in addition, subtraction, Through teacher questioning with physical mod-
multiplication, and division. Children show their un- eling of place value, students begin to understand
derstanding of place value with materials, actions, how place value works and should become comfort-
simple diagrams (e.g., Figure 10.3), and numerals. able with physical representations of larger values.
Exchanging 10 ones for 1 ten and 10 tens for 1 hun- As they progress, students discuss how many hun-
dred establishes the place-value pattern for thou- dreds, tens, and ones are in each number they cre-
sands, millions, and billions. Children notice that ate. Symbolic representation also begins. Activity
the 9 is a precursor for the next larger place. The cal- 10.4 uses children’s names as the source for count-
culator is a learning tool for place value as students ing ones, tens, and hundreds. In Activity 10.5 players
repeatedly add 1 and note changes each time 9 is have seven turns to accumulate 100 points without
reached, such as 99 ϩ 1 becomes 100. Introduction going over. The game can be played with place-
of the exchange from 99 to 100 is often celebrated value materials on a game mat or with symbols as
on the hundredth day of school to call attention to students are learning addition with regrouping.
the importance of 100. Students bring displays of
100 beans, 100 cotton balls, and 100 peanuts. Assessing Place-Value Understanding

Figure 10.3 Example of a simple diagram of base-10 Some students experience difficulty in understand-
blocks showing 237 ing place value. By working at the concrete level and
not rushing the transition to symbolic representa-
tion, most students construct place-value meaning
by the third grade. However, some students have
continuing difficulty with the meaning of tens and
ones. Kamii (1986) developed a structured interview
for place value. The interview can be done with any
number using the steps shown in Activity 10.6.

Exchanging, Trading, or Regrouping MULTICULTUR ALCONNECTION

When working with place-value materials, children English counting language may also contribute to children’s
trade, or exchange, ones for tens and tens for hun- confusion about tens and ones. In Asian and Latin-based
dreds, or the reverse. Exchanges between place languages, counting language emphasizes the place-value
values is most accurately called regrouping and structure (Table 10.1). In Spanish, for example, 16 is diez y
renaming, but trading up and trading down are seis, or “10 and 6.” Students from various cultures might be
common terms. Both are more accurate terminol- invited to teach counting words in their language. With a
ogy than “carrying” or “borrowing” because they ac- chart such as Table 10.1, students notice the patterns based
curately describe the physical actions. Many games on tens in languages such as Chinese.

166 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 10.2 Train-Car Mats

Level: Grades 1–2 Variation
• Use counting chips, tongue depressors, or stirrers with a
Setting: Pairs
place-value mat.
Objective: Students use a two-column mat as a structure for place • Have the children stack the chips or bundle the tongue
value.
depressors or stirrers with a rubber band.
Materials: Unifix cubes, two-column mat • Ask children where the bundle of 10 should go on the

• Give pairs of students sets of 12–30 Unifix cubes and a place-value mat.
two-column mat. • Discuss 11 as “10 and 1,” 12 as “10 and 2,” and so on.
• Provide task cards for students to continue work
• As children assemble trains with 10 cubes, have them
put them in the left column on the mat. Have them put independently.
extra cars in the right column. Ask students to report how
many items they have as “___ trains and ___ cars.” BEAN STICKS

• As students understand the format, work up to sets ten ones
between 30 and 100.

• Provide index cards with numerals on 1
them so that students can label the
columns: 2 tens or 20 with index cards
in one color, and 11, 12, 13 in another
color.

2
ten ones

3
ten ones

(a) 1 CUISENAIRE RODS
2 ten ones
Trains Cars 3 ten ones

(b) ten ones
UNIFIX CUBES
1 with 16 grouped by tens

ten ones
(c)

Chapter 10 Extending Number Concepts and Number Systems 167

ACTIVITY 10.3 Beans and Sticks (Representation)

Level: Grades 2–5 • The game continues into hundreds using a three-column
mat, or to thousands with a four-column mat.
Setting: Small groups of 3 or 4 students
• Give each group two spinners or two dice, and ask them
Objective: Students represent numbers in the place-value system. to decide how to use two numbers on each turn.

Materials: Beans, bean sticks of 10, and bean flats of 100 (or other • Assess student knowledge and skill with a checklist
base-10 materials); two-, three-, or four-column mat; number identifying skills: count to 10; exchange accurately, call the
spinner or number cubes correct number; write the numeral; explain tens and ones
in number.
• Play a game with beans, sticks, the place-value mat, and a
spinner or number cubes. Each player spins or rolls, counts Variation
that many beans, and puts them in the ones column. Play-
ers should take turns spinning or rolling. Play the same game with nonproportional materials, such
as yellow, blue, green, and red chips or poker chips. Label
• As each player accumulates 10 loose beans, he or she the game mat with colors, starting with yellow at the
trades 10 beans for a bean stick and puts it on the mat right, then blue, green, and red on the left. Each group
in the tens place. Play continues until all players have ten needs 5 red chips and 40 each of green, blue, and yellow
bean sticks and can trade for a bean flat. chips, one game mat for each player, and one die.

• After students have had ample time to play the game, ask
them what they noticed about playing the game. Empha-
size the trading rule and the number of chips that can be
in each color space.

• Students can experiment with other trading rules, such
as “trade 4” or “trade 9.” Younger children can also play
the game with a trading rule such as “trade 4” to practice
counting to 4.

• Players start with a red chip and trade down until one
player has cleared her or his mat.

50 ϩ 6

Working with Larger Numbers coming up with a total. Population of a city, for ex-
ample, is done by counting how many people live
Large numbers are encountered as children think in specific blocks or areas and adding for a grand
about stars in the sky, pennies in a piggy bank, total. Computer-based inventory systems keep track
or a large bag of popcorn kernels. Interest in big of large inventories in supermarkets and retail stores
numbers and their names helps students explore after inventories are taken by counting the number
the meaning and representation of numbers larger of items on the shelves of each store.
than 1,000. Numbers greater than 1,000 are seldom
counted physically in real life. Although people Manipulative work with numbers 1 through 1,000
could count to 10,000 or 1,000,000, they seldom actu- sets the stage for understanding how the base-10
ally physically count that many objects. Instead they number system works. Thousands, millions, or bil-
represent large number values symbolically. Even if lions follow the place-value pattern established with
counting a large number is necessary, people usu- ones, tens, and hundreds and extends students’
ally create many smaller amounts and aggregate the understanding that number is abstract and infinite
total. A bank teller creates rolls of pennies, nickels, and that the place-value system represents infinitely
dimes, and quarters and bundles of bills before large numbers in a compact form. Two books by

168 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 10.4 E-vowel-uation (Representation)

Level: Grades 2–5 • Ask what they notice about the values of their names
Setting: Small groups or whole group with a new rule. Some children will have the same value
Objective: Students recognize that the position of the numeral under both rules. Some will notice that their name value
switched from 35 cents to 53 cents. Ask why their names
determines its value. had the same or a different value.
Materials: Index cards
• Line up again using the new evaluation to see if the lineup
• Ask students to write their first name (or first and last is the same or different.
names) in block letters on one side of an index card.
Announce that vowels are worth 1 cent each and that Variations
consonants are worth 10 cents each. Have them count or
add up the value of their names. • Put the cards on a bulletin board or in a card box, and
order them from high to low or from low to high.
KARA THOMPSON KRTHMPSN ϭ 80 cents AAOO ϭ 4 cents ϭ 84 cents
• Using the two rules, look for names or words worth
JUAN RODRIGUEZ JNRDRGZ ϭ 70 cents UAOIUE ϭ 6 cents ϭ 76 cents $1.00.

• Ask students to line up from the lowest total to the high- • Using the two rules, look for names with large totals and
est total. Look at names that have low totals and high small totals.
totals. Ask students to suggest reasons for the low and
high totals. • Have capital letters worth a bonus of $1 each. Most stu-
dents will have a $2 bonus, but some may have $3 or $4.
• Change the rule. Consonants are worth 1 cent each, and
vowels are worth 10 cents each. Ask students if they think • Use a different system for evaluating letters, such as place
the total of their name will be larger or smaller. Let them in the alphabet (A ϭ 1, B ϭ 2, etc.) or Scrabble scoring for
calculate the new totals. letters.

ACTIVITY 10.5 Seven Chances for 100 (Reasoning)

Level: Grades 2–5 Tens Ones

Setting: Small groups or pairs 1
2
Objective: Students apply place value in a game and develop a 3
strategy. 4
5
Materials: Die, base-10 blocks, two- or three-column place- 6
value mat 7

• Organize groups of two to four children. Players take turns
rolling the die. On each roll, students can decide whether
the number will go with the tens or the ones: a 5 on the
die can be worth either 50 (5 rods) or 5 (5 cubes). Each
player will have exactly seven turns to get as close to 100
points without going over. During play, students trade 10
units for a tens rod. Remind students that they have to
take all seven turns.

• After playing for several days, ask students if they have
developed a strategy for getting close to 100 without
going over.

• After playing the game with the base-10 materials, some
students can write the scores on a tens and ones chart
and keep a running total.

Chapter 10 Extending Number Concepts and Number Systems 169

ACTIVITY 10.6 Place-Value Assessment

Level: Grades 1–4 developmental activities as needed. Concrete models for
trading are essential to illustrate the dynamic nature of
Setting: Individuals place value.

Objective: Students demonstrate an understanding of place value.

Materials: Cubes, paper and pencil

Based on the interview by Kamii (1986), ask students to
identify the value of a two-digit numeral as well as the
value of each numeral. The interview proceeds in five
steps:

1. Give a student 12 to 19 cubes, and then ask the
student to count out 16 cubes and draw a picture of
them on paper.

2. Ask the student to write the numeral (e.g., 16) for the
number of cubes.

3. Ask the student to circle the number of drawn cubes
shown by the numeral 16.

4. Point to the 6 in the numeral 16, and ask the student
to circle the number of cubes that goes with that
numeral.

5. Point to the 1 in the numeral 16, and ask the student
to circle the number of cubes that goes with that
numeral.

Evaluate the students’ understanding of the number.
The figure illustrates responses to the interview. Many

students circle the 16 cubes for “16” and 6 cubes for the
“6” correctly, but for the “1” they circle only one block
instead of ten. Based on her interviews, Kamii concluded
that misconceptions about place value persist into third
or fourth grade. Whether this problem is due to matura-
tion or inappropriate instruction is not clear, but teach-
ers should be sensitive to students and provide more

TABLE 10.1 • Counting Language: English Versus Chinese

English Chinese English Chinese English Chinese
twenty-one er-shi-yi
one yi eleven shi-yi twenty-two er-shi-er

two er twelve shi-er

three san thirteen shi-san

four si fourteen shi-si

five wu fifteen shi-wu thirty san-shi

six liu sixteen shi-liu

seven qi seventeen shi-qi

eight ba eighteen shi-ba forty si-shi
fifty wu-shi
nine jiu nineteen shi-jiu

ten shi twenty er-shi

170 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 10.7 Think of a Million (Reasoning and Representation)

Level: Grades 3–6 • Introduce Schwartz’s book, and read through it, stopping
to discuss the illustrations and questions posed in the text.
Setting: Whole group
• Ask students to suggest other things that could be a
Objective: Students visualize the magnitude of large numbers. million. Cooperative groups select a topic and determine
how to make a million. Students may wish to refer to
Materials: How Much Is a Million? by David Schwartz, illustrated Schwartz’s calculations in the back of the book.
by Steven Kellogg (New York: Lothrop, Lee & Shepard, 1985);
package of popcorn kernels or dried beans Extension

• Show students a package of popcorn kernels or dried Another book by David Schwartz, also illustrated by Ste-
beans. Ask if they think there could be a million kernels ven Kellogg, is If You Made a Million (New York: Lothrop,
in the package. Have students count the number in one Lee & Shepard, 1989). This book engages children in
package. thinking about money and the responsibilities that come
with large amounts of it.
• Based on the number, ask students how many packages
they would need to get to a million. Use multiplication by
10 and 100 as a quick way to determine the number of
packages. A rate table is useful for this.

Packages 1 10 100 200 300 400

Number 2,317 23,000 230,000 460,000 690,000 920,000

ACTIVITY 10.8 Spin to Win (Representation and Reasoning)

Level: Grades 3–6 • Spin, roll, or draw cards, and have students fill in their
forms. After four numbers have been picked, ask students
Setting: Whole group to tell the largest number anyone made. Ask for the larg-
est number possible.
Objective: Students demonstrate place-value knowledge and
develop a strategy. • In cooperative groups, have students take turns at the
spinner while other students fill in the place-value chart.
Materials: Spinner, numeral cards (0–9), standard die or a die with
numerals 0–10 for each group • Vary the game goals: Make the smallest numeral with the
four numbers called out. Make a number that is closest to
• On the overhead projector, show a transparency with four 7,000. Make a number between 4,000 and 6,000.
boxes labeled thousands, hundreds, tens, and ones. Have
students make a similar board on their paper. • The number of places can be reduced to three for younger
students and increased for older students.
Thousands Hundreds Tens Ones
Variation
• Announce the goal of the game, such as making the larg-
est four-digit number. After students are proficient with the game, the game
goal and board can be changed to include addition,
• Using a spinner, number cards, or a die to randomly subtraction, multiplication, and division. Create a game
generate numbers, call out one number at a time. After board with three blanks on the top line and two on the
each number is called, write it in one of the four place- bottom line. Place numbers in the blanks to make the
value boxes on the overhead. Once a number is placed, it largest sum, the smallest sum, the largest difference,
cannot be moved. the smallest difference, the largest product, the smallest
product, or the smallest dividend. Students can create
their own boards and game goals.

Chapter 10 Extending Number Concepts and Number Systems 171

ACTIVITY 10.9 Big City (Reasoning and Connections)

Level: Grades 3–5 • Which cities have populations of more than 5 million?
• Which cities have more population than Dallas and
Setting: Whole group
Houston put together?
Objective: Students find data with large numbers to use for iden- • Which cities are 4,000,000 larger than another city?
tifying place value, comparing numbers, ordering, rounding, and • Round the populations of all the cities to the nearest
estimating.
million.
Materials: Table of data on populations • Which cities have half the population of Chicago?

• Ask students to list large cities in North America. They Extensions
may want to think of professional sports teams.
• Find population data for the 10 largest cities and towns
• Explain that large cities are often part of a metropolitan in your state. Put the cities and their populations on index
area that includes a central city or cities plus the suburbs cards for grouping and ordering.
and surrounding smaller towns.
• Locate the 15 largest U.S. metropolitan areas on a map
• Display a table showing the populations of the 15 larg- of the United States. Are there any patterns to where the
est metropolitan areas, and have students organize the cities are located?
data. These data may be found in an almanac or on the
Internet. • Compare populations in 2000 to populations in 1950.
What cities have been added, and which have lost rank-
• In cooperative groups, have students create three to five ings? What factors might contribute to the changes?
questions to be answered with the data display.

Rank Metropolitan Area Name State 2000 Population

1 New York–Northern New Jersey–Long Island NY, NJ, CT, PA 21,199,865
2 Los Angeles–Riverside–Orange County CA 16,373,645
3 Chicago-Gary-Kenosha IL, IN, WI
4 Washington-Baltimore DC, MD, VA, WV 9,157,540
5 San Francisco–Oakland–San Jose CA 7,608,070
6 Philadelphia–Wilmington–Atlantic City PA, DE, NJ 7,039,362
7 Boston-Worchester-Lawrence MA, NH, ME, CT 6,188,463
8 Detroit–Ann Arbor–Flint MI 5,819,100
9 Dallas–Fort Worth TX 5,456,428
10 Houston-Galveston-Brazoria TX 5,221,801
11 Atlanta GA 4,669,571
12 Miami–Ft. Lauderdale FL 4,112,198
13 Seattle-Tacoma-Bremerton WA 3,876,380
14 Phoenix-Mesa AZ 3,554,760
15 Minneapolis–St. Paul MN, WI 3,251,876
2,968,806

SOURCE: http://geography.about.com/library/weekly/aa010102a.htm

David Schwartz, How Much Is a Million? (1985) and and also discuss meanings related to the location
If You Made a Million (1989), invite children to imag- and other characteristics of large cities.
ine the magnitude of large numbers. In Activity 10.7,
How Much Is a Million? stimulates investigation into Real-world settings provide context for large
ways of representing large numbers. numbers. When children understand how the place-
value system represents large numbers, they begin
In Activity 10.8 children create the largest or to comprehend such things as the size of the popu-
smallest possible numeral with four rolls of a die or lation of the United States relative to that of other
spinner. As they play, students develop strategies to countries (Table 10.2).
maximize the number and develop an understand-
ing of chance. Populations of cities provide mean- Newspapers and almanacs are also resources
ing for large numbers in Activity 10.9. Students com- for a “big number” search. When students find
pare, order, and combine the populations of cities big numbers, they write them on index cards with
what they represent and put them on a classroom

172 Part 2 Mathematical Concepts, Skills, and Problem Solving

TABLE 10.2 • Populations of Selected States is to separate each period of three numerals
Countries (2006) with commas. Internationally, spaces are often used
instead of commas to separate the period group-
China 1 313 973 700 ings, as shown here:
India 1 095 352 000
United States United States 489,321,693,235
Brazil 300 176 000 International 489 321 693 235
Japan 188 078 000
Mexico 127 463 600 E XERCISE
France 107 449 500
Italy Do you prefer using spaces or commas for separat-
60 876 100 ing the periods in large numbers? What are the
58 133 500 advantages and disadvantages of each notation

system? •••

SOURCE: http://geography.about.com/cs/worldpopulation/a/most

chart (Figure 10.4). The cards can be classified, com- After establishing the meaning of numbers with
pared, and sequenced. Big numbers with meaning base-10 materials, students are ready to record
motivate children to learn names for large numbers: numbers in various ways, including compact and
millions, billions, trillions, and so forth. expanded numeral forms. The compact form for nu-
merals such as 53,489 is most common, but it can be
Big numbers represented in several ways:

10,000,000 1,000,000 100,000 10,000 1000 0 With words: Fifty-three thousand four hundred
and larger to to to to to eighty-nine
999
9,999,999 999,999 99,999 9,999 Hundreds With numerals and words: 5 ten thousands, 3 thou-
Millions Hundred Ten Thousands sands, 4 hundreds, 8 tens, 9 ones; or 53 thousands,
thousands 4 hundreds, 8 tens, 9 ones
thousands
With numerals: 50,000 ϩ 3,000 ϩ 400 ϩ 80 ϩ 9
Figure 10.4 Classification chart for big numbers
With expanded notation: (5 ϫ 10,000) ϩ (3 ϫ 1,000)
A place-value chart to millions or billions high- ϩ (4 ϫ 100) ϩ (8 ϫ 10) ϩ 9
lights the structure, pattern, and nomenclature of With exponents: 5 ϫ 104 ϩ 3 ϫ 103 ϩ 4 ϫ 102 ϩ 8 ϫ
the numeration system. Each grouping of three 101 ϩ 9 ϫ 100
numbers is called a period; the three numbers in
the period are hundreds, tens, and ones. Students Exponential notation for large and small numbers
should be familiar with the smallest period and is written with powers of 10 or exponents: 20,000
how to read numbers having three numerals, for becomes 2 ϫ 10,000 or 2 ϫ 104. Intermediate- and
example, 379 as three hundred seventy-nine. If the middle-grade students learn scientific or exponen-
number were 379,379, the second period would be tial notation for larger numbers, such as distances
called three hundred seventy-nine thousand. The in space.
convention for writing large numbers in the United
Thinking with Numbers

While learning about the Hindu-Arabic numeration
system, students also become aware of how impor-
tant numbers are in daily living. Numbers also pro-
vide useful information for the classroom. Becom-
ing aware of the role of numbers is a first step in
number sense.

Chapter 10 Extending Number Concepts and Number Systems 173

• How much does a gallon of milk cost? used in estimation. If somebody wanted to know the
average number of passengers that traveled each day
• How far is the grocery store, and how long does it of the year, they could divide 1,212,678 by 365 days
take to walk? to drive? to take the subway? and get a computed answer that 3,322.4054 people
traveled each day. However, the computed an-
• How long will it take to do my homework? Do I swer is not a meaningful answer for several
have time before supper to finish? reasons, especially the 0.4054 person. Reasonably
rounded numbers allow ease of computation or
• If I need 12 pages of paper for my booklet, how even mental computation.
many pages does everyone in the class need for
their projects? • Think: 1,200,000 divided by 300 is 4,000.

The answers to many questions are estimations and • Think: 1,200,000 divided by 400 is 3,000.
approximations.
A reasonable estimate would be between 3,000 and
• Milk is about $3 a gallon. 4,000—maybe 3,500 passengers on average. An es-
timate communicates numerical information with
• The grocery store is about 5 miles away. It takes meaning. Students might also note that the number
10 minutes by car, 75 minutes walking, and of passengers is larger than average on the busi-
20 minutes by subway. est travel days of the year and less than average on
other days.
• I can finish this homework in 25 minutes and
still have 15 minutes to play basketball before E XERCISE
dinner.
What times of the year do most airports have the
• We are going to need about 250 pages of paper largest and smallest passenger counts? Which days
for 20 projects. might be busiest for airports in Orlando? Puerto

People learn from experience to make educated Rico? Denver? •••
guesses. Working with large numbers is a good time
to emphasize number sense, the ability to think Students learn to think about rounding and es-
about numbers in meaningful and reasonable ways. timation in the context of working with larger num-
Exact answers are not always required. If the airport bers. Teachers report that students resist estimation.
reported 1,212,678 outgoing passengers this year, is Students who have been taught about numbers with
that number absolutely accurate? Is it possible that an emphasis on accuracy and getting the right an-
somebody did not get counted? Does 1 person or swer may not be comfortable with “close enough” as
even 10 or 1,000 people make that much difference an answer. Children may also resist textbook exer-
in a number the size of 1 million? cises in rounding and estimation because the num-
bers are small enough that they can be calculated
Rounding and estimating are important nu- and understood without rounding or estimation.
merical thinking skills. Rounding expresses vital Should students estimate the sum of 34 ϩ 47 when
information about a number without being unnec- they can easily calculate it? Rounding down to 30
essarily detailed. The number of passengers in the
airport could be expressed as “more than 1 million,”
“1.2 million,” or “nearly one and a quarter million.”
Rounding emphasizes the important information
without the less important details. Different people
might need more or less precision in a number. The
airport manager and board may need the detailed
information, but the citizens only need to know “a
little more than 1 million.”

Estimation is a reasonable guess, hypothesis,
or conjecture based on numerical information. It
is more than rounding, although rounding is often

174 Part 2 Mathematical Concepts, Skills, and Problem Solving

and up to 50 for an estimated answer of 80 is more 67
trouble and work than adding the two numbers and
getting the accurate answer 81. Numbers less than (a) 10 20 30 40 50 60 70 80 90 100
1,000 may be used to introduce the processes, but 0
rounding and estimation should move quickly to
examples that illustrate their utility with larger num- 62 65
bers and their meaning. (b) 67

Rounding 60 70

Thoughts about rounding begin when children com- 65 67
pare numbers and see them on a number line or
hundreds chart. Using a number line, students visu- 62
alize the order and relative position of numbers and
discuss how numbers relate to each other. (c) 70
60
• What number comes after 38? 49? 87?
(d) 383 350
• What numbers are found between 38 and 52? Is 300 400 300 383
43 closer to 38 or to 52? Is 49 closer to 38 or to 52?
400
• What is the hundred after 365? What is the hun-
dred before 365? 834 850

• What is a number between 300 and 400? Is it 834
closer to 300 or 400?
(e) 900 800 900
• Is 385 closer to 300 or to 400? Is 307 closer to 300 800
or to 400?
Figure 10.5 Number lines for rounding
Rounding is introduced with numbers between 0
and 100. When looking at 67 on a number line (Fig- 600 and 700, the convention is to round it upward
ure 10.5a), children can see that 67 is between 60 to 700. After learning the process of rounding to the
and 70 but only three steps from 70. They recognize nearest ten and hundred, students can use the same
that 67 is nearer to 70 than to 60. On the other hand, process to round to any place value depending on
62 is closer to 60 than to 70 (Figure 10.5b). Because the precision wanted. When rounding is taught as a
65 is five steps from 60 and five steps from 70, they thinking process rather than as a mechanical one,
learn that midway numbers are usually rounded up- students ask themselves whether 74,587 is closer to
ward, so 65 is rounded to 70 and 650 is rounded to 74,000 or to 75,000.
700. Some teachers have students draw the number
line between 60 and 70 as a hump with 65 at the top • The population of our county is 74,587. Round
(Figure 10.5c). This visual clue shows that numbers to the nearest thousand and to the nearest ten
below 5 slide back and numbers above the midway thousand.
point slide forward.
• Round the population of the United States in 2006
When rounding to the hundreds, students find to the nearest million.
the numeral in the hundreds place, think next larg-
est hundred, and determine whether the tens and Some students may talk through the process or cir-
ones are more or less than 50. They can draw a cle numbers as a reminder. Having verbal and visual
number line segment, such as Figure 10.5d, to see cues helps students think about the number and
whether 383 is closer to 300 or to 400. Likewise, 834 process of rounding as they become more skilled.
is rounded to 800 because it is closer to 800 than to Once children understand rounding numbers, prob-
900 (Figure 10.5e). Because 650 is midway between lem-solving activities provide practice and applica-
tion of the process. The task card shown in Figure
10.6 provides practice with rounding numbers from
real situations.

Chapter 10 Extending Number Concepts and Number Systems 175

Figure 10.6 Round each number to the nearest hundred, then to the nearest thousand:
Problem card used to practice
rounding numbers • The baseball game was attended by 3291 fans.

• The trip from London, England, to Sydney, Australia, stopped in New York.
The total distance was 14,648 miles.

• The telethon collected $113,689 for multiple sclerosis.

Think about each situation. Does it make more sense to round to the hundreds
or to the thousands?

E XERCISE timate. Without a benchmark, students have a hard
time making a reasonable estimate and refining
What are your answers to the task card in Figure it. Students then resort to wild guessing instead of
10.6? What process did you use to round the num- thoughtful estimation. Activities with an estimation
bers? Was your process more a step-by-step pro- jar help students develop their idea of number. As
cess that you learned or more thinking about the students fill jars with different objects, they can use
benchmarks to explain their estimates.
situation? •••
• “If the baby food jar holds 943 popcorn kernels,
Estimation the jelly jar should hold about 3,000 because it is
about three times as big.”
When people do not have specific information or
need a precise answer, they often make an edu- • “The mayonnaise jar holds 4,000 popcorn ker-
cated guess. nels. It might hold 100 cotton balls because the
cotton balls are much bigger than popcorn. But
• How much time should I plan for commuting this cotton balls will squeeze up so it might hold more
morning? than 100.”

• How much money do I need for a vacation at the • “The jar holds 126 jelly beans, but I think the num-
beach? ber of marshmallows will be fewer than 126 be-
cause they are bigger; maybe 50 marshmallows?”
• Would a couch or two loveseats fit best in the
room? • “The little jar is less than half as big as the liter jar;
maybe it holds 400 milliliters to 500 milliliters.”
• How much have I spent on groceries today?
The goal is to find an acceptable range of estimates
An estimate is an educated or reasonable guess and to recognize whether an estimate is out of that
based on information, prior knowledge, and judg- reasonable range. Through activities such as Activ-
ment. Even when information is known, the situation ity 10.10 with an estimation jar, students learn the
may call for estimation. The carpenter measures the difference between wild guesses and reasonable
room for flooring as 10 feet by 1212 feet, calculates estimates. The difficulty of the estimate depends on
the area, then increases the estimate by 10% to ac- the size of the jar and the size of the objects. It can
count for waste, matching pattern, or irregularities be adapted for kindergarten so that the number of
in product. A chef decides how much meat, pasta, objects is 20 to 30 or into hundreds for older chil-
vegetable, and bread must be stocked by estimating dren. Teachers should take care not to give prizes
the number of customers who are likely to order dif- for the closest estimate but recognize all students as
ferent meals. they make reasonable estimates.

Estimation of quantity improves when a bench-
mark, or referent, is used. A benchmark gives stu-
dents a comparison unit or amount to use for an es-

176 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 10.10 How Many Beans? (Reasoning)

Level: Grades 3–5 • Discuss the first estimates: high, low, middle, cluster of es-
timates, range (high to low). Students can also talk about
Setting: Whole group how they determined their estimates.

Objective: Students estimate the number of beans in a glass jar. • Provide a benchmark for their estimate either in a smaller
jar or the same size jar with a set number of jelly beans,
Materials: Jar filled with jelly beans (lima beans, etc.), smaller jar or such as 50.
cup, extra beans, ruler
• Ask if students want to revise their esti-
• Display the jar filled with beans. Ask for first estimates, mates based on the new information.
which can be written on a chart. Tell students to use any
strategy they wish—short of removing the beans from • Post the revised estimates, and compare
the estimation jar and counting them. When students them.
estimate for the first time, they will be guessing.
• Have students count the jelly beans in
groups of 10 or 20. Post a final number.
Compare second estimates to the

posted number. Avoid a “winner,” but ask students to
decide which estimates were reasonably close to the ac-
tual number. The goal of an estimation jar is to find all the
estimates that were reasonable.

• Display estimate jars of different sizes with different sizes
of beans and different objects, such as marbles, rice, or
packing peanuts.

MISCONCEPTION Problems related to do you think will evaporate at a temperature of
other content and real life 95 degrees?
Many students feel estima- also involve estimation.
tion results in the “wrong” Estimating packages of • If the car travels 408 miles on a tank of gas, how
answer, especially if popcorn for a class party is many tanks are needed for a 1,000-mile trip?
mathematics emphasized based on past experiences
getting the right answer of popping corn and how In addition to numerical situations, estimation
rather than number sense. much popcorn each child applies to problems and situations in geometry,
Students need comfort will eat. Children’s prior measurement, statistics, probability, and fractions.
and skill for thinking knowledge and number Activity 10.11 suggests how to develop estimations
about “close enough.” awareness are essential using information from previous events.
Problems and projects to estimation and reason-
that require approximate ableness. Students can Addition, subtraction,
rather than exact answers also predict future events multiplication, and divi-
help students develop based on information. sion calculations with
skill and confidence. larger numbers are often
estimated because exact
• A population graph for the United States shows an answers are not needed.
increase from 250,000,000 to 275,000,000 between The following problems
1990 and 2000. What do you estimate it to be by can be answered by com-
the year 2008 if the growth rate stays the same? putation, by calculator, or
What if the growth rate increases; what popula- by estimation. Rounding is
tion would you estimate in 2010? useful in finding numbers
that are easier to estimate.
• If the water in a jar evaporates 1 centimeter per
day at a room temperature of 75 degrees and • If the largest city in the state has a population
5 centimeters per day at 85 degrees, how much of 2,218,899, and the second largest city has a

Chapter 10 Extending Number Concepts and Number Systems 177

ACTIVITY 10.11 Snack Stand Supply Problem (Reasoning)

Level: Grades 3–6 • Organize cooperative groups and describe their task.
Setting: Small groups 1. Have students use the information from prior years
Objective: Students estimate, using data from a classroom project. to estimate how much food they should have for the
Materials: Table with data from an earlier fundraising effort. 2007 sale.
2. Have students explain how they came up with each of
Each year the fifth-grade class of Hopewell School raises their estimates.
money for field trips by selling food at the spring parents’ 3. Ask the students to write down any question or issues
meeting. Over the past six years, they have sold cookies, that came up in their estimations that they believe
popcorn balls, and fruit punch. would improve their estimate.
4. Have each group compare its projections with those
Year 2001 2002 2003 2004 2005 2006 of other groups. Have the groups tell whether their
98 108 117 130 126 142 estimates are similar to or different from other groups’
Cookies 72 87 99 91 101 113 estimates.
Popcorn 123 143 158 160 165 172
balls • Use newspapers or almanacs to find other data that show
Fruit changes over time, such as population or crops. Develop
punch some questions for another cooperative group to answer
using the data and estimation.

population of 1,446,219, approximately how many Questions such as these can be placed on index
people live in these two cities? cards for an estimation center. Additional exercises
involving number sense and estimation are found
• The state budget is $5,251,793,723. Of that amount, in Chapter 12.
$2,463,723,192 is spent for education. What per-
cent of the budget is spent on all other expenses? E XERCISE

• The chocolate factory produces an average of Answer the previous four questions using estima-
57,123 chocolate bars each week. How many tion. How accurate do you think your estimate
chocolate bars are likely to be produced in a year? needs to be? How did you think about your esti-
mate? What other factors might be considered in
• The chocolate bars are packaged in boxes of 48.
How many boxes are needed each week? explaining your estimate? •••

Research for the Classroom •
Montague and van Garderen (2003) compared the
Recent research studies have investigated the computa- estimation strategies used by learning-disabled, average,
tional strategies used by children as well as their under- and gifted students in the fourth, sixth, and eighth grades.
standing of the strategies they have been taught. The They found that all three groups scored poorly on estima-
conclusion is that students can develop computational tion. However, the LD students used fewer strategies and
strategies, although not all students develop the same were less successful overall than the average or gifted
level of skill with strategies or from the same instruction. students.
Ainsworth, Bibby, and Wood (2002) worked with
Murphy (2004) interviewed three children to determine forty-eight 9- and 10-year-old students on their estimation
whether they were using compensation in two-digit addi- skills using a computer program that provided feedback
tion problems. She found that one used primarily counting about the accuracy of their estimation. The program
on and two employed both counting on and the associa- guided students’ estimation process using front-end
tive law to raise one number to a multiple of 10, then and truncation strategies. Students in the control group
added the remainder. She instructed the three students showed no improvement, but students with pictorial and
on compensation (adding 19 by adding 20 and subtract- numerical feedback reduced the number of errors in their
ing 1) and interviewed them a week later and found each estimation. Differences were found in student understand-
modified the taught strategy. The researcher interpreted ing of the pictorial and numerical representations.
the results as supportive of the constructivist approach
because all three had the same instructional experience,
but each had developed a slightly different understanding.