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Number relationships in preschool

Author(s): Myoungwhon Jung

Source: Teaching Children Mathematics , May 2011, Vol. 17, No. 9 (May 2011), pp. 550- 557

Published by: National Council of Teachers of Mathematics

Stable URL: https://www.jstor.org/stable/41199776

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Number relationships in preschool

Try these early-learning experiences to strengthen understanding

of number and quantity.

By Myoungwhon Jung

Showing her four- and five-year-old students a picture of six dots in a domino arrangement, Becky asks, "How many are there?"

Ethan quickly responds, "I saw a number. It looked a little more than five."

Sam, who is sitting next to Ethan, argues, "No, that's six. Here are three on the top, and here are two and one more."

Katie, the oldest child in the classroom, has a different view of the number: "It's six because

there's two plus two plus two!"

Understanding number relationships Becky is a preschool teacher in a university- affiliated program. Her primary purpose with this lesson was to develop the children's under- standing of number relationships, which goes far beyond teaching counting skills. When a child understands number relationships, he or she comprehends the meaning of numbers by developing multiple, flexible ways of represent- ing them (see fig. 1). The importance of develop- ing number relationships in the early years has been highlighted because it helps children build

| a good foundation for developing more sophis- £ ticated understanding of number and quantity 1 (NCTM 2000); therefore, teachers of young | children must provide rich experiences with 1 number relationships, which allows students to s develop flexibility in representing numbers.

Three types of number relationships (i.e., subitizing, more-less, and parts-whole rela- tionships) can be taught in early childhood classrooms. This article outlines these number

relationships, follows with the classroom activi- ties that Becky presented in her preschool class- rooms, and discusses effective instructional

approaches for teaching number relationships.

Subitizing A couple days later, Becky presented the How many dots on the picture? activity again, but she added a challenge. She showed the children a picture of five dots in a domino arrangement for about three seconds before asking them how many dots there were. Katie quickly responded,

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Children who understand number relationships develop multiple ways to represent them, as in the different arrangements below for the number 6.

• • о о ° °

• •%' • о oo[° о ° °| •• 1 and 5 2 and 4 two groups three groups

of 3 of 2

teaching children mathematics • May 2011 551

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"There is a square and one [dot] in the middle!" Clara agreed with Katie's answer, explaining,

"I will show you how she got it. There are four [at the corners], so a group of four. That one [in the middle] makes a group of one. So, one, two, three, four [shepaused', five!"

Sam pictured five in a different way and shared his view: "I grouped them, like three and two!"

To determine the number in a collection of

objects, people often depend on counting or estimation. However, if the quantity of the col- lection is less than six or seven, people often look at the set and determine the quantity without counting. In the example, Becky showed her pre- schoolers five dots for a very short time because she wanted to assess whether they could recog- nize quantities without counting. The process of seeing the number in a set without counting is called subitizing, and it is a powerful tool to foster children's understanding of number. For instance, young children who know the correct sequence of counting words may not under- stand that the last counting word represents the total quantity of a collection being counted (Fuson 1988). Their inability to understand the

cardinal property of the counting word often keeps them from answering the question, How many? According to Benoit, Lehalle, and Jouen (2004), subitizing plays a critical role in acquiring the concept of cardinality. Subitizing also pro- motes children's understanding of parts-whole relationships (Clements 1999). In Becky's class- room, Clara and Katie saw five (the whole) as a

composite of one (part one) and four (part two) and thus knew that there were five dots without

counting. From this viewpoint, subitizing is more of a mathematical process, as opposed to a perceptual process (Fosnot and Dolk 2001).

For instructional purposes, teachers must remember that children need experiences with both counting and subitizing so they can see that both methods are appropriate and that both methods give the same result. As students gain more accuracy in subitizing small quan- tities, their teachers may challenge them by presenting various number patterns (see fig. 1 on p. 551). Encouraging children to share their different perceptions of quantity is also impor- tant, because such sharing extends their ability to represent quantity (Clements 1999).

More-less relationships Although subitizing is an effective pathway to recognizing a quantity, it may not be enough to help young children develop a full under- standing of number relationships. Given a pair of collections with different quantities, many children around the age of four are able to judge which collection has more (Clements 2004). However, determining how many more (or less) one collection has than the other is more chal-

lenging than simply identifying which collection has more (or less), even when the children use

counting. The following example is from Becky's classroom.

One daily routine is to record the day's weather. The preschoolers observe the weather and record it on their weather chart to find out

which type of weather - sunny, cloudy, windy, rainy, and snowy - they have the most often.

In the middle of February, Becky asked, "What can you tell me about the weather chart?"

Jackson shouted, "Cloudy is a winner. It has seven!"

"Sunny has four! It is catching up to cloudy," added Clara.

Then Darryl exclaimed, "Yeah, we need four

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more [cloudy days]!" Jackson disagreed with Darryl's idea: "No, it

is three more. One, two, three!" [as he counted the fifth sixth, and seventh place of the sunny column on the weather chart].

Teachers often take for granted that children automatically understand more-less relation- ships as they learn to count. The example above, however, illustrates that more-less relationships are not easy for young children. Darryl knew that

the chart had seven cloudy days and four sunny ones, but he did not have a full understanding that seven is three more than four. Young chil- dren's ability to recall the correct sequence of counting words (i.e., rote counting) may help them determine which one is more or less by using the large-number principle: The later a number word is called in the counting sequence, the larger quantity it has (Baroody 2004, p. 192). However, the ability to use the larger-number principle does not guarantee that children see relationships between pairs of numbers, even when those pairs are close to each other in terms of quantity. That is, Darryl may know that seven is greater than four because it comes later in the counting sequence, but he may not realize that seven is exactly three more than four or that four is three less than seven.

Why is the relationship between pairs so challenging? According to Clements (2004), young children must arrive at the important insight that a quantity (the less) must be con- tained in the other (the more), instead of view-

ing both quantities as mutually exclusive. The concept also requires them to think of the differ-

ence between two quantities as a third quantity, which is the notion of parts-whole (Krajewski and Schneider 2009). This development takes time and can be fostered with appropriate learn- ing experiences. Unfortunately, some teachers pressure young children into memorizing more- less relationships before these relationships make sense to them. When children are not

pressured to use more-less relationships, most of them naturally develop an understanding and are more confident in their use of those relation-

ships (Carpenter, Fennema, and Franke 1996). Key information for teachers is that children need a wide variety of experience with more-less relationships that are connected with count- ing experiences long before beginning formal instruction on addition and subtraction.

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Parts-whole relationships The last type of number relationship is one in which a quantity (a whole) can be partitioned into two or more parts (e.g., five as two and three; ten as two, three, and five). Children who do not

understand parts-whole relationships will have difficulty dealing with addition, subtraction, and other mathematics problems (Baroody 2000). For example, for assessment, Becky counted five counting bears with four-year-old Daniel: "Count with me. One, two, three, four, five!" Then she said, "I am going to hide some. You have to figure out how many I am hiding."

Becky slowly removed three counting bears from in front of Daniel and hid them, leaving two on the table. After about ten seconds, Daniel

answered, "Four!" With experience, youngsters naturally gain their understanding of parts- whole relationships; however, it may not be an easy task for some children like Daniel.

Why is a child's ability to understand parts- whole relationships important? An incomplete understanding of parts-whole relationships might result in poor performance on missing- addend word problems, such as 4 +

(Resnick 1992). Also, the concept of parts- whole can serve as a basis for understanding more advanced concepts, such as place value and fractions (Baroody 2004). In a study with

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kindergartners, for instance, Fischer (1990) found that students in the intervention group, who received mathematics instructions focus-

ing on parts-whole relationships, showed greater understanding of place value than those in the control group, who received their regular curriculum on number concepts.

Finally, a chilďs understanding of parts- whole relationships provides a basis for appre- ciating continuous quantity, such as length and area. In Becky's classroom, for example, Daniel

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Teach number relationships by encouraging your preschoolers to represent a number in various ways through composition and decomposition activities.

Activity (level)

Fiveness Place five Un ¡fix® cubes (e.g., three green and two yellow) on the teacher's fingers of one hand and show them briefly. Ask students what they saw. Later, have them investigate different ways to make five (see Novakowski

Quick images Have young children use manipulatives (e.g., counters) (Pre-K to early to represent the same quantity as shown on an overhead elementary) screen. Challenge them to say a quantity that js two more (or less) than that on the screen. Place two-color counters

in various arrangements on the overhead so that children are challenged to conceptually subitize a whole quantity

Racing bears Have a pair of children play the activity. By rolling a (Pre-K to early number cube, they can move their counting bear on the elementary) board to finish a track (from zero to ten). Frequently ask

each player, "How many did you get on the number cube this time?" and, "How many more would you need to win

the race?" (See Fosnot and Polk 2001 .)

Breaking cubes Give children a long stick of ten Unifix cubes and ask (Pre-K to early them to break their stick into two parts behind their back elementary) (each hand holding one part). Allow them to see only

one of the two parts while the other part remains behind them. Ask them to figure out how many cubes are behind their backs without looking at them. Discuss all the

Building a Have students build a triangle-shaped building with pyramid with disposable plastic cups (see Andrews and Trafton 2002). cups Allow about two weeks for their free exploration. For the (Pre-K to early activity, do not give children the plastic cups. Instead, ask elementary) them to determine how many cups they would need to

build a triangle building with a specified number of levels (e.g., "Can you figure out how many cups you would need to build a four-story building even before I give you the cups?") Teachers may encourage them to draw their

expressed his intention to measure the length of a table in the classroom. With Becky's help, he selected two rulers with different lengths, put them together, and found that the length of the two rulers was slightly shorter than that of the table. Then he changed the placement order of the rulers, revealing his misconception that changing the order of two quantities would make a difference. Similar misconceptions are often found among young children when they deal with discrete quantity (e.g., failing to recognize that the sum of three and eight is the same as that of eight and three).

Many activities can promote young children's understanding of parts-whole relationships. First, instruction that allows youngsters to identify a missing addend is effective because it is an important aspect of parts-whole relation- ships. For example, an appropriate activity that connects missing addends and initial notions of subtraction involves counting out items and then having a child hide some of them under a cup. From the number left in view, another child is asked to figure out how many items are hidden under the cup. Multiple trials are appro- priate, each with a different number of items under the cup. Second, instruction should encourage chil-

dren to see all possible parts-whole composi- tions for a given number. For young children, some combinations to make a number are

easier than others. They may experience such differences even when they deal with the same parts-whole compositions. For example, some children who understand that one more than

eight is nine may not answer immediately when asked, "What number is eight more than one?" Facilitate this development by encour- aging them to represent a number in various ways through composition and decomposition (see the Breaking cubes activity, summarized in table 1).

Implications for teaching The joint statement from the National Asso- ciation for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM) noted that "to create

coherence and power in the curriculum, teach- ers must stay focused on the big ideas of math- ematics and on the connections and sequences among those ideas" (NAEYC and NCTM 2002,

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p. 8). In the previous section, I discussed the importance of developing number relation- ships, including why children's understanding of number relationships should be fostered in the early years. The following section will suggest what teachers can do to help children under- stand number relationships.

From an instructional perspective, the three features of number relationship are connected to one another and thus can be developed con- currently. For example, eight can be conceptu- ally subitized as two groups of four; represented in more-less relationships, such as two less than ten; and viewed as a whole made from sets

of five and three. To help young children make connections, teachers must present appropri- ate learning experiences that help them reflect on all key features of number relationships by exploring, examining, and communicating their understanding of number and quantity. From this view, a ten-frame is considered an

effective tool to teach all number relationships (e.g., Losq 2005). For instance, teachers might add six counters in a ten-frame and ask young children to subitize the quantity. If most of them are successful at the subitizing task, teachers might continue to encourage them to think of different interpretations of six by presenting various arrangements (see fig. 2). Similarly, teachers can use a rekenrek to teach three types of number relationships. A rekenrek consists of twenty beads in two rows, and each row has a set of five beads in red and another five beads in

white. According to Grauberg (1998), the struc- ture of a rekenrek provides easy access to sub- itize quantities because five beads in each row are separated by color. For instance, eight can be easily understood in several ways, such as five and three, four and four, and two less than

ten (see fig. 3a and fig. 3b). Recent research by Tournaki, Bae, and Kerekes (2008) shows the effectiveness of using a rekenrek. These authors found that children with learning disabilities who received number-concept instruction on addition and subtraction problems using a rek- enrek showed significantly higher achievement than those in the control group.

Table 1 summarizes other activities that

teachers can use to give instruction on all number relationships. For these activities, young children are encouraged to use different kinds of manipulatives to make sense of their

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A ten-frame is effective in teaching number relationships, as in this example of combinations that total six.

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A rekenrek is another effective way to teach different number relationships.

(a) "It is eight, because it has five reds and three more whites." "It is eight, because two whites are on the other side. Two less

than ten is eight."

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(b) "It is eight, because it has two groups of four reds." "It is eight, because it has two reds less than ten reds."

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I Allow two weeks of free exploration before asking students I to draw mental images to determine how many cups they I would need for a triangle building with a specified number I of levels.

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representation of number and quantity. Using mathematics manipulatives, such as Unifix® cubes and counters, has long been a feature of classrooms for young children. Unfortunately, in some traditional classrooms, it has been assumed teachers can "tell" students what the materials show and that

using the materials all but guarantees student learning. However, math- ematical concepts do not inherently lie in math manipulatives; children must construct the

understanding. Appro- priate tools can help, but they do not guarantee this construction (Gravemeijer 1991). For young students to construct true understanding of number relationships, their teachers must encourage them to actively work with quantity in a variety of situations, using different math manipulatives over extended periods of time.

Discussion This article provides examples of what research says about young children's ability to learn number relationships and what the implica- tions of that research are for teachers of those

children. Although youngsters have an intuitive

Mathematical concepts do not inherently lie in manipulatives; children must construct the

understanding.

sense of number and quantity, developing it into a full understanding of number concepts and mathematical reasoning takes consider- able time. Throughout the early years, teachers should provide opportunities for children to develop multiple ways of representing quantity instead of limiting instruction to teaching count- ing skills or writing numerals. Moreover, young children need to see numbers in context and

work with quantity in various situations much more than they need number-fact drills. Even more important are classroom environments in which children are encouraged to freely share their thinking about number and quantity with others. When all of the above come together, students usually surprise us with their depth of understanding.

REFERENCES

Andrews, Angela G., and Paul Trafton. Little Kids- Powerful Problem Solvers. Portsmouth, NH:

Heinemann, 2002.

Baroody, Arthur. "Does Mathematics Instruction

for Three- to Five-Yea r-Olds Really Make Sense?"

Young Children 55, no. 4 (2000): 61-67.

Number and Operations Standards." In Engaging

Young Children in Mathematics: Standards for

Early Childhood Mathematics Education, edited by

Douglas H. Clements, Julie Sarama, and Ann-Marie

DiBiase, pp. 173-219. Mahwah, NJ: Lawrence E riba urn Associates, 2004.

Benoit, Laurent, Henri Lehalle, and François Jouen.

"Do Young Children Acquire Number Words

through Subitizing or Counting?" Cognitive

556 May 2011 • teaching children mathematics www.nctm.org

This content downloaded from ����������162.195.143.199 on Wed, 27 Mar 2024 01:08:07 +00:00�����������

All use subject to https://about.jstor.org/terms

Development 19, no. 3 (2004): 291-307. Carpenter, Thomas P., Elizabeth Fennema, and

Megan L Franke. "Cognitively Guided Instruc- tion: A Knowledge Base for Reform in Primary

Mathematics Instruction." The Elementary School Journal 97, no.1 (1996): 4-20.

Clements, Douglas H. "Major Themes and Recom-

mendations." In Engaging Young Children in

Mathematics: Standards for Early Childhood Math-

ematics Education, edited by Douglas H. Clements,

Julie Sarama, and Ann-Marie DiBiase, pp. 7-72.

Mahwah, NJ: Lawrence Erlbaum Associates, 2004.

ing Children Mathematics 5, no. 7 (March 1 999): 400-5.

Fischer, Florence E. "A Part-Part-Whole Curriculum for

Teaching Number in the Kindergarten." Journal for Research in Mathematics Education 21 , no. 3

(1990): 207-15.

Fosnot, Catherine Twomey, and Maarten Dolk. Young

Mathematicians at Work: Constructing Number

Sense, Addition, and Subtraction. Portsmouth,

NH: Heinemann, 2001.

Fuson, Karen С Children's Counting and Concepts of

Numbers. New York: Springer, 1988.

Grauberg, Eva. Elementary Mathematics and Language Difficulties. London: Whurr, 1998.

Gravemeijer, Koeno. P. E. "An Instruction-Theoretical

Reflection on the Use of Manipulatives." In Realistic

Mathematics Education in Primary School, edited

by L Streef land, pp. 57-76. Utrecht: Utrecht

University, 1991.

Krajewski, Kristin, and Wolfgang Schneider. "Explor-

ing the Impact of Phonological Awareness,

Visual-Spatial Working Memory, and Preschool

Quantity-Number Competencies on Mathematics

Achievement in Elementary School: Findings from

a Three-year Longitudinal Study." Journal of Experi-

mental Child Psychology 1 03, no. 4 (2009): 516-31.

Losq, Christine S. "Number Concept and Special Needs

Students: The Power of Ten-Frame Tiles." Teaching

Children Mathematics 1 1 , no. 6 (February 2005): 310-15.

National Association for the Education of Young

Children (NAEYC) and National Council of Teachers

of Mathematics (NCTM). "Early Childhood Math-

ematics: Promoting Good Beginnings." Position

statement. 2002. http://www.naeyc.org/about/

positions/pdf/psmath.pdf. National Council of Teachers of Mathematics (NCTM).

Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Novakowski, Janice. "Developing 'Five-ness' in Kinder-

garten." Teaching Children Mathematics 14, no. 4 (November 2007): 226-31.

Resnick, Lauren B. "From Protoquantities to Opera-

tors: Building Mathematical Competence on a Foundation of Everyday Knowledge." In Analysis

of Arithmetic for Mathematics Teaching, edited

by G. Leinhardt, R. Putnam, and R. A. Hattrup,

pp. 373-425. Hillsdale, NJ: Lawrence Erlbaum Associates, 1992.

Tournaki, Nelly, Young S. Bae, and Judit Kerekes.

"Rekenrek: A Manipulative Used to Teach Addition

and Subtraction to Students with Learning Dis-

abilities." Learning Disabilities: A Contemporary Journal 6, no. 4 (2008): 41-59.

Myoungwhon Jung, mjung@niu.edu, is an assistant professor of early child-

hood education in the Department

of Teaching and Learning at Northern

Illinois University in DeKalb. His research

interests include early childhood math, professional

development for early childhood teachers, and using

technology with young children.

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