Enhance counting skills of preschoolers

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This is an intervention study that aimed at investigating the effectiveness of using computer technology and manipulatives in enhancing preschoolers' mastery of counting skills. The targeted skills were forward counting, number after, number before, counting backward, skip counting, one-one correspondence principle, cardinality principle, production of sets, and comparing quantities. A total of 48 children were recruited for the study from one kindergarten in Al-Ain City, United Arab Emirates. Two intact sections of 24 children each served as experimental and control groups. The control group was taught in the traditional way while the experimental group was taught with the use of computer technology and manipulatives for one semester. Data were collected through individual interviews with children before and after intervention. Interview questions involved items that addressed each of the above mentioned skills. Results revealed that children in both groups have improved in all counting skills. However, the experimental children outperformed their counterparts in the control group in all these skills. Implications for mathematics instruction in kindergarten are discussed.

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INTRODUCTION

Increased attention is being given to establishing a foundation in mathematics in kindergarten years in many countries such as the European countries and the United States of America (Kaufmann, Delazer, Phol, Semenza, & Dowker, 2005; Aubrey, 2003; Kilpatrick, Swafford, & Findell, 2001; Department of Education and Employment [DfEE], 1999). When speaking of mathematics in preschool, counting comes to the front. From a developmental perspective, counting is of special interest because it is the first formal computational system to be learned by children. According to Gelman and Galistell (1978), there are five conceptual principles that govern counting. Those principles are:

The stable order principle: number words must be used in a fixed order in counting.

The one-one-correspondence principle: every number word is assigned to one object in the counted set.

The cardinality principle: the value of the last number word in the counting sequence represents the quantity of the counted objects.

The abstraction principle: any type of objects can be counted.

The order-irrelevance principle: the counting result does not depend on the order in which the objects are counted.

The first three principles are viewed as the essential principles because they represent the basis for children's knowledge of counting (Geary, 2004; Gelman & Meck, 1983). Violating any of these principles will necessarily result in incorrect counting. For example, violating the stable order principle will result in stating number words arbitrarily which of course is incorrect. The same is true when violating the one-one-correspondence principle or the cardinality principle.

The other two principles viewed as unessential because violating any of them will not affect the accuracy of counting (Briars & Siegler, 1984). These two principles are used by children along with the essential principles in their early development of counting knowledge but children discard the unessential principles over time (Stock, Desoete, & Roeyers, 2009).

According to Butterworth (2004), children do not master all essential principles at the same time. The stable order principle is mastered first while the cardinality principle is mastered last. In a recent study, Le Fevre, Smith-Chant, Fast, Skwarchuk, Sargla, Arnup, et al. (2006) found that children's knowledge of the stable order principle was very good in kindergarten and equal to adults' knowledge in first grade. They also found similar results in regard to the one-one-correspondence principle. Similarly, Birars and Siegler (1984) found that children had good understanding of the one-one-correspondence principle at the age of five and that this understanding improved with age.

The mastery of the cardinality principle is much debated too. Some researchers found that the principle is mastered at the age of three (Gelman & Meck, 1983). Some others argued that the understanding of the principle begins at the age of three and a half (Wynn, 1992). Yet, some other researchers found that children could not determine quantities before the age of four and a half and that principled understanding of cardinality does not appear before the age of five (Freeman, Antonucci & Lewis, 2000).

Baroody and Coslick (1998) classify counting skills into two main categories, oral counting and object counting. Oral counting involves citing the number words. Usually children learn first the forward counting sequence (one, two, three, …) and this enables them later from citing the number after and number before. In addition, experience with counting forward helps them with counting backward. Then children become able to use skip counting by twos, fives, and tens.

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Object counting is more than citing the numbers as it involves enumeration, cardinality principle, and production of sets. Enumerating a collection of objects requires (a) knowing the needed portion of the continuing sequence, (b) adhering to the stable order principle (c) adhering to the one-one correspondence principle, and (d) keeping track of counted objects. Production of sets involves counting a subset of a set of objects. Resnick and Ford (1981) consider this skill to be harder than enumeration because it requires the child to remember how many objects are requested and to stop counting when reaching that number.

The ability to compare quantities is a natural result of the ability to count objects and counting experiences in general. Two principles - if discovered - help children succeed in comparing quantities: the same number principle and the ordering - numbers principle (Baroody & Coslick, 1998). The first principle suggests that if two or more collections show the same number name, then they are equal regardless of how they look or what they contain. The second principle suggests that later numbers in the counting sequence are larger than earlier numbers (e.g. five is larger than four because it follows four). Of course, according to this principle, if a number comes "much later" than another number in the counting sequence then it is much larger than that number (e.g. 14 is much larger than 2 because it comes much later than 2 in the counting sequence).

COUNTING AND NUMBER CONCEPT DEVELOPMENT

There were two views regarding the development of number concept, the logical-prerequisites view and the counting view (Baroody & Coslick, 1998). The first view was adopted by Piaget who believed that counting skills were rottenly learned and they had nothing to do with helping children understand numbers (Piaget, 1965). Piaget believed that number concept depended on logical thinking and that children should rely on matching, not counting, to establish equivalence and in-equivalence.

According to the second view, however, counting is the key to understanding number concepts and arithmetic (Baroody & Coslick, 1998). Many researchers agree that children construct basic number and arithmetic concepts gradually from experiences that largely involve numbers (Baroody & Coslick, 1998; Fuson, 1988; Gelman & Gallistel, 1978; Baroody, 1987). Generally, mathematics educators agree that children's success in mathematics in early grades depends largely on their counting skills. Baroody and Coslick (1998) suggest that difficulties in counting might seriously obstruct children's progress in mathematics. Stock, Desoete, and Roeyers (2009) found that mastery of counting principles in kindergarten predicted arithmetic abilities one year later in the first grade. Therefore, they asserted that improving arithmetic abilities is a developmental process that starts before formal schooling.

Some researchers confirmed the role of counting abilities in the automatisation of arithmetic facts (Aunola, Leskinen, & Nurmi, 2004; Van de Rijt & Van Luit, 1999). Beyond the automatisation of arithmetic facts, a plenty of studies confirmed the relation between mastery of counting and success in school arithmetic (Blote, Lieffering, & Quwehand, 2006; LeFevere et al., 2006; Stock, Desoete, & Roeyers, 2007). A recent study (Stock, Desoete, & Roeyers, 2009) that involved 423 children investigated the relationship between the mastery of counting principles and arithmetic abilities. The results revealed that more than 50% of children did not master the three essential counting principles by the end of kindergarten and that mastering the counting principles in kindergarten predicted arithmetic abilities in first grade one year later.

Johansson (2005) conducted two studies to investigate the role of number-word sequence skill in arithmetic performance. In one study, he asked children between 4 and 8 years old to count forward and backward on the number sequence and solve arithmetic problems. The results of the study revealed that performance on number word sequence predicted performance on arithmetic problems as well as strategies used in solving arithmetic problems.

In another study, he found that "solving doubles (e.g. 2 + 2 = ?) served as a link between the number-word sequence skill and the number of arithmetic problems solved" (p. 157). Johansson concluded that "counting on the number-word sequence may be an early solution procedure and that, with increasing counting skill, the child may detect regularities in the number word sequence that can be used to form new and more accurate strategies for solving arithmetic problems." (p. 157).

The skill of counting forward from an arbitrary point was found to predict the use of counting on procedure for addition problems (Secada, Fuson, and Hall, 1983) and performance on additive composition of number problems (Martins-Mourao & Cowan, 1998). Also, it was found that the length of counting forward sequence correlated with understanding of teen quantities and solving simple addition problems (Ho & Fuson, 1998).

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THE ROLE OF MANIPULATIVES AND COMPUTERS IN LEARNING MATHEMATICS

Young children learn best by exploring their surroundings, mostly through playing, during which they construct mental representations of the world (Hengeveld, et al, 2007). Prior to Piaget's formal operational stage, young children need concrete, hands-on experiences rather than abstract concepts to learn, develop, and think (Marshall, 2007). Learning with manipulatives was found to correlate positively with the development of mental mathematics (Gravemeijer, 1990), and produce better achievement, and conceptual understanding than do traditional teaching techniques (Sowell, 1989; Fuson & Briars, 1990; Chassapis, 1999).

Computer assisted instruction (CAI) has been widely used in education (McKethan, Everhart, & Sanders, 2001). Some of the main advantages of CAI is the presentation of the lesson in various ways [text, audio and graphics] and interactivity (Vernadakis, Avgerinos, Tsitskari, & Zachopoulou, 2005). These features help making computer an interesting and effective learning tool. Fletcher-Flinn and Gravatt (1995) reported positive effect of the use of CAI as the CAI group outperformed the traditional instruction group in a wide range of skills in mathematics, science, art, reading, and writing.

Elliot and Hall (1997) reported that children who used CAI based activities, scored significantly higher on the Test of Early Mathematical Ability (TEMA2) than their counterparts who did not use such activities. More recent studies also showed that the use of CAI had positive effect on children's learning (Chera & Wood, 2003; Segers and Verhoeven , 2002). On the other hand, other studies revealed no differences in the development of mathematical skills by computer use as opposed to traditional methods (Din & Calao, 2001; Moxley et al., 1997; Reitsma & Wesseling, 1998) . This was explained partly by the lack of maturity in children of this age at being able to develop mathematical thinking (Shute & Miksad, 1997).

A study by Shute and Miksal (1997) revealed an important finding regarding the use of computers in preschool and that was the increased attention span of children while learning with the use of computer. Klein, Nir-Gal, and Darom (2000) studied the effect of the adult- child mediation in a computer learning environment. The results indicated significantly higher achievements for preschool students who interacted with adults who were trained as mediators within the computer environment.

STATEMENT OF THE PROBLEM

Based on the discussion above, it is clear that counting skills play a central role in later development of arithmetic. Therefore, it is of high importance to help children master these skills before entering first grade. While a plenty of research studies have been conducted on assessing children's mastery of counting skills, very few have intervened with children to improve these skills. The authors of this study provided children with a learner-centered program to help them build strong counting skills through purposeful and meaningful instruction with the help of computer and manipulatives. In order to evaluate the effectiveness of our intervention, we used a quasi experimental design with pre and post assessments. Our main research question was: are there significant differences between children's performance on forward counting, number after, number before, counting backward , skip counting, enumeration, cardinality principle, production of sets, and comparing quantities due to treatment?

CONTEXT OF THE STUDY

Early childhood education in the United Arab Emirates (UAE) consists of two grade levels, kindergarten 1 (KG1) and kindergarten 2 (KG2) with one academic year for each level. Usually, children enter KG1 at the age of four. The aim of these two levels is to prepare children for first grade. In regard to mathematics, children are expected to count till 30 by the end of KG2. No emphasis is placed on skills beyond counting to 30 such as addition, and subtraction. Children are also expected to recognize and draw basic geometric shapes such as square, triangle, rectangle, and circle.

The language of instruction in public kindergartens is Arabic which is the official language in the UAE. Language and number-naming systems were shown to play a role in children's mathematical learning (Alsawaie, 2004; Miura, Okamoto, Chungsoon, Steere, Fayol,1993; Saxe, 1982). Like English, the Arabic number naming system places students at disadvantage in learning numbers, which necessitates providing students with extra help (Alsawaie, 2004). For example, numbers 11 and 12 in Arabic seem to be arbitrary. One is spoken as wahed and 2 is spoken as ithnan, 11 is spoken as ahada-ashar and 12 is spoken as ithna-ashar. As for numbers 13 to 19, Arabic replaces asharah (10) by ashar, similar to replacing 10 with "teen" in English. So 13, for example, is spoken as thalathata ashar. It is also important to note that thalathah (3) has become thalathata in 13. This is the case for all numbers 3 to 9. In English, 3 becomes "thir-" in 13 and 5 becomes "fif-" in 15. Similarly, the number of tens is not made explicit in the decades (Ishroon, 20; Thalathoon, 30; arba'oon, 40; khamsoon, 50; ...; tess'oon, 90). Except ishroon (20), in which ithnan (2) is not there at all, all other decades are made by adding "oon" (equivalent to "-ty" in English) to the single digit numbers after making some adjustment on them (thalath instead of thalathah, 3; arba instead of arba 'ah, 4; khams instead of khamsah, 5, and so on; see Table 1).

Table 1

Number Names in English and Arabic

Numeral

English

Arabic

1

One

Wahed

2

Two

Ithnan

3

Three

Thalathah

4

Four

Arba'ah

5

Five

Khamsah

6

Six

Settah

7

Seven

Sab'ah

8

Eight

Thamaniyah

9

Nine

Tess'ah

10

Ten

Asharah

11

Eleven

Ahada-ashar

12

Twelve

Ithna-ashar

13

Thirteen

Thalathata-ashar

14

Fourteen

Arba'ata-ashar

19

Nineteen

Tess'ata-ashar

20

Twenty

Ishroon

21

Twenty one

Wahed-wa-ishroon

29

Twenty nine

Tess'ah-wa-ishroon

100

One hundred

Me'ah

THE INTERVENTION PROGRAM

The intervention program aimed at enhancing children's skills in the following areas: (1) Forward counting, (2) Number after and number before, (3) Counting backward , (4) Skip counting, (5) Enumeration, (6) Cardinality principle, (7) Production of sets, and (8) Comparing quantities. Following are some examples of the types of activities used.

Forward counting

As for numbers 1-13, using power point, we had a cartoon character (CC) count and the children counted after him. Children then were asked to count individually or as a whole class. Some Arabic songs were also utilized to foster memorizing the counting sequence. In other activities the CC made some mistakes in counting and the children were asked to detect his mistakes.

For numbers 14 to 29, two CCs modeled the utilization of the pattern and the children were asked to do the same. For example, the first CC says: Arba'ata (4) and the second says: ashar (teen). Then both of them say: Arba'ata ashar (14). The same procedure is done for numbers 15 to 19. The purpose of this activity was to show children that counting from 14 to 19 is not something new. They only need to say the numbers 4 to 9 adding the word ashar (teen). The same was done for numbers 21 to 29 where the first CC says the number in the ones digit [wahed (1) for example] and the second CC says wa-ishroon (and 20), and so on. Again, error detection activities were utilized.

Except ishroon (20) which has to be memorized, decades in Arabic parallel the ones and are made by replacing "ah" in the ones by "oon". For example, thalathoon (30) is thalathah (3) with "ah" replaced by "oon". So, we utilized this pattern in teaching decades to children. As with the ashar (teen) numbers, the CCs modeled the utilization of pattern in learning these decades.

Number after and numbered before

Using numbered blocks, children played number after and number before games. Numbered cubes were made available to children. The game starts by having each player draw a dice. The player with the larger number starts the game by selecting a cube with a certain number and asking for a number after or before. The task of the other player (s) is to search for the number after or before it as determined by the first player. To make the game more challenging, the second player has to find the targeted cube before the first player finishing counting to 10 or 5. The game is flexible in terms of the number of player.

Counting backward

On PowerPoint, the CC promises the children of showing them his friend after a number of seconds. He asks them to count backward starting from that number. After reaching number 1, his friend appears. Also, children played the ordering blocks game. The game asks children to order numbered blocks forward and backward. The fastest child wins. The game is also useful for counting forward. Further, a CC counted backward and the children (in groups) detected his mistakes.

Skip counting

A number of objects (apples, marbles, …etc.) is shown one by one on the computer. The children count with the CC. Then the same objects are shown in twos and again the children count with the CC. Using concrete materials, children experienced skip counting purposefully and meaningfully through games or problem solving activities.

Enumeration (counting objects)

The CC tells children that he needs 9 apples for example and asks them for help by checking if the apples he has are enough. Because keeping track of the objects is often a problem for children, PowerPoint features were used to overcome the problem by changing the color of the objects after being counted. Some other times, the counted object was marked by X. Further, children had ample opportunities for counting animal toys, cubes, flowers, and other concrete materials and pictorial objects.

Cardinality principle

After counting a set of objects, the CC asks children: how many objects do I have? In other activities, the CC tells children that his friend claims that he has a certain number of objects and asks them to verify his claim. The teacher also implemented similar activities using manipulatives.

Production of sets

Two CCs have a pile of objects and ask children to help them in getting a number of objects for each. The teacher implements this activity in 3 different ways: (1) Children do that using the mouse, (2) the teacher has a set of concrete objects and asks the children to produce sets, (3) the teacher gives children worksheets with pictorial objects and asks the children to circle a number of objects.

Comparing quantities

Two CCs ask the children to help them decide who of them has more marbles. Alternatively, each group of children chooses a color. Then a screen with sets of marbles of different colors appears. The group with more marbles wins. So, children compare the sets of marbles by counting to decide which group wins.

Some activities were used to deliberately help children discover the same number principle and the ordering - numbers principle (Baroody & Coslick, 1998). First, children were asked to (a) count a number of organized objects (concrete or pictorial) and the same number of scattered objects and compare the two quantities, (b) count a number of flowers for example and the same number of rabbits and compare the two quantities. Second, to help children understand that later numbers in the counting sequence are larger than earlier numbers, many activities were utilized. Here are some examples: (1) using concrete materials to compare small quantities and then larger quantities, (2) comparing ages, (3) using number lines, and (4) using computer.

For example, children compared numbered objects organized in rows. Of course, shorter rows contain fewer objects. On the computer, a CC made stacks of different numbers of cubes (e.g. 2, 3, 9, and 15). After making each stack, the CC shows some signs of tiredness that increase with the number of cubes used. Children discussed the reasons and engaged in comparisons.

METHOD

Sample

Forty eight KG2 children (21 boys, and 27 girls; mean age before intervention = 62.19 months; SD = 2.27) were recruited from one kindergarten in Al-Ain City, UAE.. All children belong to the middle socio-economic class according to the living standards of the UAE. The children were in two sections, one section was randomly assigned to the experimental group (EG) and the other to the control group (CG). The EG had 11 boys and 13 girls while the CG had 10 boys and 14 girls.

Data Collection and analysis

Data were collected through individual interviews with children. Through the interview, children were tested on 9 skills. Except for counting forward, counting backward and skip counting, children were tested on 5 items for each skill. Cronbach's alpha for the test was found to be (0.87) indicating an acceptable degree of internal consistency. The test was judged by 8 experts in mathematics education and early childhood education as a valid instrument. Details for each skill follow.

Counting forward:

Children were tested on their accuracy in counting forward till 15 in the pretest, and till 30 in the post test. One point was deducted for each mistake the child made (maximum score is 15 in the pretest and 30 in the post test).

Counting backward:

Children were tested on their accuracy in counting backward from 15 to 1 in the pretest, and from 30 to 11 in the post test. One point was deducted for each mistake the child made(maximum score is 15 in the pretest and 20 in the post test).

Number After:

This part included 5 items asking children to determine the number after a number written on a flash card. The numbers ranged between 1 and 14 in the pretest and between 12 and 29 in the post test. Each correct response was granted one point.

Number before:

Same as in number after except that children were asked to determine the number before. The numbers ranged between 1 and 15 in the pretest and between 12 and 30 in the post test.

Skip counting:

Children were asked to count by twos from 2 to 14 in the pretest and from 2 to 30 in the post test. Therefore, the maximum score for the pretest was 7 and for the posttest was 15. One point was deducted for each mistake the child made (maximum score is 7 in the pretest and 15 in the post test).

One-one-correspondence principle:

This part included 5 items all asking for counting random patterns of drawings (dots, apples, pencils, birds, and diamonds). The numbers of drawings ranged between 9 and 15 in the pretest and between 12 and 30 in the post test. Each correct response was granted one point.

The cardinality principle:

The "how many" task (Wynn, 1990, 1992) was used to test this principle where the child is presented with a set of objects and asked "how many are there in total?" This task included 5 items with the number of objects presented ranging between 9 and 15 in the pretest and between 12 and 30 in the post test. Each correct response was granted one point.

Production of sets:

This part included 5 items. Children were presented with a set of objects (ranged between 6 and 15 in the pretest and between 12 and 30 in the posttest) and asked to make a group of objects. For example, presenting children with 15 apples and asking them to "put 8 apples in a box." Each correct response was granted one point.

Comparing quantities:

Children were asked to compare between two sets of objects. This task included 5 items with the number of objects presented ranging between 6 and 15 in the pretest and between 12 and 30 in the post test. Each correct response was granted one point.

Procedures

The control group was taught in the conventional way followed in the kindergarten. The experimental group however was taught through a program that involved computer and manipulatives. Both groups followed the same curriculum and covered the same content. Teachers of both groups were judged by the principal of the kindergarten and their supervisor as equivalent. Both teachers have bachelor's degrees in early childhood education from the same university; the experimental teacher has 7 years of experience and the control teacher has 8 years of experience. Both teachers had an overall evaluation by the educational district as excellent for the last three years.

The experimental teacher was familiarized with the program through six 1-hour individual training sessions conducted by one of the authors. Each two consecutive sessions were separated with 3 to 5 days. After each session, the teacher reviewed what was learned at home and brought questions to the trainer in the next session. Further, this teacher was observed by the trainer during the implementation of the intervention and given feedback on her performance. In the first month of intervention, the teacher was observed twice a week. After that, she was observed once a week for 3 months.

The first phase of interviews was conducted in the first week of the second semester (Spring 2009) and the second phase in the last week of the same semester. All interviews were conducted by one of the authors. The interviews took place in a quiet room in the kindergarten. The interviewer started by giving the child a little gift and talking to him/her for about two minutes as an ice breaker. Then, she started to ask children to perform the tasks. Tasks were presented in the same order for all children. Materials relevant to each task were made available for the child and then replaced by others depending on the type of the task.

RESULTS

As shown in Table 2, scores on all skills seem to be equivalent in the pre stage. Both groups did fairly well on counting forward but not on other skills. Performances were low on number before, cardinality principle, number after, and production of sets. Independent sample t-test showed no significant difference in the pre test on any of the skills due to group (p > .05), which assured that the two groups were equivalent before the start of the intervention program.

Table 2

Descriptive statistics on the pre and post tests for each skill by group

Experimental group

Control group

Skill

Time (max.)

M

SD

M

SD

Counting forward

Pre (15)

11.79

1.47

11.96

1.60

Post (30)

28.38

1.47

26.17

2.33

Counting Backward

Pre (15)

7.21

2.04

7.17

1.95

Post (20)

18.38

1.61

13.96

2.03

Number After

Pre (5)

2.13

.61

2.21

.93

Post (5)

4.25

.61

3.13

.90

Number before

Pre (5)

2.00

.78

2.00

.98

Post (5)

4.13

.68

3.04

.86

Skip Counting

Pre (7)

3.79

.78

3.75

.99

Post (15)

13.75

1.45

10.50

1.69

One-one- correspondence

Pre (5)

2.63

.65

2.63

.82

Post (5)

4.79

.42

4.29

.81

Cardinality principle

Pre (5)

2.08

.78

2.08

.83

Post (5)

3.63

.77

3.00

.93

Production of Sets

Pre (5)

2.13

.80

2.17

.76

Post (5)

3.50

.72

2.88

.85

Comparing quantities

Pre (5)

2.96

.62

3.00

.72

Post (5)

4.38

.50

3.50

.83

While the scores on counting forward in the pre stage were fairly high (close to 80%), they do not reflect very strong mastery of the stable order principle. Scores on counting backward were much less than those on counting forward. Low performance is also clear on number after, number before, and skip counting. Also, scores on one-one correspondence principle were so low reflecting low level of mastery of this principle. Scores on the cardinality principle were the lowest. Scores were also low on production of sets.

In the post stage, both groups have improved on all skills with the experimental group clearly improving more. To test whether the two groups have significantly improved on these skills after one semester of instruction, paired sample t-tests were carried out. Before carrying out the tests, pre scores on counting forward, counting backward, and skip counting were converted such that the maximums score in both stages are the same. For example, the maximum possible score on counting forward was 15 in the pre stage and 30 in the post stage. Therefore, the pre scores were multiplied by 2 before doing the t-test. Results of the analyses (see Table 3) revealed that both groups have significantly improved on all skills supporting the claim that counting skills improve with age (Birars & Siegler, 1984).

Table 3

Results of Paired samples t-tests for each skill by group

Group

Skill

MD

t

df

Sig.

Control

Counting forward

2.25

3.23

23

.004

Counting Backward

4.40

8.49

23

.000

Number After

.92

7.70

23

.000

Number before

1.04

6.80

23

.000

Skip Counting

2.46

6.18

23

.000

One-one- correspondence

1.67

12.82

23

.000

Cardinality principle

.92

6.26

23

.000

Production of Sets

.71

5.56

23

.000

Comparing quantities

.50

4.15

23

.000

Experimental

Counting forward

5.63

5.66

23

.000

Counting Backward

8.76

18.75

23

.000

Number After

2.13

23.22

23

.000

Number before

2.13

17.00

23

.000

Skip Counting

5.63

17.36

23

.000

One-one- correspondence

2.17

16.66

23

.000

Cardinality principle

1.54

12.84

23

.000

Production of Sets

1.38

13.62

23

.000

Comparing quantities

1.42

13.78

23

.000

To test whether the intervention had an advantage over the traditional mode of instruction, independent samples t-test was done. Results showed that there were significant differences in means on all 9 skills favoring the experimental group (see Table 4). This indicated that the intervention was successful in better improving children's counting skills.

Table 4

Results of independent samples t-tests for each skill

Skill

MD

t

df

Sig.

Counting forward

2.21

3.92

46

.000

Counting Backward

4.42

8.35

46

.000

Number After

1.13

5.08

46

.000

Number before

1.08

4.85

46

.000

Skip Counting

3.25

7.14

46

.000

One-one- correspondence

.50

2.70

46

.01

Cardinality principle

.63

2.53

46

.015

Production of Sets

.63

2.75

46

.009

Comparing quantities

.88

4.42

46

.000

Stock et al. (2009) considered the cut-off for mastery of counting principles to 3 out of 4. In this study, we adopted a little more strict cut-off, 80%. As shown in Table 5, very low percentage of children mastered each skill in the pre stage except for counting forward. About two thirds of the control group and a little less percent of the experimental group mastered counting forward before intervention. Three children (12.5%) of the control group and one child (4.2%) from the experimental group mastered the one-one principle. Only one child from each group mastered the cardinality principle.

Table 5

Percentages of children who mastered each skill by group

Skill

Experimental group

Control group

Pre

Post

Pre

Post

Counting forward

58.3

100

66.7

79.2

Counting Backward

0

91.7

0

16.7

Number After

0

91.7

8.3

29.2

Number before

0

83.3

4.2

29.2

Skip Counting

0

91.7

4.2

33.3

One-one- correspondence

4.2

100

12.5

79.2

Cardinality principle

4.2

54.2

4.2

25.0

Production of Sets

0

45.8

0

20.8

Comparing quantities

16.7

100

20.8

45.8

After intervention, the control group improved on all areas especially on counting forward and one-one correspondence principle. Improvements on other skills were not even comparable to those of the experimental group. One hundred percent of children in the EG mastered counting forward, the one-one correspondence principle, and comparing quantities. However, only a little more than half of them mastered the cardinality principle compared to one fourth of the CG children.

DISCUSSION

This study has contributed to the literature on preschoolers' counting skills. It's main purpose was to investigate the effectiveness of using computer technology and manipulatives in enhancing preschoolers' counting skills. The targeted skills were forward counting, number after, number before, counting backward, skip counting, enumeration, cardinality principle, production of sets, and comparing quantities. Comparisons of performances of the EG and CG showed that EG children outperformed their counterparts in the CG in all skills. The two groups started the semester with equivalent skills but after intervention, a considerably higher percentage of children in the EG mastered counting skills than CG children. These results support the finding of many previous studies that revealed advantages of using computer (Vernadakis, Avgerinos, Tsitskari, & Zachopoulou, 2005; Chera & Wood, 2003; Segers and Verhoeven , 2002; Elliot and Hall, 1997) and manipulatives (Marshall, 2007; Hengeveld, et al, 2007; Chassapis, 1999; Fuson & Briars, 1990; Gravemeijer, 1990; Sowell, 1989) in teaching mathematics.

Many features of the program led to these results. Most of the activities done through manipulatives took the form of playing which greatly helps children master the intended skills. This is consistent with what Hengeveld, et al. (2007) suggested in that playing help children construct mental representations of the world. Of course, playing with concrete materials was appropriate for the children because they are categorized within Piaget's preoperational stage (Marshal, 2007).

Using computer-based activities had many advantages that contributed to the results of the study. Varying the presentation of the material using text, audio and graphics made students interested and engaged in the lesson. The presence of the cartoon characters also attracted children and increased their interaction with the lesson. Children's engagement and interest were evident in the experimental class as observed by one of the authors. Similar behaviors were observed in previous studies (Vernadakis, Avgerinos, Tsitskari, & Zachopoulou, 2005). Further, one more important factor of the effectiveness of CAI is the increase of children's attention span which was confirmed by Shute and Miksal (1997). While we did not formally measure the attention span of the EG children, but the observer noticed that children were engaged in the lesson all the time and paid close attention to what was presented through the computer.

Other interesting results were also revealed by the study. These results support the idea that children do not master all essential principles at the same time. Rather, the stable order principle is mastered first while the cardinality principle is mastered last (Butterworth, 2004). The results on the stable order principle and the one-one correspondence principle (post stage) are in agreement with those of Le Fever et al. (2006) who found that children's knowledge of these two principles was very good in kindergarten. However, the results disagree with Birars and Siegler (1984) who found that children had good understanding of the one-one-correspondence principle at the age of five because participants of this study were more than five at the start of the intervention, yet their performance on the one-one-correspondence principle was too low but significantly improved after 4 months.

At the time of the pretest, the mean age of the participating children was more than 5 years (62.19 months), yet they did not seem to master the cardinality principle, only 4.2% of the children did. Even after 4 months, only 25% of the CG children mastered the principle. Some researchers argued that children master the cardinality principle at the age of 3 years (Gelman & Meck, 1983), or 3 years and a half (Wynn, 1992). However, Antonucci and Lewis (2000) found that principled understanding of cardinality does not appear before the age of five. Results of this study are consistent with the later finding. Further, the results of this study support the finding that the understanding of the cardinality principle is the most difficult among the counting principles (Butterworth, 2004; Fuson, 1983). Stock, Desoete, and Roeyers (2009) did not confirm this finding. These variations in results of different studies show that it is hard to determine an exact age at which children master the cardinality principle.

After intervention, children in the EG performed quite well on comparing quantities, counting backward, number before, number after, and skip counting but their counterparts in the CG did not even though they did well on counting forward. Baroody and Coslick (1998) suggested that experience with forward counting leads to mastering the above mentioned skills. Results of the CG on these skills seem to imply that specific activities should focus on these skills which was the case with the EG in this study.

Generally, the results of this study show that with appropriate instruction and wise use of computer and manipulatives, children can perform well on the essential counting principles and other skills before first grade. These results as well as those of previous studies show that large differences exist among children in mastering counting skills in preschool. Given that these skills are essential for future success in mathematics, this issue must be highly considered to avoid leaving some children at risk.