Abstract:

Many studies have shown that children’s experience related to the learning of number concepts at primary school level is of crucial importance in instilling their beliefs and values they associate with mathematics. If these experiences are meaningful, it will further lead to positive attitudes, values and beliefs about number concepts. On the contrary, experiences that are not mathematically meaningful will lead them to believe that mathematics learning consists of memorizing activities devoid of meaning (NCTM, 1989).

The research reported here is part of a bigger study that aims to look at children’s’ notion about number sense. The purpose of this study is to explore children’s understanding of multiple representations of numbers, that is the recognition that numbers take many different numerical and representational forms (eg. Fractions as decimals, a whole number in expanded form, or a fraction on a number line) and can be thought about and manipulated in many ways to benefit a particular purpose.

A total number of 406 11 year old children from four different schools in Malaysia were asked to answer nine number sense items related to the understanding of multiple representations in a number sense test. Results of the data analysis showed that many students have difficulties with recognizing a fraction in its decimal form and vice versa, representing whole numbers greater than 100 on a number line. Six students were chosen for interview sessions to further explore into their understanding of multiple representations of numbers. Data from the interview showed that while some students showed understanding of number sense, there are students whose representation of fractions and decimals on a number line are not meaningful and lack the understanding of number sense.

____________________________________________________________________________________________________________A Paper by Distribution for WGA 1 ( Mathematics Education in Pre-and Primary School) at the 9^th International Congress on Mathematical Education, July 31 &emdash; Aug6, 2000, Tokyo/Makuhari, Japan

Introduction

Many studies have shown that children’s experience related to the learning of number concepts at primary school level is of crucial importance in instilling their beliefs and values they associate with mathematics. If these experiences are meaningful, it will further lead to positive attitudes, values and beliefs about number concepts. On the contrary, experiences that are not mathematically meaningful will lead them to believe that mathematics learning consists of memorizing activities devoid of meaning (NCTM, 1989). Many mathematic educators are concerned that students demonstrate little understanding of numerical situations in which they solve number problems(Markovits & Sowder, 1994). The Curriculum and Evaluation Standards for School Mathematics (1989) states that "children must understand numbers if they are to make sense of the ways numbers are used in their everyday world".

Multiple Representations of numbers in the number sense framework

McIntosh et.al (1997) developed a framework for examining number sense consisting six strands that is: understanding of the meaning and size of numbers(number concepts), understanding and use of equivalent forms and representations of numbers(multiple representations), understanding the meaning and effect of operations (effect of operations), understanding and use of equivalent expressions (equivalent expressions), computing and counting strategies and measurement benchmarks. This paper will explore children’s understanding and use of equivalent forms and representations of numbers(multiple representations) strand of number sense.

Multiple representation refers to the recognition that numbers take many different numerical and representational forms (eg. Fractions as decimals, a whole number in expanded form, or a fraction on a number line) and can be thought about and manipulated in many ways to benefit a particular purpose.

Purpose of the study

The purpose of this paper is to explore children’s understanding and use of equivalent forms and representations of numbers(multiple representations) strand of number sense.

Research Methodology

A total number of 406, 11 year old school children from four different schools in Malaysia were asked to answer nine items related to the understanding of multiple representations in a number sense test. Six students were chosen for interview sessions to further explore into their understanding of multiple representations of numbers.

Discussion of the findings: understanding of multiple representations in a number sense test.

The items in the number sense test were given 1 for a correct score and 0 for an incomplete or incorrect answer. The percentage of correct answers for the children’s understanding and use of equivalent forms and representations of numbers(multiple representations) is indicated in the table below.

children’s understanding and use of equivalent forms and representations of numbers(multiple representations)	Percentage of Correct answers
Fractions as shaded regions, collection of (Q9, Q15,Q17)	34
Number density on a number line (decimals) Q12, Q13, Q 14	Q12: 4 Q13: 4 Q14: 22
Representing equivalent fractions and decimals on a number line Q10, Q11 Multiple representations of numbers (fractions or decimals)	Q10 (decimals): 45 Q11(fractions): 23
Whole numbers on a number line Q16	27

The data from this study shows that children’s understanding of fractions is good(34%). Questions 12, 13 and 14 test on students understanding of number density on a number line(decimals).

Question	Percentage correct
13. Estimate the decimal shown by the arrows on the number line   0 1	54%
14. Estimate the decimal shown by the arrows on the number line 0 0.1	22%

Students’ performance for question 13 is very good , on which 54% answered correctly. The reasearchers’ opinion is that this is because question 13 required the students to name a decimal that represent a midpoint of the number line 0 to 1. However, when students were asked to name a decimal that represent the midpoint of the number line 0 &emdash; 0.1, the percentage of correct answer dropped to 22%.

Questions 10 and 11 explores the students ability in representing equivalent fractions and decimals on a number line.

10. Place the numbers 0.1 and 0.8 in their correct positions on this number line	% correct   0 1 45%
11. Place the numbers 1/10 and 4/5 in their correct positions on this number line	23%   1

Discussion of the findings: interview

The interview items focused on exploring students understanding and use of equivalent forms and representations of numbers (multiple representations) through exploring the following concepts:

1. Students’ representation of fractions
2. Representing fractions on a number line
3. Fractions as decimals

The six students’ understanding of the above concepts are summarized in the table below:

Student	Representation of the fraction 2/4	Fractions as decimals A/10 4/5
Faiz		Converts a/10 to 0.a	Converts 4/5 to 4.5 but knows 4/5 = 8/10 therefore says 4/5 = 0.8 but still thinks a/b = a.b
Jaya		Converts a/10 to 0.a	Converts 4/5 to 4.5
Lila	The numbers that represent the given fraction	Converts a/10 to 0.a	Converts 4/5 to 4.5
Safwanah	4 marbles that is divided into 2 groups	Converts a/10 to 0.a	Converts 4/5 to 4.5
Syazwan		Converts a/10 to 0.a	Converts 4/5 to 0.8 by dividing 4 into 5
Wei Lei		Converts a/10 to 0.a	Does not know how to convert 4/5 to it’s decimal form but knows that 4/5 < 5/5, therefore 4/5 < 1.

Conclusion

. An analysis of the students’ performances in the number sense test showed that many students have difficulties understanding the concepts of number density on a number line for decimal numbers and representing equivalent fractions and decimals on a number line. However, the students in this study showed a good understanding of representing fractions as shaded regions. Moreover, from the interview conducted, it was found out that four out of the six students interviewed used a shaded figure to represent the fractions 2/4 and _. All of the six students interviewed were able to convert a fraction a/10 to its equivalent decimal form. However, four of the six students interviewed converted the fraction a/b with b not equal to 10 as a.b in it’s decimal equivalent. Two students cannot convert 4/5 to 0.8 but their arguments to the equivalent fraction for 4/5 showed that they have a good understanding of number sense. When the students were asked to represent fractions on a number line, three out of the six students interviewed thought that fractions _, 2/4 ….8/4, 9/4 and 10/4 are the fractions between the whole numbers 1 and 2.

References

Burns, M. (1989). "Teaching for understanding: A focus on Multiplication". In Trafton, P.R, & Shulte, A.P. (Ed). New Directions for Elementary School Mathematics. Reston, Virginia: NCTM; 123-134.

Leutzinger, L.P. & Bertheau, M. (1989). "Making sense of numbers". Dalam: Trafton, P.R, & Shulte, A.P. (Ed). New Directions for Elementary School Mathematics. Reston, Virginia: NCTM; 111-123

McIntosh, A., Reys, B.J., Reys,R.E, Bana,J;.& Farrell,B. (1997). Number sense in school mathematics, student performance in four countries. MASTEC Monograph series no.5, Edith Cowan University.

National Council of Teachers of Mathematics (1989). New directions for elementary School Mathematics. Reston, Va.