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research carried out over the last twenty years reported that all students constantly

invent rules to explain the patterns they see around them (A skew and William 1995).

However, when these students use incorrect rules, or use correct rules beyond their

proper domain of application, it becomes a misconception. Misconception is generally

defined as an opinion which draws to wrong conclusion due to faulty thinking and

understanding, or of incorrect facts presentation. Misconceptions often interfere with

understanding and interpreting the new recommendations on mathematics education,

and become subtle obstacles to implementing the new practices in the classrooms

(Richardson, 1996). Thus during the learning process for fractions and decimals, one

most common misconception is operating fraction or decimal numbers as if they are

whole numbers.

Difficulty is another common challenge in learning Mathematics. Developing

decimals place value knowledge and teaching mathematical operations in fractions are

the most common difficulties encountered in their process of learning. In 2005,

Gallup Incorporated, a research-based, global performance-management consulting

company, conducted a poll that asked students to name the school subject that they

considered to be the most difficult. Not surprisingly, mathematics came out on top of

the difficulty chart. Oxford dictionary defined the term difficult as a task whichrequires much effort or skill to accomplish, deal with, or understand. Unlike other

subjects, the mastery of Mathematics is not build on good memory or character

recognition, but rather on the concept of understanding and knowing how. Thus, it is a

subject that requires a lot of patience and attention to understand, as well as time and

energy to apply the concept into learning practice in order to strengthen the

foundation.

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2. Concept Maps to summarise Major Mathematical Concepts, Skills and

Relationships

Concept maps were first introduced as a research tool, showing in a special

graphical way the concepts related to a given topic together with their interrelations.

The method of concept mapping has been developed specifically to tap into a

learners cognitive structure and to externalise what the learner already knows

((Novak and Govin, 1984).

Concept map activities can reveal the underlying structure or organization of

students knowledge of a concept or constellation of concepts. These are very helpful

when the kinds of causal theories and relations among ideas are critical to them

understanding the course materials. Although concept mapping has been used as an

educational tool, above all in science, the experiences in mathematics education are

still rather seldom and not well-documented. Thus, the use of concept maps in

teaching mathematics serves as one of the potential promising tools in providing a

better knowledge for students on learning mathematical concepts.

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3. Common Difficulties in the L earning of Fractions and Decimals 3.1. Fractions

(a) Failing to find a common denominator

Students always have hard times to obtain the common denominators in a

fraction operation. This is because they do not understand that different denominators

reflect different-sized unit fractions and that operating fractions requires a common

denominator. In fact, they tend to view fractions as isolated digits, treating the

numerator and denominator as separate entities that can be operated on independently.

Such perception leads them to the use of incorrect algorithms, for instance:

(i)

X

The same underlying can lead students to make a similar error in changing the

denominator of a fraction without making a corresponding change to the numerator,

for example, converting the problem

(ii)X

(b) Reducing fractions to the simplest form

All fractions are being reduced into its simplest form by dividing both

numerator and denominator with a common factor until can no longer be divided by

the same whole number exactly or evenly (other than 1). Students may have no

challenge in providing the correct answer during mathematical operation, but most of

them have difficulties in producing the simplest fraction form, for example:

(i)X

(ii) In fact, those with pre-requisite knowledge on multiplication and division

numbers find it more reasonable, however they need to remember the abstract rule ofwhatever been done to the bottom, must be done the same to the top.

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3.2. Decimals

(a) Incorrectly adding and subtracting decimals

Students always have difficult in understanding the numbers on the left and

right of the decimal point contribute to the size of the number, for example:

(i) 3.4

+ 1.8

4.12

X

Most students will complete this operation by viewing the numbers before and

after the decimals point as independent entities, thus adding the value 3 to value 1

whereas, the value 4 to the value 8 producing to total number of 4 and 12 respectively.

Eventually they will combine these two numbers to produce the final answer of 4.12.

(b) Failing to align decimal points during mathematical operations

Just as whole numbers have place values, decimal places also have place

values. Like fractions, decimal point tells a part of the whole, such as 0.25 is a quarter

of a whole, 0.5 refers as half, and so on. However, when students that do not

understand this concept treat decimal numbers as a whole. Thus, during operations,

most students will fail to align decimal points, for examples:

(i)X

(ii)

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4. Possible Misconception in Learning of Fractions and Decimals 4.1. Fractions

(a) Treating numerators and denominators as separate whole numbers

One of the most common misconceptions in Year 4 fraction involves

mathematical operations, such as addition, subtraction and multiplication, of two or

more fractions. Gould, Outhred, and Mitchelmore (2006) noted that students

perceived fractions as parts of the sets rather than parts of the whole. Students often

add or subtract the numerators and denominators of two fractions by simply adding

the numerators and adding the denominators, for examples:

(i)

XBasically they fail to recognize that denominators define the size of the

fractional part and that numerators represent the number of whole part. In other words,

the students show a lack of understanding of the conceptual basis of arithmetic

procedures involving fractions.

(ii)

(b) The bigger the number on the bottom, the bigger the fraction.

This is another misconception in fraction which results to wrongly ordering

unit fractions, for example to think

(i) 6 is bigger than 2Hence, 1/6 is bigger than 1/2 X

Understanding that the number on the bottom tells us how many parts the

whole has been divided into. Thus, students need to understand that the more parts

there are the smaller each portion will be.

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4.2. Decimals

(a) The longer number, the bigger value

Students often lack of understanding of the place value of decimals. In this

case, when students treat decimal numbers as whole numbers, they misunderstand the

concept that a longer number is bigger, for example:

(i) 134 is bigger than 34;

Hence, 0.134 is bigger than 0.34X

Students also think that by placing a zero on the end of a number, it makes all

decimals 10 times as large, such as:

(ii) 150 = 10 X 15;

Hence, 1.50 = 10 X 1.5X

(b) The value get bigger when move to the left

However, students who understand that decimals are different from whole

number may still hold certain incorrect perceptions. For instance, they know that in

place value, numbers get bigger as they go to the left, thus in decimals they get

smaller after the decimal point, for example:

4 3 2 1 0 . 0 1 2 3 4

Such misconception indicates that students use a combination of their

understanding of place value with their understanding of a number line, but clearly

do not understand that digits after the decimal point represent parts of a whole.

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5. Suggestions on Ways to Overcome Difficulties and Misconceptions

5.1. Overcome Difficulties in Fractions

Many students consider learning fractions as difficult and too often have

difficulty understanding why a particular procedure is carried out to solve a

calculation involving fractions. Thus, in order to develop a more secure understanding

during the learning process of a fraction concept, one of the approachable methods is

through explaining procedure using visual and practical aspect of creating simple

fractions of shapes. This will assist students to generate a firm understanding of what

the denominator represents and the numerator represents through the use of visual and

kinaesthetic resources.

Apart from that, it is also important that students need to understand why the

procedures are carried out based on the rule, rather than simply learning a rule to

remember in order to solve a certain calculation. For example, when adding fractions

of a like denominator the aforementioned rule of add the tops but not the bottoms

will apply. When students understand why this is to be done, they may then apply

this to addition of fractions with unlike denominators. Understanding this rule

concerning common denominators will greatly help them to understand the next step

of altering fractions to meet a common denominator, thus enabling them to add the

fractions together or even compare the sizes and order the fractions.

Similarly using the rule of whatever do to the bottom, do to the top (and vice

versa) in explaining the concept of reducing a fraction to its simplest form would

have no meaning when students do not understand the concept of equivalent fractions

In other words, it is essential that students have this pre-requisite knowledge of

fractions in order to use and apply their knowledge within a range of different

contexts.

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5.2. Overcome Misconception in Fractions

Students often treat the numerator and denominator as separate whole numbers.

Teachers can help students to overcome the misconception that this is an acceptable

procedure by presenting meaningful problems in the classroom. For example, they

could ask: If you have 3/4 of an orange and give1/3 of the original orange to a friend,

what fraction of an orange do you have left? Under such misconception, subtracting

the numerators and denominators separately would result 2/1 or 2. Students should

immediately recognize the impossibility of starting with 3/4 of an orange, give part of

it away, and ending up with 2 oranges. After being shown why their procedure is

faulty, students should be more receptive to learning the correct procedures.

Besides that, visual representations of fractions, together with good teaching

aids, help develop conceptual understanding in fractions. From direct visual learning,

students will be able to observe fraction sizes with equal fraction numbers to enhance

their fundamental understanding on the sizes and order the fractions before proceed to

the next level of learning.

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5.3. Overcome Difficulties in Decimals

Basically when working with decimal, students appear to have an automatic

tendency to see and read the digits after the decimal point as a number, for example:

2.47 is read as two point forty -seven rather than two point four seven. Reading off

decimal numbers from a number line helps student to make links with interpretation

of previous work on graduated scales and prepares for the exercises on ordering

decimals, which some students find it difficult. In fact, reading gives a vocal

indication of where the whole numbers end, and the fractional parts begin in the same

way that the decimal point separates the whole from the parts visually. Thus,

students need to read decimal numbers correctly.

Apart from that, most teachers tend to use real world contexts such as the use

of money or sports statistics while teaching the concept of decimals. This allows

students to deal with them in the same way in which the deal with whole numbers.

They can be lined up to see which is the biggest or smallest in the same way that

whole numbers are, and can be added if line up the decimal point. This context

familiarises students with the use of a decimal point and understand the necessity of

decimal alignment when conducting mathematical operations, such as addition and

subtraction.

Placing decimal point after multiplying application is one of the reasons that

most students produce the wrong final answer. Teaching standard mathematical

procedures complying to multiplication of decimal numbers has always been a great

way to overcome such difficulty. In fact, teachers can also create simple catchphrase

for students to remember the procedure so that they will not miss any step in

calculation. Besides, it may be used to remind students to check through their work

after operation. This can help to reduce carelessness in students when providing the

final answer.

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5.4. Overcome Misconception in Decimals

There are a few common way of introducing students to decimals using visual

representation, such as rulers and place value blocks. Students can use their rulers to

show divisions of centimetres into tenths. This helps them to see that decimal

numbers come between whole numbers. When learning using the place value blocks,

students understand that each unit block plays one part of the whole. Hence, alike

fractions, students are able to develop a better understanding that any numbers which

comes after the decimal point play only as a part of the whole.

Apart from rulers, LAB (Linear Arithmetic Blocks) is a good physical or

concrete model to make decimal numbers. For examples, in using 0.2, 0.26 and 0.3

made out of pieces of pipe which represent tenths and hundredths. Students can

observe the difference between these structures built on base ten numeration system.

From that, they are able to make an ordering from 0.2 to 0.3. Besides that, adding

zeros can also reinforce their understanding between 0.2 and 0.20.

Most students often find it rather difficult to understand decimal fractions as

an extension of place value as used for ones, tens, and hundreds. Students are to be

encouraged to use sequential strategies to order a range of decimal numbers, forinstance, (1) line up the decimal points to compare the place value of the digits; (2)

order the numbers by looking at the whole numbers; (3) order by the tenths digits

when whole numbers are the same; (4) order by hundredths digits when wholes and

tenths are the same. Consequently this helps to avoid students from making mistakes

concerning the value of decimal numbers.

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6. Conclusion

Difficulties and misconceptions abound in mathematics. Obviously it has been

the results of teaching techniques that encourage the emergence of such

misconceptions. Researchers agree that most misconceptions are difficult to overcome.

Thus t he identification of misconceptions in students work is a vital part of the

process of moving towards a focus on learning rather than teaching. It is more

important for teachers to make sure that the misconceptions do not arise in the first

place while conveying the new knowledge.

Siebert and Gaskin (2006) contended that students are bound to find

mathematics computations arbitrary, confusing and easy to mix up unless they receive

assistance in understanding what these operations mean. Thus when teaching

mathematics, teachers need to be on the lookout for students' common misconceptions

that lead to errors in computation.

However, some education researchers recommended that addressing

misconceptions during teaching does actually improve achievement and long-term

retention of mathematical skills and concepts. Drawing attention to a misconception

before giving t he examples was less effective than letting the pupils fall into the trapand then having the discussion. In other words, learn from mistakes is one of the

effective ways in learning mathematics, provided that students completely understand

how and why they fall into the trap.

Understanding of their thought process is very essential to enable teachers to

use relevant teaching methods that would facilitate meaningful learning to the pupils.

On the other hand, teachers teaching approaches should be investigated with the aim

of identifying their professional training needs, if any, in developing conceptual

understanding of fractions.

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Irwin K.C. (1999). Difficulties with Decimals And Using Everyday Knowledge toOvercome Them. Research Information for Teachers set 1999: no.2.

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