J. L. NDALICHAKO

Copyright © 2013 SciRes.

nominators as two different numbers that do not constitute a

unique entity.

Question 14: Multiply the square roots of

and

.

A.

B.

C.

D.

E.

Question 14 required the candidates to find the square roots of

the numbers given and then multiply. This question was the most

difficulty item in the whole exami nation with the difficu lty index

of .1. Only 87,465 candidates out of 848,739 who attempted it

obtained a correct answer (NECTA, 2013). The majority of the

candidate s (397,841 equivalent t o 45.99%) ch ose option D which

was not a correct answer. These candidates neglected the key

requirement in the item of multiplying the square roots of the

given mixed fractions and attempted to multiply the given frac-

tions as they were given. Moreover, they used an incorrect al-

gorithm for multiplying the fraction in the sense that they mul-

tiplied the whole numbers and the fractions separately:

( )

18 1818

11

32 9329288

1 111 1111111

18 1

32 9

×=×=× =

×

×

.

Ideally , candidates should have converted the given mixed

fractions into improper fractions before they started to perform

the computation as required in the question.

Question 17: Find the product of

and

.

A.

B.

C.

D.

E.

Question 17 required candidates to find the product of the

given fractions. Only 28.62 percent of the candidates selected

the correct answer whi ch was A . Contrary to what was asked, the

majority of candidates (30.81 percent) selected option D which

was obtained by finding the sum of the fr actions given instead of

the product as required. They failed to distinguish the fact that

the sum is obtained by adding the numbers while the product is

obtained by multiplying the numbers. Their problem of not

complying with the requirement of the question was com-

pounded by the fact that even their addition procedure was

incorrect. Instead of finding a common denominator first, the

candidates treated the numerators and denominators as separate

numbers and incorrec tly adde d the fractions as follows:

152015 2035.

162116 2137

+

+= =

+

Candidates who selected this option show a lack of under-

standing of the conceptual basis of arithmetic procedures in-

volving fractions. This suggests that students tend to view frac-

tions as isolated digits, t reating the numerator and denominator

as separate entities that can be operated on independently. Such

perception leads them to the use of incorrect algorithms.

Question 20: What number should be added to

to

get

?

A. 3 B.

C.

D.

E.

This item was one among the most difficulty items as only

14 percent of the candidates got the item correctly by selecting

the correct answer E. On the other hand, 25.67 percent of the

candidates selected C

which was equivalent to the correct

answer in absolute value but could not be the correct answer

because the question required them to find a number which, if

added to the sum of the two fractions given the answer would

be ⅛. Candidates could have used the process of estimation to

realize that 3⅝ was already greater than ⅛ even before adding

the two given fractions.

Careful analysis of the options given for this item indicates

that test-wise candidates would have been able to obtain the

correct answer by elimination process because the values for all

the distracters were already greater than the sum that was to be

obtained. In order to get the fraction which was less than the

numbers that were added, a third number must have been a

negative.

Conclusion

In this paper, analysis of Primary School Leaving Examina-

tion questions related to fractions was conducted. The analysis

revealed that a considerable number of candidates could not

perform correcting operations related to fractions. They tended

to confuse fraction concepts with whole number concepts. For

instance, in questions involving addition of fractions, the y were

treating numerators and denominators as separate entities. Pos-

sible reasons may be attributed to pupils’ difficulty in solving

questions related to fractions. These include lack of understan-

ding of appropriate procedure to apply in solving a problem and

complexity of the task. Over-generalization of procedures could

also pose a problem to pupils and make them apply inappro-

priate procedures for a given problem. For instance multiplying

a number makes it bigger. However, multiplying a number with

a fraction or a decimal numbers less than one makes the prin-

ciple untenable and that could create a problem to students who

have such generalization.

This paper was limited to multiple choice items where can-

didates were only selecting the correct answer among the given

options. It is very likely that if the candidates were asked to

construct their own answers a lot of misconceptions related to

fractions would have been revealed. Therefore it is recommen-

ded that a study should be conducted in order to gain a deep

understanding of the thought process of candidates when at-

tempting questions related to fractions. Understanding of their

thought process is very essential to enable teachers to use rele-

vant 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 profes-

sional training needs, if any, in developing conceptual under-

standing of fractions.

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