Creative Education
2013. Vol.4, No.9, 69-73
Published Online Septe mber 201 3 in SciRes (http ://www.scirp.org/journal/ce) http://dx.doi.org/10.4236/ce.2013.49B014
69
Analysis of Pupils’ Difficulties in Solving Questions Related
to Fractions: The Case of Primary School Leaving
Examination in Tanzania
Joyce Lazaro Ndalichako
National Examinations Council of Tanzania, Dar es Salaam, Tanzania
Received July 2013
In this paper, analysis of the performance of candidates in Mathematics in Primary School Leaving Ex-
amination was conducted with the aim of highlighting difficulties encountered in solving fraction-related
problems. The analysis has indicated that a considerable number of candidates could not perform correct
operations related to fractions. They tended to confuse fraction concepts with whole number concepts. For
instance, in questions involving addition of fractions, they were treating numerators and denominators as
separate entities. Possible reasons for such difficulties in solving questions related to fractions include
lack of understanding of appropriate procedures to apply in solving a problem, the complexity of the task,
over-generalization of procedures even in situations which are inappropriate. It is recommended that a
protocol analysis be conducted in order to gain a deep understanding of the thought process of candidates
when attempting questions related to fractions so that teachers may use relevant teaching methods that
would facilitate meaningful learning of fractions.
Keywords: Distracter Analysis; Fraction Computations; Misconceptions
Introduction
Educators and researchers agree that most students encounter
major problems in learning fractions (Buzek & Bieck, 1993;
Newstead & Murray, 1998; Pitkethly & Hunting, 1996; Tzur,
1999). It is well documented that fractions are among the most
complex concepts that children encounter in their years of pri-
mary education (Saxe, Taylor, McIntosh, & Geahart, 2005;
Stafylidous & Vosniodou, 2004; Idris & Narayanan, 2011). It
has also been asserted that learning fractions is probably one of
the most serious obstacles to the mathematical maturation of
children (Behr, Harel, Post, & Lesh, 1993). Different views are
expressed as factors contributing to students’ difficulties in
learning fractions. Possible reasons stated for such difficulties
include the fact that there are many rules associated with the
computation of fractions which are more complex than those of
natural numbers. Moss and Case (1999) established that stu-
dents difficulties in learning fractions emanate from the fact
that most teachers devote too much time to teaching the proce-
dures of manipulating fractions and too little time to teaching
their conceptual meaning. Often students’ competence with a
rote procedure acts as a major obstacle for their conceptual un-
derstanding of fractions which in turn makes them unable to
monitor their work. They can only check their answers by re-
peating the rote procedure used and fail to judge the reasona-
bleness of their answer because they are too confident with
their approaches.
In this paper, analysis of the general performance of candi-
dates in Mathematics in Primary School Leaving Examination
was conducted with the aim of highlighting difficulties en-
countered by pupils in solving mathematical problems. More
specifically questions related to fractions have been given due
attention because review of items in Mathematics examination
indicates that candida tes did not perform well in questions from
this area.
The System of Education in Tanzania
The system of education in Tanzania is pyramidal in nature
ranging from Primary to University level. The structure of for-
mal education is 2-7-4-2-3 comprising of 2-years of pre-pri-
mary education, 4 years of secondary ordinary education level;
2 years of secondary advanced level and a minimum of 3 years
of University education. Progression from one level to another
depends on the performance in the summative examination spe-
cifically set at each respective level of education. Since the fo-
cus of this paper is on primary school leaving examination, the
following sections highlight the structure of the education at
Primary level and the nature of its final examination.
The Structure of Primary Education in Tanzania
Primary education is a seven-year cycle which is compulsory
in enrolment and at tendance for children age d 7 years. The cycle
is divided into “Standards” representing the respective year of
study (i.e. Standard I to VII) for pupils. In this paper the term
Grade will be used to imply Standard. Prom ot ion from one Grade
to the next is automatic. Repetition of a class is only allowed in
Grade I-IV. For Grades V-VII, repetition can only be allowed if
one has approval from the relevant authorities based on genuine
reasons such as ill ness. Grade VII marks the end of Pri mary e du-
cation and all pupils are subjected to Primary School Leaving
Examination which is used for selection of those who qualify to
go on with secondary education. Those who are not progressing

J. L. NDALICHAKO
70
to secondary educ ation can join vocati onal traini ng or pur sue any
other options leading into the world of work.
There are six major subject area s that are taught at Primary le-
vels namely Languages (Kiswahili, English and French); Social
Studies (History, Geography, and Civics); Science and Technol-
ogy (Science and ICT); Life Skills (Vocational Studies and Per-
sonality and Sports); Ethics and Religious studies; and Mathe-
matics (Tanzania Instit ute of Education, 2004). English and Kis-
wahili Languages as well as Mathematics are regarded as funda-
mental subjects and are give n greater weight in ter ms of hours of
teaching than other subjects. The duration of a period is 30 mi-
nutes for Grade I-II and 40 minutes for Grade III-VII. Table 1
indicates the number of periods per subjects taught in Primary
Schools.
The Primary School Leaving Examination
The Primary School Leaving Examination (PSLE) marks the
completion of primary education cycle. The PSLE consists of
five papers assessing seven core subjects in primary education
curriculum. The papers incl ude Mathemat ics, English Language,
Kiswahili, Science and Social Studies (Civics, Geography, and
History). The PSLE is set by the National Examinations Coun-
cil of Tanzania (NECTA) and administered in schools through
a close supervision of the regional and district authorities. Each
paper consists of 50 multiple choice items with 5-options where
candidates are supposed to select the correct answer.
Development of items for PSLE is done carefully by ensuring
that all questions are set within the respective subject syllabus
and conform to the key principles in setting examination items.
Experienced primary school teachers are invited by NECTA to
submit examination items in accordance with the guidelines
given. The items submitted are subjected to moderation by
Table 1.
Subject taught and number of periods per week.
S/N Subject Number of Periods per Week per Grade
I - II III - VII
1. Kiswahili 6 7
2. English 7 7
3. Mathematics 7 7
4. Science 2 4
5. Geography - 3
6. History - 2
7. Civics - 2
8. Vocational Skill s 3 2
9. Personality and Sports 2 2
10. ICT 1 2
11. Religious Studies 2 2
12. French 2 2
Total 32 42
Note: Source: Tanzania Institute of Education, (2004). Primary Education Curri-
culum.
subject officers and other education experts such as inspectors
and curriculum developers. In moderation of items, clarity of
the items, difficulty level and content coverage are among the
areas covered. Furthermore, all distracters are reviewed to en-
sure that they are equally attractive. Misconceptions and possi-
ble errors that can be made by pupils are used in developing
distracters that are likely to attract pupils who have partial
knowledge in the respective area.
Performance of Candidates in PSLE 2012
A total of 894,839 candidates registered for PSLE in 2012.
The number of candidates who sat for the examination was
865,827 which is equivalent to 96.76 percent of registered can-
didates. The overall percentage of candidates who passed the
examination was 30.72. When the performance is disaggregated
subject wise, Mathematics can be seen as one of the subjects
which contributed significantly in pulling down the overall per-
formance of candidates as only 18.74 percent of the candidates
passed Mathematics examination. Table 2 indicates the perfor-
mance of the candidates in PSLE 2012.
It can be seen from Table 2 that when compared to other
subjects, performance in mathematics is quite low. The item
analysis data for the subject revealed that item difficulty for
mathematics ranged from .10 to .75 while the difficulty level
fraction items ranged from .10 to .45 indicating that fraction
was among the most poorly performed topics in the examina-
tion (NECTA, 2013).
Overview of Challenges in Learning Fraction s
Fractions play a central role in mathematics learni ng. They are
theoretically important because they build a foundation which
helps the pupils t o succes sfully l earn topic s related to pe rce ntage,
ratios and decimal numbers. Fractions require a deeper under-
standing of comput ational proced ures than tha t typically required
with whole numbers. In Tanzania, the topic of fractions is an
integral part of pri mary school mathematic s syllabus which i s in-
troduced as early as pupils start Grade I, ye t it is one of the most
difficult areas for pupils to master. Siebert and Gaskin, (2006)
contended that c hildren are bound to find fractio ns computations
arbitrary, confusing and easy to mix up unles s they receive assis-
tance in understanding what fractions and fraction operations
mean. For instance when fraction addition and subtraction prob-
lems have the same denominator the denominator is maintained
in the answer; but that is not the case for fraction multiplication
Table 2.
Subject performance in PSLE 2012.
Subject Candidates Sat Candidates Passed
Number Per cen t
Kiswahili 865,173 354,588 41,0
English La nguage 865,176 182,145 21.0
Social Studies 865,281 247,448 28.6
Mathematics 865,221 162,078 18.7
Science 865,048 358,731 41.5
Source: NECTA, (2012). Primary School Leaving Examination Results Statistics.

J. L. NDALICHAKO
71
and division. They suggested that emphasis should be on con-
ceptual understanding rather than procedural understanding of
operations related to fractions for meaningful learning.
Gould, Outhred, and Mitchelmore (2006) in their research aim-
ed at understanding stu dent reas oni ng and misconc eption s related
to fractions, they noted that students perceived fractions as parts
of the sets rather than parts of the whole. In the parts of a set
conception of fractions ¼ can be interpreted as meaning one
object out of four; synonymous to the counting activity which
assumes that 1 represents a whole number and 4 represents a
whole number as well . Mack (1990) observe d that computati onal
algorithms involving fractions prevent students from even trying
to reason out or make sense of fraction in real life situations. For
instance, the “invert-and-multiply” algorithm for dividing frac-
tions does not develop naturally from students using manipula-
tive (Borko et al., 1992). St udents tend to not only reme mber the
incorrect algorith ms involving fra ctions but also have more faith
in them compared to their own reasoning.
Statement of the Problem
The important role of mathematics in helping the students to
develop logical thinking is well acknowledged by educators and
researchers. Recognizing that role, the education system in Tan-
zania is also putting a lot of emphasis in mathematics from pri-
mary to secondary levels of education. With the exception of
English Language, the nu mber of periods assigned to mathemat-
ics subject starting from Grade I is higher than the number of
periods in other subjects. In the Certificate for Secondary School
Examination, candidates who fail in mathematics cannot be
awarded first or second division even if they have excellent
grades in all of t he remaining subject s. The penalty exercised f or
failing mathemat ics is meant to encourage stude nts to work hard
in the subject so that in turn they establish a solid foundation to
pursue other mathematics-related subjects and enhance their lo-
gical thinking. Howeve r, the performance in mathematics at both
primary and secondary l evels of educat ion re mains unsati sfactory
for many years.
Objective of the Study
The objective of the study was to identify area s in mathemati cs
in PSLE in which the majority of candidates were not able to
obtain a correct answer and cond uct analysis of the distracters in
order to identify pos sible reasons that may have made the candi-
dates unable to obtain a correct answer. It is expected that the
analysis of items will assist educators and policy makers in rea-
lizing the difficulties encountered by candidates in attempting to
solve mathematical problems and thereby devising better ap-
proaches in training mathematics teachers so t hat they can be able
to address such difficulties in teaching a nd learning process. It i s
also anticipated that the findings will help to fill gaps in research
in this area and stimulate further studies.
Analysis of Candidates’ Responses to Fraction Items
In this section th e analys is of candidates re sponses to i tems re-
lated to fractions is presented. The fraction items ranged from
simple operations of ad ding and subtracting fractions to complex
ones involving finding the square roots of fractions. For each
item, the distracters that were selected by many candidates were
analyzed with the aim of identifying possible sources of errors
and difficulties e ncountered by students. The anal ysis of specific
fraction questions is hereby presented:
Question 7:
3
5
5
−
2
3
1
= A. 1
15
4 B.
1
2
4
C. 1
15
3 D.
14
15
4
E. 14
15
3
Question 7 was about subtracting mixed fractions. A total of
860,028 candidates attempted this question where only 283,399
(32.76%) were able to select the correct answer E. Distracter A
14
15
3
was more attractive than others as 177,216 (20.49%) se-
lected it (NECTA, 2013). These candidates used an incorrect
procedure for subtracting mixed fractions with different deno-
minators. They treated whole numbers in isolation of their re-
lated fractions. They are likely to have selected that option
through an incorrect computation approach as follows:
3
5
5
−
2
3
1
= (5
−
1)
32
53

−


= 4
9 10
15
−



= 4
1
15



The candidates who selected A
1
15
4
are likely to have used
the same incorrect procedure to arrive at their solution. Al-
though they obtained a negative fraction, they simply ignored it
and used the absolute value of the fraction. This indicates that
the candidates did not have a good understanding of procedures
required in subtracting mixed fractions.
On the other hand 18.04 percent of the candidates selected B
4½ which was not a correct answer. These candidates had a
similar problem to those who selected option A as they treated
the whole numbers and fractions in isolation of each other.
However, their problem was compounded by the fact that they
treated the numerators and denominators separately as they did
not find the common denominator first. In their incorrect
computation, th ey first subtracted the whole numbers as follows:
5
3
5
−
1
2
3
= (5
−
1)
32
53

−


= 4
32
53
−


−

= 4
1
2
Candidates’ prior learning of whole numbers and the manner
in which fractions were introduced to them at early stages of
learning of fractions may have affected their conceptual under-
standing of proper algorithms to use. Post et al. (1993) and
Moss and Case (1999) have demonstrated that children’s whole
number schemes can interfere with their learning of fractions. It
is quite possible for such a situation to have happened to the
candidates who could not answer the item correctly because in
Tanzania operations about fractions are introduced to them in
Grade IV where they ha ve a l rea dy acquired a considerable know-
ledge about whole numbers and their operations. In Grades I-III
children are only taught about recognition of different fractions.
Question 8:
1
1
2
+ 3
1
2
+
3
8
= A.
5
12
4
B.
1
4
5
C.
3
8
5
D.
5
8
5
E.
3
8
4
.
Question 8 was measuring candidates’ ability to add mixed
fractions. Only 37.4 percent of the candidates were able to se-
lect the correct answer which was C 5⅜. Among the distracters,
option A attracted more candidates than other distracters. The
22.23 percent of candidates who selected A were using an
5
12
4
incorrect algorithm for adding fractions with different denomi-
nators. They simply added the numerators and denominators as
they were provided and then added the whole numbers sepa-
rately as follows:
11 31135
13(1 3)4
22822 812
++
+ +=+=
++
Such candidates had a fundamental problem of understand-
ing the essence of fractions as they treated numerators and de-

J. L. NDALICHAKO
72
nominators as two different numbers that do not constitute a
unique entity.
Question 14: Multiply the square roots of
32
18
1
and
9
1
11
.
A.
1
22
B.
1
212
C.
5
36
D.
18
11288
E.
1
46
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
15
16
and
20
21
.
A.
25
28
B.
63
64
C.
64
63
D.
35
37
E.
25
24
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
55
24

+


to
get
1
8
?
A. 3 B.
3
4
2
C.
5
8
3
D.
1
2
3
E.
5
8
3−
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
5
8
3
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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