RESEARCH PAPERS

MATHEMATICAL KNOWLEDGE FOR TEACHING (MKT)

AMONG PRIMARY SCHOOL TEACHERS OF FIJI

By

T. N. RAIULA * VIJAYA KUMARI S. N. **

* Research Scholar, Department of Education, St. Ann's College of Education (Autonomous), Mangaluru, Karnataka, India.
** Associate Professor and Research Guide, St. Ann's College of Education (Autonomous), Mangaluru, Karnataka, India.

Date Received: 13/03/2018 Date Revised: 07/05/2018 Date Accepted: 23/05/2018

ABSTRACT
Primary School Teachers in Fiji are employed and supervised by the Ministry of Education and are expected to teach all
subjects prescribed by the Ministry with necessary teacher qualification. The objectives of the study is to examine the
current status of MKT for Primary School Mathematics Teachers in Fiji schools in terms of its components; Knowledge of
Content and Students (KCS); Knowledge of Content and Teaching (KCT); Common Content Knowledge (CCK); and
Specialized Content Knowledge (SCK), and to compare status of MKT (Mathematical Knowledge for Teaching) for
primary schools teachers, with gender. The study engages descriptive survey with engagement of Stratified Random
Sampling technique with sample size of 363 Primary School Teachers, from each stratus covering the four major
Divisions; Northern, Central, Western, and Eastern and uses a test on MKT to gauge the levels of teachers' MKT and t-test
analysis is used to analyse the data quantitatively. The relevance of the study is to give a sense of direction to the Ministry of
Education in Fiji, on how to support primary school teachers with teaching and other related assistance, to bring about
higher achievement in mathematics assessments, and more importantly, meaningful learning for students. Finally, the
study allows dialogue of different college teachers on the notions of MKT and incorporate in mathematics education
courses. The findings of the study are discussed in detail in the paper.
Keywords: Mathematical Knowledge for Teaching, Primary School Teachers, Common Content Knowledge, Knowledge
of Content and Teaching, Knowledge of Content and Students, Specialised Content Knowledge.

INTRODUCTION knowledge that teachers need 'in' teaching Mathematics

One of the key goals for education policymakers is to and 'for' teaching mathematics as defined by Schneider
improve the education system and to ensure that relevant and Plasman (2011), to enable one to teach mathematics
curriculum be in place to bring about positive outcomes in more effectively.
students' learning. For the accomplishment of the The Mathematics Knowledge for Teaching (MKT) concept
highlighted goal many researchers agree that teachers' has been initially derived from Shulman's (1986)
knowledge needs to be the focus, in terms of content and experimental model and further modified to a more inter-
pedagogical aspects of teaching (Ono and Ferreira, related components of teachers' MKT which was further
2010). Significant work has been done over the past designed by Hill et al. (2008) and been labelled as one of
decade in trying to understand teachers' content the most exceptional re-organised structures of the
knowledge, values, and skills finely woven with classroom teachers' Pedagogical Content Knowledge, within
practice, to bring about effective teaching. mathematics education arena.
Mathematical Knowledge for Teaching Shulman's Model highlighted the two major components in
Mathematics Knowledge for Teaching includes the teaching mathematics, as Procedural Content Knowledge
and Subject Content Knowledge which was later

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elaborated by Hill et al. (2008) and rephrased as et al., 2008) as teachers teach the content with
Mathematical Knowledge for Teaching (MKT), categorising conceptual understanding (Tchoshanov, 2011), using high
two major areas as Subject Matter Knowledge and quality cognitive tasks (Charalambous, 2010).
Pedagogical Content Knowledge, which are further sub- 1. Need and Importance of the Study
divided. Subject Matter Knowledge, includes Specialised
According to Muralidhar (1989), one of the reasons the
Content Knowledge (SCK), Common Content Knowledge
students do not perform well in mathematics in Fiji, is due to
(CCK), and Horizon Content Knowledge (HCK). On the other
the teaching pedagogies used by teachers; traditional
hand, Pedagogical Content Knowledge covers
approach whereby mathematics ideas are taught to
Knowledge of Content and Teaching (KCT), Knowledge of
students in exactly the same way to all students, despite the
Content and Students (KCS), and Knowledge of Content
differences in students' learning styles, which was earlier
and Curriculum (KCC).
researched by Raj (1985), as students lack mathematics
Basically SCK covers knowledge of content limited to abilities and attitudes due to how mathematics is taught,
teaching mathematics which gives opportunity for resulting to reluctance to do extra if not guided by step by
preparations for teaching using relevant resources, step procedures.
representations and making connections, attending to
In addition, as according to the one of the local
student responses and using mathematical language for
newspapers in Fiji, The Fiji Times, dated December 3rd 2016,
teaching. CCK gives way for teachers to focus on
the former Education Minister, Dr. Mahendra Reddy had
mathematical knowledge that is needed to solve
blamed the failure of the primary school system and
problems in multi-contexts to meet the needs of the
subject delivery as the main contributors to the drop in Year
community and society at large. HCK, on the other hand,
6 and 8 National Examination results as the pass rate for
expects teachers to relate mathematics topics across the
Mathematics for Year 6 in 2016 was only 29 percent, which
curriculum, and also mapping mathematical concepts
is relatively very low. The Minister added, that it is the
with other levels or grades.
opportune time to look at new strategies of subject delivery.
Alternatively, Pedagogical Content Knowledge includes
Moreover, Fiji's population is almost about 900,000 and is
KCT, which requires teachers make connections of
categorised under 'Small Island States' category, and since
mathematical concepts and ideas with pedagogical
research literature for small islands states are not plentiful,
approaches which brings about meaningful and effective
Sanga’s (2012) findings of such studies can be used as
learning in students. KCS focuses on teachers'
building blocks of local and international literature in
understanding on mathematical content interacting with
mathematics education.
how students think, and at times addressing students'
2. Resume of the Related Literature
misconceptions. KCC, directs teachers to connect
mathematical content with teaching resources. KCC and Ghousseini (2017) highlighted that rehearsals on MKT, using
HCK are normally embedded within the other highlighted repeated cycles of varied activities with related
components. instructional procedures, supports teachers' MKT, and
builds up confidence in teaching mathematics.
The use of MKT as a modified Shulman's model is seen as a
more holistic model that would measure teachers' Raveh et al. (2016) revealed that connectionist framework
mathematical knowledge for teaching from a more wider for teaching basic four operations in algorithms supports
perspective, and it provides a direct relationship between primary teachers' SCK, and could connect essential
teachers' Procedural Content Knowledge with student components of knowledge of teaching algorithms, thus
learning. could be useful in planning and unpacking teaching goals
in algorithms, making connections within the topics and
Research has proven that MKT is positively associated with
other topics in mathematics, resulting to thorough
students' mathematics achievement (Ball et al., 2008; Hill

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understanding. 3. Operational Definitions of the Terms

Turner (2012) revealed that Knowledge Quartet (KQ) 3.1 Mathematical Knowledge for Teaching (MKT)
framework supports reflection on teaching and Mathematical Knowledge for Teaching refers to what
development of teachers' MKT through collaboration of mathematical techniques and skills that teachers require
lesson observations, interviews, and reflections. Other uses to teach mathematics, and can be divided into two
of KQ is thorough reflection of collaborative dialogues; categories; Pedagogical Content Knowledge (PCK)
critical evaluation of planning and teaching; which could according to Hill et al. (2008), includes Knowledge of
also become the basis of reflective Framework. Content and Students (KCS), which refers to knowing about
Ng (2011) highlighted that teachers with better MKT were the students and mathematics; and Knowledge of
teachers with higher education attainment; and broader Content and Teaching (KCT), knowing about teaching and
range of classes taught. Recommendations included mathematics.
teachers' minimum requirement for education degrees to The other category; subject matter knowledge includes
be four years, and provision for mathematics specialists. Common Content Knowledge (CCK), mathematical skills
Charalambous et al. (2010) revealed that there is positive and knowledge, which can be used in realistic situations
relationship between teacher's MKT and teacher's and Specialised Content Knowledge (SCK), refer to
cognitive level at which the tasks were prepared and mathematical skills and knowledge, more confined to
implemented. teaching. The other two components of MKT; Knowledge at
Hill et al. (2008) highlighted the impact of MKT on quality of the Mathematical Horizon (KMHZ), and Knowledge of
teachers' instruction, which confirms positive association. Content and Curriculum (KCC) are usually embedded in
Other findings, include attitude, resources, and the impact the other categories of MKT.
of PD sessions supported teachers' MKT instruction. In the present study, the researcher expounded on Hill et
Hill et al. (2005) revealed that teachers with MKT have al.'s model (2008) of the levels of MKT with its related
acquired high quality instruction, not specifically on MKT, components for Primary School Mathematics Teachers in
but also contribute to general knowledge about teaching. Fiji, which was measured by administering a ‘Test on
Quality teachers' MKT positively contribute to quality Mathematics Knowledge for Teaching’ which was
mathematics content knowledge, resulting to higher prepared by the researcher by covering the following:
student achievement. (i) Common Content Knowledge,
After a critical analysis of the related literature findings, the (ii) Knowledge of Content and Teaching,
following questions arose in the mind of the researcher: (iii) Knowledge of Content and Students
·What are current status of MKT for Primary School (iv) Specialised Content Knowledge
Mathematics Teachers in Fiji schools in terms of its
3.2 Primary School Mathematics Teachers
components; Knowledge of Content and Students
In the present study, Primary School Mathematics Teachers
(KCS); Knowledge of Content and Teaching (KCT);
refer to male and female teachers in Fiji serving in different
Common Content Knowledge (CCK); and Specialized
schools in various locations throughout the country,
Content Knowledge (SCK)?
currently serving at 731 primary schools in Fiji, and are
·Whether MKT of Primary School Mathematics Teachers
basically divided into four main divisions, namely; Central,
in Fiji differ, with respect to gender?
Northern, Western, and Eastern division, whereby the
To find answers to these questions, the present study, titled schools are further categorised as remote, rural, semi-
“Mathematical Knowledge for Teaching among Primary urban, and urban schools. According to Ministry of
School Teachers of Fiji” was undertaken. Education Report in 2011, rural schools are classified under
10 to 20 km from a town boundary; and remote are greater

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than 20 km away; for this study, the researcher has 7. Analysis of Data and Results
categorised semi-urban, 4 km to 9 km from town boundary; The collected data were tabulated, analysed, and
and urban, within 3 km within town boundary. interpreted using Statistical Package for the Social Sciences
Primary School Mathematics Teachers are directly (SPSS).
employed and supervised by the MOE with a minimum 7.1 Objective 1
qualification of Primary Teacher Certificate and are
To study the level of Mathematical Knowledge for Teaching
required to teach all the subjects prescribed by MOE.
of Primary School Mathematics Teachers of Fiji and its
4. Objectives of the Study components; Common Content Knowledge, Knowledge
·To study the level of Mathematical Knowledge for of Content and Teaching, Knowledge of Content and
Teaching of Primary School Mathematics Teachers of Students, and Specialised Content Knowledge.
Fiji and its components; Common Content The data collected by administering the tool ‘A Test on
Knowledge, Knowledge of Content and Teaching, Mathematics Knowledge for Teaching’ was analysed by
Knowledge of Content and Students, and Specialised calculating the mean, standard deviation, and
Content Knowledge. percentage. The teachers were categorised levels:
·To compare the level of Mathematical Knowledge for High MKT Level - Teachers who have MKT score which is
Teaching among Primary School Mathematics greater than mean + 1SD
Teachers, with respect to gender.
Average MKT Level - Teachers who have MKT score which is
·To compare the level of Mathematical Knowledge for between mean +1SD to mean – 1SD.
Teaching Components among Primary School
Low MKT Level - Teachers who have MKT score which is less
Mathematics Teachers with respect to gender.
than mean - 1SD. Percentage of MKT and percentage of
5. Hypothesis of the Study MKT components, with respect to each level of teachers
·Male and female Primary School Mathematics are given in Tables 1 and 2, respectively.
Teachers differ significantly in Mathematical Knowledge Table 1 indicates that 63.9% of the teachers have average
for Teaching. level of MKT. Hence it can be concluded that Primary
·Male and female Primary School Mathematics School Mathematics Teachers of Fiji have average level of
Teachers differ significantly in Mathematics Knowledge MKT.
for Teaching components. Table 2 indicates that 72.5% of teachers have average
6. Methodology level of CCK as highest; 65.6% of teachers have average
The population covers all Primary School Mathematics level of KCT as highest; 64.7% of teachers have average
Teachers in Fiji, which has a total of 5974 teachers in 2016, level of KCS as highest; and 61,7% of teachers have
as the data given by the Ministry of Education. average level of SCK as highest. Therefore, it can be
concluded that Primary School Teachers in Fiji have an
The sample was obtained using Krejcie and Morgan in
average level of MKT components, whilst CCK being the
1970 sample size table.
highest among the Primary School Mathematics Teachers
Descriptive survey was used with engagement of Stratified
with average level of MKT.
Random Sampling technique, whereby a strata from each
Level of MKT N %
Education District is selected, from the four Education
Districts; Northern, Central, Western, and Eastern. High 63 17.4%

Average 232 63.9 %

The research tool used was validated using Content
Low 68 18.7%
Validity, whereby the tool was validated by a group of
experts in the field of educational research. Table 1. Percentage of MKT of Primary School Mathematics
Teachers in Fiji

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MKT Components Level N % hypothesis is rejected. Therefore it can be concluded that

High 54
there is a significant difference between male and female
CCK 14.9%

Average 263 72.5%

teachers in the level of MKT. From Table 4, it is also revealed

Low 46 12.7%
that the mean score for male teachers, which is 52.96, and
is higher than female teachers with the mean of 50.32 thus
KCT High 67 18.5%
it can be concluded that male teachers' level of MKT is
Average 238 65.6%
significantly higher than female teachers.
Low 58 16.0%

High 79 21.8%
7.3 Objective 3
KCS

Average 235 64.7% To compare the level of Mathematical Knowledge for

Low 49 13.5% Teaching Components among Primary School Mathematics
SCK High 69 19.0% Teachers with respect to gender.
Average 224 61.7% 7.3.1 Hypothesis 2
Low 70 19.3%
Male and female Primary School Mathematics Teachers
differ significantly in MKT components.
Table 2. Percentage of MKT Components of Primary School
Mathematics in Fiji H1: Male and female Primary School Mathematics
To compare the level of MKT, with respect to gender, the Teachers differ significantly in MKT components.
reliability of the data was calculated using Cronbach's To list the hypothesis, it was changed into null hypothesis as
Alpha. The details are given in Table 3. stated below.
7.2 Objective 2 H0: Male and female Primary School Mathematics
To compare the level of Mathematical Knowledge for Teachers do not differ significantly in MKT components.
Teaching among Primary School Mathematics Teachers The hypothesis was tested using ‘t-test’ with the level
with respect to gender. significance fixed at 0.05 level. The details are given in
The hypothesis was formulated from the objective. Table 5.
7.2.1 Hypothesis 1 Table 5 indicates that the t-value of CCK is 1.376 and P-
H1: Male and female Primary School Mathematics value of .170, which is not significant at 0.05 level, thus the
Teachers differ significantly in MKT. null hypothesis is accepted. Therefore it can be concluded
that there is no significant difference between male and
To test the hypothesis, it was changed into null hypothesis as
stated below. Gender Mean Std. Deviation t P value Result

H0: Male and female Primary School Mathematics Male 52.956 10.8576 2.25 .025 Significant at
MKT 0.05 level
Teachers do not differ significantly in MKT. Female 50.445 10.3227
The hypothesis was tested using ‘t-test’ with the level of
Table 4. 't-test' details of MKT among Male and Female Primary
significance fixed at 0.05 level. The details are given in School Mathematics Teachers of Fiji
Table 4.
MKT Mean t- value P-value Results
Table 4 indicates that the t-value is 2.25, and P value of Component N or NS
.025, which is significant at 0.05 level, thus the null Males Females

Cronbach's Alpha Cronbach's Alpha Based on No. of Items CCK 14.8177 14.4254 1.376 .170 NS
Standardized Items KCT 12.9834 12.2099 2.606 .010 S
KCS 12.7348 12.1768 1.651 .100 NS
.746 .750 8 SCK 12.4199 11.5691 2.125 .035 S

Table 3. Reliability Details of Data with Respect to Gender Table 5. 't-test' details of MKT Components amongst Male and
Female Primary School Mathematics Teachers in Fiji

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female teachers in the level of CCK. In addition, the t-value sessions on Mathematical Knowledge for Teaching
of KCT is 2.606 and P - value of .010, which is significant at and its components.
0.05 level, thus the null hypothesis is rejected. Therefore, it ·Invite resource personnel to run workshops in
can be concluded that there is a significant difference Mathematical Knowledge for Teaching.
between male and female teachers with the level of KCT.
·To review and develop resource materials in Primary
Furthermore, the t-value of KCS is 1.651 and P value is .100,
Mathematics that are aligned with Mathematical
which is not significant at .05 level, thus the null hypothesis is
Knowledge for Teaching.
accepted, which confirms that there is no significant
·Form clusters and teachers' network, to allow for
difference between male and female teachers in the level
teachers to share ideas on Mathematical Knowledge
of KCS. Moreover, the t-value of SCK is 2.125 and P-value of
for Teaching.
.035, which is significant at .05 level, thus the null hypothesis
is rejected. Therefore, it can be concluded that there is a ·Review time allocated for teaching mathematics in
significant difference between male and female teachers the class time table.
with the level of SCK. Hence, it can be concluded that ·Review class size for each class.
Teachers differ significantly with respect to KCT and SCK To increase level of Mathematical Knowledge for Teaching
components of MKT, but with respect to CCK and KCS they amongst female teachers, the Ministry of Education need
do not differ significantly. to:
8. Major Findings of the Study ·Conduct more training for female teachers on
·Primary School Mathematics teachers in Fiji have an Mathematical Knowledge for Teaching.
average level of Mathematical Knowledge for ·Allow female teachers to teach wider range of classes.
teaching.
·Encourage female teachers to build up network with
·Primary School Mathematics teachers in Fiji have an other female teachers to share Mathematical
average level of Mathematical Knowledge for Knowledge for Teaching ideas.
Teaching components. Common content was highest
·Encourage female teachers to become specialist
and Specialised Content Knowledge was the least,
mathematics teachers in primary schools.
aiming at teachers with the average level of
To increase the level of Knowledge of Content and
Mathematical Knowledge for Teaching.
Teaching and Specialised Content Knowledge, of female
·Male teachers' Mathematical Knowledge Teaching is
teachers, the Ministry of Education needs to:
significantly higher than that of female teachers.
·Conduct PD sessions for female teachers addressing
·Male teachers' level of Knowledge of Content and
Knowledge of Content and Teaching and Specialised
Teaching is significantly higher than that of female
Content Knowledge.
teachers.
·Invite resource personnel to facilitate Professional
·Male teachers' level of Specialised Content
Development sessions on Knowledge of Content and
Knowledge is significantly higher than that of female
Teaching and Specialised Content Knowledge.
teachers.
·Encourage female teachers to design Knowledge of
9. Educational Implications
Content and Teaching and Specialised Content
In order to improve the average level of Mathematical Knowledge tasks.
Knowledge for Teaching and its components among
·Encourage female teachers to test out designed tasks
Primary School Mathematics teachers in Fiji, the Ministry of
which could be part of one's action research.
Education need to:
·Allocate funds to develop resource materials that are
·Support teachers in providing professional development
related to Knowledge of Content and Teaching and

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Specialised Content Knowledge, whereby female [5]. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of
teachers could be actively engaged in such related teachers' mathematical knowledge for teaching on
projects. student achievement. American Educational Research
Conclusion Journal, 42(2), 371-406. Retrieved from http://www.jstor.
org/stable/3699380
Mathematical Knowledge for Teaching is not another
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more genuine re-organisation in the teaching of Number Operations and Fractions at the Junior Secondary
mathematics. Once Mathematical Knowledge for Teaching Level in Fiji (Unpublished M. Ed Thesis. Monash University).
is well addressed at the primary school level, there would [7]. Ng, D. (2011). Indonesian primary teachers'
be positive changes as teachers' cognitive level of mathematical knowledge for teaching geometry:
teaching task preparations and its implementation would implications for educational policy and teacher
be of high quality which results to quality teacher preparation programs. Asia-Pacific Journal of Teacher
instruction, well prepared resources and brings about Education, 39(2), 151-164. http://dx.doi.org/10.10
positive attitude in teaching mathematics. Teachers, 80/1359866x.2011.560648
therefore, would not only acquire quality instruction on [8]. Ono, Y., & Ferreira, J. (2010). A case study of continuing
Mathematical Knowledge for Teaching, but would also teacher professional development through lesson study in
contribute to general knowledge about teaching. Quality South Africa. South African Journal of Education, 30(1), 59-
teachers' Mathematical Knowledge for Teaching would 74.
also contribute to quality mathematics content
[9]. Raj, L. L. (1985). An Investigation of Mathematical
knowledge, resulting to higher student achievement. Some
Abilities and Attitudes towards Mathematics of Students in
of the strategies that could be useful to improve teachers'
the Upper Secondary Schools in Fiji (Unpublished MA Thesis.
Mathematical Knowledge for Teaching are; the use of
Suva: University of the South Pacific).
rehearsals, connectionist framework concept, and the use
[10]. Raveh, I., Koichu, B., Peled, I., & Zaslavsky, O. (2016).
of Knowledge Quartet (KQ) framework.
Four (algorithms) in one (bag): An integrative framework of
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ABOUT THE AUTHORS

T. N. Raiula is currently pursuing her Ph.D degree in the Department of Education at St Ann's College of Education (Autonomous),
Mangaluru, Karnataka, India. She is working as a lecturer at the School of Education, College of Humanities and Education, Fiji
National University, Fiji. She teaches undergraduate courses in Mathematics Education. She received her M.Ed degree from The
University of the South Pacific, Suva, Fiji. Her research interest, includes Primary Mathematics Education, Teacher Education,
and Assessment.

Dr. Vijaya Kumari S. N. is currently working as an Associate Professor and Research Guide at St.Ann's College of Education
(Autonomous), Mangalore University, Karnataka, India. Having 32 years of professional experience, research expertise, include
Research on Teaching, Teacher Education, Development of Instructional Materials, Science Education, Educational
Measurement and Evaluation, School Administration and Leadership, and Education for Sustainable Development. Dr. Vijaya’s
publications, Include Research articles in peer reviewed Journals and Conference proceedings, self instructional materials for
distance mode learners of M.A in Education and M.Ed., Handbook on science projects, and Collate Ann's Part-II – Compilation
of M.Ed. Dissertation (Editor).

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