Mathematics Teacher’s Content Knowledge and Pedagogical Content Knowledge in Learner-centred Approaches in Secondary Schools

This paper describes how mathematic teacher’s content knowledge informs teachers’ pedagogical content knowledge in the use learner-centred approaches and decision when teaching mathematics. Two trained mathematics teachers in the use of learner-centred approaches who were teaching mathematics at ordinary secondary school level in Tanzania were involved. Data was collected through classroom observation, video recording of classroom events and teacher’s self-reflections. Thematic analysis procedure both at conceptual and manifest level was employed. The findings indicated that, teacher’s weak knowledge of contents as a component of pedagogical content knowledge in the use of learner-centred approaches led to teacher’s inability to help students’ construct knowledge of the subject matter that teachers taught. The trained teachers did not teach lessons that were learners’ focused and were unable to help students discover the relationship between contents they taught with other contents in the syllabus. Teachers’ lack of content knowledge led to teachers’ communication of their misunderstanding to students through teacher-centred teaching approaches. In addition, these experienced teachers were unable to help students describe the rationale for learning contents. It is recommended that cementing on leaner-centred approaches during teachers’

in the curriculum so that students can best learn. Therefore, a mathematics teacher with well-developed PCK in the use of Learner-Centred Approaches (LCA) is likely to have strong knowledge of both SMK and have developed PCK categories in the use of LCA. With the focus on both SMK and PCK, three research questions were answered by the researchers; How does mathematics teacher's content knowledge as component of PCK in the use of LCA inform teacher's decision on the use of LCA when teaching mathematics? How does mathematics teacher's content knowledge as a component of PCK contributes to teacher's growth of PCK in the use of LCA? In what ways do mathematic teacher's content knowledge as a component of PCK in the use of LCA contributes to students' knowledge of contents?

Conceptualizing and Assessing Content Knowledge as a Component of PCK in LCA
The conceptual framework for assessing teacher's PCK in the use of LCA in this qualitative study are five as were adapted from Magnusson et al. (1999). These components are; the teacher's orientation to teaching mathematics, teacher's knowledge of mathematics contents, teacher's knowledge of students' understanding of mathematics concepts, teacher's knowledge of instructional strategies and teacher's knowledge of assessing students' learning when employing LCA. Magnusson et al. (1999) had earlier proposed five components of PCK; teacher's orientation towards teaching science, teacher's knowledge and beliefs about science curriculum, teacher's knowledge and beliefs about students' understanding science topics, teacher's knowledge and beliefs about assessment in science, and teacher's knowledge and beliefs about instructional strategies for teaching science. Since teacher's PCK is recognized as an algamation of both pedagogical and content knowledge (Magnusson et al., 1999), these construct in mind helped the researchers to assess content knowledge as a component of teacher's PCK in the use of LCA as a teacher's personal construct.
The focus on teaching approaches based on Shulman (1987) argumentation that, teacher's PCK is a knowledge base which includes teacher's decision on which approach fits to the teaching of contents. Therefore, the study assessed ways in which a trained mathematics teacher make decision on when to use LCA for specific learners, group of learners and organize contents while putting learners at the centre of teaching and learning process. In addition, Hill, Ball, and Schilling (2008) knowledge base for teaching mathematics guided the assessment of content knowledge as a component of PCK in LCA using both the SMK and PCK categories (Figure 1). Measuring PCK as a knowledge base, a PCK rubric was developed to aid assessing teachers while teaching. The PCK rubric was adapted from the work of Hume and Berry (2010), Gardener and Gess-Newsome (2011) and that of Anney and Hume (2014). In terms of credibility, the PCK rubrics have been documented to well represent teacher's PCK (Loughran, Berry, & Mulhall, 2006). KCT and KCS was captured through teacher's evidence of knowledge of LCA and teacher's practices such as ability to ask higher order thinking questions, involving students and other actions that revealed teacher's intention to put learners at the focal point while teaching. Cochran and Jones (1998) identified four elements of Content Knowledge (CT); knowledge of majour facts and concept of a discipline, knowledge of basic principles, knowledge of how validity and invalidity are established within a discipline and knowledge of connecting topic to www.scholink.org/ojs/index.php/fce

Literature Review
Previous studies have shown that, science teachers with strong PCK are successful when teaching specific science topics (Mulhall, Berry, & Loughran, 2003). These studies have also confirmed that, for effective teaching of a topic, teacher's knowledge about common students' misconceptions in that topic is necessary. It was further concluded that, teacher's development of topic specific PCK is embedded in teacher's classroom practice on the same topic. This implies that, successful teachers who have taught a particular topic not only promote students' learning but also have well developed PCK in that specific content area (Gess-Newsome & Lederman, 1999;Hill, Ball, & Schilling, 2008;Kitta, 2004;Magnusson et al., 1999;Van Driel, Verloop, & De Vos, 1998). Being recognized through a teacher's classroom practice and use of approaches to teach contents is a common view that researchers agree upon the nature of teacher's PCK. In American teacher's education programmes, teacher's content knowledge and PCK have been identified to be important components of teacher's competence (NCTM, 2000;Mewborn, 2003). In addition, there is an agreement that, all American teachers' training programmes must seek to achieve a balance between CK and PCK.
Similarly, all teachers' training programmes in Tanzania are approved by education authorities after they verify that there is a balance between CK and PCK in the programme seeking for approval. Curriculum (CBC) that could enhance learners' competency. Since then, the emphasis in the education provision is more on the improvement of students' learning, improvement in the teaching and learning process and quality of learning outcomes (TIE, 2005) rather than the teaching, content delivery and coverage (Richards & Rodgers, 2001). The CBC is in place to address more on what the learners can do rather than what they should cover during the learning process. In addition, secondary school teachers, and tutors in teachers training colleges are encouraged to the use of LCA (MoEVT, 2005).
In mathematics teaching and learning process, LCA are recommended for the purpose of raising students' learning and participation in the learning process (Philemon, 2010;TIE, 2010a). The use of LCA is expected to develop students' high level of competencies in different education levels (TIE, 2010b). Since its introduction in Tanzania, the government has been emphasizing on changing roles of teachers and tutors from that of teaching by imparting knowledge to facilitating students' learning (Vavrus, 2009 is not fully implemented by teachers in classrooms (Kitta & Tilya, 2010;Mosha, 2012;Msonde, 2011;Philemon, 2010).
Challenges to the use of LCA have included class size, curriculum being overloaded with contents, and teachers with inadequate PCK (Anney, 2013;Kitta & Tilya, 2010). Teachers' orientation to teaching has remained to be a teacher-centred pedagogy. Top-down dissemination of emphasis on pedagogy has neglected teaching environment and therefore discouraged to bring expected changes in instructional methods (Mosha, 2012;Thornberg, 2010). Curriculum overloaded with contents has not encouraged the development of students' competence through the use of learner-centred approaches (Tilya & Mafumiko, 2010). Teachers are reported to be concerned mostly with coverage of all contents in the syllabus with a belief that they have all knowledge to transfer to learners (Mosha, 2012). Researchers have also reported that, the reasons for dominant teacher-centred orientation in teaching is attributed to the fact that during the paradigm shift in education, not all teachers were educated with emphasis in CBC. This implies that, teachers who are implementing the CBC at the secondary school level in Tanzania are those who had specifically followed CBC and who did not go through a CBC during their training.
Since PCK grows with experience there are teachers in Tanzania whose PCK is underdeveloped due to lack of experience. The untrained teachers in CBC as compared to experienced teachers are therefore unlikely to effectively enhance students' learning outcomes in the context of LCA. Available findings in Tanzania on the use of LCA show that, pre-service teachers have underdeveloped PCK (Anney, 2013). A study by Anney (2013) in Tanzania did not assess PCK as a whole rather on licensed and in-service teachers' effectiveness in the implementation of learner-centred pedagogy. An action research study by Kitta (2004)  although there is an emphasis on use of LCA in Tanzania, teacher-centred teaching approaches in classroom are still dominant which suggested a need to assess teachers' PCK in the use of LCA.
Thirdly, some teachers who are implementing the CBC are untrained in the use of LCA, hence assessing trained teachers' PCK in the use of LCA was necessary.

Method
The nature of PCK as a tacit knowledge demanded to obtain deep insights into what is in the mind of the participants' teachers through a qualitative approach. The participants in this study were two trained mathematic teachers in the use of LCA. The two teachers were identified by this study as MTR1 and MTU2. The first two letters M and T were used to identify mathematics teachers while R1 and U2 are identify a teacher in a rural and urban area respectively. MTR1 was purposively selected from a group of low performing schools in rural area while MTU2 was purposively selected from a group of low performing schools in an urban area. MTR1 had a degree in mathematics with education qualification while MTU2 had had a diploma in mathematic with education qualification. Both teachers were trained in the use of LCA and had completed their studies in 2010. Both teachers had been recruited as mathematics teachers by the government and worked for at least five years in one of the political administrative districts in Tanzania. The fact that PCK is a personal construct, interpretivism paradigm guided data collection and analysis of the findings. A multiple case study design was employed whereby each individual teacher was treated as a case study. A multiple case study design ensured transferability of the findings. Data was collected through classroom observation, video recording of at least three lessons of each teacher's classroom practices and teacher's self-reflection after the classroom observation. Since teacher's PCK is a personal knowledge, the researchers agreed with the teachers on the lesson that best represented teacher's PCK before documentation. Therefore, this study presents one of the best lessons that each teacher thought it well represented their PCK. The choice of these teachers began with discussion on their teaching plans, their syllabus coverage and finally the topic that they were ready to be observed and videotaped by the researcher while teaching was later on agreed. Since the purpose of the study was to measure a personal construct, teacher's plans and teacher's choice of topic was a priority. The purpose of the observation schedule was to formulate each teacher's content representation (CoRes) and finally represent each teacher's PCK. Loughran et al. (2006) and Hume and Berry (2010) guided the identification of themes. Each teacher's self-reflection on classroom practices as guided by Gardner and Gess-Newsome (2011) helped to capture and represent each teacher's professional development Repertoires (Pap-eRs).
Due to the implicit nature of PCK and this study being purely qualitative, Braun and Clarke's (2006) and Creswell's (2009) six steps thematic analysis procedures was adapted. The analysis process in this study was data driven and allowed capturing the themes they emerged from participant teachers.
Thematic analysis allowed data analysis both at manifest and theoretical level as the data collection procedures was going on and from different sources. The themes that emerged during classroom observation and during interview were used to locate evidence of teacher's use of LCA and the level of teacher's PCK in the use of LCA. The analysis process began with transcription and describing each case with the data in order to become familiar with the data. This was followed by coding and organizing data from both video record and interview sessions into themes. FitzGerald (2012) stipulates that analysis of events in video records should be guided by research questions. The events from video records that was gauged from each teacher's classroom practices was then matched with each teacher's PCK in the use of LCA and the teacher's self-reflection during a min interview with the researcher.
This stage helped the researcher to obtain general meaning of the data and extracting supporting evidence from data. This stage was followed by description and interpretation of the themes.
After individual case study analysis, crosscase analysis followed whereby, similarities and differences in the teacher's classroom practices in the use of LCA were done. Triangulation of the data from classroom observation, teaching plans, and teacher's self-reflection helped to formulate each teacher's profile of PCK in the use of LCA. A summary of the themes that emerged from both cases was formulated before documenting each teacher's PCK as contributed by their knowledge of contents that they taught. The themes as emerged during classroom practices (lessons) are represented in italic form in each teacher's lesson as teacher's content representations (CoRes).

Results and Discussion
We begin by presenting each teacher's content representation as were captured through video recording of each teacher's classroom practices (Box 1 and Box 2). Each teacher's classroom practice is followed by each teacher's self-reflection to document each teacher's Pap-eRs.

Box 1: Lesson 1 by MTR1: Topic "Graphs of a Relation"
Lesson beginning: … "we want to see how we can make graphs of different relations". MTR1's wrote an example from the textbook. Example 1: "Draw the graph of a relation      MTR1 continued … "Look at this graph. Each graph divides the plane into two parts; the down and upper part" (see Figure 1 and the part referred). "You choose any part, either the up or down part that will satisfy the relation. This will be the part that will be shaded. Lesson continued: "After that you go to the other inequality 2  y . "Would you shade up or below?". MTR1 asked. "I am asking you … where do we shade? Is it below or up?". MTR1 asked again after a long silence. "Below" The students responded in a chorus (orientation to TCA). "Our interest is to find the part of intersection. The shading now will be as shown" (See Figure 2). After that, MTR1 pointed the intersection that was earlier mentioned by MTR1 (orientation to teaching). This was then followed by MTR1's oral questions. "Any question?" MTR1's asked. However, students were silent (knowledge of assessing students).   After about 5 minutes, MTR1 asked for a student who can show how to solve the problem (Knowledge of group accountability). A student's response was read as: "We assume that x y  is equal to x y  then make a table of values" (Students' knowledge of contents).
Using the chalkboard, the student who had been appointed by MTR1 showed how to solve the problem.
The student started by making a table of values, then drew a graph of the equation but did not shade.
"Our line must be full line and we will find the domain and range". The student explained (Students' knowledge of content). "What about shading?". MTR1 interrupted. "Now, we shade". The student incorporated the idea. "Take ) 2 , 2 (  so you get 2 2   . Is it correct? The student asked. "No". Some students replied in a chorus. "So, the upward side is not correct. We shade the downward part". The student shaded the downward part (See Figure 4). Figure 4 show how teacher's knowledge of contents is transferred to students and then reflected through students' knowledge of contents. x y y x  Lesson continued: "Is he correct? What is a mistake? Make sure you take a point which is above or below". MTR1 interrupted after a time. "Don't take a point within the line so that you cannot confuse.
After that, do a test" (MTR1 continued with explanations). "What is the mistake here? MTR1 added. In your case take a coordinate point ) 3 , 2 ( then test whether 2 3  is true then shade". MTR1 added (TCA orientation). "Is there anybody who has not understood? MTR1 asked (Knowledge of assessment). Students remained silent. "So, you don't have any question?" MTR1 added after noticing that students were silent. "Note the use of dotted and full line. When you are given either a symbol < or

MTR1's self-reflection and the components of PCK in LCA
The following extract from MTR1's self-reflection as was interviewed by the researcher revealed that,

MTR1 had adequate HCK but this knowledge did not adequately inform both KCS and KCT;
Researcher: What is the relationship between the subtopic you were teaching and other Mathematics topics or subtopics that you teach at this level?

MTR1: In drawing graphs of relation, you need to know about intercepts or the point
where the lines cuts the Y-axis and where the lines cuts X-axis. There is a relationship between the topic called Algebra whereby you will need to find the X and Y intercepts when constructing the table of values for an equation.
Researcher: How useful was the relationship in the lesson that you taught?
MTR1: The use of x and y intercepts was useful when drawing graphs.

Box 2: Lesson 2 by MTU2: Topic "Graphs of Relations"
Lesson beginning: MTU2 wrote the subtopic on the chalkboard "Graphs of Relations", later on, announced orally "today you will learn about the process of making graphs of Relations". "How do we  Figure 5). asked again (Knowledge of students and orientation to teaching). Two students raised their hands up at this moment. MTU2 picked one of the two students. "The x-intercept is 2 and the y-intercept is 2" A student answered. "Can you show us how you got 2?" MTU2 asked after the student had answered (Knowledge of teaching and students' learning). "You plus two (2) and zero (0)" The student replied. It meant to the researcher that, you add 2 and 0. "Who can show the points (0, 2) and (2, 0) on a number plane?" MTU2 asked then picked two students one after the other. The two students went to the front of the class in order to indicate the points on the number line (See Figure 6) (Knowledge of teaching and students).

Figure 6. MTU2's Student Locating a Coordinate Point on a Number Plane
After that session, MTU2 constructed a table of values, a number plane and a sketch of the graph (Figure 7). "These two lines divide the plane into a number of regions". MTU2 explained. "How many regions are there?" MTU2 asked. The class remained silent while gazing at MTU2. After about 3 minutes of such a silence moment, there was an attempt from some students who answered orally. "9, 6, 5, 8,". The students answered. The researcher noted that, the students' answers ranged between a number 5 and 9. However, MTU2 did not comment on any of the students' answers (Knowledge of teaching and students). "There are four regions". MTU2 gave an answer after the students had answered (orientation to teaching). While pointing on the graph with a ruler and without exactly showing the named regions, MTU2 gave an answer which was quite different from the students' answers (See Figure 7). (Knowledge of content and students)  Figure 8).

Figure 8. MTU2's identifying a Domain and Rrange of a Relation on a Graph
"After that, substitute the point in the equation. You get Which is now true" MTU2 continued, "We now shade the part which is true (Figure 8  y y R ". "Is there any question?" MTU2 asked. The class was silent. "I hope there is no question". MTU2's hoped that there were no questions (Knowledge of instructional strategies). "I give you two (2) minutes to finish this example 2" MTU2 wrote another example which was named example 2 on the chalkboard. Example 2; "Draw a graph of the relation then find the domain and range". None of MTU2's students managed to attempt this task. The question was written with some mathematical mistakes. However, both MTU2 and the students did not identify the mistakes. After about 5 minutes had elapsed, MTU2 noticed the mistake, made some correction on example 2 which had written then asked students to work in small groups.
Group activity: After had improved the example, MTU2 asked student to work on the new and improved example 2 in their small groups. The new example 2 was written as; "    Figure 9). MTU2 had forgotten the initial statement about dotted and full lines. Also, MTU2 had forgotten labeling and differentiating graphs of lines and segments (Knowledge of contents, Knowledge of content and teaching). teaching). The inequality shows that, 2 is greater than -3. Class, is the inequality true? or is it false?" After MTU2 had asked wrote the words; "Yes or no?" on the chalkboard. Class, is it "Yes or no?" MTU2 asked again (Knowledge of content and teaching). The class was silence. "Now, what is the domain and what is the range?" MTU2 asked again. "All real numbers x" A class replied in a chorus.
"Why?" MTU2 asked after the class had responded. None of the students tried to answer this question (Knowledge of content and teaching). "You will agree with me that the domain is all real numbers x".
MTU2 commented after such a moment of silence (Orientation to teaching, knowledge of students).
"What is the range?" Silence continued. It appeared to the researcher that, students had no clear knowledge of the two concepts; "domain and range" (knowledge of contents and students). "The range is all real number y" MTU2 answered. However, it also appeared to the researcher that MTU2 had forgotten that the Relation that was referred involved an inequality sign (Knowledge of contents and students). This part of the lesson was followed by students' work in small groups.
Small group activity session: After such initial instruction, MTU2 wrote a problem on the chalkboard then asked students to continue working in small groups: "Draw the graph of the relation , ( then find the domain and range". As students were working, MTU2 passed through different groups to see how students worked (Knowledge of content and teaching). After about 5 minutes, MTU2 discovered that some groups had failed to follow the steps or procedures. After such observation, MTU2 decided to provide an outline of all the steps on the chalkboard (Orientation to teaching, Knowledge of contents and teaching). "The first step is to make a  Figure 10). The student finished after some assistance from MTU2 (Knowledge of students, Knowledge of content and teaching).  x y y x R   Ending the lesson: To wind up the lesson, MTU2 decided to provide a home work which was found in the students' textbook. The homework marked the end of MTU2's lesson (see Figure 12).

Experienced teachers' knowledge of content as a component of PCK in the use of LCA
The participant experienced teachers in this study lacked knowledge of concepts such as; mathematical relations, domain and range which led to communicating their misconceptions to the students. For example, during a mathematics lesson, students had a misconception that domain means; "values of "x" while range "values of y". During individual activities, the students copied the same in their exercise books. MTR1 was unable to differentiate between "a line" and "a line segment", the mathematics concepts which have both similarities and differences in the study of Geometry. This finding is contrary to the expectation that when teaching mathematics, the experienced teachers are able to help students identify important ideas in the subject and lesson (Hill, Ball, & Schilling, 2008). In different occasions, there were evidences of the experienced teachers use of wrong concepts and in particular when drawing graphs of equations. Instead of employing LCA, the experienced teachers directly communicated errors and misunderstanding to their students. This implies that the experienced teachers lacked the PCK categories in the use of LCA. The curriculum developer recommended raising students' participation in the teaching and learning process (TIE, 2010a), however this expected outcome will not be achieved as the findings shows that, the experienced teachers lacked PCK categories in the use of LCA. Both experienced teachers in the study lacked questioning skills that hindered their ability to ask provoking questions so as to stimulate students' thinking. As a result, the teachers were unable to manage instructional time, provide constructive feedback to students and involve students in their lessons. Since teachers' lacked content knowledge for teaching, the teachers were forced to the use of demonstration method in order to finish what they planned to teach. Demonstration method misguided students' learning which led to students' misconception. In addition, lack of content knowledge lowered teacher's confidence which indirectly diminished teacher's growth of PCK in the use of LCA.

Experienced teacher's content knowledge and their decision on the use of LCA
The literature review is indicating that the inexperienced teachers in the use of LCA are unlikely to effectively enhance students learning outcomes in the use of LCA (Anney, 2013). However, the findings from this study show that, the participant experienced teachers had inadequate horizontal content knowledge and were oriented to TCA. These participant teachers who were oriented to teacher centered approaches were unable to decide and make use of LCA. If this situation continues on, it will discourage the government effort to have LCA implemented in classrooms. The teacher's lessons will continue to be dominated by telling facts about concepts which will finally not improve students' performance in mathematics. The experienced mathematics teachers' classroom practices which were more teachers' centred weakened their ability to help students to construct knowledge of contents that they taught. The participants' teachers made few attempts to assist students to construct, identify and understand concepts or ideas on their own. These suggested that, the participant teachers had underdeveloped PCK categories in LCA and SMK categories.

Conclusion and Recommendation
Based on the findings, it is concluded that, the experienced mathematics teacher's lack of content knowledge led to the teacher's difficulties in decision, planning and implementing learning activities that are students' focused. In addition, teacher's growth of PCK is limited by their weakness in guiding learners at the centre of their lessons. This will un address both teacher's growth of PCK in LCA and students' learning of contents. Students' misunderstanding of content will leave students' learning needs and problems unaddressed. Furthermore, this teaching and learning situation will not improve students' learning outcomes and achievements in the subject. Since teachers underdeveloped PCK in LCA hinder their decision and use of LCA when teaching of mathematics, it is recommended that, teachers' professional development programmes should cement on both teacher's content knowledge and PCK categories in the use of LCA.