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Procedía Social and Behavioral Sciences 8 (2010) 312-316

International Conference on Mathematics Education Research 2010 (ICMER 2010)

Teachers' Choice and Use of Examples in The Teaching and Learning of Mathematics in Primary School and Their Relations to Teacher's Pedagogical Content Knowledge (PCK)

Harizon binti Suffiana*, Shafia binti Abdul Rahmanb

aInstitut Pendidikan Guru Kampus Ipoh, Perak, Malaysia bUniversiti Sains Malaysia

Abstract

Examples have always played a central role in the development and the teaching of mathematics (Bills and Watson, 2008). This paper reports on teachers choice and uses of examples in elementary mathematics teaching. Selecting an appropriate example become a challenging task with teacher's pedagogical content knowledge (PCK) being a determining factor in the process of selecting and using examples. In this case study approach, two pre-service teachers from teachers training institute were selected as a participant. The nature of two different teachers choice and uses of examples in the classroom during their teaching training in primary school were documented. The lessons taught were observed, videotaped and both teachers were interviewed according to the selection and uses of examples. Data analysis were based on four categories of uses of examples identified and exemplified into variables, sequencing, representations and learning objectives (Rowland, 2008). The findings indicate that pre-service teachers need to consider carefully and raise awareness when choosing and using examples.

© 2010 Elsevier Ltd. All rights reserved.

Keywords: Examples; Pedagogical content knowledge

1. Introduction

The mathematics curriculum in Malaysia requires mathematic teachers to apply knowledge and skills obtained through teaching and learning in classroom. Every teacher needs to have content knowledge, pedagogical knowledge, students background knowledge, environment, curriculum etc. The mathematics teacher is responsible in upgrading the knowledge and personal skills in accordance with uplifting the students understanding of mathematics. The quality of the mathematical knowledge amongst teachers give us the feedback on what is taught and how it is taught. (Zaslavsky and Zodik, 2008). One of the important methods in mathematics is the use of examples in the teaching and learning of mathematics. Examples become an important part and a significant element for special reference (Rissland-Michener, 1978). A knowledgeable teacher will be able to choose and use the

* Corresponding author. Tel.:019-5556778 E-mail address: harizonsuffian@yahoo.com

1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.12.043

examples as the students would be able to understand better. Therefore, it is important to research on how the teachers' choose and use examples in a mathematics classroom.

2. The Purpose, Statement and Significant of the Study

In this study the researcher wants to know the teachers choice of mathematical examples and how the examples in the teaching and the learning process in the classroom generates questions of :

1. How teachers choose the examples for the teaching and learning of mathematics

2. How teachers use the examples in the teaching and learning of mathematics

3. Is there a relationship between the choice and use of examples with the teacher's pedagogical content knowledge (PCK)

This study is carried out to achieve the aim stated above and has theoretical and practical significance, policy and teacher training. Based on this study, the knowledge on mathematics can be enhanced by transferring the theoretical knowledge to practical use by the teacher in the classroom. Teachers should be aware of the importance of the choice and use of examples in the teaching and learning of mathematics. Teachers would benefit if appropriate examples are selected, prepared and formulated for the teaching and learning of mathematics.

3. Examples in Mathematics and Pedagogical Content Knowledge (PCK)

In the teaching and learning of mathematics, examples play an important role where the teacher uses certain examples to explain and give mathematical understanding to the students. The examples should be able to demonstrate the procedural and conceptual understanding in addition to finding the relationship and making generalization. Examples too play an important role in developing the conceptual understanding and certain examples can be significant in enhancing mathematical theory. (Alcock and Inglis, 2008). Using examples is a method practiced by teachers to explain and give mathematical understanding in the teaching and learning process. Teacher needs to have an in-depth understanding when interacting and explaining the concept and principles of mathematics. (Shulman, 1987). The teacher needs to have the knowledge of interacting between the theory and practice in the teaching and learning of mathematics.

The primary action of a teacher in the classroom includes, the choice of examples used to visualize the idea in the pedagogical content knowledge (Chick and Harris, 2007). Pedagogical Content Knowledge (PCK) is defined as the synthesizing of three areas of knowledge, ie. the content knowledge, the pedagogical knowledge and contextual knowledge (Grossman, 1992). PCK is also termed as a natural set that assists someone to transfer ones content knowledge to others and is the most effective method to deliver idea, analogy, visualize examples and explain the content that can be understood by others. (Shulman, 1987). In Pedagogical Content Knowledge, the content knowledge of mathematics and the pedagogical knowledge is blended to give rise to pedagogical content knowledge, reason being to understand the topics, problems or mathematical issue which is managed, represented, adapted to students interest and ability then presented in the teaching process (Shulman, 1987). Figure 1 shows research conceptual framework.

Example

Figure 1. Conceptual Framework

Figure 1 shows how content knowledge, pedagogical knowledge and pedagogical content knowledge (PCK) used to plan and implement examples in classroom. Teacher choose examples during planning and should be aware along the process.

4. Methodology

This method of study involves qualitative research where the researcher wants to understand the phenomena through case study. In this study, the samples are two teacher trainiees from the teacher training institute who are currently teaching in the national school. The researcher's task is to observe, take notes and do a video recording on teachers teaching. The researcher also interviews the mathematics teacher after the teaching practice. The data is then transcribed for analysis. The analysis on the content knowledge is based on the themes that has been identified based on the factors that has been categorized into four categories that is variables, sequencing, representations and learning objectives (Rowland, 2008). In the analysis of the Pedagogical Content Knowledge (PCK), the teacher is studied based on teaching techniques, strategies and approach in the teaching and learning process and the appropriate examples used for the students. Besides that, PCK looks into the students thinking aspect, for example the wrong concept, the knowledge of using teaching material and the appropriateness of using examples based on the concept. PCK analysis depends on the examples put forward.

5. Findings

Analysis is carried out on two participants that is PS (Participant S) and PF (Participant F). Below is the findings of the study. PS teaches in classroom 3J on the topic 'Addition in Real Life Situation till 10,000' using one of the

mathematical process which is problem solving. The objective is that the students should be able to solve the problem pertaining to addition. The following is an example used by PS.

PS :'I went to the market yesterday to buy some flowers. I bought 112 stalk of and 85 stalks

of daisies. I want to put all this flowers in my house. When I reached my house, my mother was so shocked and said that, there were too many flowers! Why did you buy so many flowers? (a picture showing roses and daisies in a vase). 'Are there too many? Help me to count the number of flowers I bought' (PS then writes on the white board).

112 + 85 =_

PS :"Can you count?" (While writing on the white board)

112 + 85

Pupils :'Two plus five is equal to seven, one plus eight is equal to nine, ... PS : 'So, what is the answer?'

Pupils :'One hundred and ninety seven"

PS :'Okay, as you can see, one hundred and twelve plus eighty five is equal to one hundred and ninety seven. Therefore , I have bought one hundred and ninety seven flowers".

Teaching and learning strategy using slide show attracts the students attention because of the colourful pictures and writing. The pictures of the daisies and roses did not show the number in each bouquet of 112 and 85 respectively. Instead, it was placed in a vase, not arranged and it was overlapping. In a classroom with average achievers, it is important to show the exact number accurately. After questioning, PS immediately solved the problem using the number sentence and next by writing the solution in standard written method.

There are various method that could be adapted by PS to solve the mathematical problem as mentioned by Polya (1973). Problem solving in mathematics requires the teacher to explain in detail and systematically to avoid confusion based on the needs of the question. On the choice of the examples, (after the interview session) the researcher found that the PS made a choice based on what he thought was a good choice. Below is his view.

Researcher PS

Researcher PS

Researcher PS

Researcher PS

'Why did you use flowers as examples?' 'Because I personally adore flowers like roses and daisies' 'Why did you place them in a bouquet and not in stalks?' 'I have a picture of a bouquet, and there are many flowers so I used it immediately' ' I did not see 112 or 85?'

'Too many flowers so I did not count, but I did mention the number.'

'How did you solve the problem.......Did you count straight away?'

'Changed it to standard written method, we would be able to get the Answer. Besides, the students already know addition'

PS has less understanding of the interest and need of the 9 year olds. Pictures in the questions are important as it attracts the attention and interest of the students and not only to attract the attention of the teacher. A clear and systematic solving method was not used in the process of problem solving. This is because the explanation given on how to solve the problem is important especially in understanding the need of the question. In addition to that, the number 112 and 85 is the PS's idea. The use of correct variables in the counting process using numbers that are bigger would be an advantage for the average students. For example ; 168 + 21

The next participant is PF and he teaches the year 3S consisting of weak students. The objective of the lesson is to add without regrouping. The use of his example is shown below :

PF : ' Three red apples plus four oranges plus six green apples plus eight strawberries and two pineapples, how many fruits are there altogether?

Students : (counting all the fruits orally) 'one, two, three.....(until twenty three)

PF : 'Good. Twenty three.'

Just as PS, PF is able to attract the attention of students using slides showing fruits and colourful writings. The fruits are also arranged in single order. According to PF, the choice of examples was his own idea and the selection of fruits is an appropriate example and was easily countable. Analysis show that PF uses counting strategy (Reys et al., 2001). However, the arrangement of numbers were not in order, either it starts with a smaller number added to a

bigger number or otherwise. For the weaker class the variables used would be easier if started with bigger additional numbers. (Reys et al., 2001). This can be done when addition is carried out in this order.:

8 + 6 + 4 + 3 + 2 = 23

After an interview session with PF, he stated that the selection of the fruits and the counting method based on pictures of fruits was his own idea without referring to any material. When analyzing both the participants, the researcher found that the variables and the ordering factor was not given importance when selecting and using examples. Besides that, the pictures and objects representing the examples need to be taken into consideration. This is the evidence of the lack of content knowledge while evidence on pedagogy aspects shows that the participants should use appropriate resources of examples and ask the students to participate more in class activity. The analysis of this research shows that the participants should be more careful when making the correct choice and using examples. Whatever confusion, inappropriateness and problems related to mathematical problem shows the lack of pedagogical content knowledge (blending of content and pedagogy knowledge) of the teacher himself.

6. Conclusion

The choice of examples used by the teacher affects the learning process of mathematics in the classroom. The mathematics teacher need to be aware and be careful when selecting and using examples. The PCK directly influences how the teacher selects and uses the mathematical examples. The knowledge about the selection and use of examples need to be exposed and practiced in the teacher training and the trainees need to be exposed and trained. Trainees who will be teaching mathematics in primary schools need to be exposed and given practical knowledge on the selection and use of examples at the all the teacher training institutes. This research clearly states that the participants do not have special guidance in the selection and use of examples where the examples chosen shows the PCK of the participants. This should be brought to the teachers awareness as they would be wiser and considerate in many aspects when making a choice and using the examples in the teaching and learning of mathematics.

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