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Framework of Knowledge and Teaching Skills of Expert Mathematics Teachers in Using Mathematical Examples
Available online at www.sciencedirect.com
Procedia Social and Behavioral Sciences 8 (2010) 325–331
International Conference on Mathematics Education Research 2010 (ICMER 2010)
Framework of Knowledge and Teaching Skills of Expert Mathematics Teachers in Using Mathematical Examples Mohini Mohameda,*, Faridah Sulaimanb a,b
Faculty of Education, Universiti Teknologi Malaysia
Abstract This paper presents some initial findings from a study on the use of examples by expert mathematics teachers in their teaching. The purpose of the ongoing study is to build a framework that can assist mathematics teachers to choose, use and improve mathematics examples in their teaching. This study use a multiple case studies design whereby data is gathered from observation, interviews and teacher’s note. The preliminary analyses of data from the teaching process of several expert mathematics teachers begin to uncover the framework of knowledge and teaching skills of these teachers in the use of mathematics examples. © 2010 Elsevier Ltd. All rights reserved. Keywords: Examples; Expert teachers; Mathematical knowledge for teaching
1.
Introduction
In classroom, mathematical idea is learned through examples. Examples provide experiences that help students to abstract a concept (Skemp, 1987). Examples are also used as a medium for generalization, reasoning or to show relationship (Bills et al., 2006; Watson and Mason, 2002; Zaslavsky, 2010). It is an integral part in mathematics teaching (Bills et al., 2006; Rowland, 2008; Watson and Mason, 2002; Zaslavsky, 2008). Thus, teachers play a crucial role to ensure that the examples they used in their teaching will bring maximum benefits for their students learning. However, findings from previous studies show that problems regarding the use of examples in mathematics teaching do exist. These studies indicated that the problems occur not only among novices (Crespo, 2003; Huntley, 2008; Rowland, 2008) but also among experienced mathematics teachers (Arbaugh and Brown, 2005; Henningsen and Stein, 1997; Stein et al., 1996; Ticha and Hospesova, 2006; Zodik and Zaslavsky, 2008). From these studies, mathematics teachers faced two exemplification problems, problems to choose and problems to use mathematics examples appropriately in their teaching. Both problems will limit the scope of their students’ learning.
* Corresponding author. E-mail address: [email protected] 1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.12.045
326
Mohini Mohamed and Faridah Sulaiman / Procedia Social and Behavioral Sciences 8 (2010) 325–331
The knowledge about mathematics examples is not a systematic knowledge. It is acquired through experience in teaching practice; hence, it is considered craft knowledge (Zaslavsky and Zodik, 2007). However, the knowledge about mathematics examples is a part of specialized content knowledge (Ball et al., 2008). According to Ball and her colleagues (2008), specialized content knowledge is one of the domains of Mathematical Knowledge for Teaching (MKT); it is a mathematical knowledge and skill unique to teaching. MKT is a refinement to Shulman’s (1986) notion on content knowledge for teaching. Studies have shown that MKT have a positive relationship towards students’ achievement (Baumert et al., 2010; Hill et al., 2005). This means that teachers' MKT will have an effect on their teaching and students' learning. As the knowledge about examples is a part of MKT, therefore, teachers’ weakness in this knowledge will have a negative impact towards their teaching and students learning. In order to identify the knowledge about mathematics examples, the teaching process of mathematics teachers should be studied. Although the knowledge about examples is constructed through teaching experience, not all teachers can learn from their experience (Ball and Bass, 2003; Hiebert et al., 2002; Kennedy, 2002). Thus, study should be conducted on the teaching process of effective mathematics teachers. In Malaysia, expert mathematics teachers, are teachers who are recognized by Ministry of Education as effective teachers. Studying about examples in their teaching process, will enable us to answer these research questions: How do they choose mathematics examples? How do they use mathematics examples in their teaching? How do they improve their mathematics examples? Answers to these questions will be used to construct a framework of knowledge and teachings skills of expert mathematics teachers in using examples. This framework can be used to assist mathematics teachers to choose, use and improve mathematics examples in their teaching. It is hope that, effective mathematics examples will help teachers to improve their teaching quality and boost their students’ learning. 2.
Methodology
This ongoing study uses qualitative approach with multiple case studies design. It involves expert mathematics teachers who teach in upper secondary school, in Johor. So far, data from the teaching process of three expert mathematics teachers who teach students with weak academic ability have been collected. The teaching process was divided into three phases; pre active, interactive and post active. Data were collected using five methods. They are pre active notes, observation, short interviews, post active notes and final interview. For each teacher, data were collected four times before the final interviews. Data, on how these teachers’ select mathematics examples before their teaching were gathered from pre active phase using pre active notes. These data were supported by data derived from observation and final interview. Meanwhile, in interactive phase data on how teachers use examples in their teaching were collected through observation. These data were supported by other data from short interviews and final interview. Finally, data on how these expert mathematics teachers improved their mathematics examples were collected in post active phase using post active notes. Data from observation and final interview were used to support data from post active notes. Constant comparative method was used to analyze data. 3.
Initial Findings
Initial findings reported in this paper are the result of first round analysis. E1, E2 and E3 are the expert mathematics teachers. 3.1
Choosing examples
3.1.1 Elements to be considered before choosing examples
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Three elements were considered by expert mathematics teachers before choosing examples for their teaching. (i) Curriculum specifications A curriculum specification is a supporting document to the syllabus. It contains information about learning objectives and learning outcomes for every topic in the syllabus. All three expert mathematics teachers referred to curriculum specification in order to know what knowledge and skill should be acquired by their students for a specific topic. E1: E2: E3:
“You have to refer to the specification... Analyze the topic that you are going to teach”. “This is the guidelines. It tells you what type of skills students must acquire”. “You have to understand the specification. It informs you about what you should teach”.
(ii) Rearrange the sequence of learning outcomes These expert mathematics teachers do not follow the sequence of learning outcomes in the curriculum specification. They rearrange the sequence of learning outcomes proposed by the curriculum specification according to their students’ academic ability. The purpose of the rearrangement is to help them plan their teaching in a manner that can assist their students’ learning. For examples, E1 is going to teach Motion along Straight Line to his students. In the curriculum specification, this topic has three learning objectives and 11 learning outcomes. He combines and rearranges these learning outcomes into six learning outcomes. In another case, when E2 teach statistics to her class, one of the learning outcome is students must be able to draw histogram. She adds another one learning outcome that must be achieved by her students before they can draw histogram that is, they must be able to mark horizontal and vertical axis with correct and uniform scale. Similar to E1, E3 combines and rearranges learning outcomes for topic probability to avoid confusions among his students. (iii)Previous knowledge Expert mathematics teachers will identify previous knowledge that need to be mastered by their students before they teach a new mathematical idea. They also have to know to what extent their students have mastered this knowledge. If students have not mastered the previous knowledge needed, teachers have to teach them again before they can teach a new mathematical idea. 3.1.2 Examples variation Expert mathematics teachers determine the variations of mathematical examples that need to be given to their students based on: (i) Analysis of past years exam questions Syllabus and curriculum specification provide general information on a topic that must be taught, learning objectives and learning outcomes that must be achieved, but both documents do not tell teachers to what extent certain mathematical idea should be mastered, what kind of problems that students should be able to solve by using this mathematical idea. To overcome these shortcomings, expert mathematics teachers analyzed past year exam questions. The analysis seeks to determine the scope of usage of a mathematical idea. From the result of the analysis, these teachers choose or construct their own examples according to the scope of usage. (ii) Learning problems From experience, all these expert mathematics teachers know what kind of mistakes, negligence or misconception that will occur when they teach certain topic. Therefore, they deliberately choose examples that will lead students to these problems.
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E1 is teaching Permutation. He knows that students tend to make mistakes when they are asked to arrange number in certain condition. He chooses the following examples to help his students to understand the concept. Both examples look similar. Students tend to repeat the same procedure that they have used in the first example to solve the second one. That is when the problem happens. Calculate the number of five digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6, 7 without repetition if: (a) (b) 3.2
The number is more than 45 000. The number is more than 53 000.
Using examples
3.2.1 Types of examples There are four types of mathematical examples used. (i) Concept examples This type of examples is used to introduce a concept of a new mathematical idea to students. This example is usually common things that students can relate to their daily live. To introduce basic concept of probability, E3 used this examples. “If I invite you to my open house, what are the chances that you will come?” (ii) Procedure examples This kind of examples is used to show how a procedure that involves a specific mathematical idea is carried out. This is the example used by E1 to show his students how to find the equation of velocity (v), acceleration (a) and displacement (s) when one of these equation given. (a) (b)
Find a and v when s = t3 -t2 -8t Find a and s when v = 4t - 2
(iii) Application examples This type of examples is used to introduce to the students how to use concept or procedure that they have just learnt to solve mathematical problems. This example is used by E3 to show students how the basic concept of probability is used. In a group of 90 students, 70 are girls. Another 10 boys then join the group. If a student is chosen at random from the group, state the probability that the student chosen is a boy. (iv) Reinforcement examples These examples are just like application examples, but it is used to enhance students understanding about a mathematical idea. 3.2.2 Methods of handling examples (i) Examples handled together by teacher and their students
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329
This method is used with concept, procedure or application examples. Together they try to solve problem given in an example. (ii) Examples handled by students This method is used with reinforcement examples. Students are required to do this kind of example on their own or in group. (iii) If the knowledge that is going to be introduced is advancement to prior knowledge, example is needed to show to the student the inadequacy of this prior knowledge. This example is used by the teacher as a medium to explain to the students the advantage of the new knowledge compared to prior knowledge. In lower form, students have seen and know how to use data given in frequency table but it only involves discrete data. When E2 want to introduce the concept of class interval, she used this example. The data below shows the marks obtained by 15 students in mathematics test 15 52 73
27 55 76
She then asks these questions:
31 61 78
35 67 81 (i) (ii)
50 70 89 How many students get more than 40 marks? How many students get A?
To answer these questions, students have to identify and then count how many marks meet the condition given. After they have done this, E2 introduce the concept of class interval and show the advantage of this concept compared to their prior knowledge. 3.2.3 Using examples to monitor students understanding Expert mathematics teachers used examples in three different ways to monitor students understanding. (i) (ii) (iii)
Examples are used by teachers as a base to form questions. Students’ response towards these questions reflects their understanding. Students used examples as a base to form questions to be asked to their teacher regarding the mathematical problem that they face. These questions help teachers to identify the cause of the problem. Teachers can detect their students’ understanding through their explanation or solution to mathematical problems given in examples.
3.2.4 Using examples to address learning problems From the beginning, expert mathematics teachers have identified potential elements that will cause learning problems. They deliberately incorporate these elements in examples that they use. Students are left to deal with these problems on their own. Teachers will only give assistance and explanation after their students have tried these examples. 3.3
Improving mathematical examples that have been used
(a) The effectiveness of examples To evaluate the effectiveness of examples that they have used, expert mathematics teachers will assess: (i)
The ability of their students to solve on their own the reinforcement examples in the classroom,
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(ii)
Mohini Mohamed and Faridah Sulaiman / Procedia Social and Behavioral Sciences 8 (2010) 325–331
The ability of their students to solve exercises given as their home work.
The result of these assessments will be used to determine the effectiveness of mathematical examples that they have used. (b) Recommendation to improve examples Expert mathematics teachers analyzed two elements before giving any recommendations. These two elements are the content of examples that they have used and the way they use examples in the classroom. Recommendations are made based on this analysis. E2 wrote in her post active note, she will take out the mid-point column from table below. Since her intention is to ask students to draw a histogram using lower and upper boundary as its axis, the mid-point column may cause confusion among students. Marks
Frequency
Midpoint
Lower Boundary
Upper Boundary
In his post active note, E1 wrote he will change the way he explained about possible ways of filling a space when he teach permutation next time. He thinks that his previous explanation will cause misconception. 4.
Discussion
Choosing examples are not an easy task. From the initial findings, the knowledge used by these expert teachers is in a form of craft knowledge. Knowledge constructed from their teaching experience. They use this knowledge to meet the requirement of the curriculum, students’ learning and examination. This knowledge enables them to offer the best examples that tailored according to their students’ ability and at the same time abide to the curriculum and examination needs. Each example used by these expert mathematics teachers has its own purpose. These teachers conduct their teaching according to these purposes. They are sensitive to students’ response, questions and arguments towards the given mathematical examples. This sensitivity helps them to assist and enhance their students’ learning process. This sensitivity also plays a role as a device that helps these expert mathematics teachers to improve their mathematical examples. Knowledge about mathematical examples is not a frivolous knowledge. It shows the ability of the teacher to integrate and balance a lot of important elements in their teaching. The initial findings tend to show that the framework of knowledge used by expert mathematics teacher is dynamic. It ensures the continuous improvement of the quality and the usage of mathematical examples in the classroom. This will lead to effective teaching and learning. 5.
Conclusion
These initial findings show that expert mathematics teachers choose and use examples in a consistent manner according to the learning objectives that need to be achieved. This is in contrast to the previous studies (Crespo, 2003; Rowland, 2008; Henningsen and Stein, 1997; Stein et al., 1996). Findings from previous studies show that exemplification problems occur when there is a discrepancy between the examples chosen and used with the learning objectives that need to be achieved. However, these initial findings only provided insight about expert mathematics teachers’ framework of knowledge and teaching skills in using mathematical examples. Further analysis needs to be done to refine this framework and show its relationship with the exemplification process. The second phase of this study will examine the teaching process of expert mathematics teachers who teach students with excellence academic achievement. The final result of this study is expected to contribute to the construction of
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a framework of knowledge and teaching skills of expert mathematics teachers in using examples. It is hope that this framework can be used as a basis for designing training courses to help mathematics teachers choose, use and improve mathematics examples in their teaching regardless of their students’ academic ability. References Arbaugh, F. and Brown, C.A. (2005). Analyzing Mathematical Task: A Catalyst for Change? Journal of Mathematics Teacher Education. 8, 499 – 536. Springer. Ball, D. L. and Bass, H. (2003). Towards a Practice-Based Theory of Mathematical Knowledge for Teaching. Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group. 24-28 May, 2002. Edmonton, AB, 3 – 14. Ball, D. L., Thames, M. H. and Phelps, G. (2008). Content Knowledge for Teaching: What Makes it Special?. Journal of Teacher Education. 59 (5), 389 – 407. Sage Publications. Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A. et al. (2010). Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom and Student Progress. American Educational Research Journal. 47(1), 133 – 180. Sage. Bills, L., Dreyfuss, T., Mason, J., Tsamir, P., Watson, A. and Zaslavsky, O. (2006). Exemplification in Mathematics Education. Proceedings of 30th Conference of the International Group for the Psychology of Mathematics Education. 16-21 July. Prague, Czech Republic, 126 – 154. Crespo, S. (2003). Learning to Pose Mathematical Problems: Exploring Changes in Preservice Teachers’ Practices. Educational Studies in Mathematics. 52, 243 – 270. Kluwer Academic Publisher. Henningsen, M. and Stein, M. K. (1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors that Support and Inhibit HighLevel Mathematical Thinking and Reasoning. Journal for Research In Mathematics Education. 28 (5), 524 – 549. National Council of Teachers of Mathematics. Hiebert, J., Gallimore, R. and Stigler, J. W. (2002). A Knowledge Base for the Teaching Profession: What Would it Look Like and How can We Get One? Educational Researcher. 31 (5), 3 – 15. Hill, H.C., Rowan, B. and Ball, D. L. (2005). Effects of Teachers’ Mathematical Knowledge for Teaching on Students Achievement. American Educational Research Journal. 42(2), 371 – 406. Huntley, R. (2008). Researching Primary Trainees’ Choice of Examples: Some Early Analysis of Data. Proceedings of the British Society for Research into Learning Mathematics 28(3). 15th November. Kings College London, 42 – 46. Kennedy, M. M. (2002). Knowledge and Teaching. Teachers and Teaching: Theory and Practice. 8 (3), 354 – 370. Taylor and Francis Group. Rowland, T. (2008). The Purpose, Design and Use of Examples in the Teaching of Elementary Mathematics. Educational Studies of Mathematics. 68, 149 – 163. Springer. Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher. 15(2), 4 – 14. Skemp, R. R. (1987). The Psychology of Learning Mathematics: Expanded American Edition. New Jersey: Lawrence Earlbaum Associates. Stein, M.K., Grover, B.W. and Henningsen, M. (1996). Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms. American Educational Research Journal. 33 (2), 455 – 488. Sage Publications. Ticha, M. and Hospesova, A. (2006). Qualified Pedagogical Reflections as a Way to Improve Mathematics Education. Journal of Mathematics Teacher Education. 9, 129 – 156. Springer. Watson, A. and Mason, J. (2002). Extending Example Spaces as a Learning/Teaching Strategy in Mathematics. Proceedings of PME 26. 21-26 July. Norwich, University of East Angelia, 377 – 385. Zaslavsky, O. (2008). What Knowledge is Involved in Choosing and Generating Useful Instructional Examples?. Paper presented at the Symposium on the Occasion of the 100th Anniversary of ICMI. 5-8 March. Rome. Zaslavsky, O. (2010). The Explanatory Power of Examples in Mathematics: Challenges for Teaching. In Stein, M. K. and Kucan, L. (Eds.). Instructional Explanations in the Disciplines (pp. 107 – 128). Springer. Zaslavsky, O. and Zodik, I. (2007). Mathematics Teachers’ Choices of Examples that Potentially Support or Impede Learning. Research in Mathematics Education. 9 (1), 143 – 155. Routlege. Zodik, I. and Zaslavsky, O. (2008). Characteristics of the Teachers’ Choice of Examples in and for the Mathematics Classroom. Educational Studies in Mathematics. 69, 165 – 182. Springer.
Procedia Social and Behavioral Sciences 8 (2010) 325–331
International Conference on Mathematics Education Research 2010 (ICMER 2010)
Framework of Knowledge and Teaching Skills of Expert Mathematics Teachers in Using Mathematical Examples Mohini Mohameda,*, Faridah Sulaimanb a,b
Faculty of Education, Universiti Teknologi Malaysia
Abstract This paper presents some initial findings from a study on the use of examples by expert mathematics teachers in their teaching. The purpose of the ongoing study is to build a framework that can assist mathematics teachers to choose, use and improve mathematics examples in their teaching. This study use a multiple case studies design whereby data is gathered from observation, interviews and teacher’s note. The preliminary analyses of data from the teaching process of several expert mathematics teachers begin to uncover the framework of knowledge and teaching skills of these teachers in the use of mathematics examples. © 2010 Elsevier Ltd. All rights reserved. Keywords: Examples; Expert teachers; Mathematical knowledge for teaching
1.
Introduction
In classroom, mathematical idea is learned through examples. Examples provide experiences that help students to abstract a concept (Skemp, 1987). Examples are also used as a medium for generalization, reasoning or to show relationship (Bills et al., 2006; Watson and Mason, 2002; Zaslavsky, 2010). It is an integral part in mathematics teaching (Bills et al., 2006; Rowland, 2008; Watson and Mason, 2002; Zaslavsky, 2008). Thus, teachers play a crucial role to ensure that the examples they used in their teaching will bring maximum benefits for their students learning. However, findings from previous studies show that problems regarding the use of examples in mathematics teaching do exist. These studies indicated that the problems occur not only among novices (Crespo, 2003; Huntley, 2008; Rowland, 2008) but also among experienced mathematics teachers (Arbaugh and Brown, 2005; Henningsen and Stein, 1997; Stein et al., 1996; Ticha and Hospesova, 2006; Zodik and Zaslavsky, 2008). From these studies, mathematics teachers faced two exemplification problems, problems to choose and problems to use mathematics examples appropriately in their teaching. Both problems will limit the scope of their students’ learning.
* Corresponding author. E-mail address: [email protected] 1877-0428 © 2010 Published by Elsevier Ltd. doi:10.1016/j.sbspro.2010.12.045
326
Mohini Mohamed and Faridah Sulaiman / Procedia Social and Behavioral Sciences 8 (2010) 325–331
The knowledge about mathematics examples is not a systematic knowledge. It is acquired through experience in teaching practice; hence, it is considered craft knowledge (Zaslavsky and Zodik, 2007). However, the knowledge about mathematics examples is a part of specialized content knowledge (Ball et al., 2008). According to Ball and her colleagues (2008), specialized content knowledge is one of the domains of Mathematical Knowledge for Teaching (MKT); it is a mathematical knowledge and skill unique to teaching. MKT is a refinement to Shulman’s (1986) notion on content knowledge for teaching. Studies have shown that MKT have a positive relationship towards students’ achievement (Baumert et al., 2010; Hill et al., 2005). This means that teachers' MKT will have an effect on their teaching and students' learning. As the knowledge about examples is a part of MKT, therefore, teachers’ weakness in this knowledge will have a negative impact towards their teaching and students learning. In order to identify the knowledge about mathematics examples, the teaching process of mathematics teachers should be studied. Although the knowledge about examples is constructed through teaching experience, not all teachers can learn from their experience (Ball and Bass, 2003; Hiebert et al., 2002; Kennedy, 2002). Thus, study should be conducted on the teaching process of effective mathematics teachers. In Malaysia, expert mathematics teachers, are teachers who are recognized by Ministry of Education as effective teachers. Studying about examples in their teaching process, will enable us to answer these research questions: How do they choose mathematics examples? How do they use mathematics examples in their teaching? How do they improve their mathematics examples? Answers to these questions will be used to construct a framework of knowledge and teachings skills of expert mathematics teachers in using examples. This framework can be used to assist mathematics teachers to choose, use and improve mathematics examples in their teaching. It is hope that, effective mathematics examples will help teachers to improve their teaching quality and boost their students’ learning. 2.
Methodology
This ongoing study uses qualitative approach with multiple case studies design. It involves expert mathematics teachers who teach in upper secondary school, in Johor. So far, data from the teaching process of three expert mathematics teachers who teach students with weak academic ability have been collected. The teaching process was divided into three phases; pre active, interactive and post active. Data were collected using five methods. They are pre active notes, observation, short interviews, post active notes and final interview. For each teacher, data were collected four times before the final interviews. Data, on how these teachers’ select mathematics examples before their teaching were gathered from pre active phase using pre active notes. These data were supported by data derived from observation and final interview. Meanwhile, in interactive phase data on how teachers use examples in their teaching were collected through observation. These data were supported by other data from short interviews and final interview. Finally, data on how these expert mathematics teachers improved their mathematics examples were collected in post active phase using post active notes. Data from observation and final interview were used to support data from post active notes. Constant comparative method was used to analyze data. 3.
Initial Findings
Initial findings reported in this paper are the result of first round analysis. E1, E2 and E3 are the expert mathematics teachers. 3.1
Choosing examples
3.1.1 Elements to be considered before choosing examples
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327
Three elements were considered by expert mathematics teachers before choosing examples for their teaching. (i) Curriculum specifications A curriculum specification is a supporting document to the syllabus. It contains information about learning objectives and learning outcomes for every topic in the syllabus. All three expert mathematics teachers referred to curriculum specification in order to know what knowledge and skill should be acquired by their students for a specific topic. E1: E2: E3:
“You have to refer to the specification... Analyze the topic that you are going to teach”. “This is the guidelines. It tells you what type of skills students must acquire”. “You have to understand the specification. It informs you about what you should teach”.
(ii) Rearrange the sequence of learning outcomes These expert mathematics teachers do not follow the sequence of learning outcomes in the curriculum specification. They rearrange the sequence of learning outcomes proposed by the curriculum specification according to their students’ academic ability. The purpose of the rearrangement is to help them plan their teaching in a manner that can assist their students’ learning. For examples, E1 is going to teach Motion along Straight Line to his students. In the curriculum specification, this topic has three learning objectives and 11 learning outcomes. He combines and rearranges these learning outcomes into six learning outcomes. In another case, when E2 teach statistics to her class, one of the learning outcome is students must be able to draw histogram. She adds another one learning outcome that must be achieved by her students before they can draw histogram that is, they must be able to mark horizontal and vertical axis with correct and uniform scale. Similar to E1, E3 combines and rearranges learning outcomes for topic probability to avoid confusions among his students. (iii)Previous knowledge Expert mathematics teachers will identify previous knowledge that need to be mastered by their students before they teach a new mathematical idea. They also have to know to what extent their students have mastered this knowledge. If students have not mastered the previous knowledge needed, teachers have to teach them again before they can teach a new mathematical idea. 3.1.2 Examples variation Expert mathematics teachers determine the variations of mathematical examples that need to be given to their students based on: (i) Analysis of past years exam questions Syllabus and curriculum specification provide general information on a topic that must be taught, learning objectives and learning outcomes that must be achieved, but both documents do not tell teachers to what extent certain mathematical idea should be mastered, what kind of problems that students should be able to solve by using this mathematical idea. To overcome these shortcomings, expert mathematics teachers analyzed past year exam questions. The analysis seeks to determine the scope of usage of a mathematical idea. From the result of the analysis, these teachers choose or construct their own examples according to the scope of usage. (ii) Learning problems From experience, all these expert mathematics teachers know what kind of mistakes, negligence or misconception that will occur when they teach certain topic. Therefore, they deliberately choose examples that will lead students to these problems.
328
Mohini Mohamed and Faridah Sulaiman / Procedia Social and Behavioral Sciences 8 (2010) 325–331
E1 is teaching Permutation. He knows that students tend to make mistakes when they are asked to arrange number in certain condition. He chooses the following examples to help his students to understand the concept. Both examples look similar. Students tend to repeat the same procedure that they have used in the first example to solve the second one. That is when the problem happens. Calculate the number of five digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6, 7 without repetition if: (a) (b) 3.2
The number is more than 45 000. The number is more than 53 000.
Using examples
3.2.1 Types of examples There are four types of mathematical examples used. (i) Concept examples This type of examples is used to introduce a concept of a new mathematical idea to students. This example is usually common things that students can relate to their daily live. To introduce basic concept of probability, E3 used this examples. “If I invite you to my open house, what are the chances that you will come?” (ii) Procedure examples This kind of examples is used to show how a procedure that involves a specific mathematical idea is carried out. This is the example used by E1 to show his students how to find the equation of velocity (v), acceleration (a) and displacement (s) when one of these equation given. (a) (b)
Find a and v when s = t3 -t2 -8t Find a and s when v = 4t - 2
(iii) Application examples This type of examples is used to introduce to the students how to use concept or procedure that they have just learnt to solve mathematical problems. This example is used by E3 to show students how the basic concept of probability is used. In a group of 90 students, 70 are girls. Another 10 boys then join the group. If a student is chosen at random from the group, state the probability that the student chosen is a boy. (iv) Reinforcement examples These examples are just like application examples, but it is used to enhance students understanding about a mathematical idea. 3.2.2 Methods of handling examples (i) Examples handled together by teacher and their students
Mohini Mohamed and Faridah Sulaiman / Procedia Social and Behavioral Sciences 8 (2010) 325–331
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This method is used with concept, procedure or application examples. Together they try to solve problem given in an example. (ii) Examples handled by students This method is used with reinforcement examples. Students are required to do this kind of example on their own or in group. (iii) If the knowledge that is going to be introduced is advancement to prior knowledge, example is needed to show to the student the inadequacy of this prior knowledge. This example is used by the teacher as a medium to explain to the students the advantage of the new knowledge compared to prior knowledge. In lower form, students have seen and know how to use data given in frequency table but it only involves discrete data. When E2 want to introduce the concept of class interval, she used this example. The data below shows the marks obtained by 15 students in mathematics test 15 52 73
27 55 76
She then asks these questions:
31 61 78
35 67 81 (i) (ii)
50 70 89 How many students get more than 40 marks? How many students get A?
To answer these questions, students have to identify and then count how many marks meet the condition given. After they have done this, E2 introduce the concept of class interval and show the advantage of this concept compared to their prior knowledge. 3.2.3 Using examples to monitor students understanding Expert mathematics teachers used examples in three different ways to monitor students understanding. (i) (ii) (iii)
Examples are used by teachers as a base to form questions. Students’ response towards these questions reflects their understanding. Students used examples as a base to form questions to be asked to their teacher regarding the mathematical problem that they face. These questions help teachers to identify the cause of the problem. Teachers can detect their students’ understanding through their explanation or solution to mathematical problems given in examples.
3.2.4 Using examples to address learning problems From the beginning, expert mathematics teachers have identified potential elements that will cause learning problems. They deliberately incorporate these elements in examples that they use. Students are left to deal with these problems on their own. Teachers will only give assistance and explanation after their students have tried these examples. 3.3
Improving mathematical examples that have been used
(a) The effectiveness of examples To evaluate the effectiveness of examples that they have used, expert mathematics teachers will assess: (i)
The ability of their students to solve on their own the reinforcement examples in the classroom,
330
(ii)
Mohini Mohamed and Faridah Sulaiman / Procedia Social and Behavioral Sciences 8 (2010) 325–331
The ability of their students to solve exercises given as their home work.
The result of these assessments will be used to determine the effectiveness of mathematical examples that they have used. (b) Recommendation to improve examples Expert mathematics teachers analyzed two elements before giving any recommendations. These two elements are the content of examples that they have used and the way they use examples in the classroom. Recommendations are made based on this analysis. E2 wrote in her post active note, she will take out the mid-point column from table below. Since her intention is to ask students to draw a histogram using lower and upper boundary as its axis, the mid-point column may cause confusion among students. Marks
Frequency
Midpoint
Lower Boundary
Upper Boundary
In his post active note, E1 wrote he will change the way he explained about possible ways of filling a space when he teach permutation next time. He thinks that his previous explanation will cause misconception. 4.
Discussion
Choosing examples are not an easy task. From the initial findings, the knowledge used by these expert teachers is in a form of craft knowledge. Knowledge constructed from their teaching experience. They use this knowledge to meet the requirement of the curriculum, students’ learning and examination. This knowledge enables them to offer the best examples that tailored according to their students’ ability and at the same time abide to the curriculum and examination needs. Each example used by these expert mathematics teachers has its own purpose. These teachers conduct their teaching according to these purposes. They are sensitive to students’ response, questions and arguments towards the given mathematical examples. This sensitivity helps them to assist and enhance their students’ learning process. This sensitivity also plays a role as a device that helps these expert mathematics teachers to improve their mathematical examples. Knowledge about mathematical examples is not a frivolous knowledge. It shows the ability of the teacher to integrate and balance a lot of important elements in their teaching. The initial findings tend to show that the framework of knowledge used by expert mathematics teacher is dynamic. It ensures the continuous improvement of the quality and the usage of mathematical examples in the classroom. This will lead to effective teaching and learning. 5.
Conclusion
These initial findings show that expert mathematics teachers choose and use examples in a consistent manner according to the learning objectives that need to be achieved. This is in contrast to the previous studies (Crespo, 2003; Rowland, 2008; Henningsen and Stein, 1997; Stein et al., 1996). Findings from previous studies show that exemplification problems occur when there is a discrepancy between the examples chosen and used with the learning objectives that need to be achieved. However, these initial findings only provided insight about expert mathematics teachers’ framework of knowledge and teaching skills in using mathematical examples. Further analysis needs to be done to refine this framework and show its relationship with the exemplification process. The second phase of this study will examine the teaching process of expert mathematics teachers who teach students with excellence academic achievement. The final result of this study is expected to contribute to the construction of
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a framework of knowledge and teaching skills of expert mathematics teachers in using examples. It is hope that this framework can be used as a basis for designing training courses to help mathematics teachers choose, use and improve mathematics examples in their teaching regardless of their students’ academic ability. References Arbaugh, F. and Brown, C.A. (2005). Analyzing Mathematical Task: A Catalyst for Change? Journal of Mathematics Teacher Education. 8, 499 – 536. Springer. Ball, D. L. and Bass, H. (2003). Towards a Practice-Based Theory of Mathematical Knowledge for Teaching. Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group. 24-28 May, 2002. Edmonton, AB, 3 – 14. Ball, D. L., Thames, M. H. and Phelps, G. (2008). Content Knowledge for Teaching: What Makes it Special?. Journal of Teacher Education. 59 (5), 389 – 407. Sage Publications. Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A. et al. (2010). Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom and Student Progress. American Educational Research Journal. 47(1), 133 – 180. Sage. Bills, L., Dreyfuss, T., Mason, J., Tsamir, P., Watson, A. and Zaslavsky, O. (2006). Exemplification in Mathematics Education. Proceedings of 30th Conference of the International Group for the Psychology of Mathematics Education. 16-21 July. Prague, Czech Republic, 126 – 154. Crespo, S. (2003). Learning to Pose Mathematical Problems: Exploring Changes in Preservice Teachers’ Practices. Educational Studies in Mathematics. 52, 243 – 270. Kluwer Academic Publisher. Henningsen, M. and Stein, M. K. (1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors that Support and Inhibit HighLevel Mathematical Thinking and Reasoning. Journal for Research In Mathematics Education. 28 (5), 524 – 549. National Council of Teachers of Mathematics. Hiebert, J., Gallimore, R. and Stigler, J. W. (2002). A Knowledge Base for the Teaching Profession: What Would it Look Like and How can We Get One? Educational Researcher. 31 (5), 3 – 15. Hill, H.C., Rowan, B. and Ball, D. L. (2005). Effects of Teachers’ Mathematical Knowledge for Teaching on Students Achievement. American Educational Research Journal. 42(2), 371 – 406. Huntley, R. (2008). Researching Primary Trainees’ Choice of Examples: Some Early Analysis of Data. Proceedings of the British Society for Research into Learning Mathematics 28(3). 15th November. Kings College London, 42 – 46. Kennedy, M. M. (2002). Knowledge and Teaching. Teachers and Teaching: Theory and Practice. 8 (3), 354 – 370. Taylor and Francis Group. Rowland, T. (2008). The Purpose, Design and Use of Examples in the Teaching of Elementary Mathematics. Educational Studies of Mathematics. 68, 149 – 163. Springer. Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher. 15(2), 4 – 14. Skemp, R. R. (1987). The Psychology of Learning Mathematics: Expanded American Edition. New Jersey: Lawrence Earlbaum Associates. Stein, M.K., Grover, B.W. and Henningsen, M. (1996). Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms. American Educational Research Journal. 33 (2), 455 – 488. Sage Publications. Ticha, M. and Hospesova, A. (2006). Qualified Pedagogical Reflections as a Way to Improve Mathematics Education. Journal of Mathematics Teacher Education. 9, 129 – 156. Springer. Watson, A. and Mason, J. (2002). Extending Example Spaces as a Learning/Teaching Strategy in Mathematics. Proceedings of PME 26. 21-26 July. Norwich, University of East Angelia, 377 – 385. Zaslavsky, O. (2008). What Knowledge is Involved in Choosing and Generating Useful Instructional Examples?. Paper presented at the Symposium on the Occasion of the 100th Anniversary of ICMI. 5-8 March. Rome. Zaslavsky, O. (2010). The Explanatory Power of Examples in Mathematics: Challenges for Teaching. In Stein, M. K. and Kucan, L. (Eds.). Instructional Explanations in the Disciplines (pp. 107 – 128). Springer. Zaslavsky, O. and Zodik, I. (2007). Mathematics Teachers’ Choices of Examples that Potentially Support or Impede Learning. Research in Mathematics Education. 9 (1), 143 – 155. Routlege. Zodik, I. and Zaslavsky, O. (2008). Characteristics of the Teachers’ Choice of Examples in and for the Mathematics Classroom. Educational Studies in Mathematics. 69, 165 – 182. Springer.