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## MATHEMATICAL KNOWLEDGE FOR TEACHING

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**MATHEMATICAL KNOWLEDGE FOR TEACHING**Adnan Baki Karadeniz Technical University**………………………………….………introduction**………………………………….………introduction • Studies on teacher knowledge that Shulman and his associates conducted decades ago still prove to be current (Shulman, 1986; Grossman, 1988; Magnusson et al. 1999; Ann et al., 2004; Ball, 2008). • Ann, S., Kulm, G. ve Wu, Z.(2004). The Pedagogical content knowledge of Middle School Mathematics Teachers in China and the US. Journal of Mathematics Teacher Education 7, 145-172. • Ball, D.L.,Thames, M. H. ve Phelps,G.(2008). Content Knowledge for Teaching:What Makes It Special? Journal of Teacher Education Cilt:59, Sayı: 5. (389-407) • Grossman, P, L.(1988). A study of Contrast: Sources of Pedagogical Content Knowledge for Secondary English. Unpublished doctoral dissertation, Stanford University. • Magnusson,S. (1991) The relationship between teachers’ content and pedagogical content knowledge and students’ content knowledge of heat energy and temperature. Unpublished Doctoral Dissurtation. The University of Maryland. • Shulman, L.S. (1986). Those who understand: Knowledge Growth in Teaching. Educational Researcher. Cilt:15, Sayı: 2. (4-14).**From this literature we can divide teacher knowledge into**three categories: content knowledge, pedagogical content knowledge and curricular knowledge.**………………………………………………………..………introduction**………………………………………………………..………introduction • The successful completion of the process of students’ achieving the content knowledge is linked directly to the quality of a teacher’s knowledge of teaching the content.**………………………………….…………………….………introduction**………………………………….…………………….………introduction As can be observed in the descriptive chart above, one of the most essential elements within the category of pedagogical content knowledge is familiarity with the knowledge base of one’s students. This knowledge entails making subjects that are routinely taught in mathematics easy for a student to comprehend as the teacher gets to know the student (Baki & Baki, 2010).**Knowledge of**student…………………………………..……….……… • As can be observed in the dialogs below, one of the teacher canditades launched the lesson without questioning whether the topics that would set the groundwork of the subject matter he would teach are known by the students: • T : -- You have studied the notions of compound fractions and equivalent fractions in past lessons. Today, we will find the value of the whole based on a given fraction. Kids, how many calories would be in a whole apple of which one fifth is 25 calories? • (T got an apple in his hand to be able to explain the problem and directed the following questions to the class): • T :-- Is this apple one whole piece? • Class:--Yes….. • T :Now I divid the apple into five. What does one piece represent? • Class:-- One fifth. • T :-- If the one fifth is 25, I multiple 25 by 5 to find the five fifth. • (To explain this, T drew the following model onto the blackboard): T:-- Where is 5? Class:-- In the denominator. T:-- 5x25 = 125. As you can see, I multiplied 25 by the denominator.**……………………………………………………………………..……….………findings**……………………………………………………………………..……….………findings • Above extract, it cannot be affirmed that the class fully understood why the given is multiplied by the denominator. • If Teacher had initiated the lesson with the problems answered in the previous years, this would have given him an idea about what the students already knew. • Because the students could not relate the newly learnt information to their prior knowledge, there were uncertainties in the classroom. • It was clear that the teacher had trouble pulling the class together and organizing the lesson.**……………………………………………………………………..……….………findings**……………………………………………………………………..……….………findings • Another teacher canditade prepared a lesson outline toward teaching the objective of “classifying angles based on their measures.” • She tried to motivate the students by reminding to them the content taught the previous day. • T :-- When measuring angles, as you remember, we use a protractor. Here is our protractor. Take your protractors out and look at it. What do you see on it? • Class:-- There are numbers up to 180. • T :-- You see a line right across the 90 degree mark. That is the center of the protractor. • T drew an angle onto the board and demonstrated to the class, using the protractor in her hand, how an angle should be measured. • In this classs, it was observed that the students did not understand why the line the teacher referred to as the center of the protractor was moved to the vertex of the angle. • As is evident here, Teacher, like the other teachers, did not pay enough attention in arranging the lesson to what prior knowledge the students had.**Knowledge of**student……………………………………………..……….……… • The majority of teachersrecognize that they have deficiencies in figuring out what students’ prior knowledge about a topic should be and that they cannot guess before teaching the lesson what the students knew about the subject. • The difficulty that teachers have is how to discern whether the actually students have the knowledge that they are assumed to have prior to the teaching of a topic and how this previous knowledge can be linked to the new knowledge.**Knowledge of**student………………………………………………………………..… • When data concerning the knowledge of the teachers about connecting the students’ past and present information is scrutinized, it is obvious that teachers have difficulty in referring to previous material in the course of teaching a subject and in relating the new topic to the students’ prior knowledge. • It is clear that most of the teachers usually act in an extremely unprofessional manner in such circumstances in the process of connecting the new topic to the students’ prior knowledge. • Teacher, who prepared and implemented a lesson plan intended for the acquisition of the objective of “finding the missing factor in a multiplication operation that has a product with a maximum of four digits” in the 5th grade mathematics curriculum, verified the students’ knowledge of multiplication at the start of the lesson.**Knowledge of**student………………………………………………………………. He asked the students to find the missing factor in the multiplication operation he had written onto the board: • Teacher stated that he would demonstrate a different strategy and added that they would find the number at the ones place by diving 645 by 215 and the number at the tens place by dividing 430 by 215. • It was observed that the students could not understand why they divided 645 by 215. • The teacher appeared to have presumed that the students would be able to find the omitted piece in the multiplication operation with the help of such a strategy as he believed that the students knew the connection between the division and multiplication operations. • The pre-service teacher stated that he had chosen the example in the teacher guide book and he had not foreseen that the students would have experienced such a difficulty in understanding this strategy.**Knowledge of**student………………………………………………..……….…… • Teacher was more successful than the other teachers in taking into consideration the students’ prior knowledge during the initial phase of the lesson and in conducting the lesson by relating it to the students’ prior knowledge. • Teacher prepared a lesson plan aimed at obtaining the objective of “being able to divide numbers with a maximum of four digits by three-digit numbers.” • Teacher reminded the students about the operation of division through concrete modeling during the introduction stage to the lesson. • At the beginning of the lesson, he had the students perform divisions with and without remainders with some hazelnuts he had brought to the classroom. He demonstrated the difference between divisions with and without remainders, and then reiterated the terminology of division.**Knowledge of**student…………………………………………..……….……… • However, as the lesson progressed, it was evident that the teacher had difficulty establishing a link between the previous knowledge the students should have and the new topic. As a getaway, he started to explain the operation of division on the board like it was presented step-by-step in the guide book; he assumed that the students knew division in this fashion and thus, went on with his explanations. • What Teacher should have done in this situation was to establish a connection with the students’ prior knowledge through breaking the problem down into smaller steps and finding each digit in the answer one at a time, starting with the ones place. • These limitations, as was the case in other teacher candidates’ classrooms, caused problems for the organization of their lesson as well.**………………………………………………………………..……….………conclusions**………………………………………………………………..……….………conclusions • Although the teachers were generally aware of the obligation to take into consideration the students’ prior knowledge in the process of teaching and learning, the related literature indicates that the teachers ignored the students’ prior knowledge due to the traditional belief that knowledge can transferred to the students directly by the teacher.**………………………………………………………………..……….………conclusions**………………………………………………………………..……….………conclusions • Teachers declare that they experience difficulties during the lessons, but they are unable to identify what leads to this. • The teachers justify the difficulties they experience in most lessons with statements such as these students are not our permanent students; we cannot know what they do or do not know, and this is a challenge for us. • The problem here is that teachers are not familiar enough with what the students should already know during the introductory stage to the topic, and that they are poor in determining the related topics before the teaching of a mathematical notion, topic or operation and in terms of the skill of connecting the topics.**………………………………………………………..……….………conclusions**………………………………………………………..……….………conclusions • As revealed here, the school environment provides teacher with opportunities for further learning. However, because the teacher themselves are still in the process of learning. • In this way, the teacher candidates will be aware of their weaknesses and will become better at dealing with and eliminating these difficulties. • This indicates the necessity of student’s knowledge as sub-component of knowledge of teaching.