MATHEMATICAL KNOWLEDGE FOR TEACHINGMATHEMATICAL KNOWLEDGE FOR TEACHING

Adnan BakiAdnan Baki

Karadeniz Technical UniversityKaradeniz Technical University

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Studies on teacher knowledge that Shulman and his Studies on teacher knowledge that Shulman and his associates conducted decades ago still prove to be current associates conducted decades ago still prove to be current (Shulman, 1986; Grossman, 1988; Magnusson et al. 1999; (Shulman, 1986; Grossman, 1988; Magnusson et al. 1999; Ann et al., 2004; Ball, 2008). Ann et al., 2004; Ball, 2008).

Ann, S., Kulm, G. ve Wu, Z.(2004). The Pedagogical content knowledge of Middle School Ann, S., Kulm, G. ve Wu, Z.(2004). The Pedagogical content knowledge of Middle School Mathematics Teachers in China and the US. Mathematics Teachers in China and the US. Journal of Mathematics Teacher EducationJournal of Mathematics Teacher Education 7, 145-172. 7, 145-172.

Ball, D.L.,Thames, M. H. ve Phelps,G.(2008). Content Knowledge for Teaching:What Makes It Ball, D.L.,Thames, M. H. ve Phelps,G.(2008). Content Knowledge for Teaching:What Makes It Special? Special? Journal of Teacher EducationJournal of Teacher Education Cilt:59, Sayı: 5. (389-407) Cilt:59, Sayı: 5. (389-407)

Grossman, P, L.(1988). A study of Contrast: Sources of Pedagogical Content Knowledge for Grossman, P, L.(1988). A study of Contrast: Sources of Pedagogical Content Knowledge for Secondary English. Unpublished doctoral dissertation, Stanford University.Secondary English. Unpublished doctoral dissertation, Stanford University.

Magnusson,S. (1991) The relationship between teachers’ content and pedagogical content Magnusson,S. (1991) The relationship between teachers’ content and pedagogical content knowledge and students’ content knowledge of heat energy and temperature. Unpublished Doctoral knowledge and students’ content knowledge of heat energy and temperature. Unpublished Doctoral Dissurtation. The University of Maryland. Dissurtation. The University of Maryland.

Shulman, L.S. (1986). Those who understand: Knowledge Growth in Teaching. Shulman, L.S. (1986). Those who understand: Knowledge Growth in Teaching. Educational Educational ResearcherResearcher. Cilt:15, Sayı: 2. (4-14).. Cilt:15, Sayı: 2. (4-14).

…………………………………………………………………….………introduction.………introduction

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From this literature we canFrom this literature we can divide teacher knowledge into divide teacher knowledge into three categories: three categories: content content knowledge, pedagogical content knowledge, pedagogical content knowledge and curricular knowledge and curricular knowledge. knowledge.

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………………………………………………………………………………………………………………..………introduction..………introduction

The successful completion of the The successful completion of the process of students’ achieving process of students’ achieving the content knowledge is linked the content knowledge is linked directly to the quality of a directly to the quality of a teacher’s knowledge of teaching teacher’s knowledge of teaching the contentthe content..

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…………………………………………………………………….…………………….………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).

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Knowledge of student…………………………………..……….………Knowledge of student…………………………………..……….………

As can be observed in the dialogs below, As can be observed in the dialogs below, one of the teacher canditades one of the teacher canditades launched the launched the lesson without questioning whether the topics that would set the groundwork of the lesson without questioning whether the topics that would set the groundwork of the subject matter he would teach are known by the students:subject matter he would teach are known by the students:

T T : : ---- You have studied the notions of compound fractions and equivalent fractions in 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. 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?Kids, how many calories would be in a whole apple of which one fifth is 25 calories?

((TT got an apple in his hand to be able to explain the problem and directed the got an apple in his hand to be able to explain the problem and directed the following questions to the classfollowing questions to the class))::

TT : : ---- Is this apple one whole piece? Is this apple one whole piece? ClassClass:: ----YesYes….…... T T :: Now Now I divid the apple into five. What does one piece represent? I divid the apple into five. What does one piece represent? ClassClass:: -- -- One fifth.One fifth. T T :: ---- If the one fifth is 25, I multiple 25 by 5 to find the five fifth. If the one fifth is 25, I multiple 25 by 5 to find the five fifth. ((To explain this, To explain this, TT drew the following model onto the drew the following model onto the blackblackboardboard))::

TT : -- Where is 5?Class: -- In the denominator.TT : -- 5x25 = 125. As you can see, I multiplied 25 by the denominator.

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…………………………………………………………………………………………………………………………………………..……….………findings..……….………findings

Above extract, iAbove extract, it cannot be affirmed that the class fully t cannot be affirmed that the class fully understood why the given is multiplied by the denominator. understood why the given is multiplied by the denominator.

If If TeacherTeacher had initiated the lesson with the problems answered in had initiated the lesson with the problems answered in the previous years, this would have given the previous years, this would have given himhim an idea about what an idea about what the students already knew. the students already knew.

Because the students could not relate the newly learnt information Because the students could not relate the newly learnt information to their prior knowledge, there were uncertainties in the to their prior knowledge, there were uncertainties in the classroom. classroom.

It was clear that the teacher had trouble pulling the class together It was clear that the teacher had trouble pulling the class together and organizing the lesson.and organizing the lesson.

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…………………………………………………………………………………………………………………………………………..……….………findings..……….………findings

AnotherAnother teacher canditade teacher canditade prepared a prepared a lesson outline lesson outline toward teaching the toward teaching the objective of “classifying angles based on their measures.”objective of “classifying angles based on their measures.”

ShShe tried to motivate the students by reminding to them the content taught e tried to motivate the students by reminding to them the content taught the previous day. the previous day.

T T :: ---- When measuring angles, When measuring angles, as you remember, as you remember, we use a protractor. Here is we use a protractor. Here is our protractor. Take your protractors out and look at it. What do you see on it?our protractor. Take your protractors out and look at it. What do you see on it?

ClassClass:: ---- There are numbers up to 180. There are numbers up to 180. T T :: ---- You see a line right across the 90 degree mark. That is the center of the You see a line right across the 90 degree mark. That is the center of the

protractor.protractor.

TT drew an angle onto the board and demonstrated to the class, using the drew an angle onto the board and demonstrated to the class, using the protractor in hprotractor in herer hand, how an angle should be measured. hand, how an angle should be measured.

In this classs, it In this classs, it was observed that the students did not understand why the 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 line the teacher referred to as the center of the protractor was moved to the vertex of the angle. vertex of the angle.

As is evident here, As is evident here, TeacherTeacher, like the other teacher, like the other teacherss, did not pay enough , did not pay enough attention in arranging the attention in arranging the lessonlesson to what prior knowledge the students had. to what prior knowledge the students had.

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Knowledge of student……………………………………………..……….………Knowledge of student……………………………………………..……….………

The majority of teacherThe majority of teacherss recognize recognize that they have that they have deficiencies in figuring out what students’ prior deficiencies in figuring out what students’ prior knowledge about a topic should be and that they knowledge about a topic should be and that they cannot guess before teaching the lesson what the cannot guess before teaching the lesson what the students knew about the subject. students knew about the subject.

The difficulty that teacherThe difficulty that teacherss have is how to discern have is how to discern whether the actually students have the whether the actually students have the knowledge that they are assumed to have prior to knowledge that they are assumed to have prior to the teaching of a topic and how this previous the teaching of a topic and how this previous knowledge can be linked to the new knowledge.knowledge can be linked to the new knowledge.

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Knowledge of student………………………………………………………………..…Knowledge of student………………………………………………………………..…

When data concerning the knowledge of the teacherWhen data concerning the knowledge of the teacherss about about connecting the students’ past and present information is connecting the students’ past and present information is scrutinized, it is obvious that teacherscrutinized, it is obvious that teacherss have difficulty in have difficulty in referring to previous material in the course of teaching a referring to previous material in the course of teaching a subject and in relating the new topic to the students’ prior subject and in relating the new topic to the students’ prior knowledge. knowledge.

IIt t is clearis clear that that mostmost of the teacher of the teachers usuallys usually act in an act in an extremely unprofessional manner in such circumstancesextremely unprofessional manner in such circumstances in in the process of connecting the new topic to the students’ the process of connecting the new topic to the students’ prior knowledgeprior knowledge..

TeacherTeacher, who prepared and implemented a lesson plan , who prepared and implemented a lesson plan intended for the acquisition of the objective of intended for the acquisition of the objective of “finding the “finding the missing factor in a multiplication operation that has a missing factor in a multiplication operation that has a product with a maximum of four digits”product with a maximum of four digits” in the 5th grade in the 5th grade mathematics curriculmathematics curriculumum, verified the students’ knowledge , verified the students’ knowledge of multiplication at the start of the lesson. of multiplication at the start of the lesson.

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Knowledge of student……………………………………………………………….Knowledge of student……………………………………………………………….

HHe asked the students to find the missing factor in the multiplication operation he had written onto the e asked the students to find the missing factor in the multiplication operation he had written onto the boardboard::

n TeacherTeacher stated that he would demonstrate a different strategy and added that they 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 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.tens place by dividing 430 by 215.

n It was observed that the students could not understand why they divided 645 by 215. It was observed that the students could not understand why they divided 645 by 215.

n The teacher appeared to have presumed that the students would be able to find the 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 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 believed that the students knew the connection between the division and multiplication operations. multiplication operations.

n The pre-service teacher stated that he had chosen the example in the teacher guide 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 book and he had not foreseen that the students would have experienced such a difficulty in understanding this strategy. difficulty in understanding this strategy.

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Knowledge of student………………………………………………..……….……Knowledge of student………………………………………………..……….……

Teacher Teacher was more successful than the other teacher was more successful than the other teacherss in in taking into consideration the students’ prior knowledge taking into consideration the students’ prior knowledge during the initial phase of the lesson and in conducting the during the initial phase of the lesson and in conducting the lesson by relating it to the students’ prior knowledge. lesson by relating it to the students’ prior knowledge.

TeacherTeacher prepared a lesson plan aimed at obtaining the prepared a lesson plan aimed at obtaining the objectiveobjective of “being able to divide numbers with a maximum of “being able to divide numbers with a maximum of four digits by three-digit numbers.” of four digits by three-digit numbers.”

Teacher Teacher reminded the students about the operation of reminded the students about the operation of division through concrete modeling during the introduction division through concrete modeling during the introduction stage to the lesson. stage to the lesson.

AtAt the beginning of the lesson, he had the students perform the beginning of the lesson, he had the students perform divisions with and without remainders with some hazelnuts divisions with and without remainders with some hazelnuts he had brought to the classroom. He demonstrated the he had brought to the classroom. He demonstrated the difference between divisions with and without remainders, difference between divisions with and without remainders, and then reiterated the terminology of division. and then reiterated the terminology of division.

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Knowledge of student…………………………………………..……….………Knowledge of student…………………………………………..……….………

However, as the lesson progressed, it was evident that the However, as the lesson progressed, it was evident that the teacher had difficulty establishing a link between the teacher had difficulty establishing a link between the previousprevious knowledge knowledge the students should have and the new the students should have and the new topic. topic. As a getaway, he started to explain the operation of As a getaway, he started to explain the operation of division on the board like it was presented step-by-step in division on the board like it was presented step-by-step in the guide book; he assumed that the students knew division the guide book; he assumed that the students knew division in this fashion and thus, went on with his explanations. in this fashion and thus, went on with his explanations.

What What TeacherTeacher should have done in this situation was to should have done in this situation was to establish a connection with the students’ prior knowledge establish a connection with the students’ prior knowledge through breaking the problem down into smaller steps and through breaking the problem down into smaller steps and finding each digit in the answer one at a time, starting with finding each digit in the answer one at a time, starting with the ones place. the ones place.

These limitations, as was the case in other teacher These limitations, as was the case in other teacher candidates’ classrooms, caused problems for the candidates’ classrooms, caused problems for the organization of organization of theirtheir lesson as well. lesson as well.

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………………………………………………………………………………………………………………………………..……….………conclusions..……….………conclusions

Although the teacherAlthough the teacherss were generally were generally aware of the obligation to take into aware of the obligation to take into consideration the students’ prior consideration the students’ prior knowledge in the process of teaching knowledge in the process of teaching and learningand learning, t, thhe related literaturee related literature indicates that the teacherindicates that the teacherss ignored ignored the students’ prior knowledge due to the students’ prior knowledge due to the traditional belief that knowledge the traditional belief that knowledge can transferred to the students can transferred to the students directly by the teacher. directly by the teacher.

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TeacherTeacherss declare that they experience difficulties during declare that they experience difficulties during the lessons, but they are unable to identify what leads to the lessons, but they are unable to identify what leads to this. this.

The teacherThe teacherss justify the difficulties they experience in most justify the difficulties they experience in most lessons with statements such as lessons with statements such as these students are not our these students are not our permanent students; we cannot know what they do or do permanent students; we cannot know what they do or do not know, and this is a challenge for usnot know, and this is a challenge for us. .

TThe problem here is that teacherhe problem here is that teacherss are not familiar enough are not familiar enough with what the students should already know during the with what the students should already know during the introductory stage to the topic, and that they are poor in introductory stage to the topic, and that they are poor in determining the related topics before the teaching of a determining the related topics before the teaching of a mathematical notion, topic or operation and in terms of the mathematical notion, topic or operation and in terms of the skill of connecting the topics.skill of connecting the topics.

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………………………………………………………………………………………………………………..……….………conclusions..……….………conclusions

As revealed here, the school environment provides As revealed here, the school environment provides teacher with opportunities for further learning. teacher with opportunities for further learning. However, because the teacher themselves are still However, because the teacher themselves are still in the process of learning. in the process of learning.

In this way, the teacher candidates will be aware of In this way, the teacher candidates will be aware of their weaknesses and will become better at dealing their weaknesses and will become better at dealing with and eliminating these difficulties.with and eliminating these difficulties.

This indicates the necessity of student’s knowledge This indicates the necessity of student’s knowledge as sub-component of knowledge of teaching.as sub-component of knowledge of teaching.