The lesson description:

Abdullah showed the lesson’s pages from the textbook on the board using the document camera. He started the lesson from the ‘Get Ready’ section in the student’s book which contains worked examples, and Abdullah explained how they were solved by always asking questions. The students gave answers then he put them on the board. He read out the definition of a polygon with the class, which was written in the student book as, ‘A polygon is a plane closed shape with straight sides which doesn't cross over itself’. Below the definition, there were many shapes grouped under two headings: polygons and not polygons as shown below:

After reading the definition, he examined the polygon shapes. He pointed to the triangle and asked ‘why is it a polygon? Is it a plane? Is it closed? Does it have straight sides? And does it cross over itself?’ Then he repeated the same questions for all the other shapes. During these six minutes there were some questions and answers between Abdullah and his students. For example, a student asked ‘why is this shape not a polygon’, and another student started to answer him, but then Abdullah asked the second student to stop talking and he answered the question himself.

He then wrote the formula of the perimeter of a polygon on the board as ‘it is the sum of its sides’. The class read the formula and Abdullah moved to the first example: ‘Find the perimeters of the polygon’. They spent nearly three minutes doing so.

For the next six minutes, Abdullah moved to the next part in the book which was an activity filling in a table of many squares with their sides and perimeters as described

6m 6m 9m 9m 10.5m

in the following incident. They then solved an example from the book of a picture of a square and requested finding the perimeter of a square with sides of 2 units.

Incident 1

This incident is chosen as it shows a variety of responses to students’ contributions and evidence of Abdullah’s knowledge. After finding the perimeter of a polygon, Abdullah asked the students to do the activity: ‘fill in the table below and describe the relationship between the perimeter of the square and its side length’.

Square 1 2 3 4 Side length 1 Perimeter 4 Figure ‎5.3 This dialogue happened during the activity:

1 Abdullah (A)

[Pointing to the square 1] Here the side length is how much? 2 Students Two

3 A [silent] 4 Students One

5 A Good. What is the perimeter of the first square? 6 Students Four

7 A How we find it four?

8 Student1 Because it is the sum of the side lengths

9 A That would be right for the polygon but this is a square not a polygon. _{[ He writes on the board ‘the perimeter of a square =’]} 10 Student1 I know the answer, it is because one side is one, so the others are one

too

11 A All of these squares are different [ pointing to the table]

12 Student 1 I mean the first square has four sides of one, so it is one, one, one and one.

13

A Do not add the sides because that was correct for the polygon. [He repeats the question] How do we find the perimeter equal to 4 for the first square?

14 Student 3 [Stands and goes to the board and touches the sides of the first square and says] we add one, one, one and one.

15

A No. [He then reads the formula of the perimeter of a square which is written below the table and writes ‘the perimeter of a square = 4 x’] so, what is the perimeter of a square?

17 A What does x refer to?

18 Students [The answers were] one, two, litre, millilitre, rectangle, square, angles, polygon and the length.

19

A [Listening to the answers and moving from one student to another until he reached the correct answer] It is the length, so the first square has a side of one and its perimeter should be?

20 Students Four

21 A Good, we multiply the side of one by four and get four, what about the second one?

22 Students Two times four equals eight. 23 Student4 Teacher, what is the perimeter?

24 A Right, you do the same thing to the two squares three and four and you will find that the perimeter will be?

25 Students Twelve and sixteen.

26 A Right, memorise the formula to use it elsewhere.

For the next four minutes, they moved to the next part which was about rectangles. They read the definition of the perimeter of a rectangle and Abdullah wrote it on the board as ‘2W + 2L’ under the formula of a square. There was some discussion about the definition of W and L in the formula. Then the class moved to the example of a picture of a rectangle notebook with a width of 18 cm and length of 22cm and they were asked to find the perimeter of the notebook.

Abdullah asked the students to solve the exercise of the ‘Make Sure’ section individually in the last 25 minutes. He moved around between the three tables and corrected their work until the bell rang.

5.3.2 Applying the Knowledge Quartet (KQ) and IRF

The following analysis takes into account the class observation and Abdullah’s comments in the interview.

Foundation

Abdullah’s adherence to the textbook (ATB) (students’ book) influenced his conduct of the lesson. I considered Abdullah’s use of the student book as ‘adhering’ in the light of Nicol and Crespo’s (2006) proposal for some features of using textbooks by teachers. They suggest that the characteristics of teachers who adhere to textbooks are that they are using it as their main source without any modification and they accept it as the authority of what to teach and how. He stuck to the book in every part of the lesson and followed the order of introducing the lesson’s concepts, examples and ideas. He tried to solve everything from the lesson’s pages and asked the students to complete the

exercises for homework. Abdullah’s use of the textbook affects, to some extent, some of his teaching as I illustrate in the other dimensions of the Knowledge quartet (KQ) in the following paragraphs.

Teachers use textbooks differently for many reasons. Their reasons can be categorised into two main kinds: external and internal. By the former, I mean the reasons that relate to the classroom atmosphere and the latter relates to the character of the teacher. Research shows that teachers tend to use textbooks closely as a result of a limitation of time, availability of resources, the influence from colleagues and cooperating teachers (Nicol and Crespo, 2006) and district and school policies (Sosniak and Stodolsky, 1993). The Nicol and Crespo (2006) study found that teachers who did not have time to look for information usually adhered to textbook. Also, they found that when the teacher did not find good alternative sources they stuck to textbook. In addition, they state that the cooperating teacher’s using of textbook has a great influence on the participants as the trainees did the same use of their cooperating teachers by adhering or elaborating or create their teaching materials. Moreover, Sosniak and Stodolsky (1993) found that government and school policies, among others, influence the teacher’s use of the textbook. They found that teachers used selective and variable use of textbook to achieve the educational district aims such as sticking rigidly to mathematics textbook exercises.

The internal reasons of different textbook use, according to research, could be the teacher’s belief about the nature of the subject and how it can be taught (Chavez– Lopez, 2003; Johansson, 2006; Nicol and Crespo, 2006; Sosniak and Stodolsky, 1993). In addition, the teacher’s knowledge, skills and experience shape the use of textbooks (Chavez–Lopez, 2003; Sosniak and Stodolsky, 1993). Moreover, the extent of teacher’s interest in the subject influences the teacher’s use of textbooks (Remillard, 2005).

Abdullah adheres to the textbook as a response to the Saudi educational system which requires all the teachers follow the same book for every stage and every subject. He could modify some parts of the lessons if he wanted to, but he did not do so. As he said in the interview ‘I was told by my supervisor [the cooperating teacher] to try to cover as much as I can from the book so I do not have time to bring any example outside the book’. He emphasised that the limitation of time, the authority of the

sources, the education system and the cooperating teacher, played an external role in his adherence to the textbook.

Regarding the internal reasons for adhering to the textbook, it seems that Abdullah’s belief and knowledge, and probably lack of experience, influenced his use of the textbook. He believes that mathematics is a challenging subject that required the mastering of procedures and roles and that the teacher should focus on the students’ performance. He said:

I love maths because I feel it challenges me all the time…I am keen to teach the students how to do things rather than confusing them with many details of the concepts… I sometimes prefer to give them the steps to solve the exercise before asking them to solve the exercise by themselves as this avoids them getting stuck.

These beliefs may have influenced his use of the textbook in a way that concentrated on solving as many of the exercises as possible in order to complete the majority of the Saudi mathematical textbooks content. I found that Abdullah always asked the students to use their textbooks and solved the exercises with them one-by-one, putting more emphasis on the procedure (process) rather than learning the concepts.

In this lesson, he seemed to believe that the book is the source of authority and that all of its content must be used. As he said

I found a written formula for the perimeter of a square in the student book, so I thought … they must apply the formula. I wanted to encourage the students to use the formula instead of adding the side lengths … Because it has its formula, I stuck to it.

Abdullah’s knowledge probably affected his use of the textbook as well. According to Nicol and Crespo (2006), teachers’ understanding of the content influences their use of textbooks. For example, a teacher who is familiar with the topics does not face difficulties in covering the content. Abdullah did not show some indication of overt subject knowledge (OSK) in different parts of the lessons. For instance, he always kept checking his book to see the answer and also occasionally he skipped some exercises without justification. This is indicative of the subject knowledge difficulties which led him to avoid some difficult exercises. In this lesson, Abdullah seemed not to be fully aware of the way of finding the perimeter of a square as he asked

the students to find it just by using the formula. Also, in the incident he thought that a polygon cannot be a square [9]. He made a small change in the order of the textbook to avoid his misconception of the way needed to find the perimeter of a square by adding its side length.

Transformation

Abdullah’s demonstration (TD) was influenced by his knowledge, beliefs and adherence to the text book. The concept of a ‘polygon’ was introduced by reading the definition then showing pictures of many polygons instead of, for example, showing pictures of many polygons and asking about the similarity and differences between and then suggesting the definition (Rowland et al., 2009). He introduced the concept of the perimeter of the square in a way that does not emphasize the importance of understanding it or connecting it to the perimeter definition. His explanation of the concept of the perimeter of the square was done by requiring the students to use the formula which was written in the book without explaining it well. He could have introduced it empirically by starting from the formula of the perimeter of a polygon then using that to find the formula of the perimeter of a square (Steele, 2013). He did not allow the students to use their prior knowledge about finding the perimeter of a polygon in a new situation to generate the new concept or formula. He justified his action by saying ‘I found a written formula for the perimeter of a square in the student book, so I thought that to find the perimeter of a square the student cannot add the sides; they must apply the formula’. His focus on procedural knowledge is evidence of his belief of mathematics as a set of procedures and how it ought to be taught, or as a result of his lack of knowledge. He said ‘I am keen to teach the students how to do things rather than confusing them with many details of the concept’.

Connection

Abdullah introduced the lesson as it was written in the textbook. He started with the polygon concept and perimeter, and then moved to finding the perimeter of the square and the rectangle without paying attention to the connection between these procedures (MCP). There was no link between these formulae and, as I said earlier, he did not use the procedure of finding the perimeter of the polygon as a base to build up the formula of the square. For instance, in the incident, when the students tried to use what they had just learned about the procedure of finding the perimeter of the polygon

in order to find the perimeter of the square, [8, 10, 12 and 14], Abdullah did not allow them to make this link as he probably believed that there was no connection between them.

Contingency

Abdullah responded to the students’ ideas differently (RCI). He responded to the students’ incorrect answers in the incident by ignoring them and keeping silent ([2] and [3]) then looking for the correct one (reject move followed by nomination); as he said ‘They gave me a wrong answer and I was looking for a correct one’. In addition, he kept moving from one student to another until he found the correct answer ([18] and [19]) (reject move followed by nomination). He reasoned that by saying ‘I was looking for the correct answer so I kept moving between the students until I found the right person’. Moreover, Abdullah thought some of the correct answers were wrong because he thought they should have stuck to the formulas in the book so he changed the answers himself. For instance, in the section [8] – [15], the students tried to add the side lengths (student response) as they had done with the polygon, which is a correct thing to do, but Abdullah changed their answer by saying ‘no’ (reject move) and introducing the formula after that (clue move). He justified his action by saying ‘I found a written formula…so I thought that…they must apply [it]’.

However, at times he responded to the correct answers by saying words such as ‘good’ and ‘right’ ([5], [21], [24] and [26]) (accept move). He said:

I should correct the wrong answers and praise the correct answers, for example by asking the students to clap their hands for the one who answers correctly to encourage the students to become more active in the lesson.

Also he asked for justification, such as ‘How do we find it four?’ [7] (press move). He justified that by saying ‘I want to make sure that they answered based on an understanding, not guessing’. Moreover, as a result of his adherence to the textbook, he rejected some correct answers [8] – [15] (reject move followed by comment) for his own and changed them to another form, even though they both led to the same answer.

Finally, he was faced with many questions from the students; he answered some of them and ignored others. For instance, in [23] he ignored a fundamental question

about the concept of the perimeter which is what the lesson was about (teacher’s second turn was missing). In addition, at the beginning of the lesson a student asked the teacher a question but his peer tried to answer so Abdullah asked him to stop and answered the student himself (teacher second turn was directive move followed by comment). He justified that by saying:

I am keen to stress the importance of the understanding for the students… I prefer to answer the student’s question by myself … because I believe that I can deliver the idea to the questioner better than the other student.

This tendency of Abdullah to emphasise the importance of understanding might show some special beliefs about certain aspects of learning.

In conclusion, Abdullah in this lesson tried to cover as much as he could of the lesson content, however he paid less attention to connecting the lesson’s concept and procedures. He was supposed to use the perimeter of a polygon as a starting point to work out the formula of the perimeter of the square, but he did not do so because of his knowledge, beliefs and adherence to the student book. This also affected his response to the students’ contributions as well. In general, he dealt with students’ incorrect answers by rejecting them and with correct answers by accepting them. The main exchanges in the two types of actions were reject move followed by nomination and accept move, followed by comment or press move respectively.