The third lesson (lesson 7)

The lesson of ‘Problem-Solving Investigation: Choosing a Strategy’ was introduced to Grade 4. Its aim was to check the students’ understanding of solving some written problems by choosing an appropriate strategy from several strategies, such as using a drawing, tables and rational interpretation.

7.4.1 The Lesson Description:

At the beginning of the lesson, Fahad asked the students to close their books and he asked them about the previous few lessons. They revised and summarised what they had learned about locating a point on a number line, mixed numbers and improper fractions and how to convert them. It took them nearly five minutes to do this and then Fahad drew two circles divided into 6 parts. One of them was fully shaded and the other had 4 shaded parts and he asked them to write the model as a mixed number (14₆) and an improper number (10₆). They said the answers out loud and he wrote them on the board.

Next, in the following two minutes, Fahad divided the class into two teams A and B (12 and 13 students) to start a competition. Then he asked them to listen to his instructions (e.g. do not answer without a permission) and told them that the aim of this lesson was to encourage them to think effectively.

Fahad asked team A to answer the first question, which took 10 minutes, ‘Lila can make a dish of food in 20 minutes. If she wants to make 8 dishes on the condition that she finishes at 8:00 pm, when should she start?’. He asked the students for their attention and tried to break the question down into small parts. While doing that, a student (Ibrahim) was disturbing Fahad by giving the answer ‘She starts at five twenty’. Fahad asked him to keep quiet until he had finished explaining the question. Ibrahim suggested the answer of going backwards 20 minutes 8 times from 8 pm and Fahad’s preference was to calculate the total time (160 minutes = 2h and 40 minutes) then go forward from 5:20. There was disagreement among the team about whether the final answer was 5:40 or 5:20 until they settled on 5.40. Fahad agreed that the answer was 5:40 as he thought the total time was 2 hours and twenty minutes. Then, with Ibrahim insisting on his answer of 5.20, Fahad was convinced to change his answer to 5:20. He then returned to the other student who answered 5:40 and tried to prove the incorrectness of the solution by counting forward 2h and 40 minutes from 5:40 until he

reached 8:20 not 8pm.With the class, Fahad spent the next six minutes solving the question described in the incident below.

Incident 1:

The question was ‘Saleh and three of his friends rent a boat for three hours. If the cost of renting the boat is 80 Riyals per hour, how much will each of them pay?. It was solved as the dialogue shows:

1 Fahad Can anyone from team B read the second question? 2 Student 1 [Read the first sentence and was stopped by Fahad]

3 Fahad So, if you are Saleh and you have three friends, how many persons are you?

4 Students Four

5 Fahad If they pay 80 riyals how much must each one pay for one hour? 6 Mohammed Ten

7 Students No, wrong

8 Fahad

It is ok, he is right, there is nothing wrong, [moves toward Mohammed’s table] Why did you say ten? Answer Mohammed, you are four people and you paid 80 Riyals. If you said you paid ten what is the total?

9 Mohammed Four

10 Fahad Forty you mean?. You paid forty so is it correct or not? 11 Mohammed No

12 Fahad So how much does each one pay? [Points at another student] 13 Student 2 Twenty

14 Fahad

You are right, ok Mohammed each one pays twenty Riyals so all pay eighty riyals right? [Returns to the board] They used the boat for three hours, We want to find the rent of the boat for three hours. How much?

15 Student 3 250

16 Fahad Why did you say 250, I want to know where you got 250 from? 17 Student 3 Because double eighty plus eighty… [Interrupted by Fahad] 18 Fahad Not double, we add it. [Writes 80+]

19 Student 3 Eighty plus eighty plus eighty equals two hundred and fifty

20 Fahad [Writes 80+80+80] excellent, you are right. Count with me eighty plus eighty equals what?

21 Student 3 160

22 Fahad And another eighty? 23 Student 3 240

24 Fahad You have added ten more, so how much does everyone pay? 25 Student 4 It is 220 not 240

26 Fahad [Calculates mentally] 160 plus 80 [Writes 220] 27 Students [Giving different answers] it is 240 , it is 220 28 Fahad Calm down, 80 plus 80 is how much?

29 Students 160

30 Fahad And 80?

31 Student 5 170, 180, 190, 200, 210, 220, 230, 240, it is 240.

32 Fahad [Asking the class to calm down angrily] so 240. How much does every one pay?

33 Yasser 80

34 Fahad Yasser said 80, is that right? [Writes 80+80+80+80 and shouts to some boys to be quiet].

35 Student 6 60 36 Fahad

Excellent, Yasser’s solution [Deletes 80+80+80+80 and writes 60+60+60+60] equals 120+120, so it is 240 which is right. Well done Yasser.

He then challenged the class to solve the following question within five minutes: ‘Nada bought cloths of different sizes: large (cost: 20 R), medium (cost: 15R) and small (cost: 13R). If she paid 65 riyals, how many cloths of each size did she buy?’ The rest of the lesson was spent solving other questions from the text book.

7.4.2 Applying the Knowledge Quartet (KQ) and IRF

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

Foundation

In this lesson, Fahad was not able to identify some of the students’ errors (IE). However, although they were tiny mistakes related to simple mathematical operations like addition, they made a big difference to the solving process. For instance in the first exercise, Fahad was confused and could not spot the error of converting the total time of 160 minutes into two hours and twenty minutes. Also, in the incident [26], he could not identify the error of calculating the rent of three hours (80×3) as 220 until it was noticed by some students. This inability to identify errors could be related to stress Fahad was experiencing, or his insecure subject matter knowledge. He mentioned in the interview that Ibrahim’s distractions made him confused over the correctness of the students’ answers which led him to accept any answer. He said: ‘I got confused and lost control when I was faced with two different answers from two students who appeared confident. This confusion led me to accept some answers without evaluating them’. Also, the student contribution in the incident [25] made him unsure about the answer of 240. It is clear that Fahad was not confident about his answers as he changed them immediately when a student suggested another answer. This might show some

limitation in his subject matter knowledge (OSK) as if he had secure foundation knowledge of a simple operation such as 160 + 80 is 240, he would not have been confused by 220. He also counted twenty times eight using his fingers to find out the cooking time which was 20×8=160 which may give an indication of the difficulty Fahad faced with mental calculation.

The solving problem lesson in the textbook is designed as linear steps starting with understanding the question request, understanding the problem, planning, solving and checking the answer. Fahad was not clear about this process as he started to ask questions without showing a clear plan to solve the problem. He showed a lack of knowledge (OSK) regarding the solving and checking phases in the first exercise. When he calculated the total cooking time (2 hours and 40 minutes) Ibrahim suggested the answer of starting at 5:20. Fahad then tended to use the checking phase before the solving one. Ibrahim provided the solving step as he suggested that they subtract 2h and 40m from 8:00 to reach 5:20 then Fahad asked him to explain it more. It seemed that Fahad tried to solve the question by using trial and error as he tried starting at 5:20 and later he tried 5:40 as well.

Transformation & Connection

The lesson required flexibility in solving the question and that was partly missing in Fahad’s teaching as the lesson went the same way for all the solved questions (TD). However, Fahad mentioned that the purpose of the lesson was ‘to encourage them to think effectively’ and he introduced the lesson using some short questions to help the students get started. The main role of the students was to answer Fahad’s questions while he was guiding them towards the solution, instead of thinking on their own as the lesson’s aim stated. Even though he divided the class into two teams, they did not work as groups; instead they worked individually responding to Fahad’s instructions. In addition, he showed some flexibility in changing the direction of the solution in order to extend the benefit gained from the exercises, especially the second one in the incident. He could have finished the second exercise from the utterance [13] by multiplying 20 by 3 to get 60 but he said in the interview:

In the beginning I was going to multiply 20 by 3 to get 60 …then I thought if I take the other way which was finding the total rent 80 times 3 then divide by 4 which I think contained more mathematical skills.

This shows that Fahad tried his best to support the students’ learning by combining many operations together in order to challenge the students to calculate the answers correctly. Moreover, he used an improper example in the introductory part of the lesson as not all the exercises contained mixed numbers or improper fractions and the connection between the previous lessons and this one was missing (MCC).

Contingency

Fahad accepted correct answers in different ways (RCI). He accepted some of them with hesitancy and some were accepted straightaway. When he was not sure about the correctness of the answer or felt nervous he did not accept it straightaway as he would think about it or ask for an explanation before accepting it. For example, when Ibrahim said that 160 minutes equals 2 hours and forty minutes (student response) Fahad did not accept it straightaway as he calculated it out loud by subtracting 60 then 60 then 40 (comment move). Later, he changed it to 2 hours and twenty minutes which may show some doubt about the answer. Also, he asked Ibrahim to further explain his solving strategy (press move) when Ibrahim suggested a correct answer that Fahad was probably not expecting. Ibrahim explained ‘If you subtract two hours from eight o’clock it will be six and then if you subtract forty minutes you will get five twenty’. Fahad then accepted this answer. His hesitancy in accepting this answer straightaway might have been related to the nervousness that Fahad felt when Ibrahim kept disturbing him at the beginning of the lesson, or the lack of ability to calculate mentally.

Correct answers were sometimes accepted straightaway by praising, asking for justification or repeating also, Fahad sometimes rejected them. Fahad liked to say words such as ‘right’, ‘excellent’ when the students answered correctly in order to encourage them to participate more in the lesson. For instance, he praised Yasser in the incident [36] when he provided a correct answer (accept move). In addition, Fahad mentioned in the interview that he asked for justification to check the student’s way of producing the answer. For example, when Ibrahim said ‘Five twenty’ (student response), Fahad responded ‘right, why five twenty?’ (accept move followed by press move ). He wanted to make Ibrahim’s thinking public to the class. Also, he repeated the student’s answers as a confirmation of their correctness. When Ibrahim explained what he had done as ‘from eight to six is two hours’ (student response), Fahad repeated what Ibrahim said (accept move/Re-voicing move). Moreover, Fahad rejected a correct answer, whether he

knew it was correct or not, when it appeared at the beginning of the solving process in order to avoid ending the solution early. For example, when Ibrahim gave the answer of ‘five twenty’ (student response) directly after reading the question, Fahad asked him to keep quiet until he had finished explaining the question (directive move). It seemed that Fahad believed in the importance of using the exercises as opportunities to diagnose the students’ understanding of the subject. Fahad justified that in the interview:

I did not take the correct answer and finish the exercise as we were still at the beginning of the lesson. I wanted to see the other students’ contributions to check if there was any action to be taken.

Incorrect answers were dealt with by rejection, acceptance, supporting the answerer and asking for justification. Fahad rejected incorrect answers verbally (e.g. wait) and physically (shaking head). For example, he said ‘Wait, we do not want random answers…’ (reject move followed by comment move) for the student who answered ‘Five forty’ (student response). It seemed that Fahad was sure that the answer was wrong as he described it as a random answer which naturally required a rejection. In addition, he accepted some incorrect answers with or without praising the answerer. For instance, when there was a debate between the two students who answered ‘Five twenty’ and ‘Five forty’ (student response), Fahad was confused and accepted the incorrect answer and ignored the correct one (accept move). He mentioned that he was affected by the confidence of the students who gave him the two answers. That showed some gaps in Fahad’s subject matter knowledge and a lack of confidence which did not help him to avoid being confused by the confidence of others.

Fahad showed some sympathy for some students who gave incorrect answers. Fahad supported some students by asking the class to pay attention to the answerer and saying some encouraging words such as ‘He can do it’. In addition, Fahad tried to apply the cognitive conflict strategy (Powell, 2006) to convince students, with either a good or bad mathematical background, who made errors. For example, in utterances [5]-[11] of the incident, Fahad was faced with an incorrect answer ‘Ten’ (student response) from Mohammed. Fahad then replied by supporting Mohammed and encouraging him to continue participating in the exchange. He said ‘It is ok, he is right, (evaluate move) there is nothing wrong’ (comment move). Then he asked for justification by saying ‘Why did you say ten?’ (press move). When he did not receive an answer from

Mohammed, Fahad changed his strategy and shifted to apply the cognitive conflict strategy by asking Mohammed some questions to guide him to discover his error. Fahad tried to make Mohammed aware of his mistake by showing him that if each pays 10 Riyals the total payment will be forty Riyals not eighty. The same situation occurred with student 3 in utterances [15]-[24] where Fahad accepted the answer then asked for justification and applied the cognitive conflict to convince the student that he was wrong. He justified that by saying:

I do not like to say to the student that he is wrong because he will sit down and keep silent. Instead of that I ask him to answer even if the answer is wrong, then I try to correct him by getting him to self-correct. By that I mean that he discovers that his answer is incorrect by himself. With this boy I tried to convince him that ten Riyals was not enough so that he would understand that his answer was wrong

Fahad in this lesson took the opportunity (UO) of Mohammed’s errors to increase the chance of discussing flawed solutions in front of the class which may have showed some element of confidence and secure PCK.

In conclusion, in this lesson Fahad was nervous due to some students’ interruptions which may have affected his ability to identify errors. Also, he was not flexible enough to accept the students’ solving strategy as he wanted to be the only one to give the solving instructions. These two elements may show some gaps in his subject matter knowledge and his teaching method. He dealt with most of the students’ correct answers by accepting them either immediately or after some thinking. Also, incorrect ones were dealt with by rejecting them or accepting a few of them when he was not sure whether they were correct or not. On some occasions, he showed some willingness to help the students when they faced some problems while solving questions. He did that by guiding them with questions and applying the cognitive conflict strategy to convince the students about their mistakes.