6.4 Teacher mathematical knowledge and engagement with learners’ errors in
6.4.1 Teacher knowledge and using procedural knowledge to address errors
Using the framework that underpins this study, recognition of learners’ errors lies under the domain of teachers’ common content knowledge. During data analysis codes were assigned to teachers’ statements and comments that addressed learners’ errors during instructional activities. For example, during Zafira’s lesson on algebra, after she has finished working on some examples with the learners she gave a class activity for them to try their hands at. After learners had worked for about 15 minutes, she called on one of the learners (learner H) to present her answer on the chalk board (see Figure 5.1, Chapter Five). It was evident in the learner's work that some procedural errors had been made.
Figure 6.1 Sample of learner H’s work in class activity
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From the analysis of data, it was clear that Zafira became aware of and recognised some errors in the learner’s work, and she was able to explain the steps needed to get to the correct answer.
Hence she used the learner’s work to engage the whole class on the errors identified, and she asked those who had their work wrong to make the necessary corrections. The excerpt that follows shows how the teacher engaged the class in dealing with the errors in the learner’s work during the class observation. The teacher started by drawing learners’ attention to the fact that, since the question involves fractions, the first thing to do is to find the least common dominator (LCD) of the equation, which she found to be 12. The conversation with her learners continued as follows:
Zafira: How many times does three [the first divisor] go into 12 which is the LCD?
Learner A: [Shouted from the back of the class] Four, ma’am.
Learner K: [Raised hand) Three, ma’am.
Zafira: Okay, how many of you agree on 4 and how many agree on 3?
Learners: [12 learners raised their hand for 3, 10 raised their hand for 4, and 4 of them did not raise their hand at all]
Zafira: 4 is the correct number, class. As you can see from the board, learner H wrote 2 instead of 4 so we have to change the 2 to 4 then multiply 4 by (2x). In the next term let’s see if learner H is correct.
Learners: (All shouted) Okay, ma’am.
Zafira: 2 goes into 12 how many times?
Learner M: 6 times.
Zafira: Any different answer, class?
Learner T: [Raised hand]
Zafira: Yes, learner T?
Learner T Ma’am I agree with learner M, the answer is 6.
Zafira: Okay, let’s see if learner H is correct. What did learner H write here?
Learners: [Whole class response] 3
Zafira: Is learner H supposed to write 3?
Learners: [Whole class response] No ma’am.
Zafira: Okay, change 3 to 6 and multiply 6 by (x-2).
Learners: Yes ma’am.
Zafira: 1 goes into 12 how many times?
Learners: [Whole class response] 12 times ma’am.
Zafira: Okay, then multiply 12 by 2.
Learners: [Whole class response] Yes ma’am.
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Zafira: How many times will 6 go into 12?
Learners: [Whole class response] 2 times ma’am.
Zafira: Okay, multiply 2 by (3x-2).
From the excerpt, it can be seen that Zafira’s engagement with the error was driven by the ability to solve the problem and the procedures that needed to be followed. While teacher Zafira recited these procedures, from the observation of the lesson it was evident that a fraction of the learners (for example, 12 out of 26, which represent 46% including learner H), could not make sense of the algebraic language used, e.g. how many times 3 can go into 12. While the teacher Anele continued with the explanation of the procedures with the rest of the class, it was evident that the language was not comprehended by all of the learners, for example, learner H who was puzzled by the answer.
The above scenario confirmed the data from the interview that the participant understanding of strategies to eradicate errors is anchored on doing corrections on the board rather than engaging with learner thinking. This was evident as teacher Anele, after noting the errors made by learner H, did not ask the learner to explain his/her thinking to unearth the underlying misconception, but opted to engage the whole class and started to reteach concepts to the whole class. While reteaching the concept to the whole class is important, learner H still could not work out the sum with the class, meaning that the method used has not assisted the learner to understand the concept better. Many scholars, such as Maharaj (2014), and Ndlovu and Brijlall (2015, 2016), have emphasised the importance of allowing learners to engage with what they write. While teacher Zafira used learner H’s errors to re-emphasise the concepts, she did not engage with learners’ thinking when addressing the errors made. The focus was more on re-emphasising the procedures to be followed.
Anele wrote an algebraic equation, 2x + 4 =10, on the chalk board and used a probing question to identify errors made by learners during his teaching, as follows:
Anele: Learners, what do we do when a question like this given and you are to find unknown x?
Learner G: [From the right corner] Sir you first group the like term.
Anele: Is she correct?
Learners: [Whole class response] Yes sir.
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Anele: Okay learner M, come to the board and group the like term for us.
Learner M: [Goes to the board to present answer] 2x= -10+4 Anele: Is he correct?
Learners: [Some learners raised their hands]
Anele: Yes, learner D?
Learner D: No sir.
Anele: Okay come to the board and correct it for us to see.
Learner D: [Goes to the board] 2x = 10 – 4
Anele: Okay you are correct. Explain your answer to the class.
Learner D: [Laughed loudly] Sir, I transpose the 4 which is positive to the other side of the equation to where the 10 is and the 4 turned to negative. That is why I got 2x
= 10 – 4.
Anele: Okay learners, do you understand?
Learner T: No sir.
Anele: Where are your difficulties here?
Learner T: Sir, everything on the board.
Learners: [Whole class laughs]
Anele: Okay learners, listen, remember we did integers in term one when transposing a number or when a number is crossing the equal sign and is positive, it changes to negative and positive turns to negative. From the equation here you can see that 4 is positive, therefore it will turn to negative. Therefore 10 -4 is correct on the other side of the equation. If the 4 was negative (-4) then it was going to 10 +4 on the other side of the equation.
Anele: Do you understand learner T?
Learners: Yes sir.
Anele: Yes, learner T, do you understand now?
Learner T: Yes sir.
Anele: Yes learner, what do we do next?
Learner P: Sir, you subtract 4 from 10 and get 6. So, the equation becomes 2x = 6. Then you divide 2x by 2 and 6 by 2.
Anele: Why do you divide by 2?
Learner P: Sir because we want x and 2 is Infront of x.
Anele: You are correct, sit down. Clap for him.
Learner: [All the learners clapped]
Anele: But learners, listen carefully, always we divide by the coefficient not necessarily 2 and yes in this equation the coefficient is 2. So always look for
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the coefficient and divide by both sides of the equation. Okay all of you divide 2 which is the coefficient and tell me the answer for x.
Learner J: Sir, the answer for x is 3.
Learners: [Whole class in loud voice] Yes sir x=3.
Anele: Yes, you correct x is 3. Clap for yourself.
From the above presentation, there was an engagement between Anele and his learners; it was observed that the teacher was probing the learners to get errors they made, in order to engage with them. The researcher observed that learner M could not transport positive numbers correctly. Therefore, the teacher used probing questions for learner D to pick up the error learner M had made and the teacher asked learner D to correct the error, which he did correctly.
Again, the teacher realised that even though learner D had done it correctly, some learners, including learner T, still didn’t understand, so he took time to explain.
It can be concluded that from the observation of Anele’s class that some of the learners had a problem with transposing and integers; the teacher was able to use probing questions to deal with learners' errors, as used by Brodie (2014). Drawing from the above findings about the framework of the study, the procedural understanding (Sapire et al., 2014) that the participants had in dealing with learners' errors is because of their content knowledge (Ball et al., 2008) The researcher observed from the lessons that teachers had good content knowledge of the topic they were teaching.
6.4.2 Knowledge teachers need to teach mathematics and understanding of