LMT is about anticipating learner difficulties and suggesting remediating

 What is LMT?

For Kenneth, learner errors and misconceptions are part of the discourse of engaging with LMT. He said that focusing on learner errors and misconceptions involves identifying learner error and suggesting likely strategies for remediation (LEM-IR). He says this is achieved by giving his student-teachers school mathematical questions and asks them to:

“Look at this question. What knowledge, skills, and concepts will a child need in order to do this? What strategies would you adopt in order to enable students to do this? What difficulties do you anticipate from children? And arising from that

what strategies will you arm yourself with in view of the difficulties you will anticipate?” (Extract 6.4; see extract 6.8 for more emphasis)

In this case, identifying learner errors is talked about in terms of anticipating learner difficulties. Following Peng & Luo (2009) and Jacobs et al. (2010) strategies of carrying out error analysis, Kenneth’s focus on error analysis does not include discussion of sources of errors, of why the errors are occurring. Kenneth does not raise the sources of the errors as

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something that is problematic and therefore needs to be thought about. Of course, overlooking the sources of errors could also affect the quality of remediating strategies suggested. In working with school questions student-teachers are given opportunity to think through the required knowledge in terms of skills and concepts to solve the mathematical problem, and the likely strategies for teaching for it. Only then can they think of identifying the errors learners would make and the likely strategies for remediating the errors. This suggests that student-teachers not knowing and understanding the mathematical demands of the school questions can result in difficulties in thinking of learner errors and misconceptions and maybe that is why it is emphasized.

 Where are learner errors and misconceptions (LEM-IR) learned in the courses? Error analysis which for Kenneth is about identifying the error and suggesting remediating strategies (LEM-IR) is focused on in an effort to familiarize student-teachers with the school mathematics curriculum. Kenneth states that they “… make effort to make them understand the school curriculum that they will be implementing, the mathematics syllabuses at school. In fact a component of our curriculum is to look at school questions” (Extract 6.4). This is “… dealt with in the context of what we have got on our course outline, school mathematics, from the context of specific topics that they will meet at school” (Extract 6.8). Therefore, Kenneth focuses on learner errors and misconceptions (LEM-IR) by looking at “school mathematics”, a topic in the course Mathematics Education II (MSE 332) with specific emphasis on school questions. Similar to mathematical reasoning, learner errors and misconceptions with specific focus on identifying the error and suggesting possible remediating strategies (LEM-IR) is also weakly classified in that it is not given specific focus. It is talked about when focus is on a selection of school mathematics questions.

 Positionings of student-teachers, learners, and the curriculum in mathematics education

In a situation where student-teachers experience difficulties in working out a mathematical problem also suggests that absences (A) realized in them are likely to be the anticipated absences (A) in learners as indicated by Kenneth:

“… ironically, those problems the students will make themselves as trainee teachers

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difficulties because they are making the problems, they are making the errors themselves.” (Extract 6.7)

Furthermore, for Kenneth, a sense of absences (A) in learners in terms of working with mathematics questions is a result of absences (A) in teachers:

“Even when you are examining Grade 12 examination, you could see how poorly they

attempted. It reflects on how poorly the teaching was. So we pick up such topics and

make an effort to prepare our students for the tackling of these topics.” (Extract 6.4)

It is thus possible to infer that Kenneth locates the source of errors and misconceptions in teaching. When learners are making errors, teachers should reflect on their teaching to establish what could have gone wrong and work with that. A contrary positioning of student- teachers is also observed in that Kenneth sees presences (P) in student-teachers’ suggested remediating strategies during lecture sessions in that they provide reasonable answers and a sense of absences (A) during the practice of teaching in that he does not see in them the aspect of working with errors and misconceptions. This is indicated in the excerpt below:

“Yes, in class they do attempt to come up with reasonable suggestions. …But when I saw them in the field on teaching practice it didn’t come through that they had done

sufficient homework. But they’ll give you good suggestions in a classroom context

when you are in the university discussing in the classroom. But what I saw in the field when there was school teaching practice, sometimes I was wondering, ‘Are these the students I trained?’ (laughs) … It didn’t quite come through. So I’m reflecting where did I go wrong?” (Extract 6.5)

The issue of reflecting on ones teaching is deep rooted in Kenneth in that even for his own practice if student-teachers fail to provide reasonable responses, then there could be something wrong with the way he has been putting his ideas across during lecturing. Moreover, when he is focusing on errors and misconceptions during lectures, he expects the manifestation of this skill during practice of teaching such as school teaching practice or peer teaching; and unfortunately it is missing. As discussed in Chapter 3 and following Peressini et al. (2004), this points to the issue of putting into practice through peer teaching or teaching practice what has been learned theoretically, and teaching practice is considered as one of the sites of teacher learning.

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 Strategies of how Kenneth “teaches” for learner errors and misconceptions (LEM-IR)

Similar to the two strategies identified earlier on how Kenneth “teaches” for developing in learners a sense of mathematical reasoning, he suggests the practice of teaching and the issue of lesson planning to include anticipated learner errors and suggested remediating strategies. Added to these two ways is the aspect of challenging student-teachers with more school mathematics. I discuss what is entailed in each of the two strategies suggested.

The first strategy is about the practice of teaching such as peer teaching and school teaching practice. As already pointed out in Extract 6.5, Kenneth expects to see the manifestation of what student-teachers learn about identifying errors and suggested remediation strategies in the practice of teaching. The practice is that “For now we do it theoretically then ask them to practice it” (Extract 6.5). It is during the practice of teaching that the student-teachers should demonstrate understanding of what is being taught during lectures. This means that identifying and remediating (LEM-IR) for Kenneth is still a skill that is learned in practice but there is no mention of how he would deal with it in practice other than making observations on whether what student-teachers learned theoretically was put into practice. This shows that for Kenneth there is a direct transfer of knowledge from theory to practice. However, he contemplates introducing student-teachers to peer teaching from the onset so that theory is developed out of practice. Kenneth’s option is illustrated in the excerpt below:

“And I’ve been discussing with my colleagues in the Mathematics Education section. We haven’t agreed on it but I’ve suggested that my preference would be to have peer

teaching right from the word go. Even when they have not learnt anything on

making a lesson plan. …What I think, build all the theory from the context of

whatever mistakes they may be making in their teaching. Make that as a platform for the theory that you want to do. If you want to say, ‘A lesson is planned this way’,

it should be from the context of what you have seen them doing. ‘Where did you think this, it doesn’t seem that the teacher was clear on the objective of the lesson.’ Then you go into your discussions of lesson objectives. ‘What do we need in a lesson?’… But if the theory and the practice can go hand in glove, maybe some of these can be nice.” (Extract 6.5, see also extract 6.7 for emphasis)

For the strategy of introducing peer teaching from the onset, what it would mean for focusing on errors and misconceptions for Kenneth is that he would observe the mistakes the student- teachers are making with working with errors. He would then discuss with them the importance of including in the planning anticipated learner difficulties and strategies of how they would remediate the errors. I would argue that maybe his preference of theory

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developing out of practice could address the issue of transfer earlier discussed. Moreover, his preference, as discussed in Section 3.2.4, could be one of the ways of attending to the “dilemma of experience” and is described as one of the challenges teacher-educators are faced with when working with pre-service teachers’ experiences (Doerr, 2004, p. 269). From Kenneth’s preference, there is a sense of presences (P) in student-teachers in terms of how he thinks learning how to teach should be done. This means that he sees student-teachers as coming with some understanding of what teaching entails and then the teacher-educator builds up from what they know.

Directly linked to the strategy of practice of teaching is the strategy of planning for teaching to include specific ways of remediating errors. He suggests that this could be made possible by, for example, working with learners on the identified difficulty with focus on specific strategies to solutions to mathematical problems that could address the error.

“And in your illustrations lead the children to this common difficulty that they face

and encourage a system of working which will minimize that error. Simplification of

algebraic fractions where you have maybe two terms on the numerator in one fraction, you have two terms on the numerator in one fraction and two terms as the numerator on one fraction. And you are required to subtract. A common difficulty is how to deal with the subtraction and the brackets so that the signs are not messed up. So I was citing that as an example as this is one difficulty you can anticipate from children. Maybe make sure that the first stage where brackets are maintained is

always there before they proceeded to simplify because it is a difficulty that you anticipate. So you should build in your planning some mechanism of how can I alleviate this difficulty?” (Extract 6.4)

This confirms Kenneth’s earlier point that the source of error is located in teaching: the nature of the suggested remediating strategy in the example he gives is teacher oriented. If the teacher does not emphasize the issue of maintaining the brackets in solving the mathematical problem, then the learners will always experience the identified error. However, in Hill et al. (2008) terms, and as discussed in Section 3.2.3, Kenneth brings to student- teachers’ attention the importance of KCS and how this enables the incorporation in the planning of lessons anticipated learner errors. Moreover, as argued by Even and Tirosh (2002), teacher awareness of learner errors has implications for the quality of instruction. Kenneth is also aware that errors are persistent in Smith et al.’s (1993) terms and as a result they cannot be completely eradicated by suggesting that the teacher should “encourage a system of working which will minimize that error”. In anticipating learner difficulty, there is a sense of absences (A) in

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learners in terms of subtracting algebraic fractions since they have a tendency of confusing the signs.

One of the suggested remediating strategies for errors is to include in the planning for teaching specific rather than general techniques to the task. For the identified error and suggested remediating strategy discussed above, Kenneth distinguishes between a strategy that is specific to the error and the one that is general and states that:

“First maintain your brackets until you have cleared the denominators. Then open the brackets. Then simplify.’ So you are looking at a strategy that is specific to that

task. But to say I would give them more homework, of course it’s useful, but it’s not

just useful for that topic, it’s useful generally.” (Extract 6.6)

Preference for Kenneth is on the specific strategy because it addresses the error directly while the general strategy does not. This implies that for the general strategy, more practice is required after all when learning mathematical concepts. Following Borasi (1987), Arcavi (1995) and Doerr & Wood (2004), and as discussed in Section 3.3.3, Kenneth is aware that general strategies such as giving learners more practice exercises would not necessarily remediate the errors. However, the explanation Kenneth gives as an example of a specific strategy to the error suggests what Borasi (op cit) and Arcavi (op cit) would describe as explaining the same process over and again. What is interesting about the general strategy is that Kenneth is contradicting himself by proposing challenging his student-teachers with more school mathematics as a way of making them experience more of what it means to work with learner errors as indicated in the excerpt below:

“… give them more challenges on specific questions that they would meet at

secondary school level and to keep challenging them that what skills, what concepts

do children need, what difficulties do you think pupils are likely to have …” (Extract 6.7)

There is also a contradiction here in Kenneth’s talk in that in identifying learner errors and likely remediating strategies he sees absences (A) in learners and yet at some point he sees presences (P) in learners in terms of their conceptions and that they are just limited. The teachers’ role is to work with what learners have to expand their limited framework. This is because he says misconceptions mean that “…the concepts of pupils are never wrong but they’re just operating in a limited framework and our job being expanding that framework”

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(Extract 6.9). This constructivist view is in resonance with how he views misconceptions “through the avenue of the psychology of learning mathematics” (Extract 6.9). ‘Psychology of learning mathematics’ is a topic in one of the mathematics education courses, Mathematics Education II (MSE 332), which focuses on theories of learning, more especially constructivism. The emphasis is on:

“… paying sufficient attention to what the implications are for teaching and what we know about how children learn. From what we know about the nature of

mathematics it’s a hierarchical nature and so on, how it translates to children’s learning and the children’s thought processes, and how we can minimize on the

misconceptions (if one can call them misconceptions).” (Extract 6.9)

6.2.2.1 Summary and conclusion of the analysis

What is interesting about Kenneth’s strategy of error analysis is that the focus is only on identifying the error and suggesting a remediating strategy (LEM-IR). As argued, Kenneth’s general view of the source of error is that it is an indication of inadequacies in teaching. The specific remediating strategies for an error attests to this general view of the source of error and his concern when he fails to see student-teachers model during school experience or peer teaching what they have been taught during lectures. The way Kenneth focuses on learner errors is accompanied by positioning of a teacher or a learner who does not know (A). He says learners’ poor performance in school mathematics is a sign of poor teaching or that student-teachers make similar errors learners would make in manipulating school mathematics. A contradictory positioning of student-teachers is observed in that during lectures, Kenneth sees student-teachers who know (P) in terms of suggesting remediating strategies that are specific to the errors. Kenneth also suggests developing theory out of practice as his first preference, an indication of seeing student-teachers as coming with some understanding about identifying error and suggesting remediating strategies, and so a view of the student-teachers as not simply ‘empty vessels’ (P).

Some indication of the internal contradictions to Kenneth’s expressed transmission view of knowledge and learning is also observed. He contradicts himself by putting up a constructivist view of errors and misconceptions in that they are persistent and can never be completely eradicated. Moreover, the view of learners is that they are never wrong but that their thinking is limited conceptually and it is the role of the teacher to expand their conceptual understanding. This is an indication that learners come with some knowledge to the teaching and learning situation from a constructivist perspective, hence seeing presences

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in learners (P). What then does this mean for student-teachers in terms of receiving different messages about LMT?

As previously, working with learner errors and misconceptions for Kenneth is a skill that is accomplished practically. The skill lies in the practice of teaching such as peer teaching or school teaching practice and is supported by principles for generating a lesson plan, working with school mathematical problems, and the psychology of learning mathematics where LMT is discussed. This further reinforces Kenneth’s transmission view of teaching and learning in that if student-teachers have been taught, they should be able to perform. This analysis of Kenneth’s discussion of error as important in LMT confirms that LMT with respect to errors and misconceptions is also not assigned topic status in the mathematics education courses. LMT remains weakly classified with no identity and voice of its own. As Kenneth describes, he addresses issues of learner errors when dealing with selected topics in the mathematics education courses such as ‘school mathematics’, ‘psychology of learning mathematics’, and ‘reflections on peer teaching and school teaching practice’.

In Table 13, I provide a synopsis of Kenneth’s selection and privileging of the discourse of engaging with LMT in terms of mathematical reasoning, and learner errors and misconceptions.

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Table 13: Synopsis of Kenneth’s talk of LMT with focus on mathematical reasoning, and learner errors

“What” is LMT? “How” of LMT, that is positionings in terms of presences and absences and the “teaching” of it in terms of what

student-teachers are supposed to know and be able to do. “Where” in the courses/topics LMT is focused on Elaboration of the “what” Positioning of

teachers/student teachers Positioning of learners Positioning of the curriculum (in school or TE)

“teaching” of LMT

Mathematical reasoning is relational and not instrumental

Relational thinking includes arguing, evidencing, and systematic thinking (MR-R- AES).

Instrumental thinking includes showing learners solutions (MR-I-SS) or finding answers to mathematical problems (MR-I-FA).

Absences – teachers and student-teachers do not develop mathematical reasoning in learners. A Absences in learners as a result of absences in teachers. A Presences in learners in answers they give, questions they ask, and activities lined up for them during planning. P

Absences in TE curriculum - no time to integrate theory and practice in peer teaching and school teaching. A

Through peer teaching and school teaching practice

Through lesson planning with focus on intensions of the lesson, what