Teachers’ Application of Error Analysis in the context of

Learner Interviews

School of Education, University of the Witwatersrand Funded by the Gauteng Department of Education

17 January 2012

Updated June 2013

Teachers’ Application of Error Analysis in the context of Learner Interviews

School of Education, University of the Witwatersrand Funded by the Gauteng

Department of Education

Contents page

Contents page.................................................5Tables..............................................................7Acknowledgements.........................................8Introduction..................................................10Report Plan...................................................12Section One: Activity Process.......................131.1 Process..............................................................14

1.1.1 Time line........................................................................141.1.2 The Process....................................................................14

Section Two: Methodology............................192.1 Aim of the evaluation........................................202.2 Interviews evaluated.........................................202.3 Evaluation Instruments.................................21

2.3.1 The coding template......................................................212.3.2 The five activities describe the teachers’ participation in the interview, and their respective codes...............................222.3.3 The activity that describes learner participation:.........26

2.4 Training of coders and coding process.............272.5 Validity and reliability check............................272.6 Data analysis.....................................................28Section Three: Quantitative analysis............293.1 Overall picture..................................................30

3.1.1 Time spent on interview activities.................................303.1.2 Teacher participation....................................................343.1.3 Learner participation.....................................................36

3.2 Presence of the coded teacher interview activities..................................................................38

3.2.1 Procedural Activity........................................................383.2.2 Conceptual Activity........................................................403.2.3 Awareness of error Activity............................................423.2.4 Diagnostic reasoning Activity........................................443.2.5 Use of everyday Activity.................................................463.2.6 Multiple explanations....................................................46

3.3 Quantitative Findings.......................................48Section Four: Qualitative analysis................534.1 Sample..............................................................544.2 Analysis of learner interviews...........................55

4.2.1 Grade 4 learner interview..............................................554.2.2. Grade 5 learner interview.............................................634.2.3 Grade 7 learner interview..............................................694.2.4. Grade 9 learner interview.............................................77

4.3 Qualitative Findings..........................................85Section 5: Conclusion...................................91Recommendations for professional development and for further research................................................92

Recommendations for professional development...................92Recommendations for further research.................................92

References....................................................93Appendices:...................................................95

Appendix 1..............................................................................96Appendix 2..............................................................................98Appendix 3..............................................................................99Appendix 4............................................................................101Appendix 5............................................................................105

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TablesTable 1: Setting own test and conducting learner interviews.......................15Table 2: Sample summary.............................................................................20Table 3: Percentage of time spent per activity across the interview sample set..................................................................................................................31Table 4: Percentage of time spent per activity across the interview sample set..................................................................................................................48Table 5: Percentage of time coded full over total interview time.................49Table 6: Percentage of time coded inaccurate over total interview time......49Table 7: Percentage of time coded full and partial over total interview time....................................................................................................................... 50Table 8: Activity criteria abbreviations.........................................................54

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AcknowledgementsThis report was prepared by:Professor Yael Shalem (Wits School of Education)Ingrid Sapire (Wits School of Education)Tessa Welch (Saide)Maryla Bialobrzeska (Saide)Liora Hellmann (Saide)

We would like to thank the following people for their role in the preparation of the report:Nico Molefe (Maths Education Consultant)Bronwen Wilson-Thompson (Maths Education Consultant)

The project team for DIPIP Phases 1 and 2 Project Director: Prof Yael ShalemProject Leader: Prof Karin BrodieProject coordinators: Ingrid Sapire, Nico Molefe, Lynne Manson The project team and evaluators would like to thank the Gauteng Department of Education and in particular Reena Rampersad and Prem Govender for their support of the project.

We would also like to thank all of the teachers, departmental subject facilitators, Wits academic staff and students and the project administrators (Karen Clohessy, Shalati Mabunda and team) who participated over the three years of the project.

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Introduction

Learner interviews was one of the two error-focused activities in DIPIP (Data Informed Practice Improvement Project) phases 1 and 2. The other error-focused activity was an error analysis of the evaluation data (ICAS 2006 and 2007) which included mainly multiple choice items and a few open items. Additional open items were selected from the tests that the teachers designed in their small groups in the last 6 months of the project. The results of the groups’ error analysis are reported in the Teachers’ Knowledge and Error Analysis report (Shalem & Sapire, 2012). During the final project activity (test setting and learner interviews) the teachers were given an opportunity to design an open questions test, administer it in class, and conduct an error analysis on the results. The purpose of the activity was to give the teachers experience in designing their own assessments, which they could then analyse in terms of learners’ errors and misconceptions. Groups were asked to set their tests on misconceptions associated with the equal sign or with problem solving and visuals (which related to the teaching activities done earlier in the project) (Shalem, Sapire, Welch, Bialobrzeska & Hellman, 2011). In order to deepen this exploration groups were asked to follow the error analysis with a learner interview, in which they could interact with one (or more) learner and work with the learner diagnostically to establish what the error is about. Instead of hypothesizing about the reasons for learners’ errors evident in a large scale systemic test, the teachers could explore with their own learners the reasons for their errors in the test.

The aim of the interview was to understand how teachers interact with learners about mathematical knowledge in the context of a conversation about errors. The idea of learner interview broadly follows what Young (2005) calls “the think-aloud method”. According to Young this method intends to capture human thoughts as data is collected. “The think-aloud method” can be done in two ways - while the participant is undertaking an activity or after its completion (p20). The central aim of “the think-aloud method” is to capture what the subject (the learner) is actually doing (ibid) and why.

There are two advantages of this method which are important for the DIPIP project. Unlike evaluation data whereby a teacher needs to figure out what the sample of learners might have been thinking when they wrote something down on paper, the “think-aloud method” enables the teacher to act as a researcher and through making statements, asking questions and explaining, to interrogate the learner’s way of thinking. In this way the teacher can capture the sequence of learner’s thinking and its depth, the specific errors of which he/she is not aware, their misrecognition of key conceptual links, difficulties they experience regarding visual or other problem solving skills a test question may contain, or their

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misunderstanding of the instruction in the question. In addition to the individual feedback time that a learner interview makes possible, the idea here is that through appropriate engagement with the teacher, the learner gets an opportunity to verbalise his/her thinking and the teacher gets an opportunity to understand the learner’s difficulty in relation to the acquisition of the mathematical content at hand. The main limitation of this method is that learners (or teachers) “who are less capable of thinking about their own thinking will be less capable of reporting on it” (p25). Another disadvantage (Sheinuk, 2010, p18) is “that not all thought is accessible at all times and issues of language and articulation can impede the mental processes of the participants from being accurately reported.”

Taking the above limitation into account, it is important to acknowledge the importance of these kinds of feedback situations in teaching, in particular, for ‘assessment for learning’ (Gipps, 1999 Black et al 2003; Gipps and Cumming 2004; Shepard 2000, Black and Wiliam, 2006). One of the most challenging aspects of teaching is the transmission of criteria. Researchers (Charlot, 2009) emphasise the importance of evaluative feedback for helping learners recognise what counts as a legitimate text and for helping them develop epistemic means to realise these texts in the future.1 Feedback practices work with learners’ responses to help them both understand why they are not yet meeting criteria and how they might begin to construct more appropriate responses. In this way, feedback can be used to promote what Vygotsky (1987) called a ‘Zone of Proximal Development’ by both recognising what the learner produces and articulating it in line with more powerful forms of knowledge (Shalem and Slonimsky, 2010).

1 Bernstein (2000) distinguishes between ‘recognition rules’ that refer to the Learners’ ability to classify legitimate meanings – that is, to know what meanings fall outside a theoretical model and what may/may not be put together – and ‘realisation rules’ that refer to the acquirer’s ability to produce/enact what counts a legitimate text (2000, 16 and 209).

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Report PlanSection one Activity and process: in this section we discuss the rationale for the learner interview activity, its aim, its context within the DIPIP project, and the instructions teachers received before the activity.Section twoMethodology: in this section we describe the process of the evaluation- the sample, the aspects in the activity we selected for the evaluation and the codes we developed for these, the coding instrument, the training of the coders, the coding process and lastly the validation process. Section threeQuantitative analysis: in this section we provide graphical presentation of the coded interview sample with findings presented. We address broad questions about the overall quality of the interview data. We present the quantitative data for the 13 teachers selected for the evaluation. Where appropriate we note the differences between the grouped grades – the 6 Grade 3-6 teaches and the 7 Grade 7-9 teachers. Section fourQualitative analysis: for this section we selected 4 interviews, two from each of the grouped grades. In the analysis of these interviews we intend to first demonstrate our coding so the reader understands how codes and categories were assigned to teachers’ utterances. Second we describe explicitly how the different activities in a learner interview (five of which are led by the teacher and one by the learner) coalesce and together create a successful or a weak interview. This will help us in deepening our central question for the evaluation of this activity which is: What does the idea of teachers’ interpreting learner performance diagnostically, mean in a context of a learners’ interview? We also consider other important questions relating to a diagnostic learner interview, such as, What do teachers, in fact, do when they interpret errors with the learner present? And what are the benefits for the learner.

Section fiveWe conclude with lessons to be learned from the project and recommendations both for professional development and research.

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Section One: Activity Process

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1.1 Process

1.1.1 Time line14 weeks: March to August 2010The learner interview activity was part of the final project activity which involved groups setting and marking tests for their classes, analysing the errors made by learners when they wrote the tests and selecting particular learners to interview, based on the errors they had made when writing the test. The preparation for the interviews, filming of the interviews conducted in schools and subsequent large group presentations of learner interviews took place over a period of 14 weeks between March and August 2010.

1.1.2 The ProcessIn their small groups, the teachers marked all of the written tests, looking out for interesting learner responses. Then through a discussion in the group, they selected three learners for a learner interview. The groups were requested to motivate the choice of questions for the interview (in relation to the learners’ work) by describing:

the mathematical concept the question is addressing the misconception evident in the learner’s work the way it is exemplified in the learner’s work, and what questions the learner could be asked to expose her/his thinking

more fully.

Before the interviews, the groups discussed the following: ways of creating a conducive atmosphere for the interview, such that

learners will feel safe enough to verbalise their thinking and mode of reasoning.

ways of avoiding leading questions watching one’s body language and allowing oneself to be puzzled by the

learner’s answer rather than being critical of it, and types of questions that can be used to prompt a learner to talk.

All the learner’s interviews were videotaped. The interview was divided into two parts. In the first part, the teacher and learner discussed the item, and the learner’s reasoning in addressing the question. In the second part, the teacher set a mathematical problem similar to the item already discussed.

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The last part of this process was a presentation by the small groups to the large group. Each group was asked to watch the recordings of all the learner interviews conducted in the group and select different episodes for different categories of episodes (for example, a time that a teacher was impressed by, surprised by, unsure about learner’s thinking during the interview, or when something planned/unplanned happened and how they handled it). Once the reflection was completed the interviewer from the group selected one episode from her/his own interview for presentation to the large group (an episode where it was difficult to understanding the learner’s thinking). In the presentation to the large group, the interviewer needed to justify the selection, explaining to the large group why it was difficult to understand learner’s thinking in the chosen episode.

Table 1: Setting own test and conducting learner interviews

Material Tasks Group type

Guideline on test setting

Setting a testGroups were asked to design a 6 item test (not more than 3 items to be

multiple choice).2

explain the rationale for the test (the concepts and procedures selected as a test focus, their relevance to the curriculum and the misconceptions anticipated)

specify marking criteria for the test.

11 small groups(2 Grade 31 Grade 42 Grade 51 Grade 61 Grade 72 Grade 82 Grade 9)

Readings on assessment

Readings on assessmentThree chapters from a book on assessment3 were given to all group members as a resource. The reading of these chapters was not structured and group members were asked to read them at home in order to familiarise themselves with the ideas and be able to apply them in their test setting activity.

At home

Verbal instructions to groups.

Prepare for presentation to large group.Groups were asked to prepare their tests and rationales for a

presentation to the large group at the presentations members of the groups were

make notes of comments and feedback that applied to their group.

Large groups

Verbal instructions to groups

Revision of test.On the basis of feedback the groups revised their tests. The groups were required to explain the changes they made to the test.

Small groups

Guideline on choosing a learner for an interview

Marking own test and selecting a learner for an interview.Teachers were asked to:

At school and in small groups

2 All groups designed tests with 6 questions. Several of them had questions that were broken up into sub-questions. Only one group included a multiple choice item (one).3 Linn, RL and Miller, MD (2005) Measurement and Assessment in Teaching (New Jersey, Pearson Prentice Hall), Chapters 6, 7. And 9

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Material Tasks Group type

Administer the test to one of their classes Mark all of the written tests, looking out for

interesting learner responses. Back in their groups, select three learners that

they identified for a learner interview (see below). Motivate the choice of question for the interview

(in relation to the learners’ work) by describing: the mathematical concept the question is

addressing the misconception evident in the learner’s work the way it is exemplified in the learner’s work what questions the learner could be asked to

expose her/his thinking more fully.Guidelines for learner interview

Preparation for learner interview.The aim of the interview was not to fish for the right answer but for the teacher to understand the learner’s reasoning. The groups were to consider the following: ways of creating a conducive atmosphere for the

interview ways of avoiding leading questions watching one’s body language and allowing

oneself to be puzzled by the learner’s answer rather than being critical of it

types of questions that can be used to prompt a learner to talk

Small groups

Interview Protocol Learners’ interviews were in two parts:1. The teacher and learner discuss the item, and

the learner’s reasoning in addressing the question.

2. The learner is asked to do a mathematical problem similar to the item already discussed.

The learners’ interviews were videoed.

Small groups and teachers’ classrooms.

Learner interview presentation guide

Reflections on learners’ interviews and preparation for presentations.Each group was asked to: watch the recordings of all the learner interviews

conducted in the group select different episodes for different categories

of episodes (eg a time that a teacher was impressed by, surprised by, unsure about learner’s thinking during the interview, or when something planned/unplanned happened and how they handled it )

In addition to this, each volunteer interviewer was asked to: to select one episode (from her/his own interview)

for presentation to the large group (an episode where it was difficult to understanding the learner’s thinking)

justify the selection (reasons for selection and why it was difficult to understand learner

Small groups and large groups

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Material Tasks Group type

thinking)

In this report we focus on the learners' interview. As in the error analysis of evaluation data (, we aim to answer the following questions: What does the idea of teachers’ interpreting learner performance

diagnostically, mean in a context of learners’ interview? What do teachers, in fact, do when they interpret errors with the learner

present? And what the benefits are for the learner. In what ways can what they do be mapped on the domains of teacher

knowledge?

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Section Two: Methodology

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2.1 Aim of the evaluation We aim to evaluate the quality of application of error analysis as evidenced in learner interviews carried out by the teachers. In this we aim to evaluate the quality of teachers reasoning about error when they are in a one-on-one conversation with a learner.

2.2 Interviews evaluated

For the purpose of analysis of teachers conducting learner interviews we selected a sample of 13 interviews from teachers who taught in Round 1, Round 2 and Round 3. The teachers whose interviews we selected corresponded to the teachers from the respective rounds whose classroom lessons were to be evaluated for the purposes of the report on the application of error analysis and diagnostic reasoning in classroom teaching.

Table 2: Sample summaryFull set of data to be considered SampleRound 1 teachers:Learner interviews conducted by all teachers participating in the learner interview activity.

(Five of the seven teacher groups that taught in Round 1 took part in the learner interviews.)

Five interviews – one each from Grades 3, 5, 7, 8 and 9

Round 2 teachers:Learner interviews conducted by all teachers participating in the learner interview activity.

(Six teacher groups that taught in Round 2 took part in the learner interviews.)

Six interviews – one each from Grades 3, 4, 5, 6, 7 and 9

Round 3 teachers:Learner interviews conducted by all teachers participating in the learner interview activity.

(Two of the three teacher groups (Grade 7, 8 and 9) that taught in Round 3 took part in the learner interviews.)

Two interviews – one each from Grades 8 and 9

The teachers all took part in the interview activity at the same time although their experience in the teaching activities took place over different times, according to the round in which they did their teaching. Not all teachers participated in the interview activity since two teachers were unable to arrange a convenient time for the interview within the time limits of the interviews.

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2.3 Evaluation Instruments In this section detail on the instrument used in the interview analysis is given. An overview of the coding template is followed by a more detailed explanation of each of the activities coded for, when using the instrument.

2.3.1 The coding templateSix evaluation criteria were drawn up by the DIPIP team based on the error analysis criteria (Shalem & Sapire, 2012) and in line with activities that describe the teacher’s and the learner’s participation in the interview (see Appendix 1).

A coding template was prepared for the coders so that coders could enter their codes for each minute of each interview in a similar manner. The coding template included column headers for each minute to be coded and row headers according to the criteria used for coding (see Appendix 2).

Coding Instrument overviewGeneral information on interviews Teacher name Grade Interview topic Summary – an overview of the interview content and general flow written

after the full interview was coded.

Criterion code rows to code teacher participation Procedural (Proc) Conceptual (Con) Awareness of error (Awa) Diagnostic (Diag) Use of everyday (ED)

Coding Categories4 Not Present Inaccurate Partial Full

Criterion code row to code learner participation Multiple explanations (Mult)

Coding categories

4 In the coding sheet, the categories were numbered Full (4), Partial(3), Inaccurate (2) Not present (1) 17

A simple count of the number of different explanations of errors given by learners during their interviews was used to code for multiple explanations. The following categories were coded: One explanation Two explanations Three explanations Four or more explanations

2.3.2 The five activities describe the teachers’ participation in the interview, and their respective codes Procedural activity – Interacting with learners on errors involves a

great deal of discussion of procedural activity in which statements, guiding instructions, questions and explanations are made by the teacher in order to clarify the procedure that is involved in the mathematical question. Procedural activity explanations need to be given with sufficient clarity and accuracy if the learners are to grasp the procedures and become competent in performing them. A Proc code was assigned everytime the teacher addressed the learner on how to do the mathematical working required by the question, including what steps to take and in what order. For example (Grade 8 interview):

Teacher: Ok so you’ve got a 1, a 2 and a 6; and a nine (points to 1st row); and a 4, a 5 and another 9 (points to 2nd row). And I want you to make as many different true sentences as you can using only these numbers. Now you can use different symbols, so you can use plus and minus and brackets and all that kind of stuff.Learner: Ok.Teacher: But if you can, just write down as many as you can think of, um, I don’t know, in about one minute, then let’s see what you get.Learner: Does it have to equal to 9 or can it be any of these numbers?Teacher: No, no it can be anything. You can use them in any way you like. Learner: Ok.

Four categories were selected to code the quality of the procedural activity conducted by the teachers during the interview - not present, inaccurate, partial and full. It was not expected from the teachers that they engage procedurally during every minute, and so a “time count” for the code of not present for procedural activity is not meaningful in the content of the minute-by-minute coding of the interviews. However, in this activity, for the overall interview, the category not present means that the teacher did not engage in procedural activity at all in the full course of the interview.

Conceptual activity – Mathematical procedures need to be unpacked (Shalem & Sapire, 2012) and linked to the concepts to which they relate in order for learners to understand the relations between mathematical constructs embedded in the procedure. Interacting with learners on errors involves a focused conceptual activity where by statements, questions and explanations made by the teacher conceptually open up (or

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fail to open up) the procedure that is involved in the mathematical question. The emphasis of this activity is on the teachers’ conceptual work in relation to the procedure required by the question. A code Con was assigned every time the teacher was seen to attempt to conceptually unpack the procedure or an aspect of it, in order to illuminate the background and process of the procedure. For example (Grade 3 interview):

Learner: I was trying to draw the seats.Teacher: Ah I think here you had the right idea didn’t you? (Learner nods.) Alright. What type of sum do you think it might be? Learner: It would be like a times.Teacher: You think it’s like a times. How do you think you might … do you think you might work it out differently now?Learner: You could do it as times.Teacher: You think you could do it as times. Show me how you think you might be able to do it as times. Take a pencil. Show me how you’ll do it as times.

Four categories were selected to code the quality of the conceptual activity conducted by the teachers during the interview - not present, inaccurate, partial and full. It was not expected from the teachers that they engage conceptually during every minute, and so a “time count” for the code of not present for conceptual activity is not meaningful in the content of the minute-by-minute coding of the interviews. However, in this activity, for the overall interview, the category not present means that the teacher did not engage in conceptual activity at all in the full course of the interview.

Awareness of error activity – This is a particularly important criterion which needs to be explained more fully in the context of learner interviews. Learner interviews include different activities (we included five different activities), and these activities take place relationally. They are used to reinforce each other and to clarify specific ideas expressed by the teacher or the learner during the conversation. Nevertheless, an interview in which the nature of the error does not become established in the course of the conversation has no real educational value. A code Awa was assigned specifically to statements made by the teacher that demonstrate her/his attempt/s (direct or indirect) to establish the error around which the conversation is focused. Establishing what the error is about is a delicate pedagogical matter since the aim of the interview is for the teacher to probe the learner’s reasoning and not simply to mark and state the error for the learner. This means that the code cannot be applied to every minute of the interview. Nevertheless, at particular moments in the interview and in response to what the learner verbalizes, the teacher is expected to establish what the error is about. This is particularly so in view of the general emphasis in the DIPIP project that learners’ errors are linked to general misconceptions that need to be uncovered. The emphasis of this code is on teachers’ expressions to the learner of what the error is about, in response to what is verbalized by

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the learner in the course of the interview.5 For example (Grade 9 interview):

Learner: As in for twenty-five (points to number on horizontal line), I did a two point five, to, two hundred and fifty (points to number on vertical line), because there was no twenty-five. So I used these variables (points to numbers on horizontal line) just to say this is five, even though it’s zero comma five, I said this is five, and like that…Teacher: Ok, so you ignored the zero (points to zero) and… Learner: Yes.Teacher: Actually you multiplied by what? Multiplied by ten or the…?Learner: No, I didn’t multiply it, I just ignored the (crosses out a number on horizontal line)… Teacher: You removed the zero.

Four categories were selected to code the quality of the awareness of error activity conducted by the teachers during the interview - not present, inaccurate, partial and full. In this code, not present refers to the minutes in the interview where the teacher should have but did not take an opportunity to establish the error in response to what the learner verbalized.

Diagnostic reasoning activity – The idea of diagnostic reasoning on the part of the teacher during error analysis involves the teacher going beyond the actual error to try and follow the way the learner was reasoning when he/she made the error. Through probing questions teachers can attempt to understand the way a learner was reasoning when he/she was solving the question. Much time in interviews could be spent on teachers’ probing their learner’s reasoning, but this does not mean in itself that the interview has a high quality of diagnostic activity. The importance is to understand the relation between the five activities and how together they lead the probing into a successful interview. A code Diag was assigned particularly to probing statements and questions that teachers use in order to pursue the learner’s reasoning. For example (Grade 3 interview):

Learner: I thought I had to plus over thereTeacher: yes we plus there but when we get to your working out I don’t see your subtraction symbolLearner: I did not know if I had to make the sum a plus or a minusTeacher: So you didn’t know if you had to plus or minus. Ok so for this sum you were able to say 65-19+17. So you only struggled here?Learner: yes maamTeacher: So that's why when you get here you better add the whole thing rather than showing your subtraction first?Learner: yes maamTeacher: so your problem is that you didn’t start subtracting and then only later on adding

Four categories were selected to code the quality of the diagnostic reasoning activity conducted by the teachers during the interview - not present, inaccurate, partial and full. In this code the category not present

5 The code does not tell us if the teacher is aware of the error or not (the teacher may withhold it intentionally) but rather if the teacher conveys it coherently to the learner. Notwithstanding, in some places we draw implications about teachers’ awareness of error from what they say.

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refers to the minutes in the interview in which the teachers should and did not probe or her/his probing did not engage with the learner’s reasoning behind the error

Use of everyday activity – Teachers often explain why learners make an error by appealing to everyday experiences that learners draw on and confuse with the mathematical context of the question. A code of ED was assigned every time teacher drew on everyday experiences to explain mathematical content. The emphasis of this code is on the quality of the teacher’s use of everyday, judged by the links he/she makes to the mathematical understanding he/she attempts to advance, when engaging the learner on her/his incorrect answer. For example (Grade 6 interview):

Teacher: No, but I say … if you’re saying you were supposed to stop here (points out line) and you stopped there (points out line), I need you to explain to me why? Ok…turn this around (turns page), what does it look like to you, if I turn that around? These lines, what do you think they look like?Learner: Rows.Teacher: They look like my…?Learner: Rows.Teacher: My rows, ok. All my rows now. Where else will you see those little … rows or little markings, or little demarcations? Learner: In the table.Teacher: On my…?Learner: Table.Teacher: Table?Learner: Ruler.Teacher: On my…?Learner: Ruler.Teacher: Ruler. Ok, look at my ruler very carefully and study it very carefully, where do I start with my ruler (child points at one on ruler)? What number do I start?Learner: Zero.Teacher: I start at zero (points to zero on ruler). From zero where do I go (points at one on ruler)?Learner: One. Teacher: I go to one. And then (points to two on ruler)?Learner: Two.Teacher: And then (points to three on ruler)?Learner: Three.

Four categories were selected to code the quality of the use of everyday activity conducted by the teachers during the interview - not present, inaccurate, partial and full. It was not expected from the teachers that they engage with everyday explanations during every minute, and so a “time count” for the code of not present for everyday explanations is not meaningful in the content of the minute-by-minute coding of the interviews. However, in this activity, for the overall interview, the category not present means that the teacher did not engage in everyday explanations at all. It may not have been appropriate in the context of the interview for the teacher to engage with everyday explanations, but this

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was not part of the scope of this report analysis and would need further research to be properly investigated.

2.3.3 The activity that describes learner participation:Multiple explanations stated by the learner – One of the challenges in learner interviews is for teachers to give their learners the opportunity to state explanations of the error. This is because when learners verbalise the thinking behind their errors and provide claims about the mathematics they do, they provide teachers with material with which they can work. This is the core aim of the ‘think-aloud’ method of research, as explained above. The only aspect of learner participation in the interview that was noted by this code was the number of explanations of different errors the learner stated during the interview. The learner stated her/his explanations in response to teacher probing during the interview. The number of different errors stated by the learner is linked to the number of different topics raised by the teacher in the interview. The more focused the interview, fewer different errors are discussed in the course of the interview. For example one of the best interviews was a grade 4 interview where the focus throughout the interview was on the use of place value to understand how to subtract using the vertical algorithm (see the qualitative analysis section of this report). One of the weaker interviews was a grade 9 interview where the teacher jumped from one concept to another, introducing many errors (and related explanations) but in relation to different concepts and without links between the concepts discussed in the interview (see the qualitative analysis section of this report).

A code of Mult assigned each time the learner gave voice to his/her explanation of their own error. All explanations were coded as feasible explanations of the error since they were stated by the learner and represent the thinking of the learner as he/she speaks in the interview. The teacher was not the focus in this code although the teacher’s responses to these error explanations receive attention in the other criteria. For a list of some of the explanations given by learners in the sample interviews see Appendix 4.

2.4 Training of coders and coding processTwo external coders both experts in the field of maths teacher education coded the learner interviews. The two coders watched and coded interviews after discussion with the team. They raised queries with the team leaders after which a meeting was held in which all coders sat together and watched an interview together and discussed coding decisions. After this no further combined meetings were held but individual coders continued to discuss queries as they worked through the video set. Both coders coded the full set of videos. Each interview was coded minute by minute for each of the 6 activities (five on teachers’ participation and one on learner participation). Obviously it was

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not expected that each minute will receive a code on all six activities. Some minutes received no codes, some minutes received one or more of the codes. When a code was received it was coded with its relevant category descriptor: For example, Conceptual: partial. The spread of the four categories across the interview for each of the codes enabled us to allocate an overall category per interview. There were difficulties with the minute-by-minute coding since this requires professional judgement to be made for each if these intervals throughout the course of the interview, but it did enable some insight into the types of activities the teachers engaged in during the course of the interviews that would otherwise not have been possible had the interviews only been given a global code per criteria.Coders had to enter a code of 1 to 4 (representing the four categories not present, inaccurate, partial and full) for each minute in which they noted activity on any of the criteria. The coders inserted their codes in each column, per minute, according to the given row heading for criterion coding. Cells were to be left blank when activity was not present. Coders were required to make detailed notes in which they justified their coding. For this purpose they were given another template consisting of two tables (see Appendix 3). They completed these two tables for each of the coded interviews.

2.5 Validity and reliability checkIn the overall set of coding there were similar patterns in the coding sets of the two coders, although professional judgement was not always taken in the same way and so the minute by minute codes were not always assigned in the same way. This was partly because coders did not necessarily make the same breaks between minutes when they watched the videos6. A third coder arbitrated the codes and produced a final set of coding that was used for the quantitative analysis. For purposes of alignment a final overall code was assigned to each interview for each of the interview activity criteria. These overall codes were used for the purposes of alignment between coders. Ultimately there was just over 70% alignment between the three coders which was considered adequate.

2.6 Data analysis The analysis of the coded data was done in three parts. The first is a quantitative analysis of the 13 interviews in order to get a broad picture of how the teachers handled the interviews. In the quantitative analysis section, graphs of the overall data set across all videos are given and interpreted. Graphs which give the spilt across the grouped grades (3-6 and 7-9) are included in an appendix, since there were a few differences between 6 The final interview video analysis and all our other video analysis is based on transcribed data, with minute-by-minute time intervals. Transcriptions also facilitated note taking and reporting on the interviews.

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the two groups which were note-worthy. We look at the pattern of performance that emerged from the graphs’ representing the sample data. In this we discuss the teachers’ performance on each of the five activity criteria by discussing findings on the four categories of quality for each of the criteria used to evaluate teacher participation. As part of the quantitative analysis we include a brief examination of learner participation according to the different errors which they spoke about while they were being interviewed. This is done for the total of 13 learners interviewed, looking at the spread of their statements of explanation. We conclude the quantitative analysis with a list of its main findings.

The second analysis is a qualitative analysis in which we take excerpts from four interviews and analyse the quality of teacher engagement, using the six activity criteria drawn up for the evaluation. Our qualitative analysis is guided by the findings on the quantitative analysis. This analysis enables us to demonstrate our criteria and even more importantly to show the relations between the coded activities in each interview. The last part of the analysis reflects on the project interview activity and the findings and draws recommendations.

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Section Three: Quantitative analysis

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3.1 Overall pictureAn overall code was assigned to each of the learner interviews according to the minute-by minute coding. This was done by reviewing the spread of minute-by-minute codes assigned for each interview and assigning a global code for each of the criteria for each interview as described in the methodology section above. These overall codes allow us to get an overview of the types of activities carried out in the interviews. We report the overall data first and then break it down to the different interview activities discussed in 2.3.2. We report on these activities for the sample set of 13 interviews selected for the evaluation, highlighting note-worthy differences between the two sets of grouped-grades (six Grade 3-6 teachers and seven Grade 7-9 teachers) when these arise. For simplicity of language we refer to them as Grade 3-6 teachers and Grade 7-9 teachers. 7

3.1.1 Time spent on interview activitiesThere were 103 minutes of interviews coded for the Grade 3-6 teachers and 120 minutes for the Grade 7-9 teachers. This means that a total of 223 minutes or 3 hours and 43 minutes were available for coding ion the interview sample. In Figure 1 below the total number of minutes coded according to interview activity is shown.

Figure 1: Time spent per activity, Grade 3-9 and Grade 7-9 group

Procedural Conceptual Awareness Diagnostic Everyday0

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It is important to remember that seven interviews were selected from the Grade 7-9 group for the evaluation and only six from the Grade 3-6 group (because of the nature of the sample, see Table 2), and that the number of 7 In the Error Report we referred to groups. The analysis in this report hones on the 13 teachers that were selected for the evaluation and so we refer to teachers and not to groups.

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minutes coded in each of these groups within the sample were not equal. In the next table we represent the percentage of time spent on the different interview activities in the grouped grades as well as in the overall sample. The percentages calculated for analysis are taken of the total number of minutes coded for each of the grouped grades (103 minutes for the Grade 3-6 teachers and 120 minutes for the Grade 7-9 teachers). It is important to note that because the coding was done per minute and in some minutes there could have been activity on more than one of the activity criteria (and hence received more than one code), percentages across all of the activity criteria do not make a total of 100%.

Table 3: Percentage of time spent per activity across the interview sample set

Procedural

Conceptual

Awareness

Diagnostic

Everyday

Grade 3-6 49,51% 32,04% 25,24% 89,32% 6,80%Grade 7-9 36,67% 23,33% 26,67% 83,33% 1,67%Overall 43,09% 27,68% 25,80% 86,33% 4,24%

Observations about the percentage of the total interview time spent on each of the interview activities

1. The highest amount of interview activity time was spent on the diagnostic activity, with the relative amounts of time spent by the two grade groups virtually the same.

2. The second highest amount of interview activity time was spent on the procedural activity but in this activity there was more time spent by the Grade 3-6 teachers.

3. Almost equal amounts of time were spent on the conceptual and awareness of error activities. On awareness there was virtually no difference between the relative amounts of time spent but the Grade 3-6 teachers spent more time on conceptual activities than the Grade 7-9 teachers.

4. The least amount of interview activity time was spent on the everyday activity, the Grade 3-6 group spent more time here, though still very little overall.

Whenever we compare the grouped grades the percentages of time are used to facilitate this comparison, since numbers of minutes per activity would weight the sample in favour of the Grade 7-9 group for which a higher number of minutes were coded.

In both groups the teachers spent most of the interview time on diagnostic activity. By comparison, the time spent on establishing awareness of the error is much lower. Very little time is spent on working through the everyday, which is consistent with the findings of the error analysis. In terms of teachers’ mode of engagement when the teachers make statements,

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pose questions or explain, more time was spent on procedural activity than on conceptual activity (more so by the Grade 3-6 teachers). In what follows we investigate these time-related findings further. The following two considerations guided the analysis:1. Relation between procedural and conceptual

The error analysis report shows high correlation between teachers’ procedural and conceptual explanations of the error in the ICAS 2006/7 error analysis (Shalem & Sapire, 2012, p.98). The report also shows that most of the groups’ explanations (of the correct solutions and of the error) are incomplete, missing crucial steps in the analysis of what mathematics is needed to answer the question (ibid). In view of this, our next step is to examine the quality of the teachers’ procedural and the conceptual activity in the interview and the relation between these activities.

We do this in two ways. First, we look quantitatively at the pattern across the whole sample but we refer to differences between the grouped grades when these differences are higher than 10%. This analysis will direct us to the quality of the teachers’ procedural and conceptual activity, mainly, in terms of use of time in the interviews on statements, questions and explanations that are accurate and full (procedurally or conceptually). Secondly, through a qualitative analysis of four interview, we examine what the teachers do when they engage with learners on procedure and conceptual and how they link these two activities.

2. Relation between awareness of error and diagnostic reasoning The error analysis shows correlation between awareness of error and

diagnostic of learners’ reasoning behind the error. The report also shows that groups struggled to describe learners’ reasoning behind the error. The above results on use of time in the interviews show more activity on diagnostic reasoning than on awareness of error. It suggests that teachers probe learners very often but do not always take up the appropriate opportunity to establish awareness of error.

In view of this, our next step is to examine the quality of teachers’ probing of learners’ reasoning and the ways in which teachers establish the nature of the error. To repeat, much time spent in interviews on probing learner’s reasoning does not mean in itself that the interview has a high quality of diagnostic activity. The importance is to understand the relation between all five different activities, or how together they lead the probing into a successful interview.

Our analysis of this will include a quantitative examination of the spread of the diagnostic activity across the four categories. First, we look quantitatively at the pattern across the whole sample but we refer to differences between the grouped grades when these differences are higher than 10%. This analysis will direct us to the quality of the teachers’ probing, mainly in terms of use of time on probing questions that are accurate and clearly hone in on the error. We do the same for the teachers’ use of time on awareness of error. Second, we examine in more detail, using the above four interviews, what the teachers do when they probe their learners and how do they

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take opportunities to establish awareness of error in response to what the learners bring. The latter is available to us through the data on learners’ participation (see 3.1.3).

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3.1.2 Teacher participationThe 13 teachers’ engagement with their learners in the interview was the focus of five of the six coding criteria.

Figure 2: Overall code set

Procedural Conceptual Awareness Diagnostic EverydaySubject Matter Knowledge Pedagogical Content Knowledge

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Observations about overall coding of interview data for the total number of interviews. 1. In one of the 13 interviews the teacher did not offer any procedural

activity at all. In two of the 13 interviews inaccuracies were noted in the procedural activity offered by the teachers. The majority of the teachers (eight out of 13) offered accurate but incomplete procedural activities during the interviews. Only two of the 13 teachers maintained procedural activity that is correct and includes all of the steps in the procedure.

2. In four of the 13 interviews the teachers did not offer any conceptual activity at all. In 4 of the 13 interviews inaccuracies were noted in the conceptual activity offered by the teachers. In three of the 13 interviews the conceptual activity that the teachers offered included some but not all of the key conceptual links which illuminate the background and process of the procedure. Only two of the 13 teachers (the same teachers as in the above) maintained conceptual activity that illuminates conceptually the background and process of the procedure.

3. In six of the 13 interviews the teachers did not take up any opportunity to establish the error in response to what the learner verbalised about

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her/his error. In one of the 13 interviews the teachers’ statements about the error was mathematically flawed. In three of the 13 interviews the teachers’ statements about the error were mathematically sound but did not link to a common misconception. Only two of the 13 teachers (the same teachers as in the above) offered statements about the error that are mathematically sound and suggest links to common misconceptions or errors.

4. In two out of the 13 interviews, the teachers either did not probe at all or their probing had no relation to what the learner verbalised (i.e. their probing shows that they did not listen to the learner’s mathematical reasoning behind the error). In 6 of the interviews, the teacher was found to use leading questions with little engagement with what the learner brings. In three of the 13 interviews the teachers’ probing (the same teachers as in the above and one other) was responsive to what the learner brought, but was too broad in that it was not sufficiently honed on the error. Only two teachers used probing questions that are engaging the learners reasoning and hone on the error.

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3.1.3 Learner participationThe aim was not to look at learners’ full participation in the interviews since the focus in the interview analysis was on the teachers, but we coded the explanations of errors that were given by learners so that we could analyse qualitatively the relationship between teachers’ activity and learners’ activity (see 4.3).

Figure 3: Learner multiple explanations

One explanation Two explanations Three explanations Four or more explanations0

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Observations of number of explanations stated by learners in the 13 interviews 1. Learners offered one explanation of one error in two of the 13 interviews. 2. Learners offered two explanations of two different errors in five of the 13

interviews 3. Learners offered three explanations of three different errors in three of

the 13 interviews 4. Learners offered four or more explanations of four or more errors in

three of the 13 interviews.

The overall picture of teachers’ and learners’ participation in the interviews points to the following:1. Most of the teachers (eight out of 13) use procedural activity, albeit

partial. Conceptual activity appears to be weaker; where by four teachers did not offer it at all and in other four interviews the teacher’s conceptual activity was inaccurate.

2. In six of the 13 interviews the teachers did not take up any opportunity to establish the error in response to what the learner verbalised about her/his error.

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3. Inaccuracies are more present in diagnostic reasoning. In many of the interviews (six out of 13), the teacher was found to use leading questions with little engagement with what the learner brings.

4. Most of the teachers (nine out of 13) do not connect to or draw on everyday life experiences when they engage with learners on the error.

5. The teachers focused the discussion on one or two errors in seven out of the 13 interviews. The other six of the interviews seem to be less focused.

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3.2 Presence of the coded teacher interview activitiesIn this section of the report we conduct a quantitative analysis the performance of the 13 teachers in the sample according to the six activity criteria drawn up for the interview activity evaluation. We examine the teachers’ performance according to each of the interview activities criteria. Where the performance by the two grouped grades was notably different we note and refer the reader to the relevant graph. We also include a brief examination of learner participation. We begin with the examination of the teachers’ procedural activity during the learner interviews.

3.2.1 Procedural Activity

Figure 4: Procedural explanations in learner interviews

Not present Inaccurate Partial Full0.00

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Procedural explanations in interviews

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Observations about groups’ procedural activity during learner interviews:1. In total the teachers spent 1 hour 35 minutes (43,09% of the total

interview time of 3 hours 43 minutes) on procedural activity.2. In 30% of the total interview time, the teachers gave incomplete

procedural statements, questions or explanations (partial). This means that during this time the teachers spent this time giving procedural guiding comments that are accurate but do not include all the steps needed.

3. In about 5% in the total interview time, the teachers gave statements, questions or explanations which were categorized inaccurate, procedurally. This means that there were flaws in their statements,

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questions or explanations or they were so incomplete that they might have been misleading to the learner.

4. Only in 8% of the total interview time, the procedural activity engaged by the teachers was categorised as full. On this category the grouped grades were notably different: in comparison to the Grade 7-9 teachers, when the Grade 3-6 teachers interacted with their learners on the error, they spent more time (14, 56% and 0,83% respectively) engaging the learners with statements, questions, explanations that are accurate and include all of the key steps in the procedure (see Appendix 5 for the grouped grade graph) .

5. Minute by minute coding cannot capture not present for procedural explanations. Teachers cannot be expected to be engaged in procedural activity every minute. This category is shown in 3.1.2 which compares the 13 interviews over all five teacher participation criteria across the whole interview.

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3.2.2 Conceptual Activity

Figure 5: Conceptual explanations in learner interviews

Not present Inaccurate Partial Full0.00

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Conceptual explanations in interviews

Perce

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Observations about groups’ conceptual activity during learner interviews:1. In total the teachers spent 61minutes (27,68% of the total interview time

of 3 hours and 43 minutes) on conceptual activity.2. In 9,65% of the total interview time, the teachers gave statements,

questions or explanations that were found to be incomplete, conceptually (partial). This means that during this time the teachers spent time on statements, questions or explanations that include some but not all of the key conceptual links that are needed in order to illuminate the background and process of the procedure.

3. In 9,09% of the total interview time the teachers gave statements, questions or explanations which were inaccurate, conceptually. This means that during this time the teachers spent this time giving poorly conceived conceptual links in their -statements, questions or explanations which are potentially confusing to learners.

4. In 8,95% the total interview time, the teachers gave statements, questions or explanations that included full conceptual links. On this category the grouped grades were notably different. This means that in comparison to the Grade 7-9 group, when the Grade 3-6 teachers interacted with their learners on the error, they spent quite a lot more time (14, 56% and 2,5% of their time respectively) engaging them with statements, questions or explanations that conceptually illuminate the

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background and process of the procedure (see Appendix 5 for the grouped grade graph).

5. Minute by minute coding cannot capture not present for conceptual explanations. Teachers cannot be expected to be engaged in conceptual activity every minute. This category is shown in 3.1.2 which compares the 13 interviews over all five teacher participation criteria across the whole interview.

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3.2.3 Awareness of error Activity

Figure 6: Awareness of the mathematical error in learner interviews

Not present Inaccurate Partial Full0.00

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Awareness of errors in interviews

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Observations about groups’ awareness of the mathematical error given to learners in the context of learner interviews:1. In total the teachers spent 58 minutes (25,80% of the total interview time

of 3 hours and 43 minutes) on awareness of error activity.2. In 6,52% of their total interview time, the teachers gave mathematically

sound statements, questions or explanations about the error that suggest the establishment of partial awareness of error. On this category the grouped grades were notably different. This means that in comparison the Grade 3-6 teachers spent more time than the Grade 7-9 group (11,65% and 1,66% of their interview time respectively) on giving statements, questions or explanations about the error, albeit, without linking these to common misconceptions (See Appendix 5 for the grouped grade graph).

3. Only in 1,25% of their total interview time, the teachers gave flawed statements, questions or explanations about the error, which demonstrated poor awareness of the error.

4. In 4,16% of the total interview time the teachers gave statements, questions or explanations about the error that are mathematically sound and suggest links to common misconceptions or errors.

6. In 14,02% of the total interview time the teachers received a category of not present. On this category the grouped grades were notably different. In 17,50% and 8,73% of their total interview time, the Grade 7-9 and the Grade 3-6, respectively, should have but did not take an opportunity to

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establish the error in response to what the learner verbalized. (See Appendix 5 for the grouped grade graph).

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3.2.4 Diagnostic reasoning Activity

Figure 7: Diagnostic reasoning in learner interviews

Not present Inaccurate Partial Full0.00

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Diagnostic Reasoning in interviews

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Observations about groups’ diagnostic reasoning given to learners in the context of learner interviews:1. In total the teachers spent 3 hours and 12 minutes (86% of the total

interview time, 3 hours and 43 minutes) on diagnostic activity.2. In 26,10% of the total interview time, the teachers demonstrated partial

diagnostic mode of engagement. On this category the grouped grades were notably different. This means that in comparison the Grade 7-9 teachers probed more often than the Grade 3-6 teachers (36,66% and 15,53% of their interview time respectively), in ways that were coded as too broad and not sufficiently honed on the mathematical error (see Appendix 5 for the grouped grade graph).

3. In 38,37% of the total interview time, the teachers probed learners but did not engage with what the learner brought to the conversation. In this time teachers used only leading questions with little engagement with what the learner brings. In this sense the probing was inaccurate.

4. In 10% of their total interview time, the teachers demonstrated full diagnostic mode of engagement. On this category the grouped grades were notably different. This means that the Grade 3-6 teachers engaged more often with learner’s reasoning and honed in on the errors in the interviews than the Grade 7-9 teachers (17,47% and 2,5% of their interview time respectively). See Appendix 5 for the grouped grade graph.

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5. In 11,86% of the total interview time the teachers received a category of not present. In this time the teachers should and did not probe or their probing did not engage with the learner’s reasoning behind the error.

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3.2.5 Use of everyday Activity

Figure 8: Use of the everyday in explanations in learner interviews

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Use of everyday in interviews

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Observations about groups’ use of the everyday in explanations given to learners in the context of learner interviews:1. Generally minimal use of the everyday in the interview discussions.

3.2.6 Multiple explanations Multiple explanations are coded according to the explanations stated by the learner over the full duration of the interview and hence the percentages are worked out of the total number of interviews and not the number of minutes (6 interviews in the Grade 3-6 group and 7 interviews in the Grade 7-9 group).

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Figure 9: Multiple explanations in learner interviews

One explanation Two explanations Three explanations Four or more explanations0

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Multiple explanations in interviewsN

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Observations about groups’ multiple explanations given by learners in the context of learner interviews:1. Grades 7-9 learners gave more explanations of different errors than the

Grade 3-6 learners. This reflects a stronger focus on one error in the Grade 3-6 group and a greater tendency to discuss a range of questions in the Grade 7-9 group.

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3.3 Quantitative Findings.

Main finding:

In both groups the teachers spent most of the interview time on diagnostic activity, probing the learners. By comparison, the time spent on establishing awareness of the error is much lower. In terms of mode of engagement when the teachers made statements, pose questions and explain, more time was spent on procedural activity than on conceptual activity (more so by the Grade 3-6 teachers). Very little time is spent on working with learner’s error by drawing on everyday experiences. Analysis of the number of errors raised in the interviews showed that in seven of the interviews more than three errors were mentioned by learners.

Table 4: Percentage of time spent per activity across the interview sample set

Procedural

Conceptual

Awareness

Diagnostic

Everyday

Grade 3-6 49,51% 32,04% 25,24% 89,32% 6,80%Grade 7-9 36,67% 23,33% 26,67% 83,33% 1,67%Overall 43,09% 27,68% 25,80% 86,33% 4,24%

In what follows we provide more detail on different aspects of the main finding and explain the specific finding:

1. Very little time is spent on working through the everyday, which is consistent with the findings of the error analysis, that “groups draw primarily on mathematical knowledge and less so on other possible explanations to explain the correct answer or the errors” (Shalem & Sapire, 2012, p.97).

2. The teachers spent more time on procedural activity than they did on conceptual activity. This means that in many more minutes (15,41% more) in the interview the teachers engaged procedurally with the learners. Some of the time, teachers’ engagement was procedural as well as conceptual, but at other times, the focus was on only procedural explanation. On its own this finding does not convey a judgement of quality. Not every procedural activity in an interview situation needs to be coupled with conceptual activity. A qualitative analysis is required to understand the relationship between these activities in the interview, in order to further understand if and how the teacher’s procedural engagement compromises or contributes to the learner’s conceptual understanding.

3. The pattern of quality distribution within the procedural and conceptual activities is different. Relative to each other, the teachers give more partial procedural statements, questions and explanations than they do for conceptual explanations. In other words, overall

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conceptual activity is less present in the interview, but when it is present the distribution across the three categories (inaccurate, partial and full) of quality is more even. In procedural activity, the percentage of time is distributed unevenly across the three categories, with partial being the highest.

4. In all of the five activities the teachers spent relatively little time (under 10% of the total interview time) engaging with the learners in the activity-respective way categorized as full. Notwithstanding, the Grade 3-6 teachers did better on this category of quality in three activities (procedural, conceptual and diagnostic). (See Appendix 5).

Table 5: Percentage of time coded full over total interview time

Interview Activity

Full

Procedural 8,05%Conceptual 8,95%Awareness 4,16%Diagnostic 9,99%Everyday 0%

5. Despite being the dominant activity in all 13 interviews, the diagnostic activity was found to be the one in which 40% of the time spent was categorized as inaccurate, as the teachers’ probing did not engage with the learners’ reasoning. Taken together with the 11,86% of not present , close to half of the time the teachers engaged with learners, diagnostically, was wasted on very poor quality interaction.

Table 6: Percentage of time coded inaccurate over total interview time

Interview Activity Inaccurate

Procedural 5,41%Conceptual 9,09 %Awareness 1,25 %Diagnostic 38,37%Everyday 2,50%

6. The table below looks at the time the teachers spend on each of the five activities in the two higher quality categories.

a. The table suggests that the teachers are confident in giving statements, questions and explanations that address the

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mathematical procedure required by the question. The majority of the time the statements, questions and explanations are partial and in about 8% of the time on procedural activity they are full. This is very different from the teachers’ conceptual engagement, which is done much less, but more importantly, in comparison to the procedural activity, more time is spent on poor quality conceptual activity.

b. The relation between error awareness and diagnostic is also worrying. It is worrying because the teachers spend too much time probing the learners with statements and questions that are of poor quality (close to 50% the total interview time) and which do not engage the learner’s reasoning. This constrains the learners’ ability to verbalize their reasoning – be it their confusion, misrecognition or lack of knowledge. More so, poor quality probing constrains the teacher’s ability to gather relevant information and to construct a picture of what the problem is. For teachers who have strong subject knowledge that may be less of a problem; they can rely on their understanding of the field, on their general knowledge of common errors. The error analysis report suggests that this is not the case with this group of teachers. It suggests that teachers’ content knowledge is poor, a finding that is consistent with much other research in SA. We argue that poor quality of probing takes away the focus and leads to no resolution on what the nature of the error is. It curtails the teachers’ capacity to gather evidence, to think through what the learner does verbalize or to respond appropriately. This is also evident in the high percentage of time (approximately half of the activity total time in the interview), that the teachers miss an opportunity to establish what the error is.

Table 7: Percentage of time coded full and partial over total interview time

Interview Activity

Full Partial Total of partial and full

Activity total (%of the total interview time)

Procedural 8,05% 29,63% 37,68% 43, 09%Conceptual 8,95% 9,65 % 15,60% 27,68%Awareness 4,16% 6,52% 10,68% 25,80%Diagnostic 9,99% 26,10% 36,09% 86,33 %Everyday 0% 2,50% 2,50% 4,24%

7. Arguably, the Grade 3-6 group interviews were more focused on one or two errors during the interviews than the Grade 7-9 group. This may have been the case since mathematical topics in the higher grades may have richer concept maps and thus a discussion around one error from a

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learner test may raise more than one related concept (or error in relation to this concept). Notwithstanding the many different explanations given in some of the Grade 7-9 interviews were not a result of a rich discussion around a concept and its related concepts, but rather to the number of different test items discussed by teachers during an interview.

These findings give rise to the following guiding questions for the interview:

1. In what ways does the teacher’s procedural engagement facilitate the learner’s conceptual understanding of the error?

2. What kind of probing is productive in that it gathers relevant evidence for teachers that they can use to make judgements on the nature of the error and on when to establish it in the course of the interview?

3. Given that in the context of an interview partial explanations would have a place, what is the role of a full explanation?

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Section Four: Qualitative analysis

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4.1 SampleIn the following we present four examples of learner interviews. We selected two interviews form the Grade 3-6 group and two interviews from the Grade 7-9 group. In our analysis we use the evaluation criteria (see 2.3.1) to describe the strengths and weaknesses in the ways the four teachers engage the learners with statements, questions and explanations about the error. In broad terms the analysis will focus on the quality of progression in the conversation and on the extent to which the interview increases learner’s understating of the error.The transcriptions are presented in tables, minute by minute, followed by analysis relating to each time segment of the interview. Coding allocated per minute is indicated in the second column of the transcription table. The code is indicated next to the particular line that signals it, usually for the first time in that minute, although often it also refers to other lines in the minute. Criterion abbreviations are used as indicated in the table below.

Table 8: Activity criteria abbreviations

Criterion Criterion Abbreviation

Procedural explanation ProcConceptual explanation ConAwareness of error AwaDiagnostic reasoning DiagUse of the everyday EDMultiple learner explanations

LMult

Category descriptors are abbreviated to the first letter of the descriptor:Not present (N),Inaccurate (I),Partial (P), and Full (F).

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4.2 Analysis of learner interviews

4.2.1 Grade 4 learner interviewThe learner interview focuses on the following question selected from the test set by the DIPIP grade 4 “small group”8. The teacher that elected to conduct the learner interview selected this question since the learner had done the subtraction required in the solution of this question incorrectly in a most unusual way.The question was given as a word problem. Learners needed to read and interpret the question in order to realise the sequence, the type of number operation they needed to select, and the mathematical procedure they needed to follow in order to arrive at the solution. First they needed to calculate Kim’s original mass (twice Amy’s i.e. 38 × 2 = 76 kg) and then they needed to subtract the number of kilograms she lost in the year (76 kg – 19 kg = solution).

Figure 10: Extract from Grade 4 learner test

Figure 11: Snapshot of learner working.

Extract from learner interview

8 See Process Report for the classification of ‘group’ in the DIPIP project. 50

The interview is 18 minutes long. The extract below is the first 8 minutes of the interview.

Minute 1Speaker

Utterance Code

Teacher

Good morning, you’re going to do this interview with me. About weeks ago we were writing a test.

Learner

Yes

Teacher

And you have done, this is the question 5 that you have done and you have come as far as … you have added all the um kilograms because it’s double the weight (points to paper with various numbers and words written thereon) and then, after that, you decided to subtract. Now something that I’m very interested in about is the way you were subtracting. Now can you please, while subtracting, explain to us step by step what you were doing.

Diag (P)

Analysis – Minute 1The teacher opens the interview with a broad probe describing to the learner the method of subtraction that she used answer the test question.

Minute 2Speaker Utterance CodeTeacher (points to paper) Because this is 76 take away 19, you did

it, you got 21. 76 take away 19. Can you please, step-by-step, what you are doing … explain to me. (Places pen and paper in front of L. On paper is written 76 – 19 underneath each other.)

Diag (F)

Learner Because you can’t subtract 6 (points to the 6 of 76) from 9 (points to the 9 of 19). So I took 1 away from the 7 (points to the 7 of 76, crosses out the 7 and writes 6 above it) so it becomes 6 and this one becomes 7 (crosses out the 6 of 76 and writes 7 above it). Now you still can’t subtract it, so I took away 1 again (crosses out the 6 written above the 7 of 76 and writes 5 above it) and it became 5. And I added this one (crosses out the 7 written above the 6 of 76 and writes 8 above it), so it became 8.

Mult (1)

Teacher Alright. Ok.Learner And I still couldn’t get it, so I took there (crosses out 5 on

the left side and writes 4 above it) made it 4 and then I added (crosses out 8 on right hand side and writes 9 above it) and made it 9. So 9 minus 9 obviously is going to become zero so then I took it again and 4 took away 1 (crosses out 4 on left hand side and writes 3 above it), made it 3. And I added 10 (crosses out 9 on right hand side and writes 10 above it).

Analysis – Minute 251

The teacher continues to probe, this time referring more specifically to the learner’s working. The probe hones in on the learner’s error and the teacher gives precise instructions (“Can you please, step-by-step, what you are doing … explain to me”). The learner responds in detail, giving her explanation of the way in which she had done the computation. The learner takes the teacher through each step of the “borrowing from the tens” that she did in her calculation. The teacher allows the learner all the time she needs to explain her working in full and does not interrupt her or correct her, even though it becomes clear well before the learner has finished her explanation where she had gone wrong.

Minute 3Speaker Utterance CodeLearner 10 take away 1, um 9, I got 1. And 3 minus 1 was 2.

(writes 21 in answer line) So that’s how I got my answer.Teacher Well the way you were explaining to me very well. Right,

now Jamie, I’m going to give you another sum so you can do the subtraction again, ok?

Diag (F)

Learner Ok.Teacher But this time we’re going to use something else,

something different. You going to use a little bit of blocks and you have to listen very carefully to me. (T places piece of red paper in front of L.) Right, now Jamie, ok can you please read the sum there on top for me (points to top of paper).

Learner 45 minus 14.Teacher Minus 14, ok, right. Now can you please show me where

all the units are?Awa (F)Con (F)

Learner Tens are on this side (points to right hand digits of numbers – ie 5 and 4).

Teacher Name them for me, please.Learner Units are 5, 4.Teacher And 4. So you going to subtract the units from each other.

Can you please tell me where the tens are?Learner The tens are on the left hand side (points to these).Teacher Show me.Learner It’s 4 and 1 (points)Teacher 4 and 1 right.

Analysis – Minute 3The teacher continues to probe diagnostically by introducing another example that she has chosen that will enable her to work with the learner on the error embedded in her test solution through working with concrete manipulatives – unifix cubes. The selection of the example together with the use of concrete manipulatives shows that the teacher is leading towards establishing the learner’s error: the learner does not know how to subtract two digit numbers from each other when there is an impasse. An impasse occurs when there are insufficient units in the minuend - the number from which the learner must subtract the given subtrahend - to simply subtract the units in the subtrahend -the number which the learner must subtract

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from the given minuend - without rearranging, or working in some way, with the tens and the units so that the subtraction can be made possible. The teacher works conceptually with the learner by asking her to show her where the tens and the units are in the given numbers. This further demonstrates the teacher’s awareness that the learner’s error lies in her misunderstanding of the way she needs to work with the place values of the numbers in the given subtraction problem.

Minute 4Speaker Utterance CodeTeacher Now what we going to do is we going to use a little bit of

blocks, right. And you going to help me a bit (hands blocks to L). Please can you place all the tens for me on your diagram. Just place them there. Just place them (moves blocks onto paper) there you go. You going to place all the tens together. Like you said, tens (places 4 rows of tens in front of L). How many tens do you have there?

Proc (F)

Learner 4.Teacher 4 tens is equal to how much? Con (F)Learner 4 tens equals 40.Teacher 40, good. Now I’m going to give you the units. How many

units can you count?Learner It’s 5.Teacher Do you think we should break them up a bit ne?Learner Ja.Teacher Ok there we go. Well done. Now Jamie what I want you

now to do is um you have to subtract, right? So we going to start subtracting. First we going to start subtracting the units. How many units do you have?

Learner 5Teacher How many must you take away from themLearner 4Teacher Do it for me.Learner 1, 2, 3, 4 – take them away (L removes 4 blocks)Teacher Can I give you a hand?

Analysis – Minute 4In the interaction in minute 4 the teacher gives procedural instructions and questions to the learner to direct her through the working with the blocks (unifix cubes), to help the learner work through the subtraction of two digit numbers. She also asks conceptual questions (“4 tens is equal to how much?”) to make sure that the learner is keeping in mind the place values (and the meanings of these place values) of the numbers with which she is working.

Minute 5Speaker Utterance CodeTeacher Let me give you a hand and I’ll put it all the way over Proc (F)

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there (removes the 4 blocks to her right hand side) right.Learner OkTeacher Ok so how many’s left?Learner OneTeacher Well done. Now we’re going to subtract the tens. Can you

do that for me?Learner Uh huhTeacher Ok how many tens must you take away from there?Learner One (indicates 1 with her finger).Teacher We are going to take one ten away, well done. So do it for

me. (L removes one row of tens). Ok how many tens are left?

Learner 3Teacher Altogether, what is your answer?Learner 31Teacher Ok so can you please write in there. When you take 5

away from 4 your answer was?Learner 31Teacher Ok it was 1 and then 3 ok well done. That is a star.

(Removes paper and blocks from the L.) Right we’re gonna take this and put this away. Ok so you had a little bit of practice there, we going to put it all the way over there. You are doing very well. Ok right now I’m going to give you another one. Ok. Right, there you go (places another piece of red paper in front of L with 76 – 19 written underneath each other). Ok can you read the sum for me please?

Learner 76 minus 19.Teacher 76 minus 19. How … tell me which ones are the units. Con (F)

Analysis – Minute 5The interaction continues in the fifth minute. The teacher’s activity is predominantly procedural but at all times, this carefully controlled procedure (linking hands on work with the unifix cubes to the subtraction of the given two numbers in the example) is designed to expose the learner to the correct way of working with numbers in a subtraction problem where there is an impasse. Conceptually, the teacher keeps bringing the learner back to identifying the numbers according to their place value (“which ones are the units”) since this links to the underlying misconception evidenced in her test working. The teacher finishes the first example and then moves on to a second example, which is the same as the test question, but the teacher does not say anything about this (she leaves this connection to be made later on the interview, in minute 11 which is not part of this extract from the interview).

Minute 6Speaker Utterance CodeLearner Units are 6 and 9.Teacher 6 and 9, and the tens? Con (F)Learner Is 7 and 1.

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Teacher 7 and 1. 7 tens ok, and we must take one tens away (hands blocks to L). So let’s count the tens quickly, how many tens do we have there?

Learner 1, 2, 3, 4, 5, 6, 7 (points to each tens as she counts).Teacher 7 tens, and how many units do I need?Learner 6Teacher 6 units, ok. (Hands 6 units blocks to L.) Remember what I

taught you, the same steps that we did. I want you to take … um … you’re subtracting units. How many units must you subtract?

Proc (F)

Learner 9Teacher 9. Now I want you to subtract 9 units from there.Learner I can’t because 6 is a lower number from 9.Teacher So you can’t subtract uh 6 from there. So now what do

you think we should do? Because remember we are only allowed to use this (indicates the blocks on the red paper) and only take away from 6. Right. (L nods.) You’re taking away from 76, you need to take away from there. So what do you think you should do now?

Analysis – Minute 6In the sixth minute the interaction is once again predominantly procedural linking the hands-on work to the conceptual and procedural activity involved in the subtraction of the two numbers. Unlike the previous minutes of the interaction in which the teacher guided the learner through the procedure, step by the step, in this minute the teacher asks the learner to think of the procedure (“So what do you think you should do now?”)

Minute 7Speaker Utterance CodeLearner I think you should, uh, take one ten away from 7 and put it

in front of the units.Teacher Oh so you should take one ten away (points to a tens

block) and you put it with the units. Can you do that? (L does this.) Ok does it look the same as the units?

Con (F)

Learner NoTeacher What should we do?Learner Break it up.Teacher And then if you break it up, what do you get?Learner You get uh 16.Teacher Oh ok, do it, show me. Show me how you get 16. (L

proceeds to break up the one tens she placed with the units into separate blocks.) Ok well done, I’m so proud of you. So now what you did is, did you borrow from there? Did you take from there?

Proc (F)

Learner (L nods.) Uh huh.Teacher Ok show me – how many were there in the first place

(points to the tens blocks)?Learner 7Teacher So then you took one away from them, how many is left?

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Analysis – Minute 7The teacher keeps the interview interaction focussed on the learner’s error, working slowly and systematically through the example. She does not rush the learner and she continually directs the learner to work with the blocks in order to understand the numeric calculation that she must do.

Minute 8Speaker Utterance CodeTeacher Can you please indicate (points to paper), is it still 7? Proc (F)Learner NoTeacher So indicate that it’s not 7 anymore … (passes pen to L,

who crosses out 7). And how many is it now?Learner 6 (writes this above 7).Teacher It is now?Learner 6Teacher 6. Right well done. So can you now take away? Con (F)Learner YesTeacher How many’s this (points to 6 of units) now?Learner 6Teacher Is it still 6, the units? Awa (F)Learner NoTeacher What is the units now? Diag (F)Learner 7Teacher Is it 7? Can you please count your units.Learner (Counts aloud whilst moving units blocks.) 1, 2, 3, 4, 5, 6,

7, 8, 9, 10, 11, 12, 13, 14, 15, 16.Teacher How many?Learner 16Teacher First you had 6 and then you add how much?Learner 10Teacher And it gives you?Learner 16Teacher Can you please write that six and you add what you did.

And then you get your 16 ne? Like you said. (L writes 16 above 6). Well done. Can you now take 9 away from there?

Learner Yes

Analysis – Minute 8In the 8th minute the interaction in the interview allows the teacher to work with the learner on her misconception in the written calculation, both procedurally and conceptually. Procedurally, she shows the learner the correct way to write the calculation. She asks, “Can you please indicate (points to paper), is it still 7?” and gives the learner the opportunity to write the correct procedural steps, all the time linking this activity to the concrete apparatus of the blocks. Conceptually she asks “So can you now take away?” – calling on the learner to reason about whether or not she has overcome the impasse. She also uses the blocks to allow the learner to establish the number of units from which she is now subtracting (having exchanged one

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ten for ten units). The teacher shows awareness of the learner’s error – which she is in the process of uncovering – when she asks, “Is it still 6, the units?” A little later she asks, “Is it 7? Can you please count your units.” The learner counts the units and finds out that there are actually 16 units from which she can now subtract 9 units, as required in the example. The focus and gentle pace at which the teacher carries out this explanation is remarkable. The rest of the interview continues in this way and by the end of the interview, the learner is able to subtract two digit numbers correctly, showing her working and with an understanding of the use of place value in the subtraction of two numbers when there is an impasse.

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4.2.2. Grade 5 learner interviewThis learner interview is focused on the following question selected from the test set by the DIPIP grade 5 teacher group. The teacher that elected to conduct the learner interview selected this question since she was interested in finding out the reason for the learner’s incorrect answer to the second part of the question.The test question consists of two parts. The learner did the first part correctly and the second part incorrectly.

1. The first part of the question requires the learners to write a mixed number that has been illustrated diagrammatically using numerals.

2. The second part of the question requires the learners to convert the mixed number into an improper fraction.

Figure 12: Extract from Grade 5 learner test

Question 1- fractions

1.1 Write the diagram above as a mixed number. (1)1.2 Write the diagram above as an improper fraction (1)

Figure 13: Learner working in Grade 5 interview

Extract from learner interviewThe interview is 13 minutes long. The extract below presents minutes 3-9 of the interview.

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Minute 3Speaker Utterance CodeTeacher Do you still remember that one?Learner YesTeacher Ok how did you do this problem? Can you please explain

to me? (Moves paper in front of L.)Learner This problem teacher (points)?Teacher This one (points), the first one.Learner Teacher I said one whole plus one whole plus 3 over 4

equals to 2 and 3 over 4.Teacher Good, very good. Now explain to me the second step.

How did it come about like this (points)?Diag (P)

Learner Teacher I said 2 times 3, then I said is 6 and then I said 2 times 6 is equals to, is equals to 12 teacher. Then I said 12 over 6 teacher.

Mult (1)

Analysis – Minute 3The teacher’s last question in Minute 3 (“How did it come about like this (points)?) is an open question, leading into the exploration of the learner’s error in relation to the second part of question 1 in the learner’s test. The learner starts to explain what he did, giving his first explanation of an error. It seems that the learner multiplied the whole number part of the mixed number with the numerator of the fraction part of the mixed number (getting 6), then he took this ‘product’ and multiplied it by the whole number (getting 12). This is a direct explanation of his working (shown above in Figure 13), but it does not illuminate why he followed this procedure of converting a mixed number to an improper fraction. The learner’s explanation is in direct response to the teacher asking for explanation of how he did the question. In this exchange the teacher does not build on the learner’s response, procedurally or conceptually.

Minute 4Speaker Utterance CodeLearner And then I was confused, teacher. I wanted to say 12 over

2 … 3.Mult (1)

Teacher Ok do you think that your answer is correct? Awa (N)Learner NoTeacher Why do you think that it is incorrect?Learner Because, teacher, I wrote it wrong, teacher.Teacher What made you write it wrong? Diag (I)Learner Teacher, I was confused, teacher.Teacher You were confused, ok. Eh have you ever done this sum

before? This problem before? Have you ever done a problem like this (points to paper) before?

Learner No teacher (shakes head).Teacher In the classroom, where, have you ever done these sums?Learner No teacher.Teacher You didn’t. (Sound of door opening and closing in

background.) What, what are you thinking about right now about this problem?

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Analysis – Minute 4The learner continues with his explanation of what he did, saying that he was “confused” and he tries to start saying what he had wanted to write (“I wanted to say 12 over 2 … 3”) but the teacher interrupts him and says “Ok do you think that your answer is correct?”, as a direct response to the learner’s first attempt to explain his reasoning. The teacher does not take the opportunity to probe the mathematical content of the learner’s explanation for example by asking a question like “why did you say 2x3”? This shows that the teacher is not trying to establish the mathematical reasoning behind the learner’s error. The teacher does not engage with the learner’s explanations but rather pushes him about the correctness/incorrectness of his answer and whether or not he has done such work before. She does ask another broad question, “What made you write it wrong?” attempting to engage his reasoning but does not actually follow this through or link it to his explanation given in the previous minute.

Minute 5Speaker Utterance CodeTeacher The way you have done it, why? What do you think about

it?Learner I’m not thinking right, teacher.Teacher You aren’t thinking anything. Awa(N)Learner Eh (yes).Teacher Can you explain each step as you worked through the

problem and give me reasons? You can work out this here (points to question). Work it out here (points to blank paper). Work it out.

Diag (I)

Learner (Picks up pencil and ruler and writes: R565 – R389 = …)

Analysis – Minute 5The teacher’s first question in minute 5 is another open probe, calling for an explanation of the learner’s reasoning, but the learner is unable to answer the question mathematically and simple says “I’m not thinking right, teacher.” In response to this, and despite the clear signs the learner’s provides about his plight of helplessness, the teacher says to the learner, “You aren’t thinking anything”, showing no attempt to establish the mathematical reasoning behind the learner’s error. The teacher instructs the learner to work through what he had done, to write it down step-by-step, passing him a piece of paper. The learner starts writing out the solution to the next question in the test.

Minute 6Speaker Utterance CodeTeacher No not this one. The very same one, this one (points)Learner This (points)?

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Teacher Yes. The very same one, the one that you, you said is incorrect. You said it is not correct. (L is trying to erase something.) No just leave it here. Write it here. (L writes

2 × 6 = 126 and shows teacher.) Ok this is the … you said

it’s not correct.

Diag (I)

Analysis – Minute 6The teacher shows the learner which question he should be writing out, showing no acknowledgement of his earlier explanation (given in minute 3), she just points to what the learner had done in his test. Diagnostically this still shows no response to the learner’s reasoning. The teacher’s instructions have no connection to the learner’s explanations of what he thinks the error was. All the teacher is doing is restating the incorrectness of the learner working.

Minute 7Speaker Utterance CodeLearner YesTeacher Now I want you to write it correctly. How are you going to

make it correctly? Write it correctly? (L writes 2 ) Start it from here no? Start it from here, don’t write anything that … starting from here. Do it correctly from that step. (L

inserts × after the 2, so it reads 2 × 3 = 62 . Pushes it

towards teacher.)

Diag (I)

Analysis – Minute 7In minute 7 there is no progress. The teacher uses leading questions instructing the learner to write the mixed number “correctly” although the error has not been established yet. The probing continues at a low level by way of the teacher giving repeated instructions as to from where and how the learner should re-write his working. The teacher’s probing has little engagement with what the learner brings in minutes 3 and 4.

Minute 8Speaker Utterance CodeTeacher Is that how you do it?Learner Yes teacher.Teacher Ok. You said two times..?Learner Two times 6.Teacher Here is an answer here. How did you do it here? From

here (points to top of first paper).Learner From there teacher?Teacher MmLearner I said one …Teacher No forget about that, from here (points to part of 1st line of Diag (I)

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sum). I want you to do this.Learner Here teacher?Teacher YesLearner I said 2 plus 3 over 4.Teacher Ok work it out so that you should get the answer. (L

writes 2 +34

= 104

.) Are you satisfied about the answer?

Analysis – Minute 8In minute 8 there is still no progress as the probing continues by way of the teacher giving repeated instructions as to from where and how the learner should re-write his working. The teacher refers the learner to part one of his answer, saying to him, “Here is an answer here. How did you do it here?” In doing this she has moved away from the part of the question where the learner made the error to the part where he got it right. She instructs him, “Ok work it out so that you should get the answer.” The learner responds by

writing “2 +34 = 104 ” on his sheet of paper.

Minute 9Speaker Utterance CodeLearner Yes teacher.Teacher How did you work it out? Tell me, you said what? Diag (I)Learner I said 2 times 3 Mult (2)Teacher 2 times 3Learner 2 times 3 is equals to 6Teacher Uh huhLearner Then I said plus 4Teacher Equals to?Learner 10 over 4.Teacher 10 over 4. Ok. Uh… do you think this is the correct one?

That this one is the correct answer …Learner Yes teacher

Analysis – Minute 9The teacher now has some new working from the learner, which again is not correct (but in a different way), and she probes it, but without any attempt to engage with his mathematical reasoning. She says “How did you work it out? Tell me, you said what?” The learner attempts another explanation (of the new working with the different error) and the teacher repeats after him what he says he did, and prompts him, step-by-step to write the conversion procedure. The learner explains that he has multiplied the numerator of the fraction by the whole number to get 6. The learner explains that he added this result (6) to the denominator (4) of the fraction and written the sum (10) over the denominator. This is the second learner’s explanation of an error, differently error to the error he made in the test. When the learner is finished the teacher asks, “Uh… do you think this is the correct one? That this one is the correct answer …”, again not engaging with his reasoning.

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Again a non-productive prompt (she knows the answer is wrong). Ultimately in minute 11 the teacher shows to the learner the correct procedure for conversion of a mixed number into an improper fraction. She does so without giving any conceptual links to the mathematical idea that fractions can written in different ways that are equal, so that the ‘demonstration’ of the correct procedure for the conversion between forms of fractions that she gives will not necessarily shift the learner’s understanding.

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4.2.3 Grade 7 learner interviewThe learner interview is focused on the following test question, the Grade 7 group selected from the Grade 7 test they designed. The teacher that elected to conduct the learner interview selected this test question since many learners (including the one she chose to interview) had experienced difficulty with it in the test. The question presents a shape (the top shape in the diagram below) which is to be regarded as a “whole” into which three other shapes (the three smaller shapes below, labelled A, B and C) need to be fitted. The wording of 3.1 says “What fraction of the picture would be covered by each of the shapes below?” which implies that the learner needs to find out what fraction of the bigger shape each of the given smaller shapes represents. The question is testing if the learner can work out the fractional value of parts of a whole. The activity requires that the learner checks how many of each of the smaller shapes (taken one at a time and repeated as many times as necessary) are needed to cover the whole shape above. A further instruction indicates that learners may flip or rotate the shapes if necessary in order to make them fit into the given bigger shape.

Figure 14: Extract from Grade 7 learner test

Question Three:Look at the picture.

3.1. What fraction of the picture would be covered by each of the shapes below?(The shapes can be flipped or rotated when necessary)

A) _______________ B) ________________ C) ______________

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Figure 15: Learner working in Grade 7 interview

The darker lines highlighted in this snap shot of the learner’s work show the way in which the learner thought she would fit the three different smaller shapes into the bigger shape.

Extract from learner interviewThe interview is 7 minutes long. The extract below presents minutes 1-7 of the interview.

Minute 1Speaker Utterance CodeTeacher So now, this class thing that we did in class, ok, most of it

you found very simple except for this whole question?Learner YesTeacher (zooms onto question paper) To start with, what did you

find difficult with this question?Diag (P)

Learner When I first did this question, I thought it was pretty easy, because all the shapes could fit in, like once, but then you told me they had to be these sizes. So I didn’t really know like what…I didn’t really know what to do then afterwards, because I thought just put like…it doesn’t matter what size they were as long as they fit in there. But then you said the actual size, so I didn’t know, so that’s when I got stuck, and time ran out.

Mult (1)

Teacher Ok, so then what were you actually trying to start drawing here?

Learner I was trying to see if I could fit these triangles in, but then…

Teacher Oh, these triangles [and trapeziums and parallelograms] (points to 3 shapes below top shape) to fit into here?

Awa (P)

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Learner Yes. Then I put the other two.

Analysis – Minute 1The teacher is trying to get from the learner what was she struggling with (note the language as the teacher is asking “what was difficult” and not “what do you think you did wrong”).The questions the teacher is asking are broad probes which do not hone directly in on the error, which aim to get the learner to start talking about what she had done in the test. With these broad questions the teacher is getting the learner to explain what first confused her. The learner states her first explanation of “error” which in this case is more that she was confused by the question and was not sure what she had to do. Her explanation verbalises a confusion- the question includes three different shapes which somehow could be fitted altogether into the whole. The teacher acknowledges the learner’s explanation (that she thought she needed to put all three shapes into the big shape) showing general (yet not specific) awareness of the learner’s error. The teacher points to the shapes in the question. The learner answers and it appears that they understand one another. Important to note, though, that the teacher’s understanding of the learner’s error is not yet fully developed. What the teacher has established so far is only that the learner started by trying to fit the triangle and then the other shapes into the whole.

Minute 2Speaker Utterance CodeTeacher Ok. So that’s what you obviously started doing, was

drawing the triangle part.Awa (P)

Learner YesTeacher Ok. So now, this question here said to you: the shapes can

be flipped or rotated. Alright. What did you understand by that when they told you that?

Diag (P)

Learner That these shapes themselves cannot just be put there, because otherwise they wouldn’t make the proper shape. So obviously you had to turn them or just change them so that they can actually fit inside here.

Teacher Ok. So do you think they were there for a reason?Learner YesTeacher Ok. So now, if we go back to this question (points to the

shape of ‘the whole’), it says here: each of these shapes is a part of this whole. Because this is the whole (points to the shape of ‘the whole’), and each of these (points to 2nd row of 3 shapes) is a part of that whole (points to the shape of ‘the whole’). Ok. So you’ve got to use this shape (points to first shape-trapezium, in row of shapes), and it says, what fraction is shaded? And now it tells us that we can flip it around or turn it upside down. So, if we are trying to fit…how, if you could, how could you fit this (points to trapezium) into here (points to shape of ‘the whole’)? If you want you can use a pencil or a rubber or whatever, if

Con (P)Proc (bold text) (P)

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you like.

Analysis – Minute 2The teacher again acknowledges what the learner said she did (started drawing the triangles) confirming for both of them the general awareness of the error, which does not link to any misconception at this stage. She then moves to the next instruction (in the question) that the shapes can be flipped or rotated, and probes the learner’s understanding of that aspect of the question. This probe engages with the learner’s reasoning but again does not hone in on the mathematical error. The learner’s response indicates understanding of this instruction and the teacher moves on. The teacher then gives a detailed interpretation of the question which is predominantly conceptual but ends with more procedural instructions as to how the learner should proceed. She explains that the question is drawing on an understanding of the relationship between parts and a whole. The teacher then refers to the procedural mechanics of fitting the given “part” shapes into the given “whole” shape.

Minute 3Speaker Utterance CodeTeacher How would you fit this trapezium (points to trapezium)

into here (points to the shape of ‘the whole’), to work out how many of them fit into the whole?

Proc (F)

Learner Um…I think I’d put two…maybe two there (draws line through top shape), so this is one trapezium, that’s another (outlines shape of 2 trapeziums with bottom part of top shape)…or I could do the same here as well to (draws line across top part of top shape)…

Teacher To cover the whole.Learner Yes. There would be four of them (outlines 2 more

trapeziums on top part of top shape).Teacher Ok. So you’ve worked out that four of these (points to

second row-trapezium) can fit into here (points to top shape).

Learner YesTeacher Ok. So now when you go to this question here, it says:

what fraction of the shape would be covered by each of these shapes? So if I took one of these (points to trapezium) and put it into the shape (points to top shape), what part of that whole would be shaded or covered?

Con (F)

Learner A quarter.Teacher A quarter. That’s correct. Because four of them (points to

learner’s 4 outlined trapeziums on top shape) fit into there.

Analysis – Minute 3

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The teacher completes her explanation of the question by posing a more specific question in relation to the first shape (a trapezium) using procedural language “How would you fit this trapezium … into here… to work out how many of them fit into the whole?” She then allows the learner to “do” the activity. The learner explains what she would do, procedurally (“I think I’d put two…maybe two there (draws line through top shape)…”). The teacher prompts here to give the learner the language to state what she has just done “To cover the whole”. The teacher then asks the learner some questions which call on conceptual understanding from the learner (“what part of that whole would be covered?”). The teacher also reinforces the procedural activity the learner has undertaken to fill the whole with smaller trapezium shapes (“So if I took one of these (points to trapezium) and put it into the shape (points to the shape of ‘the whole’). The learner is able to answer correctly, saying “A quarter”. The teacher explains conceptually (in relation to the shapes they are working with) that they have found a quarter, saying “Because four of them (points to learner’s 4 outlined trapeziums on top shape) fit into there.”

Minute 4Speaker Utterance CodeTeacher Ok. So now how in the beginning, did you get… …a third

(points to learner’s answer)? What did you work out that you got a third for every part?

Diag (F)

Learner Because I thought that with all, I put…in the beginning, I originally thought the trapezium here (traces another line on the shape of ‘the whole’) and the parallelogram would be here (traces another shape on shape of ‘the whole’), and there would be the triangle. And I thought that they each take up a third. So I put one third for each of them.

Mult (2)

Teacher Oh, I see! So that’s what you actually understood the question that you take each part (points to 3 shapes in 2nd row), stick it in here, and then a third, one, two, three parts, would be covered. Now fractions are obviously equal parts, is that not so?

Awa (F)

Learner Yes, which doesn’t make sense.Teacher Ok. So then it doesn’t make sense. Alright, so now we’ve

worked out that that trapezium (points to trapezium in 2nd row) is a quarter. If we take this parallelogram (points to parallelogram in 2nd row), how would you take this and fit it into there (points to top shape)? So how many can we get into there? (long pause)

Con (P)Proc (bold text) (F)

Analysis – Minute 4The teacher probes again for explanation of what the learner had done in the test – trying to get to what the learner was thinking when she answered the test as she had. She hones in on the mathematical error – calling the three different unequal shapes thirds. The learner explains much more clearly than in the first minute that what she thought was that because all three different shapes (parts) could fit into the bigger shape (whole) at the same time, she could call them thirds. This is a common misconception in

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learners who have not yet realised that thirds have to be equal in size. The teacher establishes what the error is about when she acknowledges that the learner thought she could just “stick it in here, and then a third, one, two, three parts, would be covered”. In response to this statement, the teacher then refers back to the conceptual work that they have done so far in the interview, establishing that the trapezium is a quarter of the whole, and calls on more procedural activity. The conceptual links are implicit but the procedural explanation is full – the learner must now work out how many parallelograms will fit into the bigger shape.

Minute 5Speaker Utterance CodeTeacher How many of these…like you did the trapezium (points to

trapezium) to fit in there (points to top shape), how could we draw it to try and get these (points to parallelogram) to go in there (points to top shape)? Or what would you do to try and get them into there?

Proc (F)

Learner Um…(rubs out previously drawn lines) I think…I would draw in half again (draws a horizontal line through top shape) and then there would also be (draws lines to make 4 rhombi) four that could fit.

Teacher Ok, perfect. So there’s also four that fits. So now if we go back to the question: what fraction would be covered by each of the shapes? So if I took one of these (points to parallelogram) and put it in any one of them (points to learner’s 4 drawn parallelograms), what part of the whole is coloured in?

Con (F)

Learner A quarter.Teacher Again it’s a quarter. Ok, now the tricky one. The triangle.

How are we going to work out, if I take one of these (points to triangle in 2nd row), and put it into the whole (points to top shape), what fraction that is?

Learner (rubs out lines again) I’d draw a half (draws horizontal line through top shape again),

Analysis – Minute 5The teacher continues to probe the learner to think conceptually while she works through the next step of the activity (which involves procedural activity). The procedural explanation includes all of the required key steps (“How many of these…like you did the trapezium (points to trapezium) to fit in there (points to top shape), how could we draw it to try and get these (points to parallelogram) to go in there (points to top shape)? Or what would you do to try and get them into there?”) The conceptual explanation (“So if I took one of these (points to parallelogram) and put it in any one of them (points to learner’s 4 drawn parallelograms), what part of the whole is coloured in?”) includes the background and process required in the procedure – it explains how the learner must use what she has done when she finds out how many parallelograms can be drawn to fit into the bigger shape.

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Minute 6Speaker Utterance CodeLearner …and then triangles (draws lines to make 8 triangles)…

like how I did in the beginning…and then, yes…Teacher Ok, so now if we take one of them and covered part of that

whole, what fraction of the whole would have been coloured in?

Con (F)

Learner An eighth.Teacher An eighth. Ok, so now we know what those answers are: a

quarter (points to trapezium), an eighth (points to triangle) and a quarter (points to parallelogram). Ok, so now it says here: if you only have two of these (points to trapezium), ok…what fraction of this whole picture (points to top shape) would be coloured in?

Learner A half.Teacher It’s a half.Learner Oh, I see now.Teacher Can you see?Learner YesTeacher Alright, so now you got confused because you thought you

had to take each of those (points to the 3 shapes in 2nd row) and stick them in separately (points to top shape), is that correct?

Awa (F)

Learner YesTeacher Originally.Learner YesTeacher But now how did you get a third (points to learner’s

answer)…if you took, like we drew, if you’ve got that parallelogram and that one’s coloured in (shades parallelogram on top shape), and if we took the one triangle, and coloured it in (shades triangle in top shape),

Diag (F)

Analysis – Minute 6The learner is able to correctly find the fraction part which the triangle represents of the bigger shape showing that she is able to respond correctly to the teacher’s conceptual question “if we take one of them and covered part of that whole, what fraction of the whole would have been coloured in?” The teacher reiterates the learner’s error, referring back to the learner’s original comment (”because I thought just put like…it doesn’t matter what size they were as long as they fit in “) – this comment is now made in the light of the activity that has taken place in the interview. The teacher then probes the learner’s error further by asking “But now how did you get a third (points to learner’s answer)” and linking this question to the interview activity of finding parts of the whole using the given shapes. This probe hones in on the error which was that the learner had called parts “thirds” when they were not actually thirds of the given whole because they were not all equal in size.

Minute 7Speaker Utterance Code

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Teacher …and then the trapezium that you had originally drawn, and coloured that in (shades in trapezium on top shape). Now how did you get originally the answer of a third (points to learner’s answer), because that’s not a third of the picture (points to top shape), is it?

Diag (F)

Learner Because I thought that it could be any size something to do with three, and so I just put…because all three of them fit in there even though it’s a different size, I just put a third for each of them.

Mult (2)

Teacher For each of them. Oh, ok. Perfect. Right. That’s it. (counter at 6:30)

Analysis – Minute 7The teacher completes her probing question at the beginning of the seventh minute. The learner is able to explain, with an awareness of what she has learnt in the interview (she says “Because I thought that it could be any size something to do with three”) that she had mistakenly called shapes of different sizes thirds (this is the same error that the learner spoke about in minute 4). The teacher concludes the interview by expressing her understanding of the learner’s error (“For each of them”).

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4.2.4. Grade 9 learner interviewThe learner interview is focused on the following questions selected from the test set by the DIPIP grade 9 teacher group. The teacher did not seem to have selected a particular question with a particular error for his interview, rather in his interview he went through the whole test. The teacher began the interview by asking the learner about his response to the first question and then continued to ask the learner about his answers to the rest of the questions on the test as the interview progressed. In the excerpt from the interview that follows the following three questions are addressed.

1. The first question is a simple linear equation requiring the learners to solve for x.

2. The second question presents an open equation which the learners must complete by inserting the correct operation symbols so that the equality of the equation holds true. Instructions for the completion of the equation are given.

3. The third question presents an equation. There are two parts to the question: In the first part the learners have to say whether or not the given equation is correct and in the second part they are asked to correct the equation if it is not correct.

Figure 16: Extract from Grade 9 learner test

QUESTION 1

Solve for x: 5(x + 2) = 2x

QUESTION 2

Complete this statement so it is true that the LHS=RHS.

You may use any of the following operations in place of the *'s to complete the statement.

+,-, ÷, x, ( )

6 * 8 * 4 = 24

QUESTION 3

5 + 9=14 ÷ 2=7 x 3 = 21

3.1 Is this statement correct?

3.2 If not then what is a possible solution to the problem.

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Figure 17: Learner working in Grade 9 interview

This image not very clear since the teacher has signed over the learner’s working, the learner has written the following, upon which the first part of the interview is based:5(x + 2) = 2x5x + 2 = 7x7x = 7x

Extract from learner interviewThe interview is 30 minutes long. The extract below presents minutes 2-8 of the interview.

Minute 2Speaker Utterance CodeTeacher (continues) … here, eh , we, on question 1, this is a DIPIP

planner test, hey?Learner YesTeacher Ok we were requested to solve for x. Ok and uh as we have

been requested to solve for x, you wrote 5 plus 2 … 5 into x plus 2 is going to be equal to 2x. And then on this statement here (points to paper) you wrote 5x plus 2 is equal to, is going to be equal to ..?

Learner 7xTeacher 7x. And 7x is going to be equal to ..?Learner 7xTeacher Can you tell us how did you come to the answer? Diag (P)Learner Yes I wrote the test, but the test was too difficult because I

was not prepared. The mistakes I have done in this sum it is because I said 5 times x is equal to 5x and then I said 5 times 2 is equal to … plus, uh… , plus 2x and then it equals

Mult (1)

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to 7x. And then what I’ve done is that I said 7x is equal to 7x … because I was checking my sum.

Analysis – Minute 2The teacher probes the learner’s explanation of the error on the first question by first referring to the learner’s working on his test paper and then asking an open question, “Can you tell us how did you come to the answer?” This diagnostic probe does engage with the learner’s reasoning but is broad and not yet honed in on the mathematical error in question The learner responds (giving his first explanation of an error). The learner tries to explain what he was thinking when he did the working but he pauses and stumbles through this explanation (“is equal to … plus, uh… , plus 2x”). The learner has given a first explanation with which the teacher could now engage. This learner’s explanation is not coherent and he clearly does not know how to simplify the brackets in the equation or how to solve an equation. Part of the error is that the learner has applied the distributive law incorrectly. This he explains very poorly (“The mistakes I have done in this sum it is because I said 5 times x is equal to 5x and then I said 5 times 2 is equal to … plus, uh… , plus 2x and then it equals to 7x.”). The next part of the error is that he turned the equation into a statement of the equality of 7x and 7x, he has not actually solved for x, which was required by the question (“And then what I’ve done is that I said 7x is equal to 7x … because I was checking my sum”). The learner shifts the emphasis of his explanation from the actual mathematics of the question, to the procedure of “checking”, which he says he was doing.

Minute 3Speaker Utterance CodeTeacher You were checking your sum? Awa (N)Learner YesTeacher So here (points to paper) the question was that you must

solve for x. So you said 7x is equal to 7x. What makes you think that your answer was correct?

Diag (I)

Learner It’s because, what made my sum to be correct is that I think my sum is correct because I get 7x and then I started to say : no 7x is equal to 7x.

Teacher But the question here (points to paper) was to solve for x. Say what is the value of x. That x is equal to so-and-so. So that’s why I’m asking (points to paper) do you think that your answer is correct?

Proc (P)

Learner No, Sir you tell me that my answer is not correct but me I think, I think that my answer, I thought that my answer is correct because I saw 7x and then I write 7x is equal to 7x. But I was supposed to write x is equal to the answer.

Analysis – Minute 3In response to the learner’s explanation the teacher says “You were checking your sum?”. The comment shows no attempt to establish awareness of the mathematical content of the fuller explanation that the

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learner gave. The teacher only picks up on the last part of the explanation (minute 2: “because I was checking my sum”) and when he continues to probe does not respond to what the learner has said. Instead, the teacher repeats what the question called for (“you must solve for x”) and asks “What makes you think that your answer was correct?” The teacher then repeats a brief procedural explanation of what the question was calling for, but this explanation though correct is incomplete, because it does not include an explanation of the way in which the equation needs to be simplified and the isolation of the variable required in the solution of the equation. He just says, “Say what is the value of x. That x is equal to so-and-so”. He then repeats his question, “So that’s why I’m asking (points to paper) do you think that your answer is correct?” The learner responds, “No, Sir you tell me that my answer is not correct but me I think, I think that my answer, I thought that my answer is correct because I saw 7x and then I write 7x is equal to 7x. But I supposed to write x is equal to the answer.” In his response the learner again shows little understanding of what he was meant to do in order to solve the equation (multiply out the brackets, isolate the variable and solve for x) and he repeats what he wrote in the test. But he now acknowledges that he understands that he should have solved for x (“write x is equal to the answer”). In this exchange the teacher shows no engagement with the learner’s mathematical reasoning he only gives correctional instructions.

Minute 4Speaker Utterance CodeTeacher x is equal to the answer. But you didn’t do that. Instead

you said 7x is equal to there (points to paper). So, but I want to find out. Here (points to paper) you said that … the question is 5 into x plus 2 is going to be equal to 2x. How did you get the 7 here? Because here (points to paper) you wrote the 2x here, but here it is now 7x. How did you find the 7x?

Diag (I)

Learner It’s because of I add 5x plus 2x and then I got 7x, that’s why I get 7x there.

Mult (1)

Teacher Oh ok, so you said 7x is equal to 7x, ok. Awa (N)Learner YesTeacher Alright now I see. Ok now I want us to look at question 2.

We’ll come back to this one.Learner YesTeacher And then because that is how you got the 7x (points to

paper). But I realize that you said that you could have done this sum better by saying, by saying what?

Learner 7x is equal to 7x.Teacher 7x is equal to 7x.Learner YesTeacher By the way the question was that solve for x. Proc (P)Learner Solve for x.Teacher If this is so, what is it that you understand?Learner What I understand is that they want the value of x.

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Analysis – Minute 4The teacher continues to probe the learner’s reasoning behind his error but with no engagement with what the learner has offered by way of mathematical explanation so far in the interview. The learner repeats his explanation of his error (more succinctly this time), “It’s because of I add 5x plus 2x and then I got 7x, that’s why I get 7x there”). The teacher’s response to this “Oh ok, so you said 7x is equal to 7x, ok” shows no attempt to engage with the learner’s inability to apply the distributive law. The teacher does not pick up on the problem that 5(x + 2) ≠ 5x + 2x. After this, although the teacher says he will “come back to this one” he repeats one more time that the learner had written “7x is equal to 7x”. The teacher also repeats his correct but incomplete procedural explanation that what the question actually required was to “solve for x”.

Minute 5Speaker Utterance CodeTeacher They want the value of x, ok. To say x is equal to … Proc (P)Learner Equal to the answer.Teacher So but here did you do that? Diag (I)Learner No SirTeacher Ok but you know I didn’t understand something. When

you write a test you become scared because you said you were scared.

Learner YesTeacher You were scared of what (laughs)?Learner I was not prepared, Sir.Teacher You were not prepared, so you were scared (smiles)?Learner Yes SirTeacher Ok no don’t be scared. Eh, question 2, let’s look at

question 2. In question 2 here (pulls out paper from back of pile), they say here : complete this statement so it is true that the left hand side is equal to the right hand side ok? (Hand appears on camera to remove clock which is blocking view of paper) Ok so they say: complete this statement so it is true that the left hand side is equal to the right hand side. You may use any of the following operation in the place of an asterisk or star to complete this statement. Now here they’ve given you the four basic operations … plus, minus, division, multiplication and also the brackets. Now they say 6 asterisk 8 asterisk 4 is going to be equal to 24.

Analysis – Minute 5The teacher repeats his incomplete procedural explanation (see minute 3) of what it means to “solve for x” by saying that, “They want the value of x, ok. To say x is equal to …” and follows this with a further probe, “So but here did you do that?” to which the learner responds, “No Sir”. This probing again is directed and not engaged with the learner’s mathematical explanations which have been given. The teacher seems more interested in the correctness of the learner’s answer rather than its mathematical

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content. After a brief engagement with the learner about his being scared in the test because he did not feel prepared for the test, the teacher moves on to the second test question, reading through the question’s instructions. In doing so the teacher is giving a procedural explanation of what was expected in the second question, which is that learners complete a given incomplete equation, so that it expresses a valid equality. This explanation does not include all of the key steps required in the solution of the question although it is a correct and full reading of the question itself. A fuller explanation could have included an example of how to replace an asterisk with the appropriate operation.

Minute 6Speaker Utterance CodeTeacher I can see that you have written it there. But now, all of a

sudden, you have written BODMAS and, uh, here they say, eh, the question was, uh, that you were supposed to use either plus, minus, division, multiplication or the brackets. But you’ve written BODMAS. Why did you write BODMAS here?

Diag (P)

Learner I wrote BODMAS because I want to do this sum step by step. Eh BODMAS means Bracket Of Division Must Addition and Subtraction. Now what I’m, what I do I start with the brackets and then I, I … my second step it was division. After division I get this answer (points to paper) and this answer I subtract with 8 and then give me 24.

Mult (2)

Teacher Ok now I see that.Learner YesTeacher But now the way in which you were supposed to solve the

problem which was that … 6 asterisk 8 …Proc (P)

Analysis – Minute 6The teacher refers to the learner’s test script (learner’s response to question 2) and says that, “But now, all of a sudden, you have written BODMAS […] Why did you write BODMAS here?” In this way the teacher does engage broadly with the learner’s mathematical reasoning as written in the learner’s test script. The learner responds that, “I wrote BODMAS because I want to do this sum step by step.” The teacher does not respond to this explanation by showing any awareness of the learners’ mathematical error. His response is, “Ok now I see that” which has no mathematical content. He moves on to some more procedural explanation of the question (by way of reading out the question itself) which is again accurate but incomplete since it does not expand on what is required by the question.

Minute 7Speaker Utterance CodeTeacher (continues) … you were supposed to put any of the basic

operations to uh … make sure that the left hand side is equal to the right hand side, because that is what the statement is saying.

Con (P)

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Teacher Now, tell me, why do you think that your answer is correct?

Diag (N)

Learner My answer is correct because, eh … eh … I said 6 times 4 equals to 24 ne? And then I said 8 times 4 is equals to 32. And then I said 32 minus 8, it is equals to 24. That’s proves me that my answer is correct.

Mult (2)

Teacher Ok, now, eh … why do you write equals sign in the same line? Equal to 32 minus 8 equal to 24? (Points to paper)

Awa (N)

Learner I was trying to show you, sir, that, um … I’m going step by step. I said 6 times 4 it is equals to 8 times 4. And then eh 32, eh, 8 times 4 times 4, 8 times 4 equals to 32. It equals to 32 minus 8 equals to 24.

Analysis – Minute 7The teacher expresses the conceptual aspect (the equality of the two sides of an equation) of the question, explaining that, “you were supposed to put any of the basic operations to uh … make sure that the left hand side is equal to the right hand side”. He then probes further, asking, “Now, tell me, why do you think that your answer is correct?” Diagnostically this probe shows no connection to the learner’s mathematical reasoning as expressed in the test. The learner again tries to explain his working (his main point is that he was trying to do it step-by-step, he does not pick up on the errors in the working that he has written, he just reads it out as it has been written) but the teacher does not listen to what he says. The teacher pinpoints a technical error that the learner has made in his written script, asking, “Ok, now, eh … why do you write equals sign in the same line?” This technical error underlies in part the learner’s error (he is recording incomplete mathematical number sentences and equating them) but the teacher does not follow this through. The learner responds that it was because he was trying to show the teacher that he is “going step by step”. The teacher’s response to this explanation is given in the following minute.

Minute 8Speaker Utterance CodeTeacher Ooh, ok, … alright. We’ll also come back to that ok? Awa (N)Learner YesTeacher Now let us look at our question 3. In question 3 you are

given 5 plus 9 is equal to 14 divided by 2 is equal to 7 times 3 is equal to 21. Now they say: is the following statement correct? (Looks at L.)

Proc (P)

Learner No the following statement is not correct, sir.Teacher Now why didn’t you write … because they wanted, they

wanted I think maybe, a … a … an answer. But now you didn’t give an answer whether is it correct or incorrect here.

Diag (I)

Learner Oh sir, I didn’t understand that you want a… a… answer or the answer of 5 plus 9 minus 14, eh, divide by 2 equals to 7 times 3 equals to 21. I didn’t understand that you want no or yes.

Mult (3)

Teacher Ok. But here they say: is the following statement correct?

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Analysis – Minute 8The teacher does not respond to or explain the learner’s error, he simply says “Ooh, ok, … alright. We’ll also come back to that ok?” and moves on to the next test question (“Now let us look at our question 3”). He then gives a procedural reading of question 3 followed by another probe which does not engage with the learner’s mathematical reasoning. The learner explains, “Oh sir, I didn’t understand that you want a… a… answer” and goes on to explain that he thought maybe the “answer of 5 plus 9 minus 14, eh, divide by 2 equals to 7 times 3 equals to 21” was what was needed. Here the learner is trying to explain his confusion – that he had not understood that a “yes/no” answer was required. The learner expands on the working out that he thought he was meant to have given in answer to the question. Again, the teacher does not respond to the learner’s explanation (which contains a similar error to that in minute 7 of equating unequal numeric statements) but just reiterates what the test had asked, “But here they say: is the following statement correct?” negating what the learner has said and reinforcing his own authority rather than using the interview as an opportunity for diagnosis and development of awareness of errors. The interview continues a full 30 minutes, in a similar manner to the seven minutes presented here.

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4.3 Qualitative FindingsThe central research question - What does the idea of teachers’ interpreting learner performance diagnostically, mean in a context of learners’ interview? What do teachers, in fact, do when they interpret errors with the learner present?

In order to use the qualitative data we formulate three specific questions that together describe what do teachers do when they work with learners’ error diagnostically:

1. In what ways does the teacher’s procedural/conceptual engagement facilitate the learner’s conceptual/procedural understanding of the error?

Mathematical concepts and procedures are linked – in some ways they could be seen as different expressions of the same thing – one being more practical and one more abstract. When teachers are aware of the conceptual understanding required in order to carry out certain procedures, or the procedural knowledge that when understood fluently could facilitate conceptual understanding, they can use these two as required to help learners develop their mathematical knowledge. In the example of the grade 4 learner interview, the teacher spends several minutes working very slowly and systematically through the procedure of subtraction of two digit numbers, linking the procedural explanation to the place values (conceptual knowledge) of the numbers when necessary. In the Grade 5 interview on the other hand, throughout the whole interview the teacher repeats the test question, reiterates what the learner was doing wrongly in the test (minute 6), repeats what the learner is saying, asking irrelevant question (minute 4), repeats the probing question (minute 4) but at no point does the teacher, in fact, engage with the procedure or provides the learner with the relevant concept, with which to view the procedure required by the question of converting the mixed number into an improper fraction. The overall effect of a systematic procedural explanation (made with the necessary conceptual links) is that the conceptual understanding of the learner is reinforced and the procedural knowledge of the learner is developed. In the grade 4 learner interview, in which the learner does not know how to subtract two digit numbers from each other when there is an impasse, the teacher works predominantly procedurally but at particular moments of the interview connects the procedure to the underlying number concept. Step by step the teacher guides the learner (see minutes 3 and 6), and checks that he understands the instructions (see minute 3 and 8). Whilst doing that (and whilst linking the hands on work with the unifix cubes to the subtraction of the given two numbers in the example see minutes 5 and 6), at specific points of time in the interview, the teacher brings the conceptual background of the procedure. For example in minute 3, she is asking the learner to show her where the tens and the units are in the given number of the example they are working together on, or asking her conceptual questions such as “So can you now take away?” – calling on the learner to

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reason about whether or not she has overcome the impasse (minute 8). The teacher keeps bringing the learner back to identifying the numbers according to their place value, which links to the underlying misconception evidenced in the test working of the learner. From this example, we see that the teacher chooses when to illuminate the conceptual background of the procedure and how. She does it by asking the learner questions that direct her to use the concepts (for example minutes 3, 4) or by thinking about the two concepts relationally as in “ …does it look the same as the units?” (here the concrete unifix cubes enable the learner to realise the difference between a digit in the units place and a digit in the tens place). In the Grade 5 and 9 learners’ interviews, this is not happening. In the Grade 5, after 6 minutes of interaction, the partners are locked in the error – the learner tries to explain it and the explanation gets ignored and the teacher restates the incorrectness of the learner working and keeps asking the learner to explain the error, to show it, to write it and to state the reasons. In contrast to the grade 4 learner, the grade 5 learner loses her confidence in the interaction and states: “I’m not thinking right, teacher” (see minute 5). In the grade 5 interview the teacher does not open the mathematics of the procedure (converting from an mixed number to an improper fraction let alone offer conceptual underpinning to the procedure. The grade 9 learner repeatedly acknowledges that he has made errors but this is more in defence to the teacher’s on-going questioning about the correctness of his answer. The teacher’s procedural contributions are to repeat the literal meaning of “solve for x” by saying, “They want the value of x, ok. To say x is equal to …” (minute 5). These statements do not respond directly to the learner’s mathematical explanations (which relate to his incorrect application of the distributive law). Furthermore, he keeps moving on to the next test question and so they do not make any progress in unpacking the errors which are expressed.A focused interview establishes the error by positing a relation between the procedural and conceptual, the learner is confident to articulate her/his errors to the teacher, and the teacher’s probing is focused on the mathematics.

2. What kind of probing is productive in that it gathers relevant evidence for teachers to make judgement on the nature of the error and on when to establish it in the course of the interview?

Probing with the intent to glean mathematically relevant information from learners is productive. Productive probing can be open or directed, it relates to the learner’s work under scrutiny and it is in line with what the learner is in fact saying. Teachers need to listen to their learners when they ask questions. In the example of the grade 4 learner interview the teacher allows the learner to explain her working in full and does not interrupt her or correct her (minute 2). This is very different to the Grade 5 interview in which the teacher interrupts the learner (minute 4) and even insults him (minute 5). In the Grade 9 interview the teacher asks the learner for explanation of the error, the learner stumbled attempt to explain his error

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(minute 2), which seems to be that he does not know how to simplify the brackets in the equation but the teacher ignores the learner’s verbalised mathematical confusion and only addresses (see minute 3) the learner’s last few words, in which the learner in fact shifts his explanation to the procedure of “checking”. In the DIPIP interviews teachers were told to keep their minds open and listen and respond to their learners. The qualitative (and the quantitative) analysis shows that many of the teachers seem more intent on probing in decontextualized ways (i.e. not related to what the learner verbalizes). They do that when they expose the learners’ lack of knowledge (see Grade 5 learner

interview, minute 4); they re-teach the content, as in the grade 5 learner interview where the

teacher uses leading questions instructing the learner to write the mixed number “correctly” although the error has not been established yet (see minute 7) and close to the end of the interview shows the learner the correct procedure of converting mixed number into an improper fraction; or

they repeat the test question several times but without addressing the problem as in the grade 9 learner interview where the teacher repeats a brief procedural explanation of what the question was calling (see minutes 3 , 4 and 5), and does not address the problem- the learner’s inability to apply the distributive law.

None of these ways are productive as neither of them helps the teacher to establish a focused interaction on the error. In these three instances, the teacher’s response does not shift the level of discussion from what the learner did to the nature of the error. At the beginning of an interview a broader probe is probably suitable but once the interview progresses, the teacher needs to take on the answers given by the learner and her/his own knowledge of the field and delve more deeply into the mathematical content related to the error. In this way the interview moves from a broad focus to more mathematically focused way of probing, and questions become more directed or specific. In the Grade 7 learner interview, for example, the teacher starts with a very broad question “what did you find difficult with this question?” (Minute 1) As the interview progresses the probing becomes more specific and contextually relevant to what the learner is verbalizing. First the learner says that she was confused by the question and was not sure what she had to do. This is still broad (which continues in minute 2). During this time the teacher is trying to call out the error and does it partially, in line with her broad treatment of the error at this stage of the interview. But then the teacher changes the type of questions – they become more specific in terms of the mathematical content (for example, “How would you fit this trapezium … into here… to work out how many of them fit into the whole?” in minute 3) or even more specific (“what did you work out that you got a third for every part?” in minute 4). She also relates her question to the test question (“So now when you go to this question here, it says: what fraction of the shape would be covered by each of these shapes? So if I took one of these (points to trapezium) and put

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it into the shape (points to top shape), what part of that whole would be shaded or covered?” in minute 4). This allows a process in which the specific nature of the error gets established, in response to what the learner verbalizes. This mode of probing enables the teacher to gather evidence from the learner with which she continues to establish the nature of the error, using explicit mathematical language. So in minute 4 the learner says:

Because I thought that with all, I put…in the beginning, I originally thought the trapezium here (traces another line on the shape of ‘the whole’) and the parallelogram would be here (traces another shape on shape of ‘the whole’), and there would be the triangle. And I thought that they each take up a third. So I put one third for each of them.

And the teacher responds and uses the opportunity to state the errorOh, I see! So that’s what you actually understood the question that you take each part (points to 3 shapes in 2nd row), stick it in here, and then a third, one, two, three parts, would be covered. Now fractions are obviously equal parts, is that not so?

In the first four minutes of the conversation, the interaction changes in a very controlled way. It moves from the learner’s feeling confused about the question, not sure what she has to do (minutes 1and 2), to trying out fitting the trapezium (minute 3) and lastly to stating that she thought that all the three different unequal shapes could be thirds (“each take up a third”). The last of these learner’s moves enables the teacher to define the error more fully (“fractions are obviously equal parts, is that not so?”). This is in direct contrast to the Grade 5 learner interview in which 9 out of the 13 minutes are spent on repeated instructions as to from where and how the learner should re-write his working. Throughout the interview the error is repeated and the awareness of the error does not develop as a result what the teacher or the learner says. Two minutes before the end of the interview the teacher shows learner the procedure for conversion of a mixed number into an improper fraction. Thus teacher’s probing the learner is an act of judgement about what to take up from what the learner brings, how to connect it to the related mathematical knowledge, what specifically to address in the next probing and more broadly, when and how to ask the learner for further information.

3. Given that in the context of an interview partial explanations would have a place, what is the role of a full explanation?

The progression of a discussion, if it is to be interactive, will naturally result in more partial explanations as the interview progresses. It would be nice to see a learner able to give a full explanation of the concept by the end of the interview. The learner should have reached the point where he/she is able to do so by the end of a successful interview.

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The example of the Grade 4 learner interview given above shows the way in which carefully structured, progressive partial explanations build up to a comprehensive and well scaffolded full explanation during the course of an interview. By the end of this interview, the learner had recognised the error in her initial working and was able to correctly perform the procedure required to subtract one two-digit number from another in an instance where “carrying” was required.

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Section 5: Conclusion

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Recommendations for professional development and for further researchThe interview activity undertaken during the last six months of DIPIP Phases 1 and 2 highlighted the difficulty that teachers have when dealing with learners’ mathematical errors in the context of conversation with their learners. DIPIP teachers carried out these interviews having spent more than two years in the project, working in groups with colleagues and district officials under the guidance of group leaders who were highly qualified mathematics education experts (university staff members or post graduate students). These could be considered ideal circumstances for optimal performance, and yet the teachers struggled to engage meaningfully with their learners in relation to errors they had made. This evidence necessitates a caution with regard to the potential of developing teacher competence in addressing learner errors simply through involvement in group guided discussions. DIPIP findings did not relate directly to teacher knowledge but the implications of the findings could be evidence of this knowledge. The poor overall demonstration of awareness of error (in relation to diagnostic reasoning demonstrated through the use of probing questions) could highlight general weakness in the mathematical content knowledge of the teachers.

Recommendations for professional developmentTeacher involvement in activities such as learner interviews is useful and can be undertaken as a professional development activity in the context of guided group discussions. It should be noted that the teachers’ find it difficult to engage with their learners meaningfully with regard to errors that they make. Hence careful planning is required in relation to mathematical content development necessary for meaningful discussions of errors that arise in these content areas.

Recommendations for further researchFurther research could be done into the relationship between teacher knowledge of mathematical content and their ability to engage diagnostically with learners’ errors and demonstrate an awareness of mathematical errors.

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ReferencesBernstein, B. (2000). Pedagogy, symbolic control and identity. Oxford:

Rowman & Littlefield.Black, P., C. Harrison, C. Lee, B. Marshall, and D. Wiliam. (2003).

Assessment for learning. London: Open University Press.Black, P. and Wiliam, D. (2006) Chapter 1: Assessment for Learning in the

classroom in Gardner, John (ed.) Assessment and Learning (London: Sage).

Charlot, B (2009) “School and the pupils’ work”. Educational Sciences Journal 10, pp 87-94.

Gipps, C. (1999) “Socio-cultural aspects of assessment”. Review of Research in Education 24. pp 355-392.

Gipps, V.C., and J.J. Cumming. (2004). Assessing literacies. http://literacyconference.oise.utoronto.ca/papers/gipps.pdf.

Shalem, Y and Slonimsky, S (2010) Seeing epistemic order: construction and transmission of evaluative criteria British Journal of Sociology of Education, Vol. 31, No. 6, November 2010, 755–778.

Shalem, Y., & Sapire, I. (2012). Teachers’ Knowledge of Error Analysis. Johannesburg: Saide

Shalem, Y., Sapire, I., Welch, T., Bialobrzeska, M., & Hellman, L. (2011). Professional learning communities for teacher development: The collaborative enquiry process in the Data Informed Practice Improvement Project. Johannesburg: Saide. Available from http://www.oerafrica.org/teachered/TeacherEducationOERResources/SearchResults/tabid/934/mctl/Details/id/38939/Default.aspx

Sheinuk, L (2010) Intermediate Phase Mathematics Teachers’ Reasoning About Learners’ Mathematical Thinking. A Research Report submitted to the Wits School of Education, Faculty of Humanities, University of the Witwatersrand in fulfillment of the requirements for the degree of Master of Education.

Shepard, L.A. (2000). The role of assessment in a learning culture. Educational Researcher 29, no. 7: 4–14.

Vygotsky, L.S. (1987). The collected works of L.S. Vygotsky. Volume 1: Problems of general psychology, ed. R.W. Rieberand and A.S. Carton. New York: Plenum Press.

Young, K. (2005). Direct from the source: the value of ‘think-aloud’ data in understanding learning Journal of Educational Enquiry. Vol.6 (1), 19-26.

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Appendices:

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Appendix 1

Interview Coding CriteriaCriteria Category descriptors: Learner interviews

Full Partial Inaccurate Not present

ProceduralThe emphasis of this code is on the teachers’ procedural explanation of the error. Teaching mathematics involves a great deal of procedural explanation which should be done fully and accurately for the learners to grasp and become competent in working with the procedures themselves.

When the teacher explains the error to the learner/ probes further, the teacher demonstrates a procedure.

The procedure is accurate and includes all of the key steps in the procedure.

When the teacher explains the error to the learner/ probes further, the teacher demonstrates a procedure.

The procedure is accurate but it does not include all of the key steps in the procedure.

When the teacher explains the error to the learner/ probes further, the teacher demonstrates a procedure.

Teacher’s use of procedure is inaccurate and thus shows lack of understanding of the procedure

No procedural explanation is given

ConceptualThe emphasis of this code is on the teachers’ conceptual explanation of the procedure followed in the error. Mathematical procedures need to be unpacked and linked to the concepts to which they relate in order for learners to understand the mathematics embedded in the procedure.

When the teacher explains the error to the learner/ probes further, the teacher includes conceptual links.

The explanation illuminates conceptually the background and process of the procedure.

When the teacher explains the error to the learner/ probes further, the teacher includes conceptual links.

The explanation includes some but not all of the key conceptual links which illuminate the background and process of the procedure.

When the teacher explains the error to the learner/ probes further, the teacher includes conceptual links.

The explanation includes poorly conceived conceptual links and thus is potentially confusing.

No conceptual links are made in the explanation.

Awareness of mathematical errorThe emphasis of this code is on teacher’ explanation to the leaner of the actual mathematical error and not on learners’ reasoning.

Teacher explains the mathematical error to the learner.

The explanation is mathematically sound and suggest links to common misconceptions or errors

Teacher explains the mathematical error to the learner.

The explanation is mathematically sound but does not link to a common misconception or error

Teacher explains the mathematical error to the learner.

The explanation is mathematically flawed

No explanation is given of the mathematical error

Diagnostic reasoning in feedbackThe idea of error analysis

Teacher seeks to find out the learner’s mathematical

Teacher seeks to find out the learner’s mathematical

Teacher probes but does not seek to find out the learner’s

No attempt is made to probe

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goes beyond identifying a common error and/or misconception. The idea is to understand the way the teacher goes beyond the actual error to try and follow the way the learner was reasoning when s/he made the error. The emphasis of this code is on the teacher’s attempt to engage with the learner on how the learner was reasoning when s/he solving the question.

reasoning behind error. In response to the error the teacher probes further and asks the learner to explain the steps of her/his reasoning.

Probing engages with learner’s reasoning and is open. Probing hones in on the mathematical error.

reasoning behind error. In response to the error the teacher probes further and asks the learner to explain the steps of her/his reasoning.

Probing engages with learner’s reasoning and is open but is too broad (not sufficiently honed on the mathematical error)

mathematical reasoning behind error. In response to the error he/she teaches further.

Probing is directed. The teacher uses only leading questions with little engagement with what the learner brings.

or listen to the learner’s mathematical reasoning behind the error.

Re-teaching without any questioning.

Use of everyday knowledgeTeachers often explain why learners make an error by appealing to everyday experiences that learners draw on and confuse with the mathematical context of the question. The emphasis of this code is on the quality of the teacher’s use of everyday, judged by the links s/he makes to the mathematical understanding s/he attempts to advance, when engaging the learner on her/his incorrect answer

When the teacher explains the error to the learner/probes further, the teacher appeals to the everyday.

Teacher’s use of the ‘everyday’ enables mathematical understanding by making the link between the everyday and the mathematical clear

When the teacher explains the error to the learner/probes further, the teacher appeals to the everyday.

Teacher’s use of the ‘everyday’ enables learner thinking but it does not properly explain the link to mathematical understanding

When the teacher explains the error to the learner/probes further, the teacher appeals to the everyday.

Teacher’s use of the ‘everyday’ dominates and obscures the mathematical understanding, no link to mathematical understanding is made

No discussion of everyday is done.

Multiple explanations of errorThis code examines the teacher’s probing/directing for learner explanations of their errors. One of the challenges in learner interviews is for teachers to give their learners the opportunity to voice their explanations of the error. This is because when learners give voice to the thinking behind their errors they can start to think for themselves about the flaws in their own explanations.

Four or more explanations are given by the learner in the interview

Three explanations are given by the learner in the interview

Two explanations are given by the learner in the interview

One error explanation is given by the learner in the interview

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Appendix 2

Interview Coding InstrumentVideo name (teacher and task):

Minute 1 2 3 4 5 6 7 8 910

11

12

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CriteriaProcedural

Conceptual

Awareness of mathematical errorDiagnostic reasoning in feedbackUse of everyday knowledgeMultiple explanations of errorSummary:

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Appendix 3

Interview Coding Notes templateTable 1 (these codes were quantitatively summarised and used to inform the analysis).Error focus notes (based on detailed coding in excel sheet)Teacher name and grade:

Teacher, grade 7 (see 4.2.3)

Time slot

Criterion and level assigned

Explanation, discussion and evidence

00:09 Diagnostic reasoning and awareness of error, partial

Teacher clearly indicates to the leaner which of the questions she had problems with and wants the learner to explain why she struggle with the question.

01:34 Awareness of error, diagnostic reasoning, conceptual and procedural, partial

Teacher interprets the question for the learner. The learner then finds the question easy to do, which is fitting a trapezium onto a given shape.

03:02 Awareness of error, diagnostic reasoning, conceptual and procedural, full

Teacher refers learner to her incorrect answer and asks her how she got that, which the learner explains and says she realises it doesn’t make sense. She then asks the learner to fit a parallelogram onto the given shape (03:45), which the learner finds also easy to do.

04:43 Conceptual and procedural, full

Teacher then asks learner to fit a triangle onto the shape, which the learner finds it easy to do.

05:43 Procedural , conceptual and diagnostic reasoning, full

Teacher refers learner back to her original answer again and asks her to explain how she got a third from fitting each of the pieces onto the given shape, i.e. trapezium, triangle and parallelogram. Learner says she got a third because she had three pieces to fit, regardless of the size of the pieces.

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Table 2 (this information informed the analysis but was not quantitatively analysed).General Interview summary

(see separate sheet for criteria)

Teacher name and grade:

Teacher, grade 7 (see 4.2.3)

Criterion Level Explanation, discussion and evidenceInteraction 3 Teacher created an atmosphere that is conducive

for the learner to express herself with confidenceConceptual development

3 Teacher shows an excellent mastery of the concepts embedded in the question through her interpretation of the question and guiding of the learner.

Procedural development

3 Teacher works very well in explaining how the learner should approach the question

Teacher mathematical knowledge

3 Through the interview, the teacher demonstrates that she’s got great mathematical knowledge.

Questioning 3 Teacher follows an acceptable format of posing question to the learner, and gives sufficient time to respond. She also probes the learner on explaining how she got her incorrect answers wrong.

Use of concrete material

1 No concrete materials are used in the interview

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Appendix 4

Example errors verbalised by learners in their interviewsInterview 1: Grade 3Learner words Problem and related error(s)I said 9+7 and I said plus 5Because 65-19.. I know how to do that but when I have to add 17 it is difficult. Then I said, I organised my numbers into units and tens and hundreds.

Learner was trying to work out the following “65-19+17”.Errors:Focus on units in an isolated manner/uses place value haphazardlyIgnores operation symbols/does not know how to deal with the three number string

Interview 2: Grade 3Learner words Problem and related error(s)90 + 15 equals 105It would be like a times.

Work out the number of rows needed to seat a certain number of peopleErrors:Says that the number of spectators is the number of rowsNot able to express the operation required as “division” (although if this were properly probed it could be that she is thinking about it in an inverse way and the teacher may have worked with this).

Interview 3: Grade 4Learner words Problem and related error(s)Because you can’t subtract 6 (points to the 6 of 76) from 9 (points to the 9 of 19). So I took 1 away from the 7 (point to the 7 of 76, crosses out the 7 and writes 6 above it) so it becomes 6 and this one becomes 7 (crosses out the 6 of 76 and writes 7 above it). Now you still can’t subtract it, so I took away 1 again (crosses out the 6 written above the 7 of 76 and writes 5 above it) and it became 5. And I added this one (crosses out the 7 written above the 6 of 76 and writes 8 above it), so it became 8.

Learner was trying to work out 76 – 19Errors:“Borrows” from the tens repeatedly by only carrying one across to the units each time.Does not understand relative values of digits according to place value

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Interview 4: Grade 5Learner words Problem and related error(s)Teacher by connecting all this length (indicates by circling both hands above paper), teacher.I learned measure how much.

Learner was explaining how she got the answer to the question: “If fencing is bought in lengths of 10 metres how many lengths must Mr Brown bought, buy?”Errors:Gives the perimeter as the answer, does not work out how many lengths need to be bought.Tries to apply his knowledge of how to work out the perimeter (“measure how much”) to this question.

Interview 5: Grade 5Learner words Problem and related error(s)Teacher I said 2 times 3, then I said is 6 and then I said 2 times 6 is equals to, is equals to 12 teacher. Then I said 12 over 6 teacher.

Learner was trying to convert a mixed

number (234

) to an improper fraction

Errors:Works incorrectly with the whole number and fractional partsDoes not understand that the different forms of the number should be equal (represent the same value)

Interview 6: Grade 6Learner words Problem and related error(s)I was supposed to start at zero, when I started at one…

Learner was trying to label the vertical axis of a graphErrors:Started labelling incorrectly

Interview 7: Grade 7Learner words Problem and related error(s)… because I thought just put like…it doesn’t matter what size they were as long as they fit in there.And I thought that they each take up a third. So I put one third for

Learner was trying to work out fractional parts of a wholeErrors:Not sure about the wording of the question

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each of them. and so thinks she has to fit the different shapes in at the same time.Thinks that fractional parts do not have to be the same size

Interview 8: Grade 7Learner words Problem and related error(s)Counts the shapes she has put in “(Pointing at various sections of diag A) 1, 2, 3, 4, 5, 6. Ok.” But does not count a 7th one that is also there in her drawing on the test answer sheet

Learner was trying to work out the how many diamond shaped quadrilaterals would fit into a star shapeErrors:Does not fit the shapes into the bigger shape without overlapping them – so not fitting them into the bigger shape correctly.

Interview 9: Grade 8Learner words ErrorSo I understood that the one with 2, this had to be a bigger value in order for it to equal the one thing; and using 3 those had to be smaller values in equal in order to equal to the one value.

Learner was trying to create different equations using given symbols, for which she had to work out relative values.Errors:Difficulty in understanding the wording of the questionDid not work with the given information to work out the relative values of the symbols – made assumptions about “bigger” and “smaller” values

Interview 10: Grade 8Learner words Problem and related error(s)Because on the block it showed us, what is nine minus five? So I worked it out and it gave me five.

Learner was trying to work out the missing value in the equation: “9 – 5 = - 9”Errors:Does not keep in mind the equality of both full expressions on the two sides of the equation

Interview 11: Grade 9Learner words Problem and related error(s)I made it 0,5 for 5, because the graph was not big enough

Learner needed to label the axes of a graphLearner had to find an equation using a table

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I just multiplied. I thought I might find the name of the graph

of valuesErrors:Does not work out an appropriate scale, renames the values to be written on the axes insteadArbitrarily works with tables of values with no method as to how to find out the equation they may represent

Interview 12: Grade 9Learner words Problem and related error(s)As in for twenty-five (points to number on horizontal line), I did a two point five, to, two hundred and fifty (points to number on vertical line), because there was no twenty-five. So I used these variables (points to numbers on horizontal line) just to say this is five, even though it’s zero comma five, I said this is five, and like that…

Learner was trying to label the vertical axis of a graphErrors:Reduced the sizes of the numbers so that he could fit them in according to the “scale” that he used.

Interview 13: Grade 9Learner words Problem and related error(s)It’s because of I add 5x plus 2x and then I got 7x, that’s why I get 7x there.And then what I’ve done is that I said 7x is equal to 7x because I was checking my sum.

Learner was trying to find the solution to the equation 5(x + 2) = 2xErrors:Does not know how to apply the distributive law correctlyDoes not know how to isolate x in the equationDoes not realise that he has to give a value of x in answer to the question

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Appendix 5

Grouped Grade GraphsFigure 5.1 Procedural explanations in interviews by grouped grades

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Figure 5.2 Conceptual explanations in interviews by grouped grades

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Figure 5.3 Awareness of error in interviews by grouped grades

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Figure 5.4 Diagnostic reasoning in interviews by grouped grades

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Figure 5.5 Use of the everyday in interviews by grouped grades

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Figure 5.6 Learner multiple explanations in interviews by grouped grades

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