The study investigated teacher knowledge of error analysis in differential calculus. Two teachers were the sample of the study: one a subject specialist and the other a mathematics education specialist. Questionnaires and interviews were used for data collection. The findings of the study reflect that the teachers’ knowledge of error analysis was characterised by the following assertions, which are backed up with some evidence: (1) teachers identified the errors correctly, (2) the generalised error identification resulted in opaque analysis, (3) some of the identified errors were not interpreted from multiple perspectives, (4) teachers’ evaluation of errors was either local or global and (5) in remedying errors accuracy and efficiency were emphasised more than conceptual understanding. The implications of the findings of the study for teaching include engaging in error analysis continuously as this is one way of improving knowledge for teaching.

Aside from the challenges of a variety and the complexities of students’ errors, analysing such errors is a fundamental aspect of teaching for mathematics teachers (Ball, Thames & Phelps,

‘Calculus is an important subject area within mathematics, and this underlies the argument for introducing it to non-specialists’ (Orton,

The main research question for the reported study is: What teacher knowledge of error analysis do the teachers possess? In particular:

How do the teachers identify students’ errors?

How do the teachers interpret students’ errors?

How do the teachers evaluate students’ errors?

How do the teachers remedy the students’ errors?

The four sub-questions above are in line with Peng and Luo's (

This section discusses the key ideas needed in the interpretation of the results of the study, namely error types in mathematics, students’ errors in differential calculus, teacher knowledge and teacher knowledge for error analysis.

Olivier (

Errors have been classified differently by various researchers and mathematics educators. Legutko (

A mathematical error is made by a person (student, teacher) who in a given moment considers as true an untrue mathematical sentence or considers an untrue sentence as mathematically true. Didactic errors refer to a situation when teachers’ behavior is contradictory to the didactic, methodological and common sense guidelines. (p. 149)

Mathematical errors include giving an incorrect definition of a mathematical concept and a wrong application of the definition, making a generalisation after observing a few particular cases and incorrect use of mathematical terms. Didactical errors include unsuitable selection of examples used in the formation of a concept, incoherent structure of teaching such as teaching concepts of a higher order before concepts of lower order.

Orton (

Structural errors were described as those which arose from some failure to appreciate the relationships involved in the problem or to grasp some principle essential to solution. Executive errors were those which involved failure to carry out manipulations, though the principles involved may have been understood. Arbitrary errors were said to be those in which the subject behaved arbitrarily and failed to take account of the constraints laid down in what was given. (p. 4)

Mathematical errors may also be procedural or conceptual (Eisenhart et al.,

Procedural knowledge refers to mastery of computational skills and knowledge of procedures for identifying mathematical components, algorithms, and definitions. … Conceptual knowledge refers to knowledge of the underlying structure of mathematics – the relationships and interconnections of ideas that explain and give meaning to mathematical procedures. (Eisenhart et al.,

To these we can add that procedural knowledge has two components: (1) knowledge of the format and syntax of the symbol representation system and (2) knowledge of rules and algorithms, some of which are symbolic, that can be used to complete mathematical tasks. It could therefore be argued that ‘the fluent execution of algorithm represents an aspect of procedural fluency’ (Long,

In a study by Orton (^{3} − 3^{2} + 4, that is, the points on the graph where the gradient is zero. Students in this study managed to find the gradient function (the derivative) ^{2}−6^{2} − 6^{2} − 6^{2} − 6^{2} − 6

In a differentiation task of the same study when expanding 3(^{2} students lost the middle term 6^{2} is given as ^{2} + ^{2}. Students distribute the power over the brackets as would be the case in (^{2} = ^{2}^{2}. Students should have thought in reverse by realising that in real numbers one cannot find the factors of the sum of two squares. But in complex numbers such roots would exist because (^{2} + ^{2} since by definition ^{2} = −1.

In a study by Thompson (^{2} from the first principles (i.e. using the definition of derivative); the students failed to make proper substitution for ^{2} for

The two kinds of teacher knowledge that have received much attention in the work of Shulman (

It represents the blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction. (Shulman,

This seems to suggest that SMK is necessary in teaching but it is not a sufficient condition. For SMK to operate in teaching it must be blended with pedagogy. In teaching, for example when a teacher uses a representation in explaining a concept, it means that one can now talk about the knowledge of teaching as the activity is now beyond a mere possession of subject matter knowledge.

Ball et al. (

According to Moru and Qhobela (

Shalem et al.'s (

In investigating teacher knowledge for error analysis, Peng and Luo (

In a study by Moru and Qhobela (

The sample consisted of two lecturers who are also the co-authors of this article. These lecturers were involved in the teaching of calculus in the Department of Mathematics and Computer Science at a university in Lesotho. Both lecturers (abbreviated T1 and T2) had also conducted tutorials for social science students. Their teaching experience ranged from two to three years. T1 is a subject matter specialist whilst T2 is a mathematics education specialist. Because lecturing is a form of teaching, the lecturers are sometimes referred to as teachers in this article.

The first draft of the questionnaire was constructed by the first author. The second author critiqued the questionnaire and continuous discussions occurred between them until the final draft was arrived at. The questionnaire was administered by the first author. Each research participant (T1 and T2) had one month to complete the questionnaire as it was very demanding in terms of time and thinking. The questionnaire consisted of errors committed by the majority of second-year social science students taught by the first author. In total, 103 students sat the course examination. The examination was taken in May 2013. In the teaching the students had covered amongst others the following content: concept of limits, definition of derivative and finding the derivatives of functions either by the use of the definition or by differentiation rules, geometrical and algebraic interpretation of derivatives as gradient functions and finding equations of tangent lines to the curve. The questions in which the students committed most errors in the examination are:

Question 1: Use the definition of derivative to find

Question 2: Find all the points on the curve

Almost half of the students (49%) committed errors when responding to Question 1 and 61 students (59%) committed errors when responding to Question 2. The sample student responses were chosen by the first and the second authors who further selected the scripts in such a way that there was some variation in the committed errors.

For teachers’ analysis of students’ errors the teachers were given a questionnaire consisting of a question and the corresponding student response. They were asked to study the question and the corresponding student response and then asked to perform the following tasks: (1) identify the error, (2) write the possible causes of the identified error(s), (3) show how the identified error can impact on the mathematical performance of the student either in doing or learning mathematics and (4) suggest the strategies or explanation that one would provide in remedying the error.

The follow-up structured interviews were constructed and conducted by the first author. The interviews emerged from the teachers’ responses to the questionnaire. These were conducted with the intention of seeking clarification on some questionnaire responses. The expectation was that these interviews would also start a conversation amongst colleagues about the importance of paying attention to students’ errors. The interviews for teacher knowledge of error analysis for each question were held separately with the individual lecturers (T1 and T2). On average the first stage took one and a half hours and the second just one hour. This was with the intention of achieving in-depth data with regard to the teachers’ knowledge of analysis of students’ error.

The analysis of the teachers’ knowledge of students’ errors was done by the first and the second authors. They were investigated in their ability to: (1) identify the error, (2) suggest the possible causes of the error, (3) judge how the error may impact on the performance of the student in doing or learning mathematics and (4) offer remedial strategies for the errors. After studying the teacher knowledge for error analysis, assertions for the displayed knowledge were constructed.

Permission to use students’ examination scripts was sought from the head of the Department of Mathematics and Computer Science. This is because students’ examination scripts are the property of the university. The agreement made was to conceal the students’ identities. With regard to the involvement of T1 and T2 their informed consent was sought by the first author.

Data was collected by the first author for both the questionnaires and interviews. This is because the first author had already established a rapport with the two lecturers who are not only colleagues in the department but also former students of the first author. To ensure reliability of the research instruments the second author assessed the clarity of the questions to see if they would be interpreted the same by different people for consistency of results. The suggested comments were discussed until an agreement was reached. T1's and T2's responses to the questionnaire and interviews show that the study did produce consistent results. This is because the two lecturers interpreted the questions and instructions in the questionnaire and interviews in the intended way.

In order to verify the validity of the questionnaire, it was given to the second author together with the research questions and the four error phrases by Peng and Luo (

verbatim accounts of what people say, for example, rather than the researchers’ reconstructions of the general sense of what a person said, which would allow researchers’ personal perspectives to influence the reporting. (Seale,

Data analysis of teacher knowledge of error analysis yielded the following five assertions: (1) teachers identified the errors correctly, (2) the generalised error identification resulted in opaque analysis, (3) some of the identified errors were not interpreted from multiple perspectives, (4) teachers’ evaluation of students’ errors was either local or global and (5) in remedying errors accuracy and efficiency were emphasised more than conceptual understanding. The presentation follows the order in which the assertions have been listed.

In all students’ work, the teachers managed to identify almost all the errors. A few that were left during the completion of the questionnaire were recovered during the interviews. This was a sign that failure to identify the errors in the first stage of data collection was not a sign of lack of knowledge but just a slip on the part of the teachers. The first example (see

Error analysis of S74's response.

The question asked is now followed by a response:

Researcher (R): Can we look at the minus
T2:
Oh! And the minus

This shows that an error that was left out was just an omission and not related to lack of subject matter knowledge. The same thing happened to T1 when discussing his analysis of S61's work (see

Error analysis of S61's response.

Thus, the discussion during T1's interview was directed towards wanting to see if he would recognise that he had left some errors that appeared before Line 3. The discussion with him went as follows:

R:
You have shown that the errors committed start from Line 3 downwards, so how do you think the expression in Line 3 is related to Line 2 and maybe also to Line 1?
T1:
I was thinking it is related to Line 2 but to write

The results show that the errors were omitted not because of lack of knowledge but because of human error or lack of concentration. Other errors that were identified but have not been presented include: (1)

In

R:
Here you say that every step is wrong. Is it everything or some parts?
T1:
I mean everything is wrong in the sense that in step 1 the function is equated to zero. So here the student is not answering the question and therefore everything is wrong. The student is finding the roots of the function and not the point where the gradient of the tangent is zero.

Error identification in S57's response.

T1 bases his answer on the fact that the student writes the wrong thing in step 1. So any working that results from an error also becomes an error, although not of the same kind as the first. So the problem here was that there is no step that T1 could identify as correct according to the demands of the question. A closer look at this analysis reflects that step 1 (or line 1) is the main error as everything that follows emanates from it. The discussion continues:

R:
In remedying the error you say that the whole concept of tangent has to be revisited. When you say the whole concept of tangent what do you mean by this?
T1:
The whole means to start a little bit behind. I talk about everything that will lead us here.
R:
That will lead us where?
T1:
That will lead us to finding where the gradient is zero. This student has the problem with the functions so the student has to know that a function is different from a tangent.

Though T1 has now identified the main error as that of finding the zeros of the function instead of the points where the gradient of the tangent line is zero, what T1 believes needs to be done is still not clear to the researcher. To say the whole concept of tangent means ‘to start a little bit behind’ does not explicitly give us information about where to start. Thus, the analysis has been characterised as opaque because one cannot see through the intentions of the teacher.

After identifying the errors, the teachers interpreted the errors from a single perspective instead of from multiple perspectives. This resulted in also offering remedial strategies that were confined to a single interpretation. Interpretation from multiple perspectives is important because it is unlikely that the same error could be committed by students who had the same type of thinking. This is because students are individuals whose thinking also varies. The teachers’ analysis of S25's work (

Error identification in S25's response.

T2 shows the error in lines 3 and 4 to be that the student does not understand what it means to say that the ‘tangent is horizontal’. This means the point on the tangent line where the gradient is 0 and not where

T2:
I think the student missed that part of what becomes zero for the tangent to be horizontal. What has to be zero is the derivative and not the value of

T2 suggests that the student knew that an equation involved a zero but did not know which part exactly was a zero. This is a sign that the student could not connect the procedure to the concepts surrounding the procedure.

An alternative perspective to this is that since each term of the slope function had an

In some cases, the teachers’ evaluation of students’ errors was local whilst global in others. Local evaluation in this case means looking at an impact of an error by making reference to the mathematical content that is directly related to it. Global means judging the impact of an error by not being topic specific but by making reference to an aspect of mathematics that cuts across topics (see

Identified errors with corresponding teachers’ evaluation.

Identified error | Impact of the error in learning or doing mathematics |
---|---|

S25: Tangent is horizontal where |
T1: Not being able to solve problems relating to tangent even when the skill or tools are available. |

S57: Every step. Everything is wrong. | T1: Failing to understand mathematics due to not doing work properly. |

S37: Inaccurate curve drawn. | T2: Wrong calculations. |

S57: The student confuses |
T2: The student may not see the difference between finding the intercepts and finding the slope of the tangent line to the curve especially where the tangent is horizontal. |

T2 goes global with S37 and local with S57. His evaluation of S37's error is that it will put S37 in danger of carrying out wrong calculations in mathematics in general. However, with S57 he becomes more local by relating the problem to specific concepts the student has not mastered: intercepts and slope of the tangent line. The two types of evaluation have a place in the teaching of mathematics. The local view leads to the remedying of the problem by using immediate related concepts within a particular topic and the global view cuts across topics. Some knowledge and skills in mathematics have to be acquired independently of a particular topic because they are a necessity in doing mathematics in general. However, they end up being pictured globally because they first started being applied locally. Thus, the two perspectives complement each other in this regard and are justified in the context of mathematics.

Accuracy is an important aspect of mathematics teaching and learning. However, it can only be mastered meaningfully if other methods that do not produce accurate results are appreciated and used to develop an understanding of why they are preferred in certain situations but not others. In the analysis shown in

Error identification in S37's response.

T1 suggests that the student should be shown simpler equivalent methods that are more accurate as the graphical methods make mathematics appear more difficult than it actually is. In an interview he said:

T1:
We are learning calculus so that we can have shorter ways. … In calculus we want to get the solutions quicker and faster.

The emphasis here is not on where the student is in terms of understanding or thinking. The efficiency of the method is the focus, rather than the understanding of the student. When asked if the graphical method has to be discouraged completely he said:

T1:
I am not against it (graphical method). It is just that it has lots of steps and once you have lots of steps. There is high probability that you may get lost before you get to the answer.

Although T1 says that he is not against the graphical method, he does not seem to be convinced about its usefulness. Hence, the tone set is to discourage students from using it.

Geometrical representation is a very powerful representation of mathematical concepts and thus cannot just be dismissed. What both teachers fail to do in this case is to address the problem from the student's perspective. The student is not shown how to draw an accurate graph and from there move to more accurate methods after showing that no matter how accurately one tries to draw the graph there will always be other sources of error such as the sharpness of the pencil used in drawing, the scale chosen and the parallax error when taking readings. We believe that the student could appreciate different representations of mathematical concepts better if they are taught first-hand the advantages of using one. Here the focus is on what the question requires and what the teacher values instead of where the student is in terms of understanding.

The study has shown that teachers had elements of both SMK and PCK. The teachers managed to identify students’ errors without difficulty. Thus, the teachers possessed a component of SMK called

Unlike in the study of Peng and Luo (

The study has shown that error analysis, although a necessity, is a complex process. It is complex because errors are symptoms of misconceptions (Olivier,

We would like to thank the head of the Department of Mathematics of Computer Science at the National University of Lesotho for allowing us to use the students’ examinations scripts in the study as they are the property of the university.

The authors declare that they have no financial or personal relationship(s) that may have inappropriately influenced them in writing this article.

E.K.M. (National University of Lesotho) was involved in the construction of research instruments, data collection, data analysis and the writing of the article. M.Q. (National University of Lesotho) also took part in the construction of research instruments, data analysis and the writing of the article. P.W. and J.N. (National University of Lesotho) were the sample of the study. They also took part in the validation (respondent validation) of research findings during the writing of the article.