How errors are dealt with in a mathematics classroom is important as it can either support or deny learner access to mathematical knowledge. This study examines how a teacher, who participated in a professional development programme that focused on learner errors, engaged with mathematical errors in her Grade 9 classroom. Data were collected over a two-year period in the form of videotapes and were analysed qualitatively. Our findings illustrate that this teacher dealt with four types of mistakes: slips, errors derived from misconceptions, language-related errors and errors that occurred from the incorrect usage of the calculator. She dealt with these by correcting, probing or embracing them. We found that, over time, this teacher dealt with more errors and corrected and embraced errors in different ways. We recommend that teachers use their professional knowledge to decide when, why and how it is appropriate to correct, probe or embrace errors in light of their knowledge of the content and their learners.

Errors play a central role in the mathematics classroom as they are a reflection of the manner in which learners reason and they illuminate the processes through which learners attempt to construct their own knowledge (Olivier,

The purpose of this study is to investigate the type of errors a teacher chooses to deal with in her mathematics classroom and the manner in which she deals with the errors. Since the data from this study were obtained over a two-year period, we also decided to investigate if there were any shifts in the teacher's practice in dealing with errors over time. This research is part of a larger project called the Data-Informed Practice Improvement Project (DIPIP). DIPIP works with mathematics teachers in professional learning communities so that they may develop their understanding of the importance of mathematical errors in the classroom and the learners’ reasoning behind their errors and collectively strategise how to deal with them (Brodie,

The first and second sections of this article draw on the literature to illuminate the kinds of errors that might occur in the classroom and the ways in which teachers might deal with errors. In the third section, we explain the research design and methodology used to analyse our data, which is then followed by an analysis and discussion of our results in the fourth section. Finally, we suggest recommendations based on our findings.

There are many reasons why learners may not obtain the correct solution to a mathematical problem. These reasons may include, but are not limited to, carelessness, a lack of knowledge of the mathematical concepts or the learners not understanding what is required of them in a mathematical task (Swan,

Slips are sporadic. Errors, however, are systematic. They occur on a regular basis and are pervasive and persistent, often across contexts. Errors occur at a deeper conceptual level than slips, so correcting errors is usually not enough to address these conceptual misunderstandings. The underlying conceptual framework that causes the errors is called a misconception (Nesher,

Misconceptions generate errors. But how are misconceptions generated? The theory of constructivism proposes that we actively construct knowledge by using our prior knowledge as a foundation to build new knowledge. The processes of assimilation and accommodation enable us to restructure our existing schemas to develop our conceptual knowledge (Hatano,

The word ‘error’ in the education system tends to have negative connotations. Summative assessments used widely in schools perpetuate the misconception that making errors is punishable through the system of deducting marks for wrong performances (Nesher,

Much of the research on errors and misconceptions argues that errors are a normal part of the learning process (Borasi,

Research on teachers’ dealing with learner errors in mathematics is limited, but two authors have developed frameworks for this purpose. Peng and Luo (

While Peng and Luo (

Based on the above, Brodie (

The teacher whose lessons we analysed participates in a professional learning community organised by DIPIP on an ongoing basis. This teacher is one of about 40 teachers who are part of this project and was selected for this study because she seemed to have a range of strategies in working with learner errors. The data are in the form of videotapes, which were taken before the project started as a baseline and have been collected at various points during the project over two years (2012–2013). For the purpose of this study, we analysed nine videotaped lessons in a Grade 9 class. Each of the nine lessons was categorised by the DIPIP project as either individually planned or jointly planned lessons. Individually planned lessons involved the teacher teaching a lesson planned by herself as part of her daily routine. The jointly planned lessons were planned in collaboration with other teachers in the professional learning community and aimed to deal with the possible learner errors that might emerge during the lessons^{1}

Overview of the types of lessons.

Lesson category | Number of lessons | Date | Duration (minutes) | Topic |
---|---|---|---|---|

Individually planned 1 | 3 | 20 April 2012 | 30 | Algebra: dealing with three revision questions |

25 April 2012 | 30 | Polygons | ||

4 May 2012 | 30 | Revision of algebra and properties of triangles | ||

Jointly planned 1 | 2 | 17 August 2012 | 25 | Algebra: equations and expressions |

20 August 2012 | 35 | Algebra: equations and expressions | ||

Individually planned 2 | 3 | 19 April 2013 | 35 | Algebra: laws of exponents |

22 April 2013 | 35 | Algebra: simplification of expressions | ||

23 April 2013 | 35 | Ratios | ||

Jointly planned 2 | 1 | 17 September 2013 | 35 | Revision of ratios |

The first step in data analysis was to watch each of the videos. The first author made notes on the time that an error was seen in the video, the nature of the error and the manner in which the teacher dealt with the error. To further ensure the validity of our results, she re-watched the video and documented excerpts illustrating how the teacher dealt with the error and the possible reasons that the learners provided for making such errors, if they were expressed. The first author then discussed her coding system and checked her interpretations with the second author and all disputes were resolved by discussion. Thereafter, we arranged the data into a table. We documented the error, our thoughts on the error, the manner in which the teacher dealt with the error and our thoughts on the teacher's approach to dealing with the error in light of the literature.

Initially, we intended to classify errors using the two categories, slips and errors stemming from misconceptions, extracted from our literature review. However, when watching the lessons, we realised that there were two additional types of errors which are conceptual in nature; that is, they are not slips, but are not derived from misconceptions. These conceptual errors were language-related errors and errors derived from the incorrect usage of the calculator, which we included in our framework. We then classified each of the errors in our table of results as either being a slip, an error derived from a misconception, a language-related error or an error derived from incorrect usage of the calculator. The manner in which the teacher dealt with the error was classified as correcting the error, probing into the error or embracing the error, as discussed above.

We acknowledge that the practices of one teacher cannot be generalised to other teachers in the project, or to other teachers more generally. However, this analysis has enabled us to test and refine our categories for analysis and we intend to analyse the shifts among other teachers in the future. The analysis of one teacher is useful in that it enables us to discuss in detail the different categories and how she developed her practices over time.

There were four categories of mistakes that occurred in in the classroom: slips, errors derived from misconceptions, language-related errors and errors derived from incorrect usage of the calculator. To illustrate the nature of each category of mistakes we provide an example of each category below.

An example of a

An example of an

This error is evidence of a misconception because the learner overgeneralised the addition of whole numbers to the addition of fractions. The learner could have also added the numerator and denominator based on an overgeneralisation of multiplication and division of fractions. Much research suggests that misconceptions are a result of prior correct knowledge interfering with new knowledge. However, new learning, such as the multiplication and division of fractions can also interfere with prior correct learning (Olivier,

An example of a

An example of an ^{2} − 1

Types of errors that occurred per lesson category.

Number of lessons | Lesson category | Slips | Errors derived from misconceptions | Language-related errors | Incorrect usage of the calculator | Total |
---|---|---|---|---|---|---|

3 | Individually planned 1 | 6 | 16 | 0 | 0 | 22 |

2 | Jointly planned 1 | 0 | 15 | 1 | 1 | 17 |

3 | Individually planned 2 | 1 | 20 | 0 | 1 | 22 |

1 | Jointly planned 2 | 0 | 4 | 4 | 0 | 8 |

Total | 7 | 55 | 5 | 2 | 69 |

Most of the errors that were made across these four lesson categories arose from misconceptions. This indicates that the majority of the incorrect answers made across the analysed lessons were derived from an overgeneralisation of correct prior knowledge. Nesher (

Of the 69 mistakes that were made across the four categories of lessons, 45 were dealt with by the teacher, while 24 were not. Some of these 24 mistakes were ignored by the teacher: she did not acknowledge or engage with the learners’ responses and, in some cases, she may not have heard them. Others were ignored because they were shouted out, a deliberate strategy of this teacher.

Number of different kinds of mistakes dealt with by the teacher.

Lesson category | Slips | Errors derived from misconceptions | Language-related errors | Incorrect usage of the calculator | Total |
---|---|---|---|---|---|

Individually planned 1 | 0 | 7 | 0 | 0 | 7 |

Jointly planned 1 | 0 | 9 | 1 | 1 | 11 |

Individually planned 2 | 1 | 17 | 0 | 1 | 19 |

Jointly planned 2 | 0 | 4 | 4 | 0 | 8 |

Total | 1 | 36 | 5 | 2 | 45 |

While

Number of mistakes dealt with by the teacher.

We categorised the manner in which the teacher dealt with errors using Brodie’ s (

The first example is an excerpt where the teacher corrected an error. In this excerpt, the teacher asked the learners to share 12 sweets according to the ratio 1:2:3. She asked different learners to answer how many sweets will be represented by 1, 2 and 3 in the ratio:

Teacher: What is two parts of the twelve sweets?

Learner A: Four.

Teacher: That is four sweets. And now, I’m coming to this side, Learner B? What is three parts of twelve sweets?

Learner B: Three.

Teacher: Is three. Alright, I want people on this side to help Learner B. Because Learner B is sitting on this side. So what is three parts of twelve sweets, Learner C?

Learner C: Six.

Teacher: Six.

Learner B's response to how many sweets are represented by 3 in the ratio was incorrect. The teacher then asked another learner what the correct answer was and it was established to be 6. Despite addressing the error, the teacher did not get to the bottom of why this error was made as she had another learner correct Learner B. We note here that even if the teacher gets another learner to correct the error, it still counts as correcting the error because the underlying conceptual issues are not dealt with. Borasi (

The second example is an excerpt of the teacher first probing an error and then embracing the error. In this excerpt, the teacher asked the learners to find

Teacher: If

Learner H: It might be fifteen and a half.

Teacher: It is fifteen and a half. And how did you get fifteen and a half?

Learner H: Because

Teacher: Right. How do you … Why do you get five and a half?

Learners: Sixteen and a half.

Teacher: Where do you get five and a half?

Learner H: Because five and a half plus five and a half is equal to eleven.

Teacher: Is there anything that tells us that should be solved like that?

Learners: [

Teacher: It's your mind set. Who else has got another answer for that sum? Because he, Learner H, what did he … did he divide that eleven by two. And why did he divide by two?

Learner H: Because half of eleven is five and a half.

Teacher: Because half of eleven is five and a half. So in other words, when you look at

Learner H: Because Mam, half of eleven is five and a half. And if you equal five and a half and five and a half, it gives you eleven.

Teacher: Right. Learner I, do you want to say something?

Learner I: Mam, I think its eighteen Mam, because

Teacher: Why did you say

Learners: Variables.

Teacher: They are?

Learners:[

Teacher: They are variables. What is a variable?

Learners:[

Teacher: Sssshhhh … Learner J?

Learner J: That is a letter from the alphabet that represents a number.

Teacher: That is a letter from the alphabet that represents a number. Right, so when Learner I said that

Learner K: Four.

Teacher: It would be … four. Alright. So we agree that

Learner: [

Teacher: OK, he says that because we say

Learner L: [

Learner M: [

Teacher: The answer is eleven plus

In Part A of the excerpt, the teacher constantly probed Learner H's error. She asked him how he got the answer of 15½, why he got 5½ for

By examining Part A and Part B, it is evident that the teacher uses the teaching strategy of questioning to probe the errors, for example by asking learners to justify their answers in a discussion as to why they thought the question should be solved in the manner they suggested. What makes this category of dealing with errors different from embracing errors is that she did not use these learner justifications to promote epistemological access.

In Part C, after obtaining a definition of variables from the learners, the teacher began to problematise the question. Using the definition that variables represent numbers, the teacher substituted different values that add up to 11 in place of

The above excerpt illustrates that this teacher did not only tolerate errors, but used them for epistemological purposes. The learners had learned previously that there can be a finite set of values for variables in an equation and an infinite number in an expression. Here, the teacher further supported learners in developing a conceptual understanding of variables in equations and expressions through conversation. The manner in which the teacher dealt with the error enabled learners to perceive that their errors are reasonable and are an integral part of learning mathematics (Brodie,

What is interesting about all the errors that were embraced across the lessons is that the conversations were all lengthy and required a large amount of time. Hansen (

The manner in which the teacher dealt with the errors.

Lesson category | Correcting errors | Probing errors | Embracing errors | Total |
---|---|---|---|---|

Individually planned 1 | 3 | 1 | 3 | 7 |

Jointly planned 1 | 5 | 3 | 3 | 11 |

Individually planned 2 | 10 | 6 | 3 | 19 |

Jointly planned 2 | 6 | 0 | 2 | 8 |

Total number | 24 | 10 | 11 | 45 |

There was a change in the manner in which the teacher dealt with errors across the four categories of lessons which took place over a two-year period.

Number and percentage of errors dealt with by the teacher per lesson category.

Lesson category | Total number of errors | Total number of errors dealt with by the teacher | Percentage of errors dealt with by the teacher |
---|---|---|---|

Individually planned 1 | 22 | 7 | 31.8 |

Jointly planned 1 | 17 | 11 | 64.7 |

Individually planned 2 | 22 | 19 | 86.0 |

Jointly planned 2 | 8 | 8 | 100.0 |

^{2}

In terms of correcting errors in the first individually planned and jointly planned sets of lessons, the teacher responded to errors using the initiation-response-evaluation cycle: the teacher asked a question, a learner responded incorrectly and an evaluation of the error followed (Brodie, Jina & Modau,

In terms of the teacher's shift when embracing errors, we showed that initially the process of embracing errors was time consuming. In later lessons, the teacher managed to embrace errors, but in less time. The teacher initially addressed the errors by probing learners as to whether they agreed or disagreed with an incorrect solution. After she probed learner thinking, she suggested a pathway to follow and simplified her questions to obtain the solution. This simplification of questions could be a reason that she embraced errors in less time than in the first sets of individually planned and jointly planned lessons. We were careful to check whether the simplification of questions led to what Stein, Smith, Henningsen and Silver (

In this article, we have shown that the teacher dealt with four types of mistakes, namely slips, errors derived from misconceptions, language-related errors and errors derived from the incorrect usage of the calculator.

We categorised the manner in which she dealt with the errors in three categories, namely correcting errors, probing errors and embracing errors, and showed how each of these approaches to dealing with errors provides different forms of access to knowledge. We found that most of the mistakes made throughout the four categories of lessons were conceptual in nature and that the teacher probed and embraced almost as many errors as she corrected. We also showed that the percentage of errors dealt with by the teacher across the lessons increased: from 32% and 65% to 86% and 100%, although the 100% was of a small number of errors. The shifts in the teacher's practice could possibly be due to the influence of her participation in her professional learning community, which supported her to engage with learner errors; however, confirming this conjecture requires further research.

We also argued that the manner in which the teacher corrected and embraced errors changed over time. The teacher managed to elicit and correct more errors and she managed to embrace errors in less time.

Good teaching requires using learner errors constructively in class on the basis of teachers’ professional knowledge and judgements. Embracing errors, as we have illustrated, has the potential to allow learners to develop a rich understanding of concepts. It is preferable that teachers embrace errors rather than correcting or probing errors, which provide learners with limited access to knowledge in comparison to the access afforded to learners when teachers embrace errors. However, we do not argue that teachers should always embrace errors, because as Hansen (

Teachers should be aware of the benefits and limitations of correcting, probing and embracing errors. Using their professional knowledge, teachers should decide when and why it is appropriate to correct, to probe and to embrace errors in light of their knowledge of the content and their learners. For example, it might not make much sense to embrace a slip. Probing or correcting slips would be a more suitable method of dealing with the mistake. In probing and embracing errors, teachers are likely to develop their learners’ mathematical proficiency and reasoning skills, help them become aware of their own errors and develop a sense of agency in relation to their mathematical learning.

Funding for the DIPIP project was provided by the Gauteng Education and Development Trust and the National Research Foundation.

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

A.G. (University of the Witwatersrand) and K.B. (University of the Witwatersrand) worked together on this article in discussing, drafting and refining the article. The article is based on A.G.'s research project, of which K.B. was the supervisor.

See Brodie and Shalem (2011) for a description of the DIPIP activities and how the jointly planned lessons arose out of previous error analysis.

The manner in which the teacher probed errors remained constant over time. This is because probing errors merely required the teacher to access justifications for errors.