Teaching is the profession that has many turning points and about every turning point or incident that tells the need and requirement of change or improvement required for the professional development can be termed as critical incident. It can also be called the event that may make the teacher to stop at that point and think about that. Watts et al (1997) define it as “the kind of occurrence which creates a serious pause for pedagogic and scientific thought: a ‘decision situation’; where a teacher must draw on scientific and educational rationales in order to resolve conflictual matters (p.1026).” This notion powerfully supports my understanding of “critical incident” as it makes teachers to revisit their classroom practice in terms of content knowledge and methodology of giving that knowledge to students in order to bring correction and improvement in their professional performance and outcome of their performance.

In case of analysing critical incidents the most important thing is the interpretation of observer or practitioner as Tripp (1993, p.24) says “Interpretation is important because we act according to what we think things mean.” Thus it is very important to be a good reflective practitioner to identify and analyze critical incidents and find out implications out of that. This practice will control our professional judgments or in other words our classroom practice to help us become a good teacher. As Tripp (1993) has also written, in order to go through this process, there are two expected steps. Firstly any happening is being observed and noted down or in other words we can say describing the fact and the context to make visible the real situation with reason. Secondly, that happening or incident is being explained and analyzed.


• It will be helpful in locating the causes of failures or lacks in any lesson.

• By locating those reasons teachers will be able to look for the alternatives to improve their professional performance.

• Being a teacher educator it, the person will be able to help other teachers to establish an understanding of weaknesses of their lessons and to become a good decision maker to come up with appropriate solution of their problems.


Strategies used to collect data for this purpose include observations, note taking, talking and debriefing of the collected data. In observation study of actions, sayings and doings of student and teacher was included. Note taking includes the notes taken from blackboard and students’ notebooks. For talking or discussion, the opportunity to talk with students and teacher was availed to help validating the understanding of the data that was collected. Debriefing sessions were of two types. One was the debriefing done in the group just after the observation of the class and the second was the whole class debriefing that helped to describe and analyze any incident.

Reading Tripp (1993, p.43) who says, “the simplest technique available for analyzing incidents is to ask a series of different kinds of question.” was helpful to analyze the data. Approach applied here to analyze these incidents was basically qualitative or interpretative approach in which hypothesis of the data collected from school was done. Moreover suppositions and guesses helped to see the happening from different perspectives and then come to the decision that these events need to be revisited in order to bring improvement in professional practice.


The data required to find out and analyze critical incidents in mathematics classroom was collected from a private school in Karachi. That was a school with good culture surroundings as students were provided with chances to participate in many co-curricular activities. A conducive learning environment was given to the students as they were being provided a friendly pleasant situation. Teachers were highly qualified and with excellent content knowledge in their areas of teaching.

Grade VIII was targeted to observe teaching and learning of mathematics there and look for the critical incidents happening in that class. Although teacher was well qualified and students were very much careful and attentive but it is a general saying that nothing is perfect in this world and always there is a need for improvement and perfection.


Incident 1:

To prove De Morgan’s law teacher wrote following expression on the board



Teacher wrote this mathematical statement on the board and proved it. This is a critical incident because the teacher has written mathematically wrong phrase. This equation cannot be proved as it is not specifying which sign should be considered and solved first and without this clarification one may decide to subtract ‘A’ first from ‘U’ and then make its union with ‘B’ that will not be equal to the complement set of ‘A’ union ‘B’. It tells that in mathematics there are some fixed patterns and these patterns should be dealt in a mathematically acceptable manner. The insight of this idea can be gained by reading Coburn (1995) who says, “Mathematics can be characterized as the science of patterns (p.1).” He further argues that finding patterns is a powerful problem solving strategy. It implies that to learn mathematics that is basically based on problem solving strategy, finding patterns is necessary and patterns can only be found and proved if they will be true for the given condition. In the case given here the pattern is not true because both sides cannot be proved equal as it is discussed above.

Coming to see the reason of its not being mathematically acceptable, the thing that comes out is that teacher missed the punctuation symbol “(” and its pair. While these symbols are the indicators of the process and they cannot be neglected as Pimm (1987, p.138) says “the symbolic aspect of mathematics was (is) one of the subject’s most apparent and distinctive features.” May be that was just by mistake or carelessness as she proved that law accurately and her mastery on the subject was telling that she has a sound content knowledge. When she was describing the steps of the solution it was obvious that she is very much clear about the process. Even students not only understood that law very well but they also proved the second part of the law. But it was just a mutual understanding of that teacher and her students and the only missing parenthesis were changing the meaning of the left hand side of the equal sign.

Looking at its implications, there are many negative effects that may be caused because of it. It was creating misconception with false mathematical belief that will hinder students to learn this concept. It may promote rote learning in students as when they will not be able to prove values according to this law they will try to learn things by heart. If students have got the correct process with the help of teacher’s description, they will not be able to feel the importance of missing symbol that may cause carelessness in them towards mathematical written language. This carelessness may make their whole learning unsure, as they will not be able to rationalize their own work.

Incident 2:

When teacher was taking responses from students about the notion of set she said many students “Repeat my words”.


Using such a strategy in any classroom generally and in mathematics classroom especially very much harmful for students because by saying so teacher is not eliciting students understanding of the concept. When students will not use their own words to describe or define any idea teacher will not be able to know whether they have assimilated the idea or not. William Gibbs and Jean Orton (1994, p.104) say, “there is no guarantee that ideas presented by a teacher will be received by pupils in their intended form. Some form of discussion to allow negotiation of meaning seems essential.” So that students should be given chance to make their ideas obvious in the form of discussion or responses. Barbara Jaworski (1994, p.230) underpins this view of discussion when she says, “we can not see into a child’s mind and know what sense s/he is making of what we offer. We have therefore to provide opportunities to gain access to his/her thinking.”

By saying “repeat my words” teacher was trying to develop instrumental understanding of mathematical concepts in students as according to Skemp (1989) instrumental mathematics emphasis on the handling of symbols in order to tell “this is the way to do it”. Here the learners are simply asked to accept as closely as possible how they are applied. Further Skemp (1989) says, “There is no discussion why and how the particular rules come into being or why they happen to work, and little or no attempt to negotiate meaning for them.” This tells that if there will be no discussion about rules and facts students will not be able to get the meaning of those rules and they will not be able to apply them in their real life situations. Thus it will hinder the relational understanding and make students to understand concepts instrumentally. The same thing is being discussed when Brissenden (1988) says “it is chiefly through talk and discussion that the mathematical activities in the concerned scheme become personal experience related to children (p.6).”

Another very important thing is that using such wordings promote rote memorization as students start thinking that any thing that is said by the teacher is correct and they cannot define and describe things. So that teachers should avoid usage such wordings and give students chance to have a positive contributions in the classroom discussion rather than having predetermined answers of questions asked by teachers. John Backhouse (1992, p.132) approves this idea by saying “there must be positive contribution from learners, it is not enough for them to be passively answering the teachers’ questions.”


Implications of analysis of these incidents can be seen in two different aspects. As a teacher it implies that critical incidents are always there to help us to improve our performance and to find out the gaps in our teaching. For example many of the teachers are not aware that they should pay attention even to the responses of their students as they are having messages for them about their teaching. Analysis of second incident tells that a teacher should be very much careful about his/her own wordings because these wordings can be interpreted in many different ways and these interpretations affect their performance. Analysis of mathematical pattern tells about the importance of the patterns and symbols used in mathematics.

As a teacher educator this analysis tells how should the educator help teachers to become able to locate gaps not only in their own practices but also in others’ practices as it will be supportive for them to have a deep understanding of these gaps and the impacts of these gaps on their practices.


In short it can be said that identifying critical incidents in personal performance or in any other teacher’s performance is a very helpful and constructive approach to improve the professional practice of teaching and learning enterprise. It also helps us to find out the gaps in our teaching and then to think about the suitable alternatives and substitution to help us out of the trouble as Tripp (1993) recognizes it as a support to keep our teaching on right path.


Backhouse, J. et al. (1992). Improving the learning of mathematics. London: Cassell.

Brissenden, T. (1988). Talking about mathematics. Blackwell: Oxford.

Coburn, T. G. (1995). In national council of teachers of mathematics (Eds.) Curriculum

and evaluation standards for school mathematics. Virginia: National Council of Teachers of Mathematics.

Gibbs, W. and Orton, J. (1994). The role of the teachers. In A. Orton (ed) Issues in

teaching mathematics. London: Cassell.

Jaworski, B. (1994). Investigating mathematics teaching: A constructivist Equiry.

London: The Falmer Press.

Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms.

London: Routledge

Skemp, R. (1989). Mathematics in the primary school. London: Routledge

Tripp, D. (1993). Critical incidents in teaching, developing professional judgment (pp.24

-42). London: Routledge

Watts, M. Alsop, S. Gould, G. and Walsh, A. (1997). Prompting teachers’ constructive

reflection: Pupils’ questions are critical incidents. International journal of science education. 9(19), 1025-1037

Educator, Mother