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NCETM - Working collaboratively to enhance mathematics teaching

Module for mathematics specialists in post 16 education and training

Session 6: Looking at mistakes and misconceptions

As teachers and trainers, we all know only too well that learners may have fundamental misconceptions about mathematical concepts, as well as making mistakes. If they didn't, our job would be a lot simpler! Some of the mistakes and misconceptions are very common ones that we see over and over again, but there are also some unexpected ones that arise as a result of the experience of our learners.

There are a number of ways in which we can try and deal with mistakes and misconceptions. Two common ways of reacting to learners' errors and misconceptions are:

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Task: What do you feel about these two contrasting views? Discuss this with a colleague if you are able to do so.

 

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Record your thoughts here

Please record your thoughts

You may wish to hear some thoughts from Malcolm Swan, one of the authors of Improving Learning in Mathematics, about using misconceptions. If so, go to

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Using misconceptions overview

There are a number of principles that are worth thinking about and that are supported by research. (For a summary of research see Swan M., 'Dealing with misconceptions in mathematics,' in Gates, P. (ed.), Issues in Mathematics Teaching, pp. 147-165. London, 2001, RoutledgeFalmer.)

You can find out more about these at:

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Some principles to consider.
Source: Teaching and learning QIA PD2.4 Some principles to consider, page 22.

Analysing a piece of learner's work

Here is a piece of work on fractions and decimals that has been done by a genuine learner, Saira.

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Analysing learner's work
Source: Teaching and learning QIA PD2.1 Analysing learners' work, pages 15-18.

 

 

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Task: Look through Saira's answers to some diagnostic questions on fractions and decimals. You may wish to print it out, especially if you are able to look at it with a colleague. If you do work with a colleague, you might want to start by thinking individually about the nature of the errors that have been made by each learner and the thinking that may have led to these errors, and then compare your answers.

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Record your thoughts here

Please record your thoughts

If you would like to see a commentary on Saira's work, written by one of the researchers, and compare it with your summary, then click on the following link:

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Commentary on a learner's work
Source: Teaching and learning QIA PD2.2 Commentary on learners' work, page 19.

 

How can we address mistakes and misconceptions?

Clearly, there are a number of ways in which this can be done. It may be helpful to see how one of the sessions in Improving Learning in Mathematics does this.

Depending on the particular topic area you are interested in, you can choose from the following sessions.

How to find the sessions:

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Task: Choose which session you would like to look at.. You will probably find it more convenient to print the session out

As you work through the session, think about the following. (Discuss with a colleague if you can.)

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Looking at a session

Please record your thoughts

Below is a film sequence of a session planned around sessions S2 and S3 from Improving Learning in Mathematics.It shows a teacher using these sessions with a group of adult learners and also includes the teacher thinking about the context before the session and reflecting on it afterwards.

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QIA teaching and learning site, session 2.

As you watch the video, you may wish to think about:

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Record you thoughts in your personal leaning space

Please record your thoughts

If you would like to do more

Now that you have reflected on the ways in which misconceptions can be used to enhance learning, why not try out an activity with your learners? Either use one of the activities from the resource (if it fits in with the topics you are currently teaching), or devise an activity of your own, perhaps in collaboration with a colleague.

If you would like to look at some more samples of learners' work, you can access some at:

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Analysing learners' work.
Source: Teaching and learning QIA PD2.1 Analysing learners' work, pages 15-18.

 

There is a professional development session in Improving Learning in Mathematics that looks in more detail at mistakes and misconceptions. You might want to use this with a group of colleagues. This can be found as follows: