It is a well known saying that we learn from our mistakes, but do we as teachers ensure that our pupils learn from their mistakes or do we just accept that they don't get everything right?
Research suggests that teaching approaches which encourage the exploration of misconceptions through discussion result in deeper, longer term learning (Swan M. Improving Learning in Mathematics – Challenges and Strategies)
Addressing mistakes and misconceptions (AS and A2)
Mistakes and misconceptions can be addressed in two main ways; by the questioning and teaching style of the teacher when presenting a new topic or leading whole class discussions or by planning activities and/or questions that address particular misconceptions that the teacher has identified in the work of the students or which, from prior knowledge, are known to be likely.
Effective questioning lies at the heart of allowing and maybe even encouraging pupils to make mistakes. If you, as a teacher have created an classroom where all students are prepared to give extended answers then you have also created an atmosphere where incorrect responses are accepted and are discussed.
So, take a look at your questioning and improve your methods. Doing the following is a simple and quick way of getting started:
Allow more time for the students to respond
Accept all answers without judgement
Use mini whiteboards in your classes.
Discuss wrong answers as well as the correct ones.
Effective planning should include thinking about where pupils are going to have difficulties and where they are going to make an incorrect assumption that leads to a misconception. Remember that it is far easier to learn correctly in the first place than to have to unlearn a misconception before getting to the correct solution.
To start planning for pupil errors, go through your scheme of work in a department meeting and write two extra headings in each topic area; Common errors, and Likely misconceptions. Brainstorm the most common errors and misconceptions in each area.
For example in the Use and manipulation of surds in C1 you could write the following:
Common errors:
sqrt 40 = 20 sqrt 0.04=0.02 sqrt 32 = 2sqrt 4 (not sufficiently simplified)
Common misconception
sqrt a + sqrt b = sqrt (a + b)
Effective Questioning
Contrast the two following methods of recapping on the differentiation of e2x.
Teacher 1 asks: What is the derivative of e2x with respect to x?
e2x - no not quite, you've left something out
2ex - no you're using the wrong rule there.
2e2x - excellent, well done
Teacher 2 asks What is the derivative of e2x with respect to x? Write your answers on your whiteboards.
She then writes down all the solutions in the class and a couple of extra incorrect ones without any comment including 2ex.
She then invites comments about the solutions and asks pupils how they think that a particular solution has been obtained. She draws particular attention to the answer of 2ex and says that the expression has been multiplied by the power and the power has been reduced by one. FInally the class identify the correct solution and note the correct method.
The first approach has achieved very little apart from identifying one student who knows or has guessed the correct answer. It has also reinforced the idea that the answer is more important than the reasoning (see Encouraging reasoning) and will make unconfident learners very reluctant to offer answers that may be incorrect.
On the other hand, the second approach, the whole class has to contribute and they can do so in a safe non-judgemental environment. Wrong answers and right answers are accepted. Mistakes are identified, misconceptions can be addressed and the teacher has also used prior knowledge to identify a common misconception that the rule for differentiating xn can be used for enx.
Planning Ahead
When using trigonometric identities and double angle formulae for example, many students often use incorrect formulae because the identities do not correspond to what they expect.
So, we might expect a poor student to assume the following:
sinx + cosx = 1
sin2x = 2sinx
How the teacher addresses these errors depends on the number of students who need support and how much time the teacher thinks is appropriate.
Example 1
Find an example to show that sinx + cosx = 1 is not always true.
Correct the expression to form a similar identity that is always true.
Example 2
Is sin2x = 2sinx?
Is it true for a) no values of x, b) one value of x, c) many values of x, d) all values of x?
Use the following to help you:
A graphical calculator or dynamic graphing package, a spreadsheet, a calculator
You might like to think about these questions:
What does the graph of sinx look like?
How do you transform graphs?
Does doubling the angle in a right angled triangle double the opposite side if the hypotenuse is 10cm in both cases?
How do you solve an equation with a graph?
Example 1 can be written as a comment on a piece of work to address the needs of a particular student.
Example 2 is an activity that addresses a misconception before it occurs and uses the graphing skills of the students to show that sin2x is not always equal to 2sinx. It also leads to making the difference between an equation and an identity explicit.
Both examples create a Cognitive Conflict, that is, the learner is confronted with something that is surprising and does not fit into their mathematical view of the world.
Further Reading
The CAME materials for KS3 are excellent when it comes to effective questioning and addressing common errors and misconceptions. Their home page page is at: http://www.caaweb.co.uk/
Read examiner's reports to see what the common errors are in exams - they are unlikely to vary widely from year to year or from board to board. Here is the Edexcel report for Summer of 2008:
Improving Learning in Mathematics – Challenges and strategies , Michael Swan