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Dealing with Misconceptions in mathematics.

In discussing misconceptions in mathematics we first have to define a misconception in comparison to a mistake or a misunderstanding.
Mathematic students appear to make mistakes for numerous reasons. The mistake may just be a lapse in concentration, or in trying to understand a problem, perhaps the focus of the question is lost or misinterpreted. However other reasons may be more deeply rooted and profound.
Drew suggests a loose definition for a misconception to be’

The misapplication of a rule, an over- or under-generalisation, or an alternative ‘conception’ of the situation. For example, a number with three digits is
‘bigger’ than a number with two digits works in some situations (e.g. 328 is bigger than 35)
but not necessarily in others where decimals are involved (e.g. 3.28 is not bigger than 3.5).
It is important to note that misconceptions are not limited to children who need additional
support: more able children also make incorrect generalisations.
Research by Booth showed that much early learning is based on early experiences in which a frame of reference is constructed. Since learning is seen to be a journey on which knowledge is built on, checked, reconstructed, and experience gained then it is clear that a misconception in its broadest sense can be correct within a context, or within the journey, for example the common misconception cited by the DfEE (1998:57) that the outcome of a division always gives a smaller value e.g. 4 ÷ ½, however in the context of positive whole numbers the apparent misconception is correct. In contrast other misconceptions cannot be so easily contextualised e.g. lining up columns of numbers for operations against a left hand margin, irrespective of the position of the decimal point.
The mathematics website (excellence/qia/org.uk) suggests that
misconceptions should not be dismissed as...