Cloud
Pupils’ Misconceptions in Mathematics
One of the most important findings of mathematics education research carried out in
Britain over the last twenty years has been that all pupils constantly ‘invent’ rules to explain the patterns they see around them.
(Askew and Wiliam 1995)
While many of these invented rules are correct, they may only apply in a limited domain. When pupils systematically use incorrect rules, or use correct rules beyond the their proper domain of application, we have a misconception. For example, many pupils learn early on that a short way to multiply by ten is to ‘add a zero’. But what happens to this rule, and to a child’s understanding, when s/he is required multiply fractions and decimals by ten? Askew and Wiliam note that
It seems that to teach in a way that avoid pupils creating any misconceptions … is not possible, and that we have to accept that pupils will make some generalisations that are not correct and many of these misconceptions will remain hidden unless the teacher makes specific efforts to uncover them.
(1995: 13)
According to Malcolm Swan
Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalisation that the pupil has made. It may in fact be a natural stage of development.
(2001: 154)
Although we can and should steer clear of activities and examples that might encourage them, misconceptions cannot simply be avoided (Swan 2001: 150).
Therefore it is important to have strategies for remedying as well as for avoiding misconceptions. This paper examines a range of significant and common mathematical mistakes made by secondary school children. Descriptions of these mistakes are followed by discussions of the nature and origin of the misconceptions that may explain them.
Some strategies for avoiding and for remedying these misconceptions are then suggested. The paper ends by relating some general features of the recommended strategies to the educational
One of the most important findings of mathematics education research carried out in
Britain over the last twenty years has been that all pupils constantly ‘invent’ rules to explain the patterns they see around them.
(Askew and Wiliam 1995)
While many of these invented rules are correct, they may only apply in a limited domain. When pupils systematically use incorrect rules, or use correct rules beyond the their proper domain of application, we have a misconception. For example, many pupils learn early on that a short way to multiply by ten is to ‘add a zero’. But what happens to this rule, and to a child’s understanding, when s/he is required multiply fractions and decimals by ten? Askew and Wiliam note that
It seems that to teach in a way that avoid pupils creating any misconceptions … is not possible, and that we have to accept that pupils will make some generalisations that are not correct and many of these misconceptions will remain hidden unless the teacher makes specific efforts to uncover them.
(1995: 13)
According to Malcolm Swan
Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalisation that the pupil has made. It may in fact be a natural stage of development.
(2001: 154)
Although we can and should steer clear of activities and examples that might encourage them, misconceptions cannot simply be avoided (Swan 2001: 150).
Therefore it is important to have strategies for remedying as well as for avoiding misconceptions. This paper examines a range of significant and common mathematical mistakes made by secondary school children. Descriptions of these mistakes are followed by discussions of the nature and origin of the misconceptions that may explain them.
Some strategies for avoiding and for remedying these misconceptions are then suggested. The paper ends by relating some general features of the recommended strategies to the educational
References: APU (1980) Mathematical Development: Secondary Survey Report (London: HMSO) Askew, M. and Wiliam, D. (1995) Recent Research in Mathematics Education 5-16 (London: HMSO) Berk, L.E. (1997) Child Development, 4th edn. (Boston: Allyn and Bacon) French, D Gates, P. ed. (2001) Issues in mathematics teaching (London: RoutledgeFalmer) Hart, K.M. ed. (1981b) Children’s Understanding of Mathematics: 11-16 (London: John Murray) Küchemann, D. (1981) “Reflections and Rotations” in Hart, ed Morgan, C. (1998) Writing Mathematically (London: Falmer Press) Tirosh, D., Ruhama, E Swan, M. (2001) “Dealing with misconceptions in mathematics” in Gates, ed