Mathematical misconceptions
While at a conference in Prague 4 years ago an Italian asked me, ‘Why is zero
plural in English? Why do you say “one dog” but “zero dogs”?’ To be honest I
had never thought about it but it really got me thinking about mathematics,
language and culture. So much so in fact that, a year later, I had set up an
international project funded by the British Academy investigating mathematical
misconceptions. The project involved a team of 9 academics from 4 countries
(UK, Italy, Czech Republic and Israel) each working with a team of teachers
from across the primary age range. The aim of our work was to compare and
contrast common mathematical misconceptions observed in the wide range of
primary classrooms available to us. We explored the likely sources of these
misconceptions and how teachers might diminish their occurrence. Our
conclusions suggest that,
a)
Despite significant differences in the way teachers are trained and
schools operate in each of the project countries (e.g. in Israel primary children
are taught by specialist mathematics teachers rather than generalists and in
Italy teachers teach the same group of children for several years) our data
suggest that some misconceptions seem to be very common across the cultures
(suggesting that some aspects of primary mathematics may be intrinsically
difficult).
b)
fundamental mathematical misconceptions often originate in the earliest
years of schooling but frequently do not become manifest for several years
and, indeed, may persist at university level. For example many children
develop the idea that adding a nought when they multiply by 10, always
starting with the bigger number when you subtract etc. which do not hold true
in all circumstances.
c)
there is a need for us all to gain a more in-depth understanding of
mathematics across the full primary age range, before and beyond it. So, for
example, we need to appreciate how extremely complex it is for children to
grasp the idea of ‘0’ associated to the empty set one minute and being of
great significance the next as in, for example, £ 1000 . Without such
understanding we seriously obstruct the learning of algebra in the future.
d)
there are interconnections between mathematics and linguistic learning
which may exacerbate the development of mathematical misconceptions.
These may vary from country to country but are often similar in type. Thus,
for example, in the UK children may confuse 16 and 61 (both beginning ‘six…’)
while in Italy are more likely to confuse 13 and 31 (both begin with ‘tre…’)
e)
we noticed some considerable differences between classes. For example
the year 7 children in the research classrooms of the Czech Republic had great
difficulties in dealing with questions such as 999 + 8 whereas less able pupils in
a year 6 UK class did not.
f)
the vast majority of primary children in the research study were logical
when responding to mathematical problems but this did not guarantee success
(e.g. 1 = 4 + 3 was a common response as was 5 = 4 – 1)
The research process highlighted the importance of teachers having sound
mathematical knowledge and the value of using misconceptions as an
educational tool. It also highlighted the fact that focusing on common issues –
in this case mathematical misconceptions – across an entire school age range
was a very effective form of in-service development. I would be most
interested to hear your thoughts on our findings. Do you think, for example,
that there tend to be common mathematical misconceptions across your
school? Why/why not? Can misconceptions ever be useful? How do you spot
them?
Anne Cockburn,
School of Education and Lifelong Learning,
University of East Anglia