Misconceptions as overgeneralisations

Filling the pail

A few years ago, I read a book on misconceptions in mathematics. The authors, Julie Ryan and Julian Williams, contend that many misconceptions result from overgeneralising. Ever since, I have noticed these kinds of errors on an almost daily basis in my teaching.

What do they look like? I might be teaching function identities, for example, and demonstrate that for the function

$latex f(x)=log _{e} x $

The following is true

$latex f(x)+f(y)=f(xy) $

However, a student might then assume that this is true for all functions or for a larger class of functions.

Similarly, all maths teachers have encountered students making the following mistake

$latex (a+b)^2=a^2+b^2 $

And this can be understood as an overgeneralisation of the distributive property of multiplication, something that is true i.e.

$latex 2(a+b)=2a+2b $

I have recently noticed a similar misconception. The distributive property is only true for a linear factor, yet some…

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