Yesterday, I attended #MathsConf5 in Sheffield. Amongst many
interesting and thoughtprovoking moments, two related things particularly
stood out for me, both about the dangers of the ways in which students form
incomplete generalisations of mathematical concepts, how this can lead to later
misconceptions and how this can be avoided in designing the learning
experience.
Overgeneralising
Firstly, Peter Mattock (@MrMattock) ran a session on ‘Concrete
Approaches to Abstract Maths.’ At one point in this session, he made an
impassioned plea… that students should not learn that the mean average is ‘all
the numbers added together divided by how many there are’, as this causes
confusion when students encounter frequency tables and are required to find the
mean – adding all the visible numbers in this case is simply nonsense.
The trouble here is that students have overgeneralised
their entire concept of mean from a subset of similar cases. Are they aware
that there are cases outside of this subset where this rule does not apply?

Peter proceeded to demonstrate how he introduces the mean
average to students, by first weighing a bag of apples and asking for the mean
weight of an apple, then weighing a bag of oranges and asking for the mean
weight of an orange, then weighing a bunch of bananas and asking for the mean
weight of a banana. Does the rule apply here? There wasn’t an addition sign in
sight! Rather, he contended, the rule should be understood as something akin to
‘the total divided by the number of things.’
It strikes me that the early examples students encounter
when first learning about a new concept have a great impact on the way they
embed a generalised rule. If students are solely presented with lists of
numbers where the mean is to be found (using exactly the same steps each time),
it is no surprise that they internalise a rule which optimises solving those
cases – but this can cause problems later.
I can see exactly why for students, remembering the concept
of mean as ‘all the numbers added together divided by how many there are’ is
appealing. This algorithmic definition leaves little thinking for the students
to do – simply put in the ingredients and follow the simple recipe. However,
this makes the student passive in the process of doing mathematics, a slave to
the algorithm – not the active, thinking, problemsolving mathematics student
we are trying to develop. Generalising a concept as an algorithm that solves particular cases can limit students’
flexibility of thought when faced with a problem in which a different, more
efficient approach can be used. For example:

I have written before about the mean and the idea of
‘levelling out’ the numbers.
If we want students to be good problem solvers,
they need to be able to think flexibly about concepts, rather than being
constrained by an overgeneralised rule. To this end it is important that
students:
 Focus on the ‘idea(s) behind a concept’ rather than an algorithm for solving a particular subset of cases. (See previous blogpost.)
 Encounter different types of cases as soon as possible in learning about a concept, to avoid incorrect overgeneralisations from being embedded.
Undergeneralising
Secondly, Bruno Reddy (@Mrreddymaths), Craig Jeavons
(@craigos87) and Matt Fox (@MFx15) ran a session called ‘Sh****ai is not a
dirty word’, in which they shared their recent experiences observing maths
lessons in Shanghai.
One of the things explored was the different approaches taken
to introducing the law of indices for multiplication in the UK and in Shanghai.
Typically in the UK, students will see the rule applied in a
simple case, and practise using the rule in many similar cases – here is an
example of the sort of questions students might initially encounter on this
rule:

“Fair enough, students are getting practise of applying the
rule” one might think. However, comparing this to the first few questions students would encounter on this concept in a
Shanghai classroom gives pause for thought:

Right from the outset, students are exploring a much broader
example space. The difference in how students are likely to mentally generalise
the rules are implicit:

From this, we can see that in this scenario UK students are undergeneralising the rule. They are
only aware of its application to a subset of the example space:

Would UK students even be aware that there are cases outside
of the narrow subset they have encountered? When they come across a case from
outside the subset at a later point in their learning, will they be able to
transfer the idea?
The human mind is keen to generalise ideas. When a pattern
is spotted, the tendency to find a generalised rule for it is compelling –
indeed, this may be said to be at the very heart of mathematics. However, we as
teachers need to think carefully about how we introduce concepts to students in
such a way that they do not internalise either overgeneralised rules (to find the mean, add the numbers and
divide by how many there are) or undergeneralised
rules (the multiplication law of indices applies where a, m, and n are
single numbers or letters.) Once these rules are embedded, it can be hard to
change ideas later on, and misconceptions often arise. Exploring the whole example
space as fully as possible and as early as possible may be the key to this.
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