Yesterday, I attended #MathsConf5 in Sheffield. Amongst many interesting and thought-provoking 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.
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 over-generalised 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, problem-solving 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 over-generalised 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 over-generalisations from being embedded.
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 under-generalising 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 over-generalised rules (to find the mean, add the numbers and divide by how many there are) or under-generalised 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.