Mastering the common problem types in maths

I’ve been wanting to combine some of the thinking I’ve been doing into cognitive overload, worked and partially completed examples, and the bar method as a pictorial representation of mathematical problems.  The lesson described below is what I did, with a substantial eye on the expectations of the new national curriculum and the idea of mastery.

A worked example

A good explanation clearly takes children through the steps needed in order to solve a problem, but these steps should be rooted in the deeper structure of a problem and not the superficial.  As such, the success criteria that the teacher works from and that children refer to should support that expectation of analysis.  In the example, I showed my class how they could solve a comparison problem where both the total of the parts and the difference between them is known, but the value of each part is not.  To solve this type of problem, they need to be able to pick out totals and differences, as well as get an idea for which part is bigger/smaller.  The success criteria I used was this:

Then, I took one question and walked them through each of these three steps:

A further few questions can be prepared and ready if further modelling is necessary.

Partially completed examples: scaffolding for mastery

Children will need to deliberately practise representing the information pictorially.  Work into cogntive overload suggests that when children are overloaded, a number of things can happen:

• They’ll complete the first or last instruction only
• They’ll lose their place in the sequence of instructions
• They’ll abandon the task

In an effort to prevent this things from happening, the work I gave most of the children in my class included partially completed examples.  These aim to reduce the cognitive load while still providing the opportunity for deliberate practice.  The first few questions had some information already transeferred onto the pictorial representation.  Gradually, there was less and less of this until children were solving problems with just the basic structure of the problem given.

Undoubtedly, some children will need more practice with heavy scaffolding before it is removed and some will need much less.  Working in this way makes adapting the scaffolding for different children easy to manage.

Trickier problems

There will be children who are already able to solve this kind of problem, but the process modelled and the success criteria will still be of use to them to solve trickier problems.  In the tasks that I chose to give to my class, I gave them three parts to compare:

Although the basic structure of the problem is sketched out for children, in this example I did not give them any labelled sections, as the children I was intending this work for I felt would not need it.  Were I to do this again, I’d have another variation with some labels provided so that the questions are partially completed.  Hopefully, this would enable some children to work with questions this complex that otherwise might not have been able to.

Sure, this is but one lesson and it will take time for children to master the underlying patterns of problems so that they can solve them efficiently.  So it got me thinking about when different problem types should be introduced or when they should be mastered by.

Mapping out the problem types

With all this is mind, I set out to allocate problem types to phases for when they shoud be introduced.  The idea is that children in KS1 will master the basic additive and multiplictive probem types;  lower KS2 will master more complex additive and multiplicative reasoning problems as well as multi step problems based on the basic additive and multiplicative types; and upper KS2 will master still more complex additive and multiplicative reasoning problems as well as a wider variety of multi step problems.  Of course, children in each phase wlll need to revise older problem types, the goal being that eventually, they can see the undelrying pattern of a problem, thinking their way clearly to an accurate answer.  There are not many problem types for each phase, so they can be practised over time in a variety of contexts.  Each time they are revised, the scaffolding provided can be gradually removed so that towards the end of the phase, children are solving the problems with no scaffolding at all.

Introduction of bar models for additive and multiplicative reasoning Year 1-6

The expectations here are high, but achievable.  Mastering the common problem types by the end of KS2 will set children up very well for the next stage of their education.