5 Review: the hero’s return
The review phase is where the real learning takes place.
The review phase is where we reflect on the whole process.
In some cases, our review might reveal that more work is to be done and we are forced to return to the entry and attack phases. At a base level, we need to check that our solution is consistent and reasonable given the problem posed, however we also might need to see whether or not the method will be valid for different values or under different conditions. We may find that our model leads to unreliable results, requiring us to start the process over again or go back a few steps. We also might think about whether we could go about similar problems in the future in a more efficient or simpler way.
Our thoughts at the review stage should be to:
Check
- Calculations
- Arguments to ensure that computations are appropriate
- Consequences of conclusions to see if they are reasonable
- That the resolution fits the question
Reflect
- On key ideas and moments
- On implications of conjectures and arguments
- On your resolutions and justifications: can it be made clearer
Extend
- The result to a wider context by generalizing
- By seeking a new path to the resolution
- By altering some of the constraints
This stage provides the opportunity to really understand the problem and know that our solution seems reasonable. Beyond that, the review phase also provides us with an opportunity to think about alternative strategies and approaches along with their relative “elegance”.
Checking that you’ve done things correctly is a fine strategy in an exam or on completion of a project – but remember that this is your main chance to become a ‘better’ problem solver! Reflect on why you found something difficult – what was the missing knowledge or false assumption that got in the way? Or was this a problem that is hard to solve? Now that you have obtained an answer – can you see a pattern or method that makes sense of it? Reflect on why you found something easy – is there an extension to the problem? Would a slight change make it harder?
In short, the review reflects back on the attack and broadly asks:
- does my solution make sense?
- do I have proof that the answer is correct?
- could I find a simpler solution?
- did I get all possible information out of my attack?
- can I explain my reasoning?
This stage is often ignored with answers given and checked as ‘right’ or ‘wrong’.
Stop and think
Think about your problem solving experiences when you try to explain something or teach someone.
The process of problem solving, conjecturing, and coming up with models allows us to continuously challenge and then revise or reject our ways of thinking about a situation. The ‘model’ one develops can then be used to reinterpret the situation. Some might argue that this is what is at the heart of learning mathematics. Don’t forget how important ‘the struggle’ is in helping you to really learn and absorb something – the journey and the Aha! moments are what reinforce the learning that we return home with, in the process becoming better and wiser problem solvers.
The following video recaps the concepts of generalising and specialising.
A last note on problem solving
In solving problems we need to reflect back on the process and make sure that our solution is reasonable, however it can also help to reflect on the entire process of problem solving in order to cultivate modelling abilities in learners. There are a number of mathematical strategies that students can be introduced to for use in problem solving, however in order to help build their mathematical intuition, it is better not to be prescriptive in which strategies to use for which situations. Through more and more practice at creating models to solve non-routine problems, students can become aware of the underlying mathematical similarities that exist between problems and cultivate this intuition.
