>>> Introducing conceptual change programs
Let’s face it – nobody likes being wrong. Not kids, and not us. So, when maths becomes all about the right answers, that makes it tricky for kids to want to try.
One simple adjustment in thinking will go a long way towards fixing this:
The basic idea is to pose problems for kids to experiment with before explaining how to solve them. Explain that this is new learning, not something that you expect them to know already, and that you are going to make predictions and then try them out until you find a method that works. Students will usually express a degree of nervous excitement at this prospect – it will probably feel a little bit novel, but also feel like they might be exposed to ridicule if they express an idea that turns out to be wrong. It is essential to set the correct tone from the outset. I like to do that by telling stories of great discoveries that only came about after many failed attempts. One of my favourites is Andrew Granville, who ultimately solved Fermat’s last theorem after finding it as a 10 year old and having it bug him for 30 years before finding an idea that might work.
- Launch with the problem
- Explain the problem and ask for predictions. Everyone should make their own prediction anonymously before starting.
- Record all the different predictions on the board, or on post-its etc. so that they are all given equally validity, and so that kids can see the variety of different ideas.
- Prove or disprove
- Make some conjectures together about possible ways forward – what could we try that would prove or disprove some of the ideas? Refer to the ideas on the board as a stimulus.
- Form groups and try out all of the different approaches, with the aim of proving or disproving any of them. Encourage students to try out their own idea first, then work with others on a different idea. The focus for this time is on trialing, proving and disproving ideas. Even if a student finds one that works, try to find another approach that works as well or try to disprove a different approach. As each idea is trialed, encourage students to record their findings on the board.
- Encourage students to ask each other questions and discuss what they are finding.
- Teacher’s job: Ask questions to draw attention to any disparities (evidence that disprove an idea), enabling a student to disprove their own idea and then try out a new one.
- Find connections and explain why
- Try to find 2-3 different methods that all work
- Ask students to evaluate the solutions and “find the maths” – the principles underlying the approach that make it mathematical – and also identify the similarities between the approaches that were successful. Why do they work?
- Ask students to pare down the steps to just the most important or efficient ones, creating a “proof”.
- Generalise the process
- A good process has multiple applications – how else can you use your proof?
- A good process is adaptable when the circumstances change – adjust the question and see what happens.