This post is the fourth in a series (one, and two) and follows directly on from my previous post Great Ways To Introduce Topics
Some topics are difficult to introduce well. Concretes can’t be used with everything and some topics have visual representations that are complicated (damn you dividing fractions). Some topics are very difficult to understand why they work and the process for demonstrating why doesn’t link to the process of how you answer the questions. Maybe they can’t be built up from what they currently know as they involve much more necessary knowledge than arbitrary (Go and watch “Why Learning Maths Is Hard” by Stuart Welsh, another great session).
This post isn’t about bad explanations, the best place to find out more about those is “Nix the Trix” which is here and FREE. This is about situations where starting with the “how” is probably more useful as the explanations for why doesn’t help pupils to understand the topic in the beginning.
Explanations that are just designed to show the pupils that they can trust you and that you know what you are talking about
Examples of this would be:
- Tearing up the corners of a triangle and sticking them together to show it is a straight line. This isn’t a proof and is hard to recall at a later date, especially if you do the same method for angles in a quadrilateral
- Putting it into a calculator and showing it is true. Also not a proof and doesn’t demonstrate any of the underlying maths.
Explanations that don’t resemble how pupils will answer the questions and are never seen or heard from again
Examples of this would be:
- Cutting a circle into sectors and making a “rectangle” is nice but it’s quite a lot of investment in time and pupils are not likely to use this to help them recall it in the future.
- Showing a long process once and then just telling pupils to do the shortcut from then on without ever using it again. If you are showing a longer process you should stick with it for a little bit to help retention and give an opportunity for pupils to spot the quicker process themselves. For example this could be dividing fractions by showing pupils that we can turn it into a big fraction and then multiply by the reciprocal to make the denominator 1. So it is just the numerator! Hey, let’s just always from now on always multiply by the reciprocal. Let’s not bother writing it out the long way let us just always remember to keep the first fraction, flip the second fraction and change it to multiply.
Topics where the level of understanding required to access an explanation is too high and isn’t worth starting with
As a general rule of thumb you shouldn’t start a lesson with a proof as most of the time pupils aren’t familiar enough at the start of a topic to follow each step well. For example the area of a trapezium, Pythagoras’ theorem or the Sine and Cosine rule are all very difficult proofs and don’t add much to answering questions.
Discovering the process by pattern spotting
This is discovery maths and usually doesn’t help pupils see the underlying maths. It’s mostly an attempt to try to avoid just starting with How. It can also be an inefficient use of time. Craig Barton suggested method that is quicker than just giving pupils a worksheet to help them discover a process in his game changing book “How I wish I Taught Maths”. In section 3.4 Craig suggests introducing numbers to the power of negative 1 by having an excel sheet that produces answers and having pupils try to spot patterns and predict answers. This is a better way of doing discovery maths but I would still rather have a process that demonstrates the underlying maths, my alternative approach for this would be:
This could be seen as a bit of a faff and making it more difficult than just telling them the process. Yes, it is more difficult but my hope is that by not being explicit with a process in the beginning the pupils are concentrating more and spend more time trying to connect the threads of what is happening than if I just told them what to do. Those pupils that struggle with this are also more likely to create misconceptions for themselves and I like to think that this reduces that in the long run. They also would not struggle for long as the direct instruction happens anyway.
There are a few exceptions to this which I discuss in my next post.