This blog post follows on from my last one The Best Kind of Maths Teacher and my thoughts on Kris Boulton’s session during #mathsconf23, “Always teach what before why”.
After arguing that I personally would not teach “what” before “why” I thought for completeness I should give a counter argument of how I would instead introduce these topics. For Kris’s first example he looked at multiplying terms with indices and compared his current style against what I would also consider a subpar explanation. Kris said that for simplifying indices when multiplying terms he would originally start out with
He would then have pupils write the full working out for a bunch of questions leading to pupils working slowly, not appreciating the point of the working out and not answering enough questions.
I agree, this is a poor way of starting the topic. You have not started at the beginning and writing the working out is unnecessary if they can picture it mentally. Pupils should only be writing the working out (in this topic) in response to getting a question incorrect and being instructed to by the teacher to aid support.
Let’s first be explicit about what we want our pupils to understand in this topic and why that knowledge is important. Thanks to Tom Manners and his session on “The importance of the words we use” I rewrote this many times...
By the end of this unit pupils should know/understand
· The index conveys a list of products of the base
· The index only refers to a single base
· We don’t use the multiplication symbol in our final answer
· Numerical coefficients go first
· Mr Marks is really happy when the bases are in alphabetical order but this is not sufficient to be correct
Why this is important
· knowing that you can only simplify when they have the same base
· knowing this process only works for multiplication
I hope you read the list of knowledge that pupils should gain from this topic and realise that they should already know all of this from prior units. We aren’t really teaching them anything new, just exposing them to types of questions they haven’t seen before. This isn’t an isolated unit. We could have carried on and learnt it previously, but we instead switched to doing some statistics for a bit.
I also understand the temptation to start this with bases that are numbers as it is less abstract, I think though that it leads to cognitive overload and distracts the focus away from the indices. Another benefit of starting with the bases as numbers could be that you can plug it into a calculator to show that it works but
a) kids don’t care
b) it’s not a proof
It’s just making you feel better about it as you hope the pupils will trust you now.
So this is how I would plan the learning journey. On the left is what I would show on the board and on the right what I would be saying, with the instructions around routines cut out. [Square brackets denote explanations for why I have made any pedagogical choice].
When I can’t be bothered to finish a think/pair/share script I trail off like this…
Obviously, there are a multitude of ways lessons can be derailed by pupils not being able to access the work so this isn’t a fully detailed tree diagram of how to address each possible issue. In most cases the process to deal with not having enough pupils with correct answers would be me writing another superficially different question after my summary. This is just an outline of what, in my experience, is most likely to happen.
So, have I not wasted time faffing around if I was just going to tell them what to do anyway? I believe that there are two benefits to this approach
- the knowledge is more flexible. These types of errors are less likely to happen as if you are picturing it correctly in your mind they just look weird.
It is also easier to introduce the idea of questions in the style of
as it isn’t challenging our current knowledge, it’s challenging how we read the maths.
- pupils are more likely to remember the process as it is better connected
I’ve not taught them a standalone piece of information. I’m teaching them to read maths properly and that is harder to undo.
Both reasons also mean that we save time in the future (hopefully) when “new” questions are easier to access and more pupils retain the knowledge and need less correction.
For Kris’ second example he looked at solving quadratics. I agree that it is best to start with already factorised examples and I also think that teachers usually avoid the more complicated questions with fractional answers for too long. I disagree though about hiding the fact that we’re using a mathematical trick by having it equal 0. That is just too beautiful to ignore.
At this point I switch to explicit direct instruction to add clarity to their thoughts and proceed with ever changing examples. Kris had a great selection of them.
The benefit of this start is that I can also drop in a trick question like (x+1)(x-1) = 8 much earlier and reinforce at a much deeper level that if we are going to solve quadratics then they must equal 0.
When Kris suggests starting with the how he is proposing that before doing any of the work the pupils won’t be able to access a level of understanding that makes it useful. On reflection my explanations also started with “how” it is just that I started at an earlier point. Kris is correct to say that pupils struggle to understand why before they have been exposed to the work, in me building up their knowledge in this manner it means that pupils were getting a feel for it first so that they could understand the explicit explanation better.
In my next posts I will have a more general look at great ways to introduce topics and less great ways.