Negative numbers, Pythagoras, and automaticity in teaching…

A bit haphazard this post, but I was doing my usual Sunday evening thing of reviewing my Twitter favourites, catching up on blogs and resources, and came across a couple of things that I wanted to share some thoughts on. The first was this post from @cavmaths on the topic of damaging short cuts – or as I call it in my lessons, cheap and dirty maths. I have a few pet hates – but number one is teachers who encourage students to ‘just add a zero’ when multiplying by 10 (to which my immediate response is “so if I see something in a shop for £4.99, does that mean I can buy 10 of them for £4.990?”). But I digress – the reason for me discussing this post is not to turn it into a rant about my pet hates, but because the original post was about negative numbers, which I am half way through teaching with my year 7 groups.

Negative numbers is a really important concept to get a grasp of early – my year 11s have shown some difficulties early this term in the context of negative terms when expanding brackets, to name but one example. So, knowing I was due to teach it early on in the year, I was intrigued to see this explanation given to an answer on (the full thread can be found here):

“The key idea is that negative numbers represent changes, not amounts. It doesn’t make sense to say that you have -4 slices of bread. It does, however, make sense to say that you ate 4 slices of bread, and therefore the change in the number of slices you have is -4.” (My emphasis)

This was a revelation for me – a really clear and succinct example that I was desperate to use in my teaching. I was lucky I had an upcoming opportunity. I knew from their pretests (see my summer posts) that my year 7s were not confident on negative numbers, so I set up a slide to show a game scenario where I received a point each time I won, but my opponent gained a point when they won. I expressed this as my receiving -1 when my opponent won. We modelled a variety of situations, focussing on the total. The initial results seem encouraging, with the students telling me they have a much clearer understanding about negative numbers. Tomorrow we move on to multiplying and subtracting after a quick recap, but I now can’t wait to do the post-test in a couple of weeks to see if it really DID make a difference. I suspect it did, and I suspect I have a ‘banker’ method for teaching a key topic.

Pythagoras – even simpler than a, b, c!

Another of my pet hates is the teaching of Pythagoras’ Theorem. Our year 11s are taught identikit lessons put together by our SLT maths link, which states that a^2 + b^2 = c^2. WHY???? My immediate response (I always seem to have one….!) is “well how do you know which is side a, b and c?” Clearly this method is based on labelling the hypotenuse as c. Here’s a thought, though: if you know that c is the hypotenuse, why not call it, I don’t know, h for shorthand? Or hyp? After all, Pythagoras’ Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. By using the a, b, c method, I fear students assume that the square of one side is equal to the sum of the squares of the other two sides. And if they DO know which side is c, surely it makes sense (and is more correct mathematically) to give that side its proper mathematical name!

There is, of course, a question to be asked about how to label the other two sides. Not having an angle in Pythagoras situations (certainly not when introducing the topic), it would be unhelpful to use adjacent and opposite, or any shorthand thereof. In my own teaching, I refer to the other two sides simply as ss1 and ss2 (ss meaning ‘shorter side’). I explain that because the hypotenuse is the longest side, the other two sides are both shorter. It is irrelevant to distinguish further between the two.

I have yet to receive any negative feedback about this approach.

Automaticity in teaching

The second tweet was this from @oliverquinlan, which posed the question “How much can you systematise your teaching..?” The post is of little use to me (it is effectively an advert for a book), but the question resonated with me greatly. Systematising teaching – making it as predictable as possible is my interpretation – is something I have been trying to do with regard to my planning and my timetabling of my non-contact time, and which I wrote about during the summer. The reason is cognitive and expressed in ‘Why Don’t Students Like School?’ By getting into habits, reducing the number of decisions we make, and becoming robotic in certain aspects of our practice, we open ourselves up to creativity. As Willingham says, creative thoughts occur most when we think. By making things systematic, we think less, and our creativity increases. I can certainly point to many examples already this year of this in action. I can honestly say I feel like a different teacher. Systematising my teaching has definitely worked for me!


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