Hyperspace Hopping

The Resource

This is a worksheet which introduces some of the ideas contained within the 3-d geometry section of A-level maths: vector equations of lines and planes and testing whether or not points lie on these planes.  The latter questions also give students an idea of linear dependence, although this understanding isn’t strictly necessary for the A-level course.

It’s in the context of getting from one planet (coordinate) to another using the buttons (direction vectors) in a spacecraft.

I’ve used this as an initial introduction to vectors, as was the norm at my previous school (from where I stole the idea – I think it comes almost directly from Robert Wilne). More recently I’ve used it as an introduction to 3-d vectors, after previously teaching 2-d vectors and this has been more effective as the pupils have a better toolkit of terminology to help them solve the problems, but the context seems to make the leap from lines to planes more intuitive.

And it’s called hyperspace hopping, so it’s fun! Enjoy.

Flippin’ Heck

I’ve very much enjoyed several of Craig Barton’s podcasts recently and they are the highest quality audio source of ideas for Teaching and Learning I can find. I also like radio 4’s the educators but it doesn’t seem to be producing new programs, or at least they are very rare. If you know of any other podcasts that you think are similarly great, please comment below.

The title of this post refers not only to Craig’s most common response to his guests’ ideas (which makes me feel at home as a northerner!) but could also describe my overall response to the most recent episode with Greg Ashman.

Firstly, I love the sound of Greg’s department. It sounds very similar to the department at Highgate, where I worked for seven years. From my experience of other schools, I think it’s pretty rare to find a department where all the teachers work together so closely and come to agreement on the best approaches to teaching, but I think it’s a great idea and wish it was more common. From my own experience, it gives the students a very consistent year-to-year experience even if they change teachers, and encourages levels of discussion and debate amongst staff that just don’t happen without it.

I’m also a big fan of Greg’s behaviour management strategy of pointing out students that are doing what you’ve asked, instead of those who are not. I encountered this idea a couple of years ago in the form of “doing the politician” (when politicians come on stage they often point out their supporters) and it has been incredibly effective.

Then we come to discovery learning. Flippin’ Heck!

I’ve never heard of Cognitive Load Theory before reading about it through Greg’s blog, so I’m no expert, but this application of it seems relatively intuitive: students ‘working memory’ is limited and quickly becomes overwhelmed. I definitely agree with this and have regularly witnessed the problem he describes: students working memory is taken up by sub-tasks required as part of the discovery and so they fail to make the required discovery.  This can definitely be frustrating for me as a teacher and some pupils find it stressful.  It’s for this reason that I spend much of my planning time providing a variety of different levels of scaffolding (best case scenario, I admit!) and design resources which aim to reduce the cognitive load during discovery. Furthermore, I have put a lot of effort recently into improving the clarity of my follow-up explicit instruction and I’ve reduced the proportion of lesson time spent on discovery tasks, as I am to some extent convinced by the research evidence to which Greg refers. I also enjoyed the toilet fixing / beer tap installation anecdotes, although I personally very much enjoy discovery DIY!

Why am I not willing to stop entirely?

I’m a little worried that I’m suffering at the hands of the backfire effect, but I have another major reason for using discovery: motivation.  I find that a large majority of pupils are excited by the joy of discovery and feel empowered as mathematicians; my lessons almost never contain the games, frequent changes in or variety of tasks used by many teachers to gain attention but as one of my heroes Michel Thomas said,”it’s the learning process itself that motivates these kids, not the material used”. Also see Dan Meyer’s blog: “if math is basketball, let students play the game.” Greg does go on to talk about problem solving, and how he uses it after explicit instruction, but is that really playing the game of maths?!

I’d also say that as a team of 20 teachers at Highgate, we found that our students seemed to gain a better understanding using these approaches, as measured by teaching them in later years. Although this isn’t hugely scientific, they also did better in the UKMT maths challenges (the best test of problem solving skills I know) relative to students of similar ‘ability’ (Midyis scores) elsewhere. NB: sample size = 2 schools.

Overall, I like to think that my approach isn’t actually as different from Greg’s as my former blog title (discovery maths) suggested, as I am a proponent of more research-based approaches in schools, would like to do further study myself and am very envious of the quality of his writing.

Equations as Balances

The Resource

I think this is a fairly standard introduction to equations, attempting to embed the idea that the rule is that you perform the same operation to both sides in order to keep it balanced.

This works particularly well if you have some balance scales in your department with which you can demonstrate an example of finding an unknown mass by removing objects from both sides etc.

At Highgate, we used to follow it up by taking the model further, to cover other operations / situations which don’t really fit the traditional balance model. This worksheet does that, but I’m not sure I’d particularly advice it unless your whole department is working together to get the pupils thinking about equations as balances in this way.