A few years back there was an ad campaign for Snickers where people would be shown throwing a wobbler and then became calm when someone gave them a chocolate bar.

That’s me with lovely maths questions. Could literally be fewmin and stomping around but chuck me a nice circle theorem question or a chunky bit of algebra and I’m all good.

So I’m thinking about maths questions a lot.

TLDR: Here’s some lovely questions. Fill your boots. (Reasoning With Bearings)

The two things I’m focusing on in my questioning at the moment are:

- Directing student attention to the conditions and conceptual structures underneath the maths; and
- Making the implicit explicit

Below are a three pairs of questions that do this for bearings.

These questions are designed to combat the misconception of north being ‘up’ on a page, and to make explicit the directional plane as layered on top of a grid plane. The first pairs of questions are designed to address this and reason with half turns as well; whereas the second and third pairs are about the impact and conditions two out of the three aspects of these questions (point A, point B and north) affect each other when changed.

In all question pairs, part 2 is the same to see the way this changes given the conditions of each question.

Exploring this further; each picture below shows variations of each of the three questions above.

Here I find it interesting how it’s questions 3 and 5 that have the same answer and not 1 and 3. Again, these can be used to make predictions before doing them, or to discuss what we notice after.

Here we can start sneakily linking up to gradients and perpendicular lines on grids. I’ve recently started describing slopes as compound measures between our two dimensions (a metaphor I think has legs but need to explore further before I commit) and this sequence of questions ties into links with slope nicely too.

In this set of questions, parts 1, 3 and 5 stay the same while there’s some lovely geometric reasoning going on questions 4 and 6. Or problem solving. I’m becoming more convinced the definitions between fluency, reasoning and problem solving during the course of teaching (assessment is a different thing) depend more on the reception of the learner than it does the intention of the teacher.

Thinking about interleaving/reasoning/procedural fluency; the following angle-maze style question on the right is a good example to pick apart some key aspects of these bandied about terms.

Here on the right, question 1 and 2 are difficult in terms of getting pupils to filter their attention and work their way through lots of ‘skills’/’core knowledge’/’whatever you’re calling it’ and could be an example of interleaving key content throughout angles, or could be reasoning as pupils need to decide and filter what they’re looking at.

Question 3 is actually about the properties of parallelograms; and a neat way into talking about conditions of where things hold and what aspects need to change for certain things to be true.

Question 4 here does two things. Firstly, it acts as a explicit prompt to extend lines, something we as experts do when answering questions (looking around at everyone’s scribbles when we do a geometry problem in our department meetings is a fascinating study in how each of our minds show up on the page unfiltered first). Secondly, the resulting quadrilateral pupils have to find the missing angle for has a transversal line splitting one of the angles, so pupils have to add together 4 angles and take the total from 360, which makes explicit what we mean by interior angles of a polygon (something appreciated when learning about them in detail, but is often a sticking point in ‘seeing’ much later, particularly at the top end of higher GCSE)

The picture below breaks down all the things pupils need to know to do the four questions (these are they way I would do it, which is probably not the most elegant).

When is it appropriate to use a question like this? It could be used as a synoptic task bringing everything together, or it could be used mid-statistics unit like a cat amongst pigeons, but the value in tasks like this (and in the bearings tasks above) goes beyond being able to do the maths and are a jumping off point for what comes later.

These took ages to think about and describe the subtleties of what’s going on in each part. Individual teachers and departments cannot be doing this for every single bit of maths and its glorious web of connections between content areas.

All the pictures above are in the powerpoint linked at the top of the post.

As always, I’m interested in any feedback or similar resources people know of that tackle the things I’m trying to address in these questions so please get it touch via twitter or here; but not until I’ve had a brew and a Snickers obviously.