Maps and Maths

Wow, it has been a while since I have written anything other than posts to link sessions to, but I had something in my head today which I just had to write about.

Recently Ofsted have produced a report, "Coordinating mathematical success" where they outline what their inspections have found about what maths teachers are doing well, and not so well. The Association of Teachers of Mathematics (of which I am a member just to make sure any biases are public) have written a response to this which I think is very balanced - supportive in some areas and rightly critical in other. There is one section of this response to highlight:

"It is of course possible to create a journey of ‘small steps’ with the hope that these steps become connected, but we challenge the prominence of these small steps in the report. When undertaking a journey, the primary focus shouldn’t be on each step but rather on an awareness of the landscape and the multiple possibilities within it. "

I have seen these arguments before and also seen people before liken this to a map - with some advocating providing a map of the journey through mathematics, with strict adherence to the planned route and a focus on each "step" of the journey and where it takes us (the "small steps" indicated above), whilst others advocate for just getting out there and exploring, creating our own sense of where things are in the landscape, and looking out for interesting landmarks to go and see. 

The reason this came to mind is that today I was travelling to the lovely High School Leckhampton in Cheltenham to work with the GLOW Maths Hub LLME community on using Cuisenaire rods. As it was a nice day I decided to walk the (little over half an hour) journey from Cheltenham Spa train station to the school. On my way, I used Google Maps walking feature in the way I often do, which is to get it to show me the journey, but not to start the navigation so that I can retain an overview of the whole trip and my progress on it. Now, of course, there is plenty of analogy already there in terms of making sure that pupils retain a sense of the journey and where they are within it, but it wasn't this that prompted my reflection. What did prompt my reflection were three incidents along the journey:

1) I took a wrong turn at a place where the map wasn't complete. I was supposed to follow a footpath that appeared to be the only way I could go on the map. However, when I actually arrived there was a smaller path (that turned out to be the path I should be taking) and a larger path (which I assumed would be the one I would take). I walked for about a minute down this larger path before glancing at my phone and realising that I wasn't following the path laid out for me, and that the path that I was following didn't appear to be marked on the map. Could I have got from where I actually was back on course? Perhaps, but not being familiar with the area I didn't know this (or know how) and so I back tracked the way I had come to get back on the correct path. This cost me a small amount of time, but fortunately I had factored extra time into my journey.

2) A little later I came out of the correct path onto a road on which I was supposed to turn left and follow around an arc to continue my journey when I noticed that, across the road, a new path seemed to continue on in the direction I was headed, potentially cutting out the arc. This path was also absent from the map. In that moment I faced a choice - follow the route as programmed or divert off on this path which I was fairly sure was going in the right direction, and possibly quicker than the route shown on the map, However, I couldn't guarantee that this was the case, that the path wouldn't veer off and take me in an unwanted direction. I made the choice that I suspect quite a few people would make which was to forgo the new path and continue on the route laid out. When I got around the bend a little later, I noticed that there was a path exiting onto the road again, and noted that this was likely the path I had noticed before.

3) On my way back from the school to the station, I didn't really need my map. I had it on, as a safety net, and my have glanced at it once or twice, but I didn't need it to tell me all of the twists and turns I needed to take - I knew the way I had come and was quite comfortable back-tracking along that route. Except I didn't really back-track along that route. I was able to adapt it to make it more comfortable for me. This time I did take the path I alluded to in point 2, and it was indeed quicker than following the road around. I had also noticed on my journey to the school that the small path I had come down in point 1, was connected back to the road I had walked down before joining it at a point further along the road, and that I could probably save myself an uncomfortable and a little muddy walk by cutting out onto the road earlier when I was coming back. So I did that (and this is one of the points where I did glance at the map again to confirm that this would be an acceptable alternative.

Why did this prompt me to think about maths teaching/curriculum (as if the reader couldn't guess!)? Well it suggested to me a few things:

1) The map was ultimately important. Because I was under a time pressure going in both directions (one to get to the session on time, and one to get to my return train on time), I likely wouldn't have got to where I needed to be when I needed to be without the map. Given that school level maths has its own time pressures built in, I think the idea of having the map and a planned journey within it is important, which translates directly to a well-sequenced curriculum that takes learners on a journey.

2) The usefulness of the map, and sticking more rigidly to the planned route on the way to the school was, in part, down to the fact that it allowed me see places where, on my return journey, I could perhaps be a little more flexible. As I mentioned, I didn't use the step by step navigation because I like to retain the overview but I wonder if I had, whether I would have noticed those same places where I could adapt the route as I did in point 3 above. I wonder about whether we are doing a disservice to pupils in situations where we take them through very small steps, particularly for those pupils (like me) who would get frustrated if they couldn't see where that step fell in the journey, and also if this deprives all pupils of the opportunity to notice places where they could take an alternative. However, this is linked to the next reflection/suggestion...

3) It was only because I had followed the route provided on the way there that I felt comfortable making adaptations when I followed the route in reverse. I don't think, it is the reverse thing here that is important - I think it was familiarity with the journey between the two. I feel like I could walk a very similar route again from the station to the school without the map, or only having to reference it occasionally rather than follow it completely (at least whilst it is fresh in my memory). But it was only the familiarity I gained that has given me this confidence. Now I am sure that I could eventually get the same confidence if I took the time to stroll around the space between the two venues, but that time wasn't available to me given the aforementioned time pressures. I also think that the map is useful to help with my explorations were I to have the time to explore, which leads to the next point...

4) The route I took wasn't the only route I could have taken. When I first put my destination into Google Maps, it offered me three possible routes to complete that journey. Being able to look at the map suggested several others (mainly slight deviations or possible shortcuts based on the three main routes). On the route I followed I saw several landmarks - a church, a Texaco garage, street names etc. that were useful checks that I had for when I made the return trip and would be useful were I to make the trip again. But there were lots of things I didn't see because of the route I chose. Some of those things may have been more interesting than the church or the garage. I suspect had I taken a more westerly route my views would have been better (judging by the glimpses of the countryside I saw in that direction whilst following the route I did, and my subsequent scrolling of Google Maps to see what might lie in that direction). 

So what do I take away from all of this?

1) A planned sequence is ultimately useful, but having the step by step navigation would perhaps draw attention away from the journey itself, frustrate those that need to see how their progress fits into the journey as a whole, and might mean those following the journey miss some things that could build their confidence in being adaptable when encountering the same or similar content in the future - so part of our job is making sure that they don't miss these things and that they have the opportunity to recognise the important aspects of the landscape.

2) No route or map is perfect, no matter how expert the map maker/route planner, so it is important to build in time to allow for the occasional wrong turn, even with those who are generally quite good at following the journey - a new journey is still a new journey.

3) For some at least, the confidence to explore and adapt will be enhanced by the security of "the map", the pre-existing knowledge of the landscape that can be accessed, and knowing that they are never too far away from a familiar space.

4) We must build in time for that exploration and adaptation once the map is laid out - if we are only ever moving onto a new journey with every step then, for most, the only thing they will feel like they can do is follow the route exactly.

5) When we plan a journey through a curriculum for pupils there will always be different competing considerations - time, which route allows the "best" experiences, the attitudes of the learners towards possible wrong turns or deviations etc. and we need to evaluate our chosen journey to look at what we sacrifice as much as what we gain.

Of course, the big difference in this scenario is the pupils can't really "see" the map until they have experienced it. I was able to call up a map that someone else had created, but in learning mathematics the "map" exists inside everybody's head, I have my map, you have yours. Learners create their own maps when we take them on a journey through a mathematical idea, and whilst we can communicate something about what that journey looks like, it only becomes real once experienced - I can't simply transfer my "map" into my pupils' heads. I think this reinforces point 4 above about building in time for exploration in areas that pupils have already travelled, revisiting ideas not just to see if they can remember the exact same journey as before, but to give them chances to take shortcuts, amend the route, or explore the consequences of going off on a new path.

My final thought is that I wonder how much of this is specific to maths? It seems like a lot of it might apply to many subject areas but, not being overly familiar with many of their maps, I cannot say for sure.