I have spent much of my teaching career trying to work out what limits pupils’ progress in mathematics. Why is it that some pupils seem to make less progress? Of course, there is no one single reason, but it has made me reflect on some of the factors which contribute, and vitally, if there is anything that we can do to mitigate these. This blog focuses on the importance of understanding the wider mathematical journey in order to find the right starting point for pupils and to prioritise securing crucial milestones.
As a teacher, I feel as though my understanding of the mathematical journey has widened over time and with experience. Initially, I probably only considered the journey through the curriculum of the year group that I was teaching and to be honest probably saw this as a variety of endpoints that I raced headlong towards. It has been over time that I have come to understand the significance of the small steps, which need to be secured, in order to advance to the year-end destinations and over an even longer time that I have come to realise how gaps further back in learning can limit progress so acutely.
Of course, I have always known that mathematics is a hierarchical subject. I appreciate the huge schema of interconnected steps that lead to secure mathematical learning. I think what I hadn’t fully appreciated is that a gap in any one area can have such far-reaching impact and that this impact may not be immediately obvious. Often deficits and learning insecurities, what I now refer to as “shaky learning” can be masked by pupil behaviour to hide this or overlooked through ineffective focus on seeking it out. This “shaky learning” sometimes holds for a bit, it sometimes last for years and it can last forever. The problem is that any weak foundations can be the start of much bigger issues: problems making sense of what follows, a reliance on memorising “tricks” as underlying concepts are not understood and dependence on inefficient strategies.
When training, I often relay my own mathematical journey; how I presented as a secure mathematician at primary school. I worked hard, I followed the pattern of the lessons, I learnt some facts and I remembered some strategies. It was only as things became more and more abstract and letters started to fly around the pages during my secondary maths lessons that I started to feel lost. I suddenly felt as though I did not understand what everyone else seemed to know.
A relatively simple equation such as 3x + 1 = 2x + 4 meant little to me and on reflection; the problem was that I didn’t fully understand equality and the equals sign as a balance. In fact the “shaky learning” probably began somewhere in my infant years and had never been picked up. It was at this point that my own mathematical progress slowed significantly. Not only did I not understand the maths that my peers did, I also started to lose confidence and stopped pushing myself into new challenges instead seeking reassurance that what I had done was correct.
I am sure you will have children in your class of whom you have said, “they just aren’t confident in their maths”. My question would be – what has led to that lack of confidence? What is their “shaky learning”? Of course, that question is hard to answer due to the very nature of all the interconnected ideas that are continually being built on in mathematics. However, the more we can seek these out, the more we can accelerate progress because as we fill the gap not only do we secure that mathematical understanding, we also provide firm foundations for all other relevant connections to be built from.
So what does this mean for teachers? I think there are two key things that teachers need to consider: what are the crucial milestones and how can we find ways to check that pupils have reached these before moving into learning which builds on them?
Get to know the crucial mathematical milestones
Remembering that new learning will build on multiple previous elements and that gaps may be further back in learning than expected, this might seem an impossible task. In fact, it feels like it could be a lifetime of subject knowledge training and actually, I would suggest that this is something teachers do not get enough of – particularly the cross phase kind of training that would allow teachers to understand how learning builds.
Other practical suggestions could be to use the resources around you to learn more about the maths journey. Teachers in other year groups and phases can provide excellent support in helping to unpick likely barriers to learning and also in knowing where the learning will have come from or where it is going to. Make time for some subject knowledge development to support understanding of the progression through looking at how the same concepts are taught in different year groups. If you are an ESSENTIALmaths user, remember to read the front page of each learning sequence, where the links forwards and backwards are often clearly identified and use the tracking back documents.
Once the maths journey is more clearly identified, it is possible to see what the crucial milestones might be. I think it is important to note here that in considering what is crucial; sometimes it is only through knowing where the learning is going next that the significance becomes clear.
For example – by providing children with lots of opportunity to regroup numbers in different ways…
…allows a deeper understanding of decomposition as a method of subtraction.
Find ways to check that pupils are building on secure foundations
Assessment for learning will be key in establishing whether foundations are secure and I would urge teachers to plan ahead where possible to give themselves the best chance of being able to respond to pupil needs. Remembering that “shaky learning” is often well hidden by pupils, there is a need to be absolutely clear about what you are looking for and being a real detective in the classroom.
One useful resource could be a diagnostic assessment. I really enjoy writing these to support teachers to identify barriers to learning and to make the link to teaching so that they can plan to meet pupil needs. However, I know that teachers sometimes feel as though the questions writer is just being mean. “They would have got that one, if it wasn’t for…” That is the point. That mistake that the child just made – that is what I was looking for. That tells me that there may be some “shaky learning” here which is great because now I can do something about it.
Of course, diagnostic assessments are just one example of making use of good questions. To help find out whether foundations are secure, I might suggest teachers think about what it is that they are trying to find out and the question that would expose that. Now when I say expose, I do want to know the gap, but also recognise that pupils can become very good at hiding their weaknesses and so sometimes actually listening in on the talk between pupils can be provide more than asking for a recorded response when pupils may rely on the resources around them!
Once a gap has been identified it is, I would suggest that the more you can be specific about it the better as this will help you identify the appropriate starting point more effectively. Rather than saying “they don’t know their multiplication facts”, I will want to dig deeper and ask “Which ones? Which ones do they know? Which strategies/understanding of the arithmetic laws have they got? Can they build me 3 x 4?” It is this further diagnosis that will lead to more chance at getting to the root of the problem and without getting to the root of the problem, you will probably only sticky plaster the cracks in the foundations.
When I am doing work around closing learning gaps, I often use the analogy of the rose bush. Gardeners know that to get the best rose blooms, they must cut the bush right back. It will look a little ugly at the time and can feel quite scary, but when the spring comes you will see the benefits. And so it is with learning, cut right back to the last good growth – the point at which the foundations were secure (to mix analogies). Be brave and you will see the results. I know teachers feel pressure to “get through” the curriculum, but this is pointless if it doesn’t lead to successful learning. Remember that a starting point is just that. You don’t stay there. It is just the point from which you start – the point from which you can accelerate.
Why is this all so important?
Weak foundations are not easily built upon and are much more likely to collapse. Weak mathematical foundations do not lend themselves to strong connections being made and are less likely to lead to learning being retained.
Put simply – as pupils move further along in their mathematical learning journey with gaps, the gap gets bigger and then students start to fall into a dangerous zone where the gap becomes a chasm which we assume can never be closed. It can absolutely be closed but will need time and great commitment from teachers to do so - if not the chasm will keep widening until, hopefully, that child finds the teacher who can. Will it be you?