Reader you probably won’t be surprised to hear that I find myself working with schools to improve provision for children still referred to as the ‘more able’. This is not least because Ofsted reports are coming back with commentary that these children are not being challenged enough. In some reports the commentary is more specific citing instances where children are not being encouraged to think deeply enough about the maths that they're learning. Another reason is that schools would obviously like to increase the number of children reaching the higher standard in the Year 6 SATs.
And this is interesting me greatly because of course ‘greater depth’ and reaching the ‘higher standard’ in mathematics are not necessarily quite the same thing.
Let's start with greater depth. At the end of Keystage One, children are assessed against the teacher assessment framework. Some children may meet all of the secure fit criteria for age related and also the criteria for working at greater depth within age related expectations. This must be achieved independently by the child. They are then assessed as working at ‘greater depth’. Schools continue to talk about children working at ‘greater depth’ beyond KS1 and into KS2.
So, what do I want ‘greater depth’ judgement to look like in our children?
I think those children that are consistently working at ‘greater depth', as schools talk about it, are those children who frequently thrive on thinking that little bit more. They are able to appreciate viewpoints other than their own and accommodate other ways of thinking into their own approach. They also have multiple strategies and therefore can solve the same problem in a number of different ways. These children are also prepared and able to convince and prove using different resources or representations. As a result, these children are often able to deal with more complex renditions of the maths they are learning. It can be served up differently and it won't throw them at all. And these children are often the children who are able to deduce what's coming next from the clues that you've already showed them. For example, when provided with two connected multiplication facts such as 18 ÷ 3 and 180 ÷ 30 and asking, “What's the same and what's different?” children working at greater depth will be able to suggest further related examples where the quotient is 6 and reason why with less hand holding along the way.
They may then go on to consider examples for the quotient 60, keeping a weather eye on the relationships between new facts and the original fact as they go. Reasoning is precise and, rather than being descriptive or a recount of what they did to solve the problem, really nails the why. Moreover, these confident thinkers are likely to display curiosity about what they are learning, asking their own questions as they tinker around the edges of the examples the teacher has provided.
But shouldn’t all children be thinking at ‘greater depth’ about their maths?
Well, yes. I strongly feel that all children need to be thinking more deeply about the maths that they're doing at the stage they are at. This is different to the overall judgement of 'greater depth' I hear used in schools to denote a group of children. I see some pretty poor proxies for judging whether children are working beyond the expectations of the curriculum. One of these is speed. Children are often sat on the ‘top’ table because they're quick at the maths they do. They are inevitably keen calculators who thrive on pages of long multiplication for example, where they can happily continue to follow a procedure and get the answer right. Yet these children are not always the ones who are doing the most thinking. They are often happy to follow rules and procedures to get the right answer they don't necessarily want to think outside of that.
A good example was one such young lady I met recently in Year 6 who was very happy with her method finding equivalences and adding and subtracting her fractions but who did not want to estimate or develop fractional benchmarks. Being asked to do this slowed her down and she just wanted to show everybody how quickly she could solve the examples - she also required the answer to be absolute. Don’t get me wrong I have no problem with children being automatic with procedures (see my comments on the higher standard), but I really would like them to have more of a sense of what they doing and why. Accommodating new views and strategies is not always easy for these children. But consider her approach to the following questions where her love of finding common denominators will neither prove efficient or necessary.
Now in some ways the shift in language from more able to working at greater depth is helping to shift the culture of challenge in the curriculum from acceleration to a more robust approach that fosters connections and relationships. But in practical terms, I see groups of learners who never get to think more deeply about their maths. They're stuck in the fluency section and are not allowed to move onto reasoning or problem solving until they are finished. As though they are not capable of deeper thought. It's often my aim to dispel this myth.
One such way is to develop the culture of the classroom so that the teacher gets the space to listen to the learners.
Anne Watson, Professor Emeritus for Maths Education at Oxford University, talks about making the private maths in our head public and by taking time out to listen to the thinking our children are engaged in and that by sharing this we are developing a culture of valuing and encouraging deeper thinking.
Classrooms where great ideas can be shared and deep thought is valued are a joy to visit. A really good recent example of this happened while team teaching in a Year 4 classroom. The children were counting in sevens and learning all about which numbers are multiples of seven. We counted on the counting stick and removed some numbers, discussing ways we could derive the missing multiples from the multiples we could still see. Next the children used beadstrings to find the 4th multiple and the 8th multiple of seven themselves.
As I was walking the tables listening to the children, one young man, called Sean, shyly shared with me that the 4th multiple of fourteen was equal to the 8th multiple of seven and he showed me how he knew on the beadstring. In the opposite corner of the room was a gaggle of fast finishers, so I shared Sean's thinking with them. They spent a minute or two deciding whether Sean was correct and why he was correct before suggesting other multiples of fourteen that were equal to multiples of seven. Here, without being taught, the children were beginning to explore common multiples outside of the usual times tables facts. They began to make generalisations about the relationship between multiples of fourteen and multiples of seven.
This cost me nothing, and it's not the kind of thing you can predict will happen when planning. It’s an opportunistic little gem that just needs sharing and, for me, this entirely encapsulates what Anne Watson was talking about. The realisation could have stayed private in Sean’s head, but by listening and sharing we made it public and other people had the opportunity to think to think about it and to accommodate it into the way they were thinking about multiples. The really lovely thing about this story is that Sean was not one of those children that the teacher thought of as a ‘greater depth child’, but boy you should have seen his body language afterwards. You could see how proud he was in the straightness of his back. In the end the whole class was thinking more deeply, initiated by Sean.
However, I believe that working at greater depth within age related expectations is one thing and the higher standard is quite another. I think the idea of promoting maths at greater depth is about learning, whereas, reaching the higher standard is most definitely about performance over three tests. Bluntly, the higher standard judgement is about how many marks your children can score on the Key Stage 2 SATs papers. This means schools have to prepare children in a slightly different way. For a start if they're going to get all of those marks there going to need to get through all of each paper. Consequently, they're going to need to be able to sustain their attention long enough to get to the end. We can support this in classrooms by ensuring that children have an adequate amount of practice that they undertake over the whole of Key stage Two ideally. They also need to be automatic with fundamental knowledge and procedures. They’re less likely to become overloaded if they have this at their disposal, and are likely to be quicker. They also have to be able to deal with the range of responses, language and variation of presentation thrown at them in the test. This takes experience over time.
A teacher on training reported that her subject leader was extremely frustrated about this tension. She was bemused that, although it was clear through books and in lessons that children were being challenged appropriately through 'greater depth' examples, the school was not increasing the numbers of children reaching the higher standard at the end of Year 6. One thing was not leading to the other.
This made me think of my son, who is in Year 6 now. At a parents’ evening, I could see he was solving all kinds of complex problems across all of the maths curriculum. He moved through fluency, into reasoning and problems solving and into red hot chilli examples pretty much daily. His reasoning was relatively secure and he was proving in different ways and dealt really well with the wide range of examples provided. And then his teacher commented that my son could be ‘ponderous’ when working though examples. Aha I thought, you’re worried he won’t reach the higher standard. And perhaps he won’t reach the magic threshold in May, but I know that he finds maths very satisfying and enjoys thinking about it immensely, possibly a reason why he is ponderous. Or could it be that he is not quite automatic with some fundamental knowledge and this slows him down? I tend to think the latter is most likely the case. He simply was not completing enough practise at each stage to be automatic with procedures and core knowledge. He may be a fantastic problem solver but he is at times quite inefficient as a result of this lack of procedural fluency. The opposite, perhaps, of the young lady from earlier in the blog.
So we can have children, who consistently think deeply about their maths, who may not have the procedural fluency or enough facts at their fingertips to 'perform' well. And we have the procedurally fluent but intractable pupils for whom deeper thinking and understanding the maths has become a threat to their sense of success. Is it possible to gain enough marks to reach the higher standard without ever thinking deeply? Well there you have me, reader. I wonder. Sadly, I think it is to reach the expected standard. Perhaps we should think instead of the mathematical learners we would like to leave Year 6 prepared well for the future. It would have to be a pretty amazing test to get it right for every single child. Is any single assessment that good?
Barton, C (2018) Anne Watson and John Mason: Variation, questioning, visualising and developing mathematical thinkers [Mr Barton's Maths Podcasts 14th March] Available at: www.mrbartonmaths.com/blog/anne-watson-and-john-mason-variation-questioning-visualising-and-developing-mathematical-thinkers/ (Accessed: 3rd June 2018)
STA (2018) Key stage 2, Mathematics Paper 2: Reasoning: Crown publishing