What do we mean by 'Greater Depth' in maths? What would a child working at greater depth be doing? How can we support children to work at greater depth? With a little detective work we can piece together a good idea of what we might be talking about.
At first, we might think that to be working at greater depth in maths children should be fluent in their mathematical ability, and that they should be able to solve problems and reason well. But that can't be it as the National Curriculum states that those are the aims for ALL pupils:
So whilst children working at greater depth will be fluent and will solve problems and reason mathematically, we can't use those indicators to define 'Greater Depth' in maths. The National Curriculum document does give us another clue, however:
We might define children who work at greater depth as still working within the expected standard but at a deeper level (as the 2016 Interim Teacher Assessment Framework did). These children will most likely be children who 'grasp concepts rapidly' - let's assume the two are synonymous. For these children, the ones working at greater depth, we should provide 'rich and sophisticated problems' and we shouldn't just be getting them to move on to the next year group's work - this is made clear in the NC document: working within the expected standard. So, as an indicator, those working at greater depth should be able to access 'rich and sophisticated problems'.
But what about 'mastery'? A word mentioned only twice in the National Curriculum document (in relation only to English and Art) but one which has been bandied about a lot since its publication. If a child demonstrates mastery, could they be considered to be working at greater depth? In a word: no. The NCETM have this to say: "Mastery of mathematics is something that we want pupils - all pupils - to acquire, or rather to continue acquiring throughout their school lives, and beyond." Again we see that word 'all'. The NCETM say that "at any one point in a pupil’s journey through school, achieving mastery is taken to mean acquiring a solid enough understanding of the maths that’s been taught to enable him/her move on to more advanced material"- mastery is something which allows children to move on to be taught new content (c.f. to the NC) whereas working at greater depth pertains to working on current content, but at a deeper level. Notice those words 'solid enough' - a child working at greater depth won't just have 'solid enough' understanding - they'll have something more than that.
In her article 'Greater Depth at KS1 is Elementary My Dear Teacher', Rachel Rayner, a Mathematics Adviser at Herts for Learning, identifies that for pupils to be working at greater depth they should confidently and independently be able to deal with increases in complexity, deduction and reasoning. Please do read her article for more information about, and examples of, these three areas.
Complexity is not about giving children bigger numbers, nor is it necessarily giving them more numbers (for example, giving children more numbers to add together, or order). Complexity needs to be something more as, based on curriculum objectives, giving bigger numbers is just a case of moving children onto the content of a following year group.
So, how do we provide more complex work which will challenge those children identified as working at greater depth? One consultant advises that "in order to provide greater challenge we should keep the concept intact while changing the context." And, anyone who has witnessed a year 6 class doing their SATs will know that if there's one thing that throws them more than anything, it's the context of the questions. The test writers come up with endless ways of presenting maths problems but children working at greater depth are very rarely phased by these, whereas children working at the expected standard will come up against a few that they cannot answer.
The best bet for increasing the complexity of the maths but continuing to work within the expectations for the year group is to present the problems differently, and in as many ways as is possible. The more children are exposed to problems presented in new ways, the more confidently they will approach maths problems in generally - gradually, nothing will phase them and they will have the determination to apply their maths skills to anything they come across.
The NCETM Teaching for Mastery documents, although designed for assessment purposes, contain a wide range of complex problems under the heading 'Mastery with Greater Depth'. Organised under the curriculum objectives, these provide a great starting point for teachers to begin thinking outside the box with their maths questioning. Here's an example from the Year 1 document:
Here's an example from the London South West Maths Hub, taken from their year 3 documents:
It's also worth looking at the KS1 and KS2 tests to get an idea of the question variety. The mark schemes will help you to decide which year group's content is covered in each question. When picking a question from the tests, decide whether or not it could be considered as an example of greater depth, rather than just mastery. Here's an example (from an end of KS2 test) of how different the questions can look:
Reasoning is defined in the NC document as "following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language."
As already discussed, reasoning is a skill that we want every child to have. But the greater depth exemplification makes more of reasoning than the expected standard exemplification so we need to be able to differentiate between those who are reasoning at the expected standard and those who are reasoning at greater depth. When it comes to assessing children on their level of depth in reasoning, NRICH have a very useful progression of reasoning:
I would suggest that those working at greater depth would be able to work at at least step 4: justifying. The NRICH article gives excellent examples and analysis of children's reasoning work so it is a must read to become more familiar with recognising reasoning at these five different levels.
For further discussion of reasoning skills, please read this article, also on NRICH, which discusses when we need to reason and what we do when we reason.
Deduction (and asking mathematical questions)
Making deductions, a key part of reasoning, is similar to making inferences when reading and is all about looking for clues, patterns and relationships in maths. Once they have found clues they need to make conclusions based on them, and to then test them out. To be able to make conjectures, generalisations and to follow a line of enquiry, children need to ask their own questions. They need to look a sequence of numbers and ask themselves, 'Does the difference between each number in the sequence is the same?' - this is all about wonder: 'I wonder if...'.
In order for children to ask questions about maths, so that they can begin to deduce things such as patterns and rules they need to be provided with activities that encourage them to do this. But even more importantly, initially they need to have these questioning skills modelled to them by an adult. They need to be taught and shown that maths can be questioned because many children think that every maths problem just has one set answer to be found.
NRICH is the go-to place for such activities, but don't just give children a problem and expect them to be able to get on with it on their own - they need to have had much practice in questioning mathematically. Only when children are asking questions about maths, testing out their hypotheses and following lines of enquiry that they themselves have set, will they be able to reason at those higher levels set out by NRICH.
Confidence and Independence
In order for children to be working at greater depth we would expect to see a certain confidence not seen in all children. We would also want to see that they were working independently on the three areas outlined above. As already mentioned, children may need plenty of modelling before they become confident and independent - especially those children who are currently working at the expected standard who could work at greater depth with some extra help. A key indicator of whether or not children are working at greater depth will be their levels of confidence and independence (especially the latter, as some children are of a more nervous disposition yet are still highly capable).
To answer our original questions we would hope to see that children who are working at greater depth would confidently and independently:
access maths problems presented in a wide range of different, complex ways;
be able to justify and prove their conjectures when reasoning;
ask their own mathematical questions and follow their own lines of enquiry when exploring an open-ended maths problem.
In order to make provision for children working at greater depth we must:
model higher-level reasoning skills (justification and proving) and encourage children to use them;
model mathematical questioning during open-ended maths problems and encourage children to ask them;
provide complex maths problems (open and closed) with a variety of contexts and support children initially to access these, until they can do them independently;
motivate children to be confident and resilient enough to do the above.