Showing posts with label greater depth. Show all posts
Showing posts with label greater depth. Show all posts

Friday, 2 February 2018

General Principles Of Challenge For Higher Prior Attaining Pupils

General Principles Of Challenge For Higher Prior Attaining Pupils

The National Curriculum states that 'teachers should set high expectations for every pupil. They should plan stretching work for pupils whose attainment is significantly above the expected standard.' My question is: are there general principles that underpin the setting of stretching work?

In this blog post I propose 6 general principles (there may well be more) to consider when thinking about how to challenge higher prior attaining pupils:
  1. More (and More Complex) Knowledge
  2. Provide Opportunities For Children To Demonstrate And Do Something With Their Knowledge
  3. Teach Masterclasses
  4. Remove Scaffolds and Support To Encourage Independence
  5. Communicate High Expectations
  6. Personalise The Challenge
Teachers can often identify the kinds of activity that they provide for pupils who are achieving highly, but it's more difficult to put a finger on identifying the general principles which might be applied when trying to challenge children. It's not too hard to call to mind a particular pupil and then recall the ways in which they've been provided for: writing an alternate viewpoint, encouraging debate in response to texts they've read or solving complex open-ended problems covering a range of maths domains, for example. But are there similarities between the subject-specific challenge activities that children are given? And is it actually all about activity, or something more?

Image taken from ronbassett.com
Bloom's Taxonomy and Knowledge

When teachers' thoughts turn to the general principles that might apply to challenging higher prior attainers (I'm avoiding the other 'a' word), they might often think of Bloom's Taxonomy and the so-called Higher Order Thinking Skills. And it's not a bad starting point; the taxonomy certainly has some implications here. David Didau and Doug Lemov have both written interesting articles about the issues with the taxonomy, particularly in its pyramid form. They both point out that without knowledge at the bottom none of the other skills can be exercised. The pyramid might suggest that reaching the top is the goal but at the same time the fact that the foot of the pyramid (knowledge) is the largest section, the foundation on which the rest is built, points towards the fact that a secure knowledge base is crucial. This in turn points us towards what might be the most important way to challenge higher prior attainers (HPA).

General Principle 1: More (and More Complex) Knowledge

Someone (maybe just a person on the internet) said "The more you know, the more you know you don't know." For those with a thirst for knowledge, this realisation that there is more to know spurs them on to learn more. If you have an HPA who has lapped up all the facts and information you've taught them so far, then surely a simple way to challenge them is to provide them with more information on the subject, or linked to the subject. It might be that this information is harder to understand, or relies on a good understanding of previous information (it will), but that's where the challenge lies for the child. I also think that the second category in the taxonomy should never be separated from the first: knowledge and comprehension should always be hand in hand - what good are facts, information and knowledge if not comprehended?

Contrary to popular belief the National Curriculum does support this principle. For example, in the maths curriculum it says this: 'The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage.' Although the focus is on NOT teaching more, it does acknowledge that for some children, this is a legitimate way forward. This then becomes an issue of pace - sometimes children will need to accelerate past something they are already capable of doing, other times they will need to work at something more slowly in order to ensure it really is excellent.

Bloom's Taxonomy and 'Higher Order Thinking Skills'

Where knowledge is definitely the keystone of any model relating to different thinking categories, I think it is probably arguable that the other categories don't need to be set out so hierarchically, or at least that children don't need to work through them in a linear fashion in every context. What I wouldn't argue with is that it is a good idea to develop thinking and learning in the other categories; I just don't think they need to be thought of as higher order skills. However, what might make someone a higher order thinker is the ability to demonstrate their knowledge securely across the range of categories in the taxonomy:


Image taken from getsmarter.com
General Principle 2: Provide Opportunities For Children To Demonstrate And Do Something With Their Knowledge

In that heading I mean to sum up the other categories of Bloom's original taxonomy: application, analysis, synthesis, evaluation. If knowledge (and comprehension of that knowledge) is what the other categories are built on, then the other categories are all ways of showing that the knowledge is understood. Having said this, when the skills outlined by the rest of the model are exercised there will be additional outcomes - they all provide opportunities for knowing to inform actions.

When it comes to providing opportunities for children to exercise a range of skills in these categories the verb-based revised taxonomy can be quite inspiring when it comes to designing challenging work. Taking just one verb, such as 'dramatise', can spark inspiration in the mind of the teacher who is planning work for HPAs. Every time a child is asked to do something with their knowledge in a different way there is the opportunity to take them out of the comfort zone and into the learning zone (or the optimal performance zone).

In addition to teaching new and more complex knowledge the National Curriculum suggests, in the maths aims section, that 'pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content'. This is an example of this second general principle.

General Principle 3: Teach Masterclasses

Whilst another principle of challenging HPAs might be encouraging greater levels of independence (more about this next), it is very difficult for us to become independent when doing new things. If we are going to provide children with new and more complex knowledge, and if we are going to expect them to do new and varied things with this knowledge, then we will need to teach them more. It might be that we teach either know-what (knowledge) or know-how (knowledge of how to do something, which leads to children learning a new skill) in these masterclasses.

Image taken from What Does This Look Like
In The Classroom? by Robin Macpherson
and Carl Hendrick
In their book Robin Macpherson and Carl Hendrick summarise that 'independence does not seem to be the best way to become independent' and that 'independent learning looks different at different stages, and develops as a continuum rather than having the nature of a spectrum'. They go onto point out that 'independence is something that is achieved and exercised by experts, and our students will have a long way to go to reach this level of ability. What we do as teachers along the way is vital, and requires a lot of scaffolding'. Our HPAs may have attained a certain level of independence (i.e. they may be able to easily do the work that the rest of the class are doing) but will need to be taught the things they need to know in order to become independent at the next level. Becoming independent in one thing does not automatically mean a child will be able to work independently at something more difficult.

General Principle 4: Remove Scaffolds and Support To Encourage Independence

The assessment frameworks for Year 2 and Year 6 use an interesting phrase with regard to our higher prior attaining pupils - they refer to 'working at greater depth within the expected standard'. Where children are still working within the expected standard independence is a good goal for them. They know what they need to know (know-what and know-how) and before being exposed to new material, or even material with some depth, they should be able to work without the aid of adults, scaffolded worksheets and so on. As discussed above, as soon as new knowledge is introduced, they will need scaffolding and support before they reach independence with the new knowledge and skills that they have learned. An analogy: the independent mastery of the making of a boiled egg doesn't mean that the preparation of a carbonara can be achieved independently without some form of instruction. This general principle definitely comes with a health warning: don't throw your HPAs in at the deep end just because they've demonstrated they can take a bath safely.

General Principle 5: Communicate High Expectations

Differentiation by outcome: my old favourite. But can we expect children to produce work of high quality if they don't know what it looks like to produce something that distinguishes it from work produced based on general expectations? No. Even if a child produces an exceptional piece of writing it is probably because they have set high expectations for themselves, expectations which have been defined by high quality literature that they have read themselves. In this example we see that the child has seen an example of how good something might be and have drawn up a set of criteria, albeit subconsciously, that they want to adhere to.

Most of us have to be intentional about producing something that is excellent - children will benefit from having high expectations clearly communicated to them so that they have a standard to work towards. Even if the idea is for them to produce something independently, perhaps especially in this case, it is necessary for them to know what the hallmarks of excellence might be in that piece of work. A model (it should be this good) or a list of criteria (it should contain or demonstrate these things) will probably suffice.

When it comes to high expectations, not only should they be communicated but children should be held accountable to them. Analysis of their own work (e.g. editing and revising) should lead to further improvements - they should not expect to produce excellence in a first draft. Indeed, this process of improving might be one criterion that is communicated to and required of HPAs.

General Principle 6: Personalise The Challenge

We often think of personalisation for lower prior attaining children. The curriculum even prioritises them saying that 'teachers have an even greater obligation to plan lessons for pupils who have low levels of prior attainment'. But we often forget that personalisation, tailoring if you like, might be a great principle for challenging HPAs.

Personalisation can come as a result of assessment - we find out that they know something or can do something already, and adjust their future learning accordingly. But personalisation could be more than this. We could take into account their interests, strengths and weaknesses, personality traits and so on. In order to engage them and encourage them to produce something excellent we might tap into their hobbies and interests. Conversely, in order to challenge them and to encourage them out of their comfort zone, we might not allow children, for example, to write yet another story if they are already demonstrating excellence in this area. In this case the child might be challenged to produce a piece of written work within another genre that they are less comfortable with.

Often, children who are working at greater depth will know what they find easy (and often what they find boring as a result) so it may be worth consulting the child about this in order to determine ways of working that might challenge the child. Sometimes a child might have ideas about how they want to challenge themselves which could be tapped into in classroom work.

Challenge for all

In reading this you may have been struck by the same thought as I had whilst writing it: shouldn't we think about these principles for all pupils? I think the answer is yes - we should seek to challenge all pupils, regardless of levels of prior attainment. With this in mind principle 6 becomes very important: the better we know the children, the better we can challenge them. Robert Bjork refers to desirable difficulty which is what we might think more simply of as pitching work correctly: the correct pitch will mean that work won't be too easy or too hard for the pupil who is attempting it. When work is set at this optimal point children will experience the correct amount of struggle which in turn leads to better retention and higher future performance.

If all we ever do is teach only HPAs new and more complex knowledge or ask them to do a wide variety of tasks when other children are doing similar tasks over and over, there will always be a set number of higher prior attaining pupils (most likely the ones who entered school already at an advantage due to other factors). If we want all pupils to have the chance to attain highly then we should ensure appropriate challenge for all, meaning that all the above principles apply to every child so long as principle 6 is taken into consideration first.

The challenge then for teachers is of course how they do this for each of the 30-or-so pupils in their class without falling into the old trap of three-way differentiation, attempting to provide 30 different activities or just giving up and asking all the children to do the same thing with no thought given to tailoring at all. Ongoing assessment and dynamic grouping is probably the simplest answer here, although within those dynamic (ever-changing) groups it is important to acknowledge that there are differences in need which might be better addressed by differing levels of adult support. This is still a real challenge, one which has workload implications, but taking on this challenge, most of us would agree, is absolutely central to our job as teachers.

Providing challenge for HPAs, and challenge for all children, is the greatest challenge teachers face.

Monday, 25 September 2017

What Does 'Greater Depth' Look Like In Primary Maths?


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; this is how the Interim Teacher Assessment Framework (ITAF) classifies them. 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 and the language of the ITAF: 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.

The Key Stage 2 ITAF does not contain any information about what a children working at greater depth should look like by the end of year 6 so we have to look to the Key Stage 1 ITAF for more clues. Thankfully Rachel Rayner, a Mathematics Adviser at Herts for Learning, has done a great piece of work on this already. Her article 'Greater Depth at KS1 is Elementary My Dear Teacher' identifies three key differences between the statements and exemplification material for working at the expected standard and working at greater depth within the expected standard: she says 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

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:


A working group from the London South West Maths Hub have also begun putting together some similar documents, focusing initially on number, place value, addition and subtraction and again categorised under NC objectives - those documents can be downloaded here. Here's an example of one of those, taken from the 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 last year's year 6 test) of how different the questions can look:


Reasoning

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).

In Summary

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.