Rachel Rayner is a Primary Mathematics Adviser at Herts for Learning. The team are currently engaged in designing a mathematics curriculum for schools and teachers. In this blog she considers how curriculum design impacts on learners. This will be the first of a series of blogs on progression and design.
As a maths team we are currently writing every sequence of learning from Year 1 week 1 Autumn term to Year 6 final week Summer term. More on that later. But I don’t mind telling you that it’s raised quite a few questions on the team about what a great curriculum for maths should look like. Curriculum ‘14 for mathematics raised age-old debates – acceleration versus breadth and depth, knowledge versus engagement – let the twitter set debate. Furthermore, the new curriculum is being regularly referred to as a ‘mastery curriculum,’ heralding a bewildering array of products stamped with the ‘mastery’ brand all claiming to revolutionise your curriculum and behave rather like you might imagine a magic wand to work. And yet, and yet…still we battle to build a secure curriculum framework and schools are desperately seeking something (even after they have discovered concrete-pictorial-abstract). One in which, age-related expectations become the norm for all pupils irrespective of their prior attainment – though we know there are a few children for whom added provision is ever needed irrespective of the curriculum. On top of that OfSTED are looking at how curriculum design supports learning for all pupils even beyond the mathematics lessons. So where do we even begin? In this the first of a series of blogs I want to set out the current landscape as I see it (and sorry but no, I don’t possess even a modicum of fairy dust or a magic wand) before focusing in further in future blogs.
I consider myself privileged to be able to visit so many schools in the many roles an adviser carries out. Obviously I do draw commonalities from them – human nature I suppose to want to sort and group. And through them I see teachers and leaders valiantly trying to make sense of the landscape…so let my tale begin…
…by considering the three knowledge bases exemplified by the shapes below and considering the following questions:
- As a teacher, which knowledge base would you rather build future learning on?
- What are the benefits and weaknesses of each knowledge base?
- What does challenge look like for each of these knowledge bases?
The knowledge base
Now I have used the term knowledge base deliberately. The more I think back to when I had my very own class, I realise that perhaps I didn’t understand the importance of knowledge as much as I thought. It wasn’t that I didn’t pursue it with my pupils or value it, rather that I didn’t pursue it enough or consider well enough the core knowledge needed in order that pupils could build on it.
Without core knowledge, what is there to build on, to compare or connect to and to apply? What is left in the long term memory for later revisions of learning? I could see, and can see in groups of pupils in classrooms today that they have no base facts to use or to apply to find other facts or knowledge. Our fluency research project bore this out cohort after cohort. A little bit of knowledge goes a long way, whereas, those that have none are disadvantaged.
The ‘Goldilocks Principle’ when designing the mathematics curriculum
On the wooden kitchen table, there were three bowls of porridge. Goldilocks was rather peckish. She tasted the porridge from the first bowl.
“This porridge is just too hot!” she howled.
Next, she tasted the porridge from the second bowl.
“This porridge is just too cold,” she shivered.
So, she tasted the last bowl of porridge.
“Yum, this porridge is just right,” she said with a cheeky grin as she winked and ate it all up.
Extract from Goldilocks and the Three Bears (my retelling)
To be just right, the ‘Goldilocks Principle’ states that something must fall within certain beneficial parameters. When the effects of the principle are observed, it is known as the Goldilocks effect.
Perhaps one of the most interesting uses of the principle is in astrobiology where the Goldilocks zone refers to the zone around a star where life could form. The ‘Rare Earth Hypothesis’ uses the Goldilocks principle in the argument that a planet must neither be too far away from, nor too close, to a star and galactic centre to support life. Either extreme would result in a planet unable to support life. A planet that falls within these parameters is known as a “Goldilocks Planet”.
But what has that to do with the curriculum design in primary mathematics?
The mathematics curriculum in schools has its extremes as well – some being ‘just too narrow’ and some ‘just too broad’. Below we outline the characteristics of each curriculum.
The characteristics of a mathematics curriculum that is just too broad (A)
Where I see the curriculum is too broad, it can be characterised by the tendency to ‘skip’ around the curriculum from one set of knowledge to another over days. An example of this might be, written addition on Monday and Tuesday, and geometry for the next two days followed by problem solving Friday. It’s just too hot and can leave pupils unable to deepen their understanding and so only surface level learning is possible for many. We know that children who have relatively poor working memories need time to develop skills and knowledge more deeply so that cues and connections can be made and facts secured in long term memory, allowing them to be far likelier to access the knowledge more efficiently at a later date. Even those with good working memories need a little more time to allow knowledge to move from there to their long term one. Too broad can also be a lot of lovely activities with no cohesion between them, just a lot of experiences on the same theme. This leads to a broad curriculum that is essentially only an ‘inch deep’. For me, it would be difficult to get to grips with pupils’ learning and next steps because the ‘skipping around’ disrupts the depth with which I could assess pupils – there’s nothing long term for them to get their teeth into and really build. I found when I was asked to teach like this myself pupils had forgotten what had been covered when they came back to the learning again at a later stage. Challenge is likely to be an nrich problem or a problem from the brilliant Mathematical Challenges for the More Able book (nowadays I question why other pupils couldn’t have a go too).
The characteristics of a mathematics curriculum that is just too narrow (B)
This is definitely how I planned the curriculum early in my career, when I truly believed maths was only about getting the answer right with as few mistakes as possible – and that’s just too cold. Narrow curriculum design is typified by a focus solely on procedural mathematics such as pages of calculations and activities where pupils follow rules and practise skills (which are sometimes tricks) to get to an answer. Whilst it seems that there is precedence given to factual learning, this is often trumped by the emphasis on performance of skills e.g. the rehearsal of numberlines has never in my experience built up recall of complements to landmark numbers. The culture in these classrooms is often on speed, rehearsal of procedures, precision (teaching children about how not to make errors) and performance activities which are seen as indicators of learning. Indeed some performances in children look so convincingly like learning, I’m sure I was fooled more than a few times. Acceleration through performing procedures is often quicker for children with great working memories, well developed processing speed and the ability to remember the sequence of a set of steps. However, fluency by pupils is more likely to be procedural and as a result pupils do not necessarily develop connections in knowledge needed to become adaptive. Challenge within a narrow curriculum is likely to be seen as more of the same – making the numbers bigger (“so you can add with 3 digits now try 4”), and increasing speed or by increasing the number of steps in procedures. Many pupils are likely to perform well over time using written methods but are not necessarily able to understand varied representations.
The characteristics of a mathematics curriculum that is just right (C)
Teaching is rooted in the development of all pupils’ understanding of important concepts and progression through the lesson and over time. More time is devoted to developing each area of knowledge systematically before moving into another area and the teacher is more likely to build a linked curriculum where knowledge can grow and connect to other areas. A balance of the direct teaching of knowledge and skills and the exploration of concepts in a variety of relevant contexts enables pupils to make connections to prior learning within mathematics and use appropriate mathematical tools. The culture is less about pupil performance and more emphasis is given to the learning taking place and knowledge and understanding being accrued. Time is given to thinking about a range of possible answers and the focus is on an increase in evaluating strategies and trying new ones. Incorrect answers are valued as highly as correct answers and teachers successfully confront the misconceptions alongside the class through marking and high quality questioning and spend less time focusing on clerical errors – these become the pupils’ responsibility. Problem solving, reasoning, discussion and investigation are integral to pupils’ knowledge of mathematics in these classrooms and these are given as much daily emphasis as procedural skills. Learning in books and lessons demonstrate that pupils are confident to communicate their understanding through symbols, language, pictorial representation and concrete resources as appropriate to their age and the learning. More pupils in these classes demonstrate curiosity about mathematics and are likely to ask what ifs. Pupils with less developed working memories, processing speed and sequencing skills have more time to develop the connections needed to develop knowledge and understanding. Challenge is evidenced through quality questioning, rehearsal of skills through tasks that require both appropriate retrieval of knowledge and higher-order thinking, the expectation for precision in explanation, the evaluation of multiple strategies or proofs, reasoning and a wider range of contexts in which to apply the mathematics – though I would argue that all pupils should have access to those! Yum this curriculum is just right.
How Goldilocks right do you believe your curriculum is?
Is it cohesive and progressive?
Does it identify non-negotiable knowledge in each year group as well as the high value learning?
Is it weighted to the non-negotiable knowledge and high value learning?
Is it inclusive?
Summer may be a good time to take a step back and evaluate all that has been taught over the year…and begin to plan for the next. Or contact me firstname.lastname@example.org to find out all about our new planning resources available from the end of August on our HfL e-Shop.
In the next blog I aim to begin thinking about re-branding teachers as artisans.
Coming soon:- Planning Essentials from Herts for Learning Primary Mathematics Team.
Herts for Learning is a not for profit organisation that provides a wide range of training and CPD courses, events and conferences to support teachers and school staff in their professional development and also offers an extensive range of resources to support their offering through the HfL e-Shop. Please visit the website for more information.