This is the second of three posts reflecting on my first term as Curriculum Lead for Maths. The last post, on our new post-levels assessment system, can be found here.
A New Key Stage 3 Curriculum
The curriculum is so much more than a statement of what is to be taught and when. It embodies a school’s vision for its students and its philosophy of learning. I can look at a school’s mathematics curriculum and tell you all about the person who wrote it – their expectations of students, their hopes for their futures, their beliefs about how to get there. The curriculum is the embodiment of all these things, and it is crucial to get it right.
To begin writing a curriculum you must begin from a vision of the mathematicians you want your students to become. Mine is that I want our students to become “knowledgeable problem-solvers who relish the challenge Mathematics offers”. They should, at the end of their time with us, be able to independently tackle an unfamiliar mathematical problem and create a meaningful solution to it.
I am mindful when considering this vision, of the roaring debate around discovery and project-based learning, and how it can fall foul of WIllingham’s novice-expert distinction. My view is this:
Knowing that students begin novices, the purpose of education is to make them into experts.
The curriculum needs to train students. We cannot assume expert qualities of them from the start, and plunge them into investigations where most or many students will fail to learn. Similarly, we cannot dogmatically write off any activity involving discovery, investigation or project work. I emphatically want my students to be capable of expert investigation when they leave school, and so our curriculum must explicitly prepare them for that. The last part of this post in particular looks at how we manage this in a practical way.
From the vision of how students should leave school, I drew up three design principles:
1) The curriculum must develop fluency.
2) The curriculum must develop conceptual understanding.
3) The curriculum must teach students to solve problems.
Principle 1: Developing Fluency
When I wrote the last iteration of our school’s KS3 Maths curriculum, I abandoned the traditional spiral structure and opted for a depth before breadth approach. We probably halved the amount of content covered in a year, as we wanted to give students the time they needed to develop fluency. This year we’ve cut it again. Each of the six terms covers a maximum of three ‘topics’, most of which are closely linked. Terms in Year 7, for example, look like this:
- Mental addition and subtraction; Decimal addition and subtraction; Rounding
- Mental multiplication and division; Decimal multiplication and addition; Factors and multiples
- Understanding fractions; Operations with fractions
- Generalising with algebra (expressions and functions only)
- Properties of 2D shapes; Angle rules
- Equivalence between fractions, decimals and percentages
Smaller concepts that tie closely with big ones above are taught alongside them. For example, perimeter is taught in Term 1 alongside addition, area and the mean average are both taught in Term 2 alongside multiplication and division.
Since we give so much time to teaching each mathematical skill, we expect a high degree of fluency. To take the National Curriculum’s definition, fluency is students’ ability to “recall and apply their knowledge rapidly and accurately to problems“. It means not just being able to do something, but being able to reliably do it well and quickly. I would add that a necessary condition for fluency in a skill or operation is that it is embedded in your long-term memory.
This is exceptionally valuable in mathematics. A student may learn to be able to multiply decimals, but not become fluent in it. When multiplying decimals they have to slow down, to stop and think, and may make mistakes. This means that in Term 4 when they are learning to substitute into formulae with decimal numbers, they will face two severe problems. Firstly, their working memory will be occupied thinking about multiplying decimal numbers together, and not about substituting into formulae. Secondly, their reduced pace will mean that they have less exposure to substituting into formulae in each lesson. Overall they will spend less time thinking about the new concept they are supposed to be learning, and will learn it less well as a consequence (after all, “memory is the residue of thought”).
Developing fluency then, means lots of practice time with well thought out problems. Practice has got a bad reputation in mathematics, with too many people having been turned off maths by pages of repetitive textbook questions. My response is that practice need not mean making maths dry or uninspiring. Our students appreciate the value of practice as something that gives them the skills to do fun maths, and to achieve things they are proud of. Practice is invaluable, but can be dangerous if not used alongside meaningful and motivating problems.
If fluency is about rapid and accurate recall of knowledge, then Kris Boulton will tell you that fluency depends on high storage strength and high retrieval strength. A depth-focused curriculum gives us storage strength, but could easily sabotage retrieval strength if knowledge is not revisited. This is probably our biggest area to work on. The curriculum includes notes about what content to revisit when (thanks Kris!) and our assessments presume previous content as mastered prior knowledge. However we haven’t yet found a more structured way of revisiting content consistently across classes.
Principle 2: Conceptual Understanding
It is not enough for a curriculum to say what to teach. A meaningful curriculum also says how to teach it. At WA we’re big believers in the Singaporean approach of concrete-pictorial-abstract (CPA), and use this to structure our teaching. One of the reasons mathematical understanding in Britain is historically so poor is because students have been immediately confronted with abstract representations, representations that are well separated from any concrete reality, and not been given enough support in understanding them.
A favourite example of mine is ratio. I meet strikingly few students who can answer a question of the following type correctly:
“Bill and Ben share sunflower seeds in the ratio 3:2. If Ben has 20 sunflower seeds, how many does Bill have?”
I’d love to do some research and rely on less anecdotal evidence, but I’d guess that more British 16 year olds would say 12 than would say the correct answer of 30. Why? Because they were taught ratio in a completely abstract way, where they learned to apply a method but didn’t every receive the support needed to understand the concept of ratio.
In our curriculum, however, the pictorial bar model is central to teaching ratio. In fact I don’t teach my students an abstract method (they’re perfectly capable of coming up with it for themselves by doing the bars mentally and writing down calculations). For the unfamiliar, a bar model to represent the above problem would look like this:
Students draw the ratio, label what they know, work out the size of each block and then the size of Bill’s bar. I am yet to find a student who doesn’t understand this method, and who can’t do considerably harder problems using it. This is the benefit of having a pictorial representation to help students understand the concept they are learning, and to soften the jump into pure abstract. Every topic in our curriculum comes with CPA guidance to develop strong conceptual understanding in all students.
Also key to developing conceptual understanding are links between areas of mathematics. I am eternally frustrated by how students see maths as broken down into small discrete chunks that have little or no relationship with one another. Even when we have topics that are just different representations of identical concepts (sequences and linear graphs, for example), few British students will ever see them as linked. At the core of our curriculum then is a sequence carefully designed to make every concept learned useful to a later one. More than this, it guides teachers to make links, and uses assessment to make sure students are comfortable making these.
Principle 3: Problem-solving
Mathematics is essentially the study of problem-solving. The process of mathematical abstraction has been followed for millennia because it is so useful for generalising and solving what the National Curriculum calls “some of history’s most intriguing problems”. If our students are to become the experts we want them to be when they leave, we need to train them in problem-solving now.
For me, problem-solving is a skill to be taught, and it should be taught like any other. Adept problem-solvers have not come to be so through innate talent, but because they have seen the solutions to many problems before and are able to spot similarities and apply familiar techniques. Our curriculum aims to teach students the most powerful problem-solving techniques by exposing them to a carefully selected sequence of problems, some of which are taught and some of which are independently worked on.
Each term has a problem-solving focus. For example, Term 1 was “Working systematically”. Students began with a problem where they had to work out how many different possible orders there were for a two course and then a three course set menu at a restaurant. They began using ordered lists to write out combinations, before speculating on general rules and checking them on new possibilities. Through a range of different problems in the term students learned (a) how to work systematically in different contexts, and (b) the value of doing so.
Our curriculum has definitely met the three design principles set out, and is working well for our students. Depth before breadth has given them time to become fluent, to develop conceptual understanding and to solve problems. They see the value in mathematics as they’re exposed to interesting and meaningful problems. However this is done in a deliberate and structured way to make sure they are learning throughout. By applying the concrete-pictorial-abstract principle throughout we make sure that all students can interact with the concepts they’re learning and develop their understanding to a deeper level.
For me, we have two key things to work on after Christmas. Firstly, the revisiting of prior knowledge. We need to keep retrieval strength high, and must find a more structured way of doing this. Secondly, developing the guidance we give for teaching, particularly around drawing links between areas of maths. Although this happens well it is not yet a big enough part of our formal curriculum documents, which risks it slipping away in future.