Tag Archives: curriculum

How exams took the joy, and the learning, out of our classrooms

Originally published on the Parents and Teachers for Excellence blog here.

Two trends have dominated how British exams have changed over recent decades: they have become more high-stakes, and they have become more skills-based. The two have combined to create a perfect storm that slows down learning and makes school less joyful. School leaders are under pressure to achieve good exam results, and so orient their schools around exam performance. They measure pupils in all year groups against the assessment objectives from exams, and expect teachers to teach to these objectives too. Every piece of work is a mini-GCSE exam.

This would make sense if the assessment objectives could be taught directly, but they can’t because they’re based on generic skills. Skills can only be acquired indirectly: by learning the component parts that build up to make the whole such as pieces of contextual knowledge, rules of grammar, or fluency in procedures. These components look very different to the skill being sought – just as doing drills in football practice looks very different to playing a football match, and playing scales on a violin looks very different to giving a recital. Yet in these analogies exam objectives would be something like “play with flair”, “keep possession” or “hit notes accurately”, and the instruction given to teachers is to directly teach these skills. Not to spend time on passing drills and scales, but to spend time on “having more flair”.

Most teachers see that the emperor isn’t wearing any clothes. Consider the plight of a typical English teacher. They’re told that their pupils aren’t good enough at understanding the author’s purpose, so as a result they need to teach more lessons on understanding the author’s purpose. They’re given lesson plans that tell their pupils to identify words which illustrate the author’s purpose, and to write paragraphs explaining why they do so. Maybe they include a handy mnemonic for remembering the model “author’s purpose” paragraph. But it doesn’t work. And it doesn’t work because you cannot teach generic skills directly.

To become better at understanding the author’s purpose you need to know more words, so you can understand the fullness of what the author has written, and you need to know more contexts, so you can understand the significance of those words to the author’s life and times. If you know that the gunpowder plot happened in 1605 and that Macbeth was first performed to an audience in 1606, then The Scottish Play becomes a warning against regicide. If you know that “to twist” was Victorian slang for “to hang”, then Oliver Twist becomes a tale about a boy destined for the gallows. If you know that Dickens first came up with the plot when appalled by the experience of attending a young criminal’s public hanging, then it becomes a campaign for social justice. You cannot infer this from practising to understand the author’s purpose. You can only infer it if you have the knowledge.

Becoming a better reader requires investing time in learning a wider vocabulary and building deeper contextual knowledge, but it would be a brave teacher who puts this maxim wholly into action in today’s schools. With the pressure of high-stakes exams there is no room to teach anything except the assessment objectives being examined, and the assessment objectives only measure generic skills. Instead of exciting lessons where pupils learn knowledge that opens up new worlds of history and literature, their teachers are pressured to push them through yet more rounds of dry and soulless skills practice. Pupils and teachers suffer with frustration as they try to become better at inference by doing lots of failed inferring. They rarely have the chance to learn the knowledge they’d need to imagine what was in the author’s head. Both pupils and teachers leave school unhappy as a result.

The same problem occurs in mathematics. Pupils fail exam questions involving problem-solving, so their teachers are told to teach them problem-solving. They’re expected to make their classes discover Pythagoras’s Theorem at the start of the lesson, as if the great breakthrough of a pioneering mathematician could be reliably and spontaneously reproduced by every fourteen year old on a given Thursday afternoon. Having to do this gives them less time to teach Pythagoras’ Theorem, and so jeopardises their pupils’ chance of successfully solving a problem about it in the future. Once again the pressure to teach generic, skill-based exam objectives directly undermines teachers’ attempts to make their pupils better at their subjects – and better in exams as a result.

We now need to realise what high-stakes, skills-based exams have done to our schools and how to recover from it. This will involve moving away from trying to teach skills directly, and from focusing on measuring them at every juncture. Instead we should plan the knowledge (e.g. vocabulary, historical context) and specific micro-skills (e.g. recognising whether the result of an addition will be negative, or re-writing a sentence to be active not passive) that our pupils need to learn in order to perform at a high-level in their exams. We can still target strong exam performance, but we should do so without expecting every lesson to resemble a mini-exam task. Doing so will mean creating schools where pupils learn more tangible things they can go home proud of, and where teachers teach more of the exciting content that brought them into teaching in the first place.

Why Nicky Morgan needs to set a curriculum for teacher training

In many ways, this will be a Parliament of consolidation at the Department for Education. The policies of the last five years are coming into force, and Nicky Morgan will need to put her political energy into seeing them through. But there is one area that does need reforming, and it needs it now. It is possibly the biggest opportunity to improve education in this Parliament, and one that would last well beyond 2020. It doesn’t sound glamorous or exciting, and won’t make the headlines. But its potential should not be underestimated. Nicky Morgan should use this Parliament to set a curriculum for teacher training.

Teacher workload is already extremely high, as Morgan has publically recognised. This means that government can’t improve outcomes in a way that puts pressure on schools – there are no more gains to be made from making teachers work harder. Instead, government has to look for ways to help teachers be more effective; and it should start by making sure every new teacher gets the training they deserve.

When I did my teacher training we spent laughably little time learning about learning. We discussed what made a good lesson (in the lecturer’s opinion…) but rarely why those components were good. We were often given quasi-moral justifications, like the assertions that “it is better to discover things for yourself” or “children learn better when they work in groups”, but I cannot recall a single time I heard something explained in terms of how a child’s brain would be responding.

Read the rest of this article at Conservative Home.

In defence of mastery learning

This defence of mastery learning was written in response to this article by Steve Chinn, and is published below it here.


 

Mastery learning is the belief that students should master a skill before moving on to learn a new one. In contrast to the classic spiral curriculum, where students raced between topics without properly learning any of them, a mastery curriculum gives students the space to learn a skill, understand it conceptually, and practise until it’s automatic.

This approach matters because of its effect on working memory. Students who have mastered previous skills have their working memory freed to learn new ones, while students who haven’t get bogged down in the basics and don’t have the working memory space to learn something new.

There are some important subtleties of definition that Steve Chinn picks up on. What it means to have mastered a topic must be clearly defined from the outset, or confusion will ensue. As understanding improves when students develop their conceptual map of maths and draw links between topics, we know that mastery early in school will not mean perfection. For me, mastery means two things:

  1. The student can demonstrate or explain the concept orally, concretely, visually and abstractly.
  2. The student can apply the concept automatically, so that it is not dominating their working memory.

Chinn does not engage with these fundamentals of mastery learning.

His first criticism is that mastery learning  will not help children catch up, and that they should instead be taught with an emphasis “on understanding maths concepts”. Given that Singapore Maths and its mastery model is renowned for its focus on developing understanding, this seems like an odd criticism. Conceptual understanding is at the heart of mastery learning, especially of Singapore Maths and its concrete-pictorial-abstract model of learning mathematical concepts.

His second criticism is that mastery learning is flawed because the ordering of skills for teaching is imperfect. This is true – there is no universally accepted hierarchy of all skills. This does not detract from the obvious fact that some skills are dependent on others, and that these dependencies are important for the order in which we teach. Adding fractions requires a knowledge of lowest common multiples, which requires a knowledge of times tables. We may disagree on whether we should teach names of shapes or bar charts first in the gap between them, but we know they have to come in that order.

The next criticism is that mastery learning is flawed because some people, for unknown reasons, appear to learn things differently. Even if we accept this argument, I cannot see where it leads. Is the implication that we therefore don’t need to care about the order in which we teach topics, and should pull them from a hat? If order doesn’t matter for some people, why deprive the others of being taught in a logical sequence?

It is particularly dangerous to support such arguments with anecdotal success stories like the dyslexic maths student whose times table recall was not perfect. Anecdotes do not a policy make. This anecdote seems compelling precisely because it is so rare, and it is so rare because it is an exception to a large body of well established research. This student succeeded in spite of imperfect times tables, not because of them. That they succeeded against the odds is not a reason for us to stack the odds against everybody else.

A Curriculum That Works

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:

  1. Mental addition and subtraction; Decimal addition and subtraction; Rounding
  2. Mental multiplication and division; Decimal multiplication and addition; Factors and multiples
  3. Understanding fractions; Operations with fractions
  4. Generalising with algebra (expressions and functions only)
  5. Properties of 2D shapes; Angle rules
  6. 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. 

Conclusion

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.

Lightbulb Moments

I love lightbulb moments. They’re the real reward for teachers. When thinking about a definition, I’d say that a lightbulb moment is when a pupil jumps from Remembering to Understanding. These moments are incredibly valuable. They increase students’ ownership over their learning, and are mammoth motivators. They make students feel successful in a qualitatively different way from just being successful on a test. They feel like their brain ‘gets it’. The associated dopamine release helps addict them to learning, and raise students who want to study and learn well after they leave my classroom.

Yet there’s a real problem with relying on these moments for your students’ learning. A lightbulb moment is analogous to the slipping of tectonic plates. It’s the unpredictable culmination of a process. Changes in conditions – heat, pressure, other movements – increase the probability of the slip occurring, but the actual moment of slipping is random and unpredictable. The same is true of lightbulb moments. The conditions you create can alter the probability of a given student having a lightbulb moment at a given time, but nothing you do can make it it happen, or predict exactly when it will.

Lightbulb moments are the joy of teaching.

This is a huge problem. I am not willing to rely on the good favour of probability for my students’ success. I cannot run my classroom solely on a model of improving conditions to maximise the probability of lightbulb moments, because there will be some students who the probability does not favour. I need to teach in a way that guarantees each student’s mastery of our key concepts. Teaching solely for lightbulb moments clearly doesn’t do this.

Yet neither am I willing to write off lightbulb moments completely. The promise of the motivation they create, the positive attitude they cultivate, and the increased feeling of self-efficacy for students is just too much.

This term I’ve been consciously working on a trade-off. I need an uncompromising approach to making sure every student succeeds, that still has adequate conditions to keep the probability of lightbulb moments high.

I can safely say that this term I’ve seen more lightbulb moments in one class than I have in the rest of the year. Why?

Sequencing Concepts

Think about what students say when they’re explaining a lightbulb moment. Most of the time it goes like this:

“Oh I get it! It’s like [insert other concept they’ve learned] but just [insert key difference]”

Understanding is all about links. Part of the reason students are slow to understand is that we start from scratch. We begin a “new” topic like it’s new. But it’s not. The curriculum just happens to be split up that way. Lightbulb moments come when students understand something new in the context of something already understood.

Maths is about links between a vast array of connected concepts.

My way around the trade-off was not to change my teaching, but to increase the proximity of related, already understood concepts. It’s all about sequencing.

My Year 7 class began this term with sequences – something they’ve all seen before. Everyone was comfortable with the idea of a sequence, and describing a term-to-term rule. We spent a lot of time working on constructing sequences, first physically (with Numicon, multilink or matchsticks), then pictorially, and then abstractly with numbers. They understood that a term in a linear sequence was composed of n differences and a constant, because that’s how they’d built each sequence. Cue dropping in something about writing this down in algebra.

The first, second and third terms of a Numicon sequence.

Throughout this time we’d also been playing games about functions and substitution that we’d learned earlier in the year. I needed to increase the proximity of these previously understood concepts.

The next concept was linear graphs. It’s always irritated me that students (and curricula) don’t see the connection between linear sequences and linear graphs. Part of the problem is that we insist on writing them in different forms, and treating them completely differently. Lightbulb moments demand that students make this often hidden connection.

We spent our time plotting sequences on coordinate axes, filling tables of values, and investigating gradients and intercepts. I taught linear graphs in the same way I normally do – just in close proximity to linear sequences.

It didn’t take long. One lesson in particular I remember – I was bombarded with students sharing their Eureka moments, one of whom did leap up like Archimedes from a bath!

“The gradient is just like the difference, it’s how much it changes”
“The y-intercept is where it starts from. Like the constant in a sequence”
“Steeper lines have bigger differences”

and even

“Sir this is the same as sequences. Why did you make it seem so difficult?”

The next thing we did was scatter graphs, and the connections didn’t take long. We finished the term with a lesson of Barbie Bungee. One of my top all time teaching moments came when a student, NC Level 4c, had estimated the equation of his line of best fit and used it to predict different bungee heights. I know students in top set Year 11 who couldn’t have done that.

Dropping a thrill-seeking Barbie makes for some excellent mathematics.

Proximity to previously understood concepts had fundamentally changed the conditions of the classroom in favour of lightbulb moments. Not everyone had them. Some students didn’t really figure out all the connections, and don’t have as strong an understanding. But those students weren’t sacrificed – I hadn’t given up on traditional teaching in the hope they’d find a lightbulb moment.

My key takeaway here is that although there often seems like a harsh dichotomy between discovery learning and direct instruction, this doesn’t have to exist in practice. You can encourage discovery and deep understanding by tweaking direct instruction. Knowing that discovery relies on making links with sound prior knowledge re-prioritises a classroom that values both.