7 Keys to Classroom Habits That Stick

Last week, in the 1 Big Secret to Good Behaviour, I looked at how habit is at the heart of good behaviour in schools. Students, particularly those from the most challenging backgrounds, suffer from depletion of their self-control. Each bit of stress they face, every time they resist temptation or make a challenging decision, their ability to deal with the next challenge diminishes. This means that sometimes they just don’t have the self-control to delay gratification and make good decisions.

Habit is the solution to this problem. Habits bypass decision-making and go straight to action. Regardless of the state of self-control, habits get to work and manage your behaviour.

Classroom habits that stick
Good behaviour in lessons is largely about classroom habits. Classroom habits govern all those routines and behaviours you want students to display in the room, such as entry and exit routines, holding whole class discussions, listening to explanations, using mini-whiteboards, etc. I’ve recently been working to add habits for silent work and practising grit to the mix as well. But it doesn’t matter what the habit is, if you want it to stick, you need to set it up right. Here are 7 keys to doing just that.

1. Plan meticulously
And I mean meticulously. This is where you shape the routine of the habit cycle (see this post if that’s unfamiliar). Vague plans let unintended behaviours creep into the routine. A meticulous plan gives you complete control. Here’s an example of a vague plan for an exit routine:

      1. Complete exit question on exit card
      2. Close folder, put pencil case, planner and homework in bag
      3. Put folders in the cupboard
      4. Stand behind chairs
      5. Dismissed table by table

This is a meticulous plan:

      1. Three minutes to complete exit question in silence. Timer projected on the board.
      2. At the end of the timer: close folder, put pencil case, planner and homework in bag. Sit upright and wait.
      3. Once the whole table is ready, teacher instructs tables (one at a time) to put folders in the cupboard.
      4. Return and stand behind chairs silently.
      5. Teacher dismisses table by table.

Map out your transitions: know whether they happen automatically or whether an instruction is required.
Plan your environment: know if you’re expecting silence, quiet or talking.
Prepare your red lines: know what behaviour is breaking the routine so it can be challenged immediately.

2. Design the cue
What is going to prompt students into this routine? A cue doesn’t have to be complicated, but it does need to be consistent. If you’re going to rely on a verbal cue, make sure you use the same sentence or phrase every time. Better still, incorporate other aspects of the environment into your cue. Think about the physical elements – for example, always standing in the same place, or including an action like opening the door, or closing your laptop lid. Visual elements can also be useful. Before using mini-whiteboards I have a slide with a picture of a whiteboard and a 15 second countdown. The more immersive the cue, the more clearly it will be heard.

3. Design the reward
What happens at the end of your routine? The reward is just as important as the cue for making a habit stick. A reward doesn’t need to be significant, but it does need to be reliable. A good exit routine could end with:

“Have a great afternoon, and thank you for working so well on those exit questions. Now I know what your best work on this topic looks like I can make sure your next lesson is designed right for you.”

Recognising that the routine was completed well and reminding students of its benefits will often be enough.

4. Teach it
Students won’t get the habit by osmosis, they need to be taught. Set aside some classroom time to explain the cue, routine and reward. Explain the rationale (helps with the reward later on), and all the steps you meticulously planned. Know that students won’t remember all the steps the first time, so have them written up as a guide and gradually remove them over time.

5. Practise to consistency
This is for both you and the students! Students will need some practice when you first teach the routine; enough to be able to follow it next lesson with a bit of prompting. Build practice time into lessons until the habit is embedded and the routine becomes automatic. This is a necessary and worthwhile investment. Time invested at this stage will save much more time later as your class runs smoother, behaving perfectly out of habit.

You’ll need to practice as well. Practise your cue so that it as consistent as possible. Mentally rehearse the classroom situation so that you know exactly what you are expecting. Think up situations where the routine goes wrong and rehearse your response. Your behaviour needs to be as reliable as possible for the habit to stick. An Assistant Head once told me that you can get students to do absolutely anything, as long as you ask and enforce consistently. They were right!

6. Do It Again
This is one of Doug Lemov’s top techniques from Teach Like a Champion. If students haven’t followed the routine properly, get them to Do It Again until they get it right. As soon as you allow some sloppiness into the routine, you’ll let sloppiness into the habit. Your meticulous plan needs to be followed to the letter every time, otherwise it just won’t stick.

7. Don’t give up
Do not give up on a habit because it doesn’t work at the start. Research suggests that habits take anywhere from 18 to 254 days to form – with the average taking 66 days. Easy habits take the least amount of time, but we’re still looking at a good few weeks. As a rule of thumb, you should wait at least a half term before adjusting something because “it’s not working”. You’ve probably just not had enough practise.

Further Reading
Joe Kirby on using habits to build school ethos
Charles Duhigg (author of The Power of Habit) on how habits work
Doug Lemov on academic habits

The 1 Big Secret to Good Behaviour

Good behaviour is all about self-control. It’s about the self-control to delay the gratification of having a chat/staring out of the window/arguing with Mr Smith/stabbing Jamie with a compass, in favour of the much less immediately appealing orderly learning environment. Students with self-control can resist these temptations to follow rules and learn.But self-control isn’t that simple.

It’s not a fixed trait that you either possess or don’t. Self-control is more like the power in a rechargeable battery: it empties with usage, before replenishing when plugged in. One ingenious experiment showed this by getting university students to attempt some impossible geometry puzzles. However before the puzzles they sat in a waiting room, and on the table was a bowl of radishes, as well as a bowl of freshly-baked chocolate cookies. Some students were allowed the cookies. Others were instructed to resist the cookies and eat the radishes instead.

The cookie-eaters attempted the puzzles for an average of 20 minutes before giving up. The cookie-resisters only held out for 8 minutes. Resisting the cookies used up their self-control.

Stress depletes self-control.

This means that a salesman who was stuck in a traffic jam will be less able to deal with a tricky customer, or a parent with a wailing baby will be less able to deal calmly with an overly playful child. It also means that often we will face students whose self-control is already significantly depleted by the time we see them. This might be because it’s the end of the day, and Period 4 French used up the last bit; or it might be because home is a chaotic hothouse of stress at the moment, and there’s no space to recharge. Either way, their ability to make good decisions has been depleted.

Depleted self-control means poor behaviour.

The tragedy is that the students who most need to value each minute of their education are often those whose self-control has depleted the most. Yet they will be the least able to resist temptations and behave well. It’s not that they don’t have self-control – they’ve just used it all up.

Students need to bypass the self-control system.

There’s no way of fixing their depleted reserves of self-control. They need to bypass the decision-making system, the one that sets up two options and asks them to choose. This system demands self-control, and there’s none of that left.

Habit is the cheat that unlocks good behaviour.

When you act out of habit you don’t need to stop and think, or to weigh up options and make a decision. You just act. Habits are driven by a different part of the brain (they’re tucked away in the basal ganglia, just above the top of the brain stem), and by a different neurological system. If students behave out of habit, then depleting self-control stops being a problem.

Habits are formed of cues, routines, and rewards.

Charles Duhigg’s book, the Power of Habit, teaches us about the habit loop. He says that every habit starts with a cue from the environment, is followed by a routine of behaviour, and culminates in a reward for completion.

For example, a habit of entering the classroom might begin with a cue of being greeted at the door by your teacher, followed by a routine of fetching your folder and immediately beginning the Do Now, and then finished with a reward of being verbally recognised and of completing the first task successfully. A student could choose to do this by using up self-control, or they could launch into autopilot as soon as they are greeted at the door – when habit kicks in and takes over.

The 1 Big Secret to good behaviour is to build habits.

In all schools, but particularly the most challenging, students will come to you with their self-control depleted. They will choose a course of bad behaviour, unless they have a habit of good behaviour that takes over. But there is no one habit called good behaviour – it’s a set of lots of small habits that deal with different cues.

One type of habit is the classroom routine.

Unless you work in an exceptionally well-organised school, you build this yourself. Decide the routine you want the class to have, then decide on a simple and clear cue as well as an appropriate reward. Once established, this habit will make sure your students perform standard tasks just as you wish, regardless of their self-control situation at that time.

Another is the ‘coping strategy’.

Because school isn’t all about predictable situations, students need habits that give them routines for the unexpected. A classic example is ‘counting to ten’ (cue = anger, routine = stop and count to ten, reward = increased calm/reduced risk of regretting a rash action). These are harder to design and teach, particularly for the lone teacher. However if done well they are the most transferable habits, and most useful for students’ futures.

Self-control depletes, habit rescues.

The more stressful situations a student has been through, or difficult decisions they’ve had to make, the lower their self-control will be. This makes it harder for them to behave well if relying on them to make good choices. It is this phenomenon that lies behind much bad behaviour. Habit can rescue students from this problem. It takes over their behaviour, avoiding the need for delayed gratification or tough decisions. If the right habits are in place, depleted self-control is no longer a problem.

How to build habits of good behaviour – coming soon…

Next week’s blog will be on building habits for classroom routines. Then in the coming weeks I’ll cover some different behaviour habits for more general situations, to help students around school and outside of it.

For more reading:

Trying is Risky

This blog is about the most powerful pedagogical lesson I’ve ever learned.

In my first year of teaching I had to write an essay about two under performing students I taught. I chose two Year 9 boys, both of whom had potential but whose behaviour was stopping them from achieving. I followed the behaviour policy, experimented with all the standard behaviour advice, and had great support from more senior staff, but their learning just wasn’t good enough. In my frustration with the lack of help from the recommended education literature I turned to a reliable old friend: game theory.

The Model

When coming into a lesson students can make one of two choices: to exert effort, or not to exert effort. In a school with a solid behaviour policy the students who choose not to exert effort may avoid work, complete only the bare minimum, or not spend enough time thinking to remember. In a school without a solid behaviour policy they may cause carnage.

The lesson they are coming into can be one of two things: it can be a good lesson, or it can be a bad lesson. A good lesson is one where a student will learn if they exert effort; a bad lesson is one where they may not.

These two sets of options give us a two by two matrix like this:



For each pair of inputs there are two outcomes, the student’s level of academic and social success.

Consider the student’s choice. If they choose to exert effort, they will get either the best or the worst outcome. If the lesson is a good one then they will be both academically and socially successful, having learned in class and appeared capable/talented in front of their peers. However if the lesson is a bad one then they will be both an academic and a social failure. They will not only have failed in learning, but by trying and failing they will be embarrassed as an incapable or unintelligent person.

If a student chooses not to exert effort they receive a certain outcome – academic failure and social success. They have no chance of succeeding academically as they do not try to learn, however their rejection of learning guarantees that they never try and fail – their social status is secure.

So how does a student make their choice? It depends on how likely they think the lesson is to be a good one. Call the student’s perceived probability of the lesson being good p. If p is high, then they’re more likely to choose to exert effort, as it’s more likely they will get the best available outcome.

Risk Aversion

Imagine p = 0.5; that is the probability of a lesson being good was 50%. In this case would a student choose to exert effort (gambling between the best and worst outcomes) or to not exert effort (accepting a certain, albeit mediocre, outcome)? Most students would, quite rationally, opt to not exert effort. The reason for this is that they’re risk averse. They’d much rather choose a strategy that guaranteed them an okay outcome than a strategy that gambles between a good outcome and a bad one.

Because students are risk averse, p will have to be a high value before they would consider taking the risk of trying in class. Otherwise they’d rather settle for the poor yet certain outcome of academic failure complemented by social success.

The goal for teachers is making p as high as possible so that all students, no matter how risk averse they may be, exert effort in school.

What makes p?

Remember that p is the student’s perception of the probability that the lesson will make sure they learn, if they exert effort. It’s not a measure of how good the lesson actually is, or anything to do with the actual quality of teaching. All that matters for the decision to exert effort is the student’s perception. This can be affected by a huge number of variables way beyond the teacher’s control. A very non-exhaustive list is:

  • the student’s self-esteem (p is low if “I can’t do it”)
  • the student’s prior experience of the subject (p is low if “I’ve never been able to learn this”)
  • stereotypes around learning (p is low if “people like me don’t do well at this”)
  • the school culture (p is low if “our school’s no good at this”)

Teacher quality plays a part (p is also low if “this teacher’s rubbish”), but is by no means the whole picture, and is often not the dominant factor.

Raising p

Students reason by induction. Just as they believe that the sun rises tomorrow because it has always risen before, they believe that they’ll do badly in Maths because they’ve always done so before. Raising p is about breaking this damaging chain of reasoning, and the only way to go is by forcing them to experience success. This means that you plan your lesson to make sure that if they exert any effort at all, they will have some measurable success.

A personal tale

At the start of January I took over a new class, who were pretty disengaged about Maths. Our first lesson wasn’t great – they came in expecting to do badly, and largely met their expectations. p was low. Our lessons since then have been an all out war of attrition to raise p, and make sure they believe that if they exert effort they absolutely will succeed. My p-raising lessons have a very distinct structure:

  1. Clearly defined, ambitious lesson objective that seems daunting and will be rewarding if met.
  2. Sub-skills or steps broken down, almost list-like.
  3. Super-clear, often rehearsed explanation of the first step.
  4. Guided practice on mini-whiteboards until everyone can do it.
  5. Independent (timed) practice in books.
  6. Short assessment to prove to them they have achieved that step.
  7. Repeat 3-6 for next steps.
  8. Final assessment to prove to them they have achieved the whole skill.
  9. Repetition of my p-raising mantra – that everything in Maths looks scary and confusing at first, but easy once you’ve learned it.

If this looks remarkably like archetypal Direct Instruction, that’s because it is. The aim of these lessons are not to excite or engage in the popular sense. The aim is to convince all students, that if they try then they will learn. Discovery and inquiry have their place, but not when building confidence in fragile learners. Right now, I can’t risk any student not understanding at the end of the lesson.

I worry that too often teachers are encouraged to deal with disengaged classes by engaging them in expert-type activities that leave them too open to the risk of failure, and entrench many students’ pre-existing beliefs that they will not learn even if they try. I emphatically aim to build up to meaningful mathematical inquiry with all my students, but only when they have the confidence to cope with the very real prospect of failure in this.

A Warning

Teaching a student whose p is low is very different to teaching a student whose p is high. The former needs nurturing, confidence-building treatment where they are protected from failure and practically forced to succeed. The latter need to build their confidence by trying, failing and trying again. Where one type of student needs a tight structure, the other often needs a more open one. The trick is in identifying each type of student, and teaching appropriately to both of them.

Conclusion

Trying is risky. Lots of students quite rationally decide not to bother in their lessons, because the evidence they have tells them the probability of them doing well isn’t high enough. They’d rather take the certain path of failing academically, but with the social kudos of never having tried. To tackle this disengagement we need to take the risk out of trying. Turning around disengagement means relentlessly ensuring that every lesson ends in success, until confidence is built sufficiently high that trying no longer seems risky.

Why homework is bad for you

Laura McInerny’s third touchpaper problem is:

“If you want a student to remember 20 chunks of knowledge from one lesson to the next, what is the most effective homework to set?”

After a day of research at the problem-solving party, I came to this worrying conclusion:

Setting homework to remember knowledge from one lesson to the next could actually be bad for their memory.

So stop setting homework on what you did in that lesson – at least until you’ve read this post.

Components of Memory

Bjork says that memories have two characteristics – their storage strength and their retrieval strength. Storage strength describes how well embedded a piece of information is in the long-term memory, while retrieval strength describes how easily it can be accessed and brought into the working memory. The most remarkable implication of Bjork’s research surrounds how storage strength is built.


Storage and Retrieval strength – courtesy of Kris Boulton


Retrieval as a ‘memory modifier’

Good teaching of a piece of information can get it into the top left hand quadrant, where retrieval strength is high but storage strength is low. Once a chunk of knowledge is known (in the high retrieval sense of knowing), its storage strength is not developed by thinking on it further. Rather storage strength is enhanced by the act of retrieving that chunk from the long-term memory. This is really important. Extra studying doesn’t improve retention. Memory is improved by the act of retrieval.

The ‘Spacing Effect’

Recalling a chunk of knowledge from the long-term memory strengthens its storage strength. However for this to be effective, the chunk’s retrieval strength must have diminished. ‘Recalling’ a chunk ten minutes after you’ve studied isn’t going to be very effective, as your brain doesn’t have to search around for such a recent memory. Only when a memory’s retrieval strength is low will the act of recall increase storage strength. This gives rise to the spacing effect – the well-established phenomenon that distributing practice across time builds stronger memories than massing practice together. 

Rohrer & Taylor (2006) go a step further and compare overlearning (additional practice at the time of first learning) with distributed practice. They find no effect of over learning, and ‘extremely large’ effects of distributed practice on future retention.

Optimal intervals

There is an optimal point for recalling a memory, in order to maximise its storage strength. At this point, the memory’s retrieval strength has dropped enough for the act of retrieval to significantly increase storage strength, but not so much to prevent it from being accurately recalled. Choosing the correct point can improve future recall by up to 150% (Cepeda, et al., 2009).

There has been a common design of most studies into optimal spacing. Subjects learn a set of information at a first study session. There is then a gap before a second study session where they retrieve learned information. Before a final test there is a retrieval interval (RI) of a fixed time period. Studies such as Cepeda, et al (2008) show that the optimal gap is a function of the length of the RI, and that longer RIs demand longer gaps between study periods. However this function is not a linear one – shorter RIs have optimal gaps of 20-40%, whereas longer RIs have optimal gaps of 5-10%.

Better too long than not long enough

Cepeda et al’s 2008 study looks at four RIs: 7, 35, 70, and 350 days. The optimal gaps for maximising future recall were 1, 11, 21 and 21 days respectively, and these gaps improved recall by 10%, 59%, 111% and 77%.



Perhaps their most important finding is the shape of the curves relating the gap to the future retention. For all RIs these curves begin climbing steeply, reach a maximum, and then decline very slowly or plateau. The implication is that when setting a gap between study periods it is better to err on the side of making it too long than risk making it too short. Too long an interval will have only small negative effects. Too short an interval is catastrophic for storage strength.

Why homework could be bad

Homework is usually set as a continuation of classwork, where students complete exercises that evening on what they learned in school that day. This constitutes a short gap between study sessions of less than a day. We know that where information is to be retained for a week, the optimal gap is a day, and that where this is not possible it is better to leave a longer gap than a shorter one. For longer RIs, the sort of periods we want students to remember knowledge for, the optimal gap can be longer than a week.

Therefore, if you want students to remember information twenty chunks of knowledge for longer than just one lesson to the next, the best homework to set is no homework!

Setting homework prematurely actually harms the storage strength of the information learned that day by stopping students reaching the optimal retrieval interval. In this case, students who don’t do their homework are better off than ones who do!

Why I might be wrong, and what we need to do next

There is not enough good evidence of how to stagger multiple study sessions with multiple gaps. For example, we do not know where it would be best to place a third study session, only a second. However we do know that retrieval is a memory modifier, and so additional retrieval should strengthen memories as long as the gap is sufficiently large for retrieval strength to have diminished. Given we know that retrieving newly learned information after a gap of one day is good for storage strength, it may be that studying with gaps of say 1, 3, 10 and 21 days are better for storage strength than a solitary study session after 21 days, where the RI is long (350 days or greater). In this case for teachers who only have one or two lessons a week, homework could help them make up the optimal gaps by providing for study sessions between lessons.

The optimal arrangement of multiple gaps is a priority for research. We need to better understand how these should be staged, so that we can begin to set homework schedules that support memory rather than undermine it. Until then, only set homework on previously learned knowledge, and better to err on the side of longer delays. My students will be getting homework on old topics only from now on.

Bibliography

Joe Kirby on memory this weekend
EEF Neuroscience Literature Review
Dunlosky, et al., 2013. Improving Students’ Learning With Effective Learning Techniques: Promising Directions From Cognitive and Educational Psychology
Rohrer & Taylor, 2006. The Effects of Overlearning and Distributed Practise on the Retention of Mathematics Knowledge
Cepeda, et al., 2009. Optimizing Distributed Practice
Cepeda, et al., 2008. Spacing Effects in Learning: A Temporal Ridgeline of Optimal Retention
Everything Kris Boulton writes

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.