*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:

- Clearly defined, ambitious lesson objective that seems daunting and will be rewarding if met.
- Sub-skills or steps broken down, almost list-like.
- Super-clear, often rehearsed explanation of the first step.
- Guided practice on mini-whiteboards until everyone can do it.
- Independent (timed) practice in books.
- Short assessment to prove to them they have achieved that step.
- Repeat 3-6 for next steps.
- Final assessment to prove to them they have achieved the whole skill.
- 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.

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mathsfeedbackThank you for this insight. You reminded me that we can really raise the confidence level of our students if we show them they can learn maths. Hopefully they can then experience the joy of understanding something new. I sometimes remind my sixth form students of this when they are stuck. “When you first saw Pythagoras’ theorem, remember how you felt? You didn’t think you would understand it. But now you are the master of Pythagoras! You will look back on integration in the same way.”