At which stage in a topic do you stop modelling examples? I’m going to go out on a limb and suggest the answer is probably….too soon. I’m going to go further and suggest that the rationale you have for doing so is probably wrong.
Here are examples of conversations I’ve had, or resources I’ve seen shared, where what was suggested sounded superficially sensible until a moment’s reflection made me think they had it completely backwards.
- (paraphrasing) I’ve prepared the lesson on multiplying proper fractions, and set them an extension homework on doing it with mixed numbers to really stretch them, since it’s set 1.
- (paraphrasing) I’m going to model how to expand and simplify (a+2)(a+3) and (b+3)^2 and (a-5)(a-7). They need to be able to do it with coefficients in front of the variable as well, and when the order or variables are different, such as a (3+a)(5-b), or three terms in one of the brackets, but I don’t want to show them too much so I’ll put those ones in the worksheet.
- This sequence of tasks in a lesson on TES (I’m not going to say the creator as it is obviously an act of generosity on their part to have shared and I am not seeking to ridicule them). First the example they planned for sharing with the class:
The tasks for the pupils to undertake following this example:
I don’t know the author for the TES resources, but I can certainly attest that the first two were intelligent colleagues who were making those judgements as a result of serious thought.
They are decisions I would have taken earlier in my career as they fit with not wanting to spoon-feed pupils and I wanting them to enjoy a few head-scratching moments. Both are laudable aims and are important considerations. It is boring for pupils if they never have to think, and it makes it less likely that they’ll remember what you teach them.
With the benefit of (some) experience, the examples above now strike me as badly misguided. They reflect a common problem in maths instruction: we model examples that are too easy, then leave pupils to ‘discover’ methods for the harder ones. I think this has several causes:
- Modelling seems like a boring thing to do. They are just sitting and (hopefully) listening. Pupils will get bored and restless.
- Modelling takes ages, especially if you’re checking they understand as you go and are asking questions to get them to tell you the next step, and so on. If you modelled every example, they’d never get any practice, surely!
- We want to feel that they’re getting a chance to figure things out for themselves and really enjoy maths (i.e. feel like mathematicians, trying out strategies and reflecting on them).
- It feels a bit cold and exam-factory-ish to explicitly model every form a procedure or problem can take. Where is the room for surprise?
- They only need a basic grasp of it, they’ll get a chance to look at the harder ones again in Y9/10/11, right?
- Homework should be interesting and new. Researching and learning about more advanced or complex versions of a problem or procedure is way more interesting than more practice of what they’ve already done.
- (I suspect this is truer than we’d like to admit) The difficult examples are really difficult to model. We’re not sure how we know how to do them, they’re just….you just can see it, right?
Here are some counter-arguments:
- Modelling isn’t thrilling, but it is the most efficient and effective way to get pupils to be able to do procedures. Procedural mimicry isn’t the goal of our instruction, but it is a crucial foundation. It is also the bulk of what we are doing in secondary. It is almost impossible to get a (typical) child to understand a concept when they have no procedural fluency with the topic.
- Modelling wouldn’t take so long if you didn’t ask so many flipping questions during it. We do so much talking that isn’t simple, concise narration during an example that pupils – understandably – think procedures are much longer and more complex than they actually are. Think about how long it takes you to solve ‘4 + 5a = 17’ if you aren’t showing it to anyone. I assume 5-20 seconds (since you’re a teacher…), maybe 30 if you show every line of working. They should see this! Taking several minutes to model one example sets them up to think it is fine to think they can spend more than 2 minutes per question on questions that are, frankly, trivial.
- Letting them have head-scratching moments is fine, but it should be in response to a question that synthesises what they have been taught, and doesn’t require them to invent new knowledge (well, new to them). Leaving the hardest content for them to try on their own is just crazy: it disadvantages the weakest pupils and those who haven’t had good teachers on this topic in the past, it makes ‘giving it a go’ seem incredibly hard and adds to the perception that teachers are just refusing to tell them how to do things to be annoying. They should try hard problems on their own, but you should have first equipped them with requisite techniques and knowledge.
- You should still show them the elements of surprise in maths. When showing them how to form algebraic proofs, we first look at the propositions with real numbers and form conjectures (e.g. for four consecutive integers, what is the relationship between the product of the 1st and 3rd and the square of the 2nd? Does this always happen? How will we prove it?). They should get hooked in, and make conjectures, and experience the pleasing smugness of having applied what they already know…BUT YOU SHOULD ALSO SHOW THEM, REALLY CLEARLY. Ten may have figured it out before you finished modelling it, but twenty didn’t, and they’re relying on you. Well-designed questions in their independent work will still force them to think and synthesise what they know and test methods, but won’t ask them to stab in the dark with no sense of if they are right or wrong. We tend to overestimate how much pupils can tell if their work is correct or not.
- Lucky Y9/10/11 teacher. This increases the gap they need to bridge as they approach the end of Y11, and also means that they have simplistic ideas around a topic (and probably many misconceptions). They need to see the complex versions, and grapple with them, to really grasp a topic. Once you have taught them to reflect in the line y = x, reflecting a simple shape in a given line seems like a breeze. Often the fastest route to mastering the basics is tackling the tough stuff (at least, that seems to be true for my weaker classes!).
- What a way to make homework seem annoying and pointless. If they struggled in class, there is no way they will be able to do the homework. If they got it in class, they MIGHT be able to do the homework…but probably only if there is an older person in the house who can help them, or a really good CorbettMaths video on exactly that procedure (so it will only work for procedural work anyway). I’m all for letting the most able fly ahead, but it shouldn’t be so cynical as this. I cannot see the logic of presenting the most challenging work at the point of lowest support (i.e. you’re not there, and they are tired and probably rushing it). It also makes you seem like a rubbish teacher: “I can’t do it because she didn’t teach it to me.” They wouldn’t be wrong for thinking this.
- Explaining how to enlarge a fiddly shape by a negative scale factor is annoying and difficult. Proving congruence with annoying parallel lines is really difficult to do, let alone explain in a clear way. I just ‘see’ the answer to those long locus questions and feel like if they just read the question more carefully then maybe…maybe they’d just kind of realise? If we struggle to explain it, and are relying on our instincts to know what to do, then goodness help them trying to do it with no guidance or instruction from us.
What we should be doing instead:
- Think of all the variations a problem or procedure can have, and plan examples that cover all these eventualities.
- Pair every modelled example with a highly similar one for pupils to do. This holds them accountable for listening to, and thinking about, what you are showing. It also gives you feedback on if that small segment of instruction made sense to them.
- Make examples challenging. Yes, start with (a+4)(a+6) to help them see the simplicity of a grid. Unless you haven’t secured the basics with multiplying and simplifying, you should be able to then show (2a+5)(4+3b) and then an example with three terms in one of the brackets. I was struck by this in a talk at MathsConf last year, where teachers who had visited Shanghai were showing the kind of examples that teachers there used with pupils. They were doing LOADS of examples, but briskly and covering a wide range of applications, which meant that pupils were in a better position to see the common threads and critical differences, and not jump to incorrect conclusions. I shudder to think how many pupils in the UK think that the terms in ‘the diagonal’ in a grid for double brackets must be the ones ‘that go together’, having only seen examples in the form (x+a)(x+b).
- Just model it! Make it quick, make it concise. Something that works well for my weakest pupils is to have a silent model (i.e. they see the physical motions to enlarge a shape, or expand two brackets, or do the same to both sides), then one which I narrate (no questions or clarifications), then one where I check that they can mimic (i.e. they do a very similar one on whiteboards so I can see if they can recall and follow the steps – this is much more important than them being able to articulate the steps…at least at the early stage).
- Start at the basics, but move as fast as you can to the harder content, and you will reap rewards. They will also ask much more interesting questions and feel smart. You’re also not saddling their future teacher with too much to get through.
- Just use homework to practise and consolidate things they already know how to do. Set homework where you can expect 100% of pupils can get close to 100% (provided they are putting effort in). Otherwise you are setting up a grey area where you can’t tell if non-compliance is your fault or theirs, and you have lost the homework battle. You also save class time by not practising things where they need your help less.
- The harder it is to explain, the greater the imperative that we spend time together planning and refining (and ideally practising) those explanations. The return on planning and rehearsing complex examples is huge: it will illuminate the key features for simpler cases, help you see the crucial things they must do from the start, and also streamline your language for easier questions. Pupils always rate highly a maths teacher who can explain well, and lose faith in those who can’t. If you can’t explain it well, they’ll think the problem is that they are stupid. That is a tragedy.
- Plan hard examples
- Practise explaining hard examples