The Lesson

I feel like 'the lesson', or 'the teaching bit' is something that gets lots of attention, and definitely should do, too! The thing is, we can't address the differing needs of each individual mathematical idea in one blog post. Instead, this will serve as a place for some general thoughts around the main body of a lesson, and the things we need in place in order for learners to be successful - both in the lesson, and over time.

The first thing necessary for learning is that learners have a solid understanding of the prerequisites for the idea they're about to encounter. If this doesn't exist, there's no point in teaching the new mathematical idea.

Don't know their square numbers? Don't look at square roots. 

Don't know what factors are? Forget about highest common factor.

Can't plot quadratic graphs? Don't bother with solving quadratic equations.

How can we know that learners hold this knowledge? Well, check for it.

We can check in the lesson by using mini whiteboards to question the whole class, asking questions about the ideas that we expect learners to know already. But what if it isn't there? That's a big pivot to do in the moment, having planned a lesson on the Pythagorean Theorem and finding out that learners struggle with squaring numbers or finding square roots.

We could always check ahead of time? At the end of the lesson before? That way we can prepare for where learners are, not where they're expected to be on a scheme of learning that was designed 10 years ago.

Better still, curricula should adapt to reflect the current strengths of our learners and allow us to teach the 'next' maths based on what they already know well. This data should be available to teachers, so they can be confident about what their class have been successful with, directing them to the 'next' maths to support long-term learning.

Once we're confident that learners are ready to approach the new maths, we can move on, safe in the knowledge that they stand every chance of success in maths over time. This is better than scaffolding to support learners in the moment, as performance is a poor proxy for learning, and long-lasting learning is more likely to develop if learners are making connections between new learning and their existing knowledge.

Performance isn't a bad thing in itself. Learning cannot occur if learners aren't successful with the maths in today's lesson - if they can't do it it now, there's no chance of it sticking around a week, a month or a year later. Performance today means that learners stand a chance of being able to recall it next week, next month or next year.

This is where example-problem pairs come in - show learners what and how to do, and have them mimic the steps with a minimally different question. Once everyone has shown that they can do what's expected, we can give them some independent practice without concern that they'll encounter significant difficulties.

Considering learners have just shown that they can complete the steps necessary to be successful, this is then an opportunity to begin to develop automaticity with the method. Ideally completed independently, in silence, learners are working on questions similar to those they've just been successful with. This means that learners develop confidence in their abilities, feeling more and more successful.

A golf podcast I listened to had a guest on who said something that resonated with me a lot. Speaking about a golf coaching app, he said "If you suck at something, you get a lot of blocked practise".

I don't think that there should ever be a chance of learners struggling with a concept in a lesson. The content should be selected such that it is just beyond their current level of understanding. If this is selected well, learners don't need too much blocked practise to solidify their understanding. However, if a learner's prior knowledge isn't close to that required to bridge effectively to new knowledge, it follows that they'll need a lot of blocked practise to learn the new content. If learners can't confidently identify the factors of a number, they might need 20 questions on finding the highest common factor to develop confidence. However, learners who can confidently find the factors of a given number see finding the highest common factor as a small step, and might only need 5 questions.

Suppose that the content has been selected so that the learners' chances of learning the new content are as great as possible. Example-problem pairs followed by a short period of independent practise should be sufficient to begin to develop automaticity, and spaced practice will make this permanent. The time that we save without giving learners lots of blocked practise can be given across to mixed practise - an opportunity to develop method selection.

We could do this within the same domain, revisiting prerequisites and highlighting links between related concepts, or across domains, mimicking an exam paper. Either way, learners now have the opportunity to engage with retrieval practice - revisiting material that they have been successful with previously, increasing the chances that learning will occur over time.

In a 50-minute/one-hour lesson, time might be scarce, so this might need to be spread out over a couple of lessons, but 100-minute lessons offer the opportunity to develop method selection with a significant portion of each lesson handed over to mixed practise. Rosenshine stated that 80% of each lesson should be spent working with known ideas, and I see mixed practise as a way to engage with content in this way, as well as by engaging with known content before bridging to new knowledge, revealing the unknown maths as an extension of that which is already known.

After writing down the LO, checking prerequisite knowledge, bridging to new information, checking for understanding using example-problem pairs, and engaging in a period of independent study followed by mixed practise, we're likely to be out of time in my upcoming 50-minute lessons. A learning episode is likely to take more than one timetabled lesson in many cases, and that's absolutely fine. Not all maths fits nicely in 50-minute, one-hour or 100-minute blocks.