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:

- The student can demonstrate or explain the concept orally, concretely, visually and abstractly.
- 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.

steveDavid,

Cards on the table here – as a HoD the KS3 curriculum that we designed was probably something that some would call a spiral curriculum – although we spent 2-3 weeks on each unit there was inevitably times where you moved on from, say, looking at four rules with fractions with a class to looking at finding the area of a circle without 100% of the class being fluent at their fractions work.

The approach served our students well with results and value added being the match for any mixed ability school in the country. Of course that could be despite and not because of our curriculum and perhaps even greater achievements would have been possible with a different curriculum structure.

To go back to my example from my own scheme – fractions and then area of a circle – how, in practice, would the teacher progress from one topic to another?

Does the teacher work with the class on fractions and then keep doing that topic until the entire class are adept at using the 4 operations with fractions? If that is the case then are some students not ‘standing still’ waiting for others to catch up?

Do the class work on different topics at different times (so they progress through the scheme at their own rate)? This is how I was taught maths at KS3 using some SMP card sets that we worked through – the problem was that the teacher spent all the time keeping his records of who was where in the scheme up to date and did little actual teaching!

It’s always difficult to get tone right when responding to blogs but please be assured these are not “smart” questions – I am genuinely trying to enhance my understanding of how it might work.

Cheers

Steve

dmthomas90Post authorHi Steve,

Thanks for your comment – the tone is just fine! I’m all about having proper discussions we all learn from. For a start it sounds like you weren’t running a ‘standard’ spiral curriculum. Two/three weeks on a topic is a good amount of time, and wouldn’t be rushing everybody. If you were getting great results then clearly you were leaving time for a lot of students to master the content.

The difference with a mastery approach is the explicit and public expectation that everyone will have mastered the content by the end of the unit, and that action will be taken if they haven’t. This is hard – you don’t want to hold everyone back, but you can’t leave some people behind either. Inevitably what this means is that students falling behind need to find extra time. This might be a catch-up club or extra homework, depending on their need. We will run extra sessions on specific topics for students to attend until they’ve mastered them.

So keep the class together and focus on teaching. If and when some students fall behind, put extra support in place to catch them up. Once students know they really have to master everything you’d be surprised how their attitudes change!

David.

steveThanks for the prompt response David.

What you suggest sounds like a more formal version of what we (sometimes) did – although we called them clinics and attendance was optional. Quite a number of students asked to repeat their tests (afterschool but under our supervision) following extra work in the clinics.

I suppose the difference is that our clinics ‘caught’ the basically diligent kids who just had not quite grasped everything whereas your approach would capture more students. I imagine there are some interesting discussions with students/parents about what some may see as a punishment/detention because their son/daughter failed to grasp a given topic. It’s all in the sales pitch I suppose!

dmthomas90Post authorExactly! We have the same sort of system where students get out of catch up lessons by passing a re-test to prove their mastery. These sessions are part of their timetable, and sold as essential extra tuition to help rather than to punish.

Primary prowessAre they in maths sets though. Can’t see it working in a mixed ability primary class with a range from low level 2 to level 6

dmthomas90Post authorNope – our KS3 classes are mixed ability. Not willing to let their KS2 performance dictate how they see their mathematical potential.

Primary prowessHow???? Serious question as I can’t quite get my head around how you can teach to mastery in a mixed ability class spanning 4 curriculum levels. Eg ratio- inappropriate for the level 2’s and 3’s, too easy for the level 6’s. So what does mastery look like?

dmthomas90Post authorHave you ever taught ratio with bar models? Not inappropriate for Level 2/3, and easily extended with problems to challenge any Level 6/7.

Primary prowessYes. We use bar models extensively. But level 2 child needs to work on consolidating place value and adding multiples of 10- ratio work could be made accessible but it’s not what she needs to know right now. Ditto lower level 3’s. Place value, confidence in 4 ops, number facts must come first – everything else is a distraction. Do send me a link to hard ratio questions for my level 6 child- he could do the gcse ones we found.

dmthomas90Post authorGCSE ratio questions are way too easy! Try getting hold of the extension books from the My Pals are Here series to stretch the Level 6s, or there are some good secondary extension books I can give you links to. Agree that four operation and place value are the priorities, but using other concepts as a way to practise these can be beneficial. These are the challenges of transitioning to a mastery system – if you start with one it’s much easier!

Primary prowessThanks. I recently bought the yr7 mymaths textbooks but even the top tier one was a bit easy for our top students- we’ve got my pals but not the extension ones. But as for starting with a mastery curriculum already within our reception class we have children who can (using manipulables) add two two digit numbers and then again those without basic 1 to 1 correspondence. We are doing maths mastery next year in yr 1 and we’ve been told to ‘ keep them altogether’…..mission impossible. Am going to hide the brightest from them and put them straight into yr2. Otherwise they’ll be sticking pins in their eyes with boredom. I agree with low threshold high ceiling problem solving but there is only so much you can do reasoning about numbers 1-5 with children who are well into level 2

dmthomas90Post authorThe extension ones are great – at least the ones at the older end of the spectrum. We use the extension alpha, beta and gamma series for KS3 extension, as most normal textbooks aren’t actually that challenging. I wish I knew more about KS1 teaching, but I think making suggestions here would be extrapolating a bit too much!

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