Adventures in Mastery 1: Starting Our Journey

When you’ve been in education a while you see plenty of fads come and go and you become carefully cynical about the latest big pronouncement or the new product that’s going to “transform” your practice.  And so it was that I responded (in my mind) when everyone started to talk about mastery.  I teach in a high-performing secondary comprehensive school, where regularly 90% or more of our cohort have achieved A*-C in mathematics at GCSE, and 40-50% have achieved A*-A.  We always felt that we were getting it pretty right and, by many standards, we were.  We are fairly traditional in our approach, our students sit in rows and work quietly (on the whole), but we are not afraid of the odd investigation.  Maths is regularly praised by students in SLT monitoring, who generally say that they feel successful in maths, that they know they are improving, and that they enjoy their maths lessons.  Why would we want to change something so successful?  What could we learn from Shanghai or Singapore?  They only teach two lessons a day, don’t you know?  They plan and prepare and mark the rest of the time, we could never do that.  The parents probably all employ private tutors as well.  They’re in a completely different culture to us, it’s hardly going to be transferable.

The thing is, all those objections are pretty irrelevant to the real question here: is there any reason I cannot learn something from the way other people teach mathematics?  Moreover, if the way a whole country of teachers teaches mathematics is quite (fundamentally, even) different to our way, there will obviously be something I can learn.  What if there is a way for even more of our students to achieve higher grades?  Politicism aside, the opportunity to learn from others and to potentially enhance my practice very quickly beat my natural cynic.  After all, 10% of students in my school still don’t reach a “good pass” at GCSE.  Over half of them don’t actually learn enough compulsory mathematics properly, if we consider that only those with an A or an A* have really learnt what we want them to learn.

So, I’ve spent a lot of time over the last year researching mastery in mathematics, reading about methods in Shanghai and Singapore.  I bought samples of the Secondary textbook produced by Maths – No Problem! to see what the questions and sequencing of learning were like.  I’ve read blog posts by a number of proponents of the approach, spoken with teachers already employing some form of mastery, listened to podcasts, attended conferences, read as many refutations as I could find and discussed it all at length with colleagues.  As you will have guessed from the title of this post, I became convinced that mastery is more than just a fad, and have started our department on a new journey.  I am fortunate to work with a team of great maths teachers, many experienced, some in their early years, all with seemingly endless enthusiasm; we know a lot about teaching maths between us, but the greatest thing about the department is everyone’s desire to do even better by our students.  They are simultaneously open to new ideas and ready to critique them in the light of their expertise, so I know they would tell me if they thought I was talking rubbish.

The one thing I noticed throughout my year of research is that there are so many brilliant ideas around, and such interesting work going on in a number of schools, but all the information is scattered over the net.  I have written a scheme of work (still in beta) and am planning to share what I’ve read, what I’ve learnt and what I’ve already produced on this blog soon.  There are contributions from members of the department as well, without which much of the work done would still be in even earlier infancy.

For today, I’ll start with the obvious question.

What is mastery?

First of all, let’s get rid of the elephant in the room: it’s true, mastery can never be achieved.  No one is ever a master of something in the strictest definition, as we can all, always, improve.  Instead we are talking here about a philosophy of curriculum design, where a level of fluency acceptable to a group of teachers is classed as “mastery”.

There are various definitions floating around of ‘mastery’ in an educational context.  The idea was first brought into play by Carleton Washburne in the early twentieth century who, writing his Winnetka Plan, decided that children should be given as much time as they needed to master content before moving on.  A version of the ideas was named the behaviourist movement in the 50s but something more similar to what we are talking about today was formalised in the late 60s by Bloom; the model proposes that students achieve a certain level of mastery (often determined by a test score) before moving on to learn more.

In theory, we would give every one of our students as long as they needed on a topic, with materials carefully designed to take them to the required level of understanding before moving on.  After all, what is the point in learning to expand brackets if our students cannot multiply or work out if the shopkeeper’s given them enough change?  When they’ve had two lessons on negative numbers, straight into two lessons on scatter graphs, can we really say they understand it?  Have they spent enough time thinking about it for the ideas to move into long-term memory?  Mathematically, the first pair of lessons in my example here is inordinately more important than the second!

In practice, we cannot take every student at their own personal pace, so we must design a curriculum that will allow enough time to take as many students as possible to the required point.  Those who reach that point sooner can go deeper into the topic area.  As Mark McCourt points out in Mr Barton’s brilliant podcast, every topic in school mathematics can be deepened to go past undergraduate level, so there’s no problem in stretching those who need it.  I will talk more about this in another post.

A mastery model also requires the most careful sequencing in curriculum design.  It is important that topics follow a natural progression and build upon each other, so that nothing stands in isolation.  Again, I will go into more detail in another post.

The NCETM outlines five features of teaching for mastery in south-east Asia, which are very helpful in beginning to think about the model:

  • Teachers reinforce an expectation that all pupils are capable of achieving high standards in mathematics.
  • The large majority of pupils progress through the curriculum content at the same pace. Differentiation is achieved by emphasising deep knowledge and through individual support and intervention.
  • Teaching is underpinned by methodical curriculum design and supported by carefully crafted lessons and resources to foster deep conceptual and procedural knowledge.
  • Practice and consolidation play a central role. Carefully designed variation within this builds fluency and understanding of underlying mathematical concepts in tandem.
  • Teachers use precise questioning in class to test conceptual and procedural knowledge, and assess pupils regularly to identify those requiring intervention so that all pupils keep up.

In my next post I will talk about how I designed my mastery scheme, including the reading and thinking that informed it.


This is the first post in a series, the others are:

Adventures in Mastery 2: Writing a Scheme of Work

Adventures in Mastery 3: Prognostic Practice

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7 comments

  1. Do you have any thoughts as to how this might work in a mixed age (2 or 3 year groups together) primary class? My (Year 5) kid is working at about Year 7 level, while some of the others in her class are maybe working at Year 4 level or slightly below. If she waited for the others to catch up, she would be doing the ‘depth’ stuff for an awfully long time. Also, my assumption would be that this would lead to more setting in secondary maths, rather than less, to limit the range of attainment within each class. Would that be a fair assumption?

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    • Hi Sue, that’s a good question. It’s worth noting first of all that Bruno Reddy at KSA did have mixed ability classes when he started the mastery route (http://mrreddy.com/blog/2015/01/peer-tutoring-king-solomon-academy-tips-and-resources/). If anything, it appears to me that mastery is more compatible with mixed ability as everyone works on the same topics at the same time.

      I can see why you would be worried for someone like your daughter. The idea of mastery is not that you give more advanced students “fillers” while you wait for others to catch up, but that they do more advanced mathematics within the same topic area until the teacher is happy that as many students as possible have reached a certain desirable point. In practice, this is built in to the scheme of work. If you look at my scheme on the second post in this series I have allocated a predetermined amount of time to each topic. What different groups of students will do in that time will be very clear, so no one will be sitting doing “aimless problem solving” or something they don’t learn from while they wait for others. It will be the case that many go deeper than my minimum requirement, some won’t quite reach it and will have to have intervention, but it is all purposeful and useful and not “filler”. Does that answer your question?

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  2. Thanks for your reply. I’m wondering what is meant by the ‘same topic area’ and what happens if some children are ready to access a different higher order concept (algebra, for instance) well before the others? (Again, I’m thinking from the perspective of mixed age classes in primary which are surprisingly common outside of urban locations, e.g. Year R, 1 and 2 all in the same class.)

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    • Be aware, Sue, that higher order concepts exist throughout mathematics. Something as “simple” as arithmetic can be extended to beyond undergraduate level conceptually. If a teacher knows their stuff they can stretch and teach a child plenty of new things within one broad topic area (like arithmetic). The child is still learning new mathematics, it’s just within the same domain as what others are learning. Does that make sense?

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