*Mastery in Primary Mathematics*Garry takes a simple conception of that and uses it as a foundation for a tightly focused guide to primary teaching.

Garry narrows mastery down to five key elements. The first four
of these are fairly general: working out what students know,
planning what to teach next, continuous teaching and assessment,
and encouraging fluency and depth. The last, and the one central
to a mastery approach, is "ensuring a high degree of success for
all pupils before moving on to new learning", coupled with "high
expectations for *all* pupils".

As background to this, Garry takes as given that almost all children are capable of learning school mathematics. And he touches on the controversial topic of setting, where his suggestion is to "set pupils where necessary but err towards mixed-ability teaching", especially in lower years.

Getting underway, part one looks at how to design an instructional sequence: identifying learning steps to go into it, selecting representations that will be useful in it, connecting it to "big ideas", identifying its prerequisites, planning assessment to check understanding, and actually delivering it.

Part two focuses on individual learning steps: the importance of reviewing previously studied material, conceptual variation (and ensuring the boundaries of concepts are clearly delineated), procedural variation (more about varying questions and test cases than about varying methods), the use of language and terminology and encouraging talking about mathematics, and how best to use "interventions" for children who lack prerequisite knowledge or are having trouble keeping up.

And part three zooms in on activities — "every learning step should have some kind of activity for pupils to complete" — with chapters on designing them to build student fluency, problem solving skill, reasoning ability, and depth of understanding, and on the use of "scaffolding" to support individual students. A final section touches on being a maths lead, professional development, and school culture.

"'Acceleration' involves moving pupils quickly on to new content when they've initially been successful with a particular learning step or concept. The problem with acceleration is that it doesn't give time to explore a concept further, it doesn't give pupils a chance to consolidate what they're learning and it increases the chances of moving on before all pupils have fully grasped a new concept. For these reasons I'd strongly advocate against the use of acceleration ...

Depth is the ability to answer questions which are not standard in style, which require the application of mathematical thinking as well as mathematical processes, which may require pulling together knowledge of multiple mathematical domains, and which require a clear grasp of the underlying mathematical concepts to solve them successfully."

This is all tied to actual classroom practice and school realities. When discussing interventions, for example, Garry considers not just their design, and the differences between "keep up" and "catch-up", but also who should best deliver them and the logistical problems with fitting them into timetables.

Garry doesn't delve into the detail of any particular
mathematics, using examples only to illustrate general points:
a page showing different triangles (and "non-triangles") to
illustrate conceptual variation, for example, or four different
manipulative representations of 8 + 7. And there's little that
is curriculum-specific, so while *Mastery in Primary Mathematics*
broadly assumes an English school context, it would be perfectly
useful for teachers in any similar system.

Garry is succinct and to the point. His division of the material
into eighteen short chapters, while unavoidably awkward in a few
places, works effectively in breaking a complex and multi-scale
challenge into manageable pieces. He doesn't argue the case for
his approach, or evaluate it against alternatives, but rather
describes the conclusions he has come to and his own practice.
(There are, however, some references at the back of *Mastery
in Primary Mathematics*, along with two pages of further reading
suggestions.) His is, then, a mastery approach to teaching mastery
teaching, designed to give all primary teachers the basic knowledge
required to teach mathematics effectively.

So, what's missing from *Mastery in Primary Mathematics*?

Garry touches on the need for staff to maintain a positive attitude to mathematics, but is largely silent about motivation. There is, it is true, no point telling children that they are capable of achieving if they aren't provided with teaching that actually enables them to do that — what one might call "Mindset without Mastery" — but motivation still matters. It might even be quite high priority for a teacher facing (say) a Year 5 class with a significant number of children who have actively hostile feelings about mathematics. And ideas about "gifted and talented" students and negative sentiment towards mathematics are unfortunately ubiquitous in the community.

There's nothing about homework, or more generally about working with (or managing) parents. This may be outside the remit of ordinary class teachers, but Garry is also addressing leaders who may have to make decisions about how much and what mathematics is set as homework. And there's nothing about the use of games, either in explicit gamification of curriculum learning or in games that incorporate general mathematical thinking.

Finally, there's no suggestion that children learn anything *about*
mathematics, its structure and history, what mathematicians actually
do, or the role mathematics plays in the world and its strengths and
limitations. Whether this works as motivation or otherwise assists
nuts and bolts learning seems to me beside the point — it is,
even if that is only hinted at in the national curriculum, important
general knowledge. And in a world obsessed with quantitative data,
broader numeracy is vital, and something that I think can usefully
be started on in primary school.

In *Mastery Mathematics for Primary Teachers* Newell covers a much
broader range of material, beginning with a much more diffuse
conception of what mastery means. So he spends quite some time
looking at East Asian mathematics teaching, at differences in
pedagogy and curriculum, the relative strengths of UK teaching,
and so forth. He includes a bit of theory, with references to
Piaget and Vygotsky. There is a lot of detail about different
manipulatives. And he gives a lot of space to case studies, of
individual teachers and classes as well as of schools and programs.

This is interesting — and there is a lot here that Garry doesn't consider at all — but it is also quite scattered and not always clearly connected to anything else. If I found myself actually teaching, Garry's book is definitely the one I'd turn to first.

September 2020

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*Mastery in Primary Mathematics*

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*Mastery Mathematics for Primary Teachers*

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