Teaching for mastery
Robert Wilne
Director for Secondary
NCETM
8 December 2015
robert.wilne@ncetm.org.uk
@NCETMsecondary
Everyday “mastery”
What does “mastery” mean?
• I know how to do it.
• It becomes automatic and I don’t need to think about it.
• I do it confidently.
• I do it well. (Does “well” mean “quickly”? Sometimes?)
• I can do it in a new way, or in a new situation.
• I can now do it better than I used to.
• I can show someone else how to do it.
• I can explain to someone else how to do it.
“Mastery” … what?
“Mastery
teaching”
“Mastery”
“Mastery”does
is
not
to what
therefer
hard-fought
the TEACHER
outcome
for ALL
knows
and does
the PUPILS
“Mastery
curriculum”
“Teaching of
mastery”
Mastery of mathematics (after Holt)
I feel I have mastered something if and when I can
• state it in my own words CONCEPTUAL UNDERSTANDING (CU)
• give examples of it CU
• foresee some of its consequences CU
• state its opposite or converse CU
• make use of it in various ways PROCEDURAL FLUENCY (PF) &
KNOW-TO-APPLY (K-TO-A)
• recognise it in various guises and circumstances PF & K-TO-A
• see connections between it and other facts or ideas CU & PF
& K-TO-A
John Holt, How Children Fail
Evidence of mastery?
a
c
a +c
+
=
b +d
b
d
a
c
a ×c
×
=
b ×d
b
d
a
c
a ÷c
÷
=
b ÷d
b
d
Insufficient mastery?
Mr Short is as tall as 6
paperclips. He has a friend
Mr Tall. When they
measure their height with
matchsticks, Mr Short’s
height is 4 matchsticks and
Mr Tall’s height is 6
matchsticks.
What is Mr Tall’s height
measured in paperclips?
About ⅔ of pupils said
“8”. They interpreted 4 to
6 as an additive increase
“+ 2”, not a multiplicative
increase “× 1.5”
Teaching FOR mastery: ALL pupils develop …
Really? Low
Yes!
“Ability”
ability as well?
factual
knowledge
Confident,
secure, flexible
and connected
procedural
fluency
conceptual
understanding
Fixed ability or attainment hitherto?
I do believe that
• some pupils grasp mathematical concepts more rapidly
than their peers do;
• some pupils get better marks in tests and exams than their
peers do;
• some pupils have the cognitive architecture that means
they see patterns and structures more quickly, and
remember them more readily, than others do.
• But I don’t know reliably who will
• and I can’t predict with certainty who will
• and my pupils’ brain pathways will change over time, as
they grow up and in response to their environment.
I ONLY know a pupil’s
attainment hitherto.
I NEVER know a pupil’s
ability or potential or
“bright”-ness
Rotten to the core or a one-off lapse?
Teaching FOR mastery
If ALL pupils’ thinking and reasoning about the concrete is
going to develop into thinking and reasoning with increasing
abstraction, they need us their teachers to help make this
happen. Crucial to this happening successfully are
• the choice of the representation / model with which we
introduce a (new) concept
• the reasoning we cultivate and sharpen through the
discussions we foster and steer
• the misconceptions we predict and confront as part of the
sequence of questions we plan and ask
• the conceptual understanding we embed and deepen
through the intelligent practice we design and prepare for
the pupils to engage in and with.
The vision of the new curriculum
All pupils become fluent in the fundamentals of mathematics, including through varied and
frequent practice with increasingly complex problems over time, so that pupils
develop conceptual understanding and the ability to recall and apply
knowledge rapidly and accurately; reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical
solve problems by applying their mathematics to a variety of routine and non-routine
problems with increasing sophistication, including breaking down problems into a series of simpler steps and
persevering in seeking solutions. The expectation is that the majority of pupils will move through the
programmes of study at broadly the same pace. However, decisions about when to progress should
language;
always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp
concepts rapidly should be challenged through being offered
rich and sophisticated problems
before any acceleration through new content. Those who are not sufficiently fluent should consolidate
their understanding, including through additional practice, before moving on.
The vision of the new curriculum
All pupils become fluent in the fundamentals of mathematics, including through varied and
frequent practice with increasingly complex problems over time, so that pupils
develop conceptual understanding and the ability to recall and apply
knowledge rapidly and accurately; reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical
solve problems by applying their mathematics to a variety of routine and non-routine
problems with increasing sophistication, including breaking down problems into a series of simpler steps and
persevering in seeking solutions. The expectation is that the majority of pupils will move through the
programmes of study at broadly the same pace. However, decisions about when to progress should
language;
always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp
concepts rapidly should be challenged through being offered
rich and sophisticated problems
before any acceleration through new content. Those who are not sufficiently fluent should consolidate
their understanding, including through additional practice, before moving on.
Good models / representations …
• can at first be explored “hands on” by ALL pupils
irrespective of prior attainment
• arise naturally in the given scenario, so that they are salient
and hence “sticky”
• can be implemented efficiently, and increase ALL pupils’
procedural fluency
• expose, and focus ALL pupils’ attention on, the underlying
mathematics
• are extensible, flexible, adaptable and long-lived, from
simple to more complex problems
• encourage, enable and support ALL pupils’ thinking and
reasoning about the concrete to develop into thinking and
reasoning with increasing abstraction.
A good model …
• 12 ÷ 3 = 4 …
• … because four sticks of length 3 fit into a gap of length 12.
• 12 ÷ 2.4 = 5, because …
• 12 ÷ 8 = 1½ because …
… leads to fluency …
• 12 ÷ 3 = 4 (4 sticks of length 3 fill a gap of length 12)
• … so 12 ÷ 1.5 =
• 3÷⅔=
… and supports generalisation
• So 12 ÷ 0.3 =
• and 120 ÷ 0.03 =
• and 12 ÷ 0 =
• and 24 ÷ 6 =
• and 12x ÷ 3x =
• and 12a ÷ 3/b =
“BODMAS”… waste of oxygen!
•
•
•
•
3238 + 5721 + 1762 – 5721
5a – 3b – 5a + 3b
731 ÷ 13 × 91 ÷ 7
8a ÷ 6 ÷ 4a × 9 ÷ 3
•
•
•
•
44 × 175
37 × 17 + 17 × 63
(6a + 9b) ÷ 3
(6a × 9b) ÷ 3
Pupils read
“BODMAS” left to
right, and so make
slow or no progress
with these
“BODMAS” is
utterly silent here:
it says NOTHING
about associativity
or distributivity
A Shanghai lesson
When you design a lesson, what do you think about?
What is the sequence of your thinking?
Where do you get ideas from?
Where do you get help from?
How do you decide what’s the correct design?
A Shanghai lesson
A lesson from Shanghai
Teaching FOR mastery: N (all!) QTs need …
factual
knowledge
… confident,
secure, flexible
and connected
procedural
fluency
conceptual
understanding
through …
Time to
think
Time to
prepare
Time to
develop
Time to
reflect
Time to be
observed
And their selfcommitment
to do so
Time to
discuss
Time to
observe
Raise the water, raise the boats
… and what’s
in the tap?
What’s the boat?
What’s the water?