Asking “why?” without
asking “why?”
Reasoning: little and often
Robert Wilne
Director for Secondary(ish)
NCETM
30 January 2016
Holmes or Watson?
Which calculation is the odd one out? Explain your reasoning.
753 × 1.8
75.3 × 20 – 75.3 × 2
753 + 753 ÷ 5 × 4
750 × 1.8 + 30 × 0.18
(75.3 × 3) × 6
7.53 × 1800
Developing pupils’ reasoning is NOT …
Why?
Why?
Why?
Why?
Why?
Why?
Why?
Learning …
through
of
through
of
through
of
through
of
through
of
through
of
through
of
… reasoning
Odd one out
9
1
2
=
+
20
4
5
13
1
2
=
+
20
4
5
3
1
2
=
+
9
4
5
3
1
2
=
+
20
4
5
True or false?
5 ÷ 5
5
=
8 ÷ 8
8
5 × 2
5
=
8 ÷ 2
8
5 - 2
5
=
8 - 2
8
5 × 0
5
=
8 × 0
8
“Always, sometimes, never”
a
c
a +c
+
=
b +d
b
d
a
c
a ×c
×
=
b ×d
b
d
a
c
a ÷c
÷
=
b ÷d
b
d
Mastery in 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
“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”
Teaching FOR mastery: ALL pupils develop …
Really? Low
Yes!
“Ability”
ability as well?
factual
knowledge
Confident,
secure, flexible
and connected
procedural
fluency
conceptual
understanding
Rotten to the core or a one-off lapse?
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
Conceptual understanding …
3, 5, 8: three numbers four sentences
3+5=8
5+3=8
8–5=3
8–3=5
… supports procedural fluency …
•  + 17 = 15 + 24
17
22
24
15
• 99 –  = 90 – 59
90
9
9
59
?
… and leads pupils to generalisation
• 90 – 59 ≡ 99 – 68 ≡ 102.4 – 71.4
3.4
9
3.4
9
90
59
?
• 10 – 4 ≡ 17 – 11 ≡ 6 – 0
• 10 – 4 ≡ 5 – -1 ≡ -3 – -9
Pupils can generalise
subtraction into the
negative numbers
BEFORE being taught
how to do so
Generalisation …
Pupils can generalise with confidence that
• whenever
B+A=C
A+B=C
it will always be true that
C–A=B
C–B=A
… leads to abstraction
Conceptual understanding …
3, 5, 15: three numbers four sentences
3 × 5 = 15
5 × 3 = 15
15 ÷ 3 = 5
15 ÷ 5 = 3
… and leads pupils to generalisation
Pupils can generalise with confidence that
• whenever
A×B=C
B×A=C
it will always be true that
C÷A=B
C÷B=A
A “good” representation …
6×⅓
1
… support conceptual understanding
6 × ⅓ = 2, and what ELSE do I know?
1 1
A “good” representation …
4×⅗
1
… support conceptual understanding
4 × ⅗ = 2⅖, and what ELSE do I know?
1 1
Pupils can consider
non-integer rational
divisors BEFORE
being taught how to
do so
Conceptual understanding …
?
Distance
Speed
Time
?
Not long after …
?
?
?
... supports factual recall
Snails and cheetahs
Does +, –, × or ÷
capture the intuitive
relationship of speed,
distance & time?
Distance
Speed Time
Ever after …
“BODMAS”…
•
•
•
•
3238 + 5721 + 1762 – 5721
5a – 3b – 5a + 3b
731 ÷ 13 × 91 ÷ 7
8a ÷ 6 ÷ 4a × 9 ÷ 3
• (6a + 9b) ÷ 3
• (6a × 9b) ÷ 3
• (6a2 ÷ 9a) ÷ 3a
Pupils read
“BODMAS” left to
right, and so make
slow or no progress
with these
“BODMAS” is
utterly silent here:
it says NOTHING
about distributivity
“BODMAS” is kryptonite
Because the mnemonic “BODMAS”
• is unhelpfully (and inaccurately) dogmatic
• is arbitrary (why not OBMDSA?)
• often leads to cumbersome and inefficient calculating
• masks the actual axiomatic structure of arithmetic
• leads to thinking there are exceptions and “special cases”
• doesn’t help pupils move from (concrete) arithmetic to
(abstract) algebraic manipulation.
“BODMAS”
Pupils recall that × and
÷ are distributive over
+ and − because × and ÷
are over + and −
Olympic Podium
Learning maths through reasoning
If 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.
Convince yourself
In each pair of fractions, what mathematical symbol could you
insert between them? Explain your reasoning.
1
3
11
31
33
93
>
>
9
3
33
11
1
5
2
7
<
2
5
Convince me
In each pair of fractions, what mathematical symbol could you
insert between them? Explain your reasoning.
1
3
>
1
5
• I have ⅓ of a bag of a sugar and ⅕ of a bag or flour. I notice
that they weigh the
THEsame.
SAME.IsIsa afull
fullbag
bagofofflour
flourheavier
heavieroror
lighter or the same weight as a full bag of sugar?
“Which is
a radius?”
Ah-ha! The “top
side” is not always
the “top” side
The choice of model matters a lot!
If I SHARE 13 objects
between 4 people, each
gets 3 objects and there’s
1 object left over.
13 ÷ 4 = 3r1
If I GROUP 13 objects into
groups of size 4, there are
3 full groups and one
quarter-full group.
13 ÷ 4 = 3¼
Conceptual understanding …
• 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 …
… supports procedural 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 =
How would your pupils tackle this?
•
I share some sultanas between Alice and Bob in the ratio
3:5. Alice gets 28g fewer sultanas than Bob. How many
grams of sultanas does Bob get?
Alice
Alice
Alice
Bob
Bob
Bob
28g
Bob
Bob
28g
14g
Bob gets 70g
Box / Bar representations
1. Sam has 9 fewer sweets than Sarah. They have 35 sweets
altogether. How many sweets does Sam have?
Sam
13
?
= 35
Sarah
13
?
9
2. Sam and Tom share 45 marbles in the ratio 2:3. How many
more marbles does Tom have than Sam?
Sam
9?
9?
= 45
Tom
9?
9?
9?
Box / Bar going deeper
In Year 4, 45% of the pupils are boys. There are 14 fewer boys
than girls in Year 4. How many girls are there in Year 4?
If you use different values other than 14, will the answers you
get to the question “How many girls are there in Year 4?”
always, sometimes or never make sense?
Box / Bar challenge
•
On Saturday I opened a new packet of sultanas and gave
Charlie 40% of them. On Sunday I gave him 25% of the
remaining sultanas. Today I have given him the remainder
of the sultanas, which weigh 27g. What was the weight of
the packet when it was full?
Full weight is 60g
12g
12g
12g
36g
Sunday
9g
9g
9g
Saturday
Saturday
“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.
But PF is not the same as CU
•
•
•
•
I share some sultanas between Alice and Bob in the ratio
3:5. Alice gets 28g fewer sultanas than Bob. How many
grams of sultanas does Bob get?
I share some sultanas between Alice and Bob in the ratio
6:10. Alice gets 28g fewer sultanas than Bob. How many
grams of sultanas does Bob get?
I share some sultanas between Alice and Bob, so that Bob
gets ⅝ of all the sultanas. Alice gets 28g fewer sultanas
than Bob. How many grams of sultanas does Bob get?
I share some sultanas between Alice and Bob, so that Alice
gets 60% of what Bob gets. Alice gets 28g fewer sultanas
than Bob. How many grams of sultanas does Bob get?
Even good models have limitations
•
Drawing boxes / bars focuses pupils’ attention on the
additive structure of the Alice and Bob problem: Alice has
2 boxes / bars fewer than Bob, Bob has 2 boxes / bars
more.
• The risk is that we don’t ensure that pupils also acquire
deep conceptual understanding of the multiplicative
structure of the problem: that
A A A
• Alice’s share is ⅗ of Bob’s share
B B B B B
• Bob’s share is ⅝ of the total
• Alice’s share is Bob’s share reduced by 40%
• the scale factor from Alice’s share to Bob’s is 1⅔.
The CSMS project (Kath Hart et al.)
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”
Aims and ethos of the UK curriculum
All pupils must
become FLUENT in the
fundamentals of mathematics,
through varied and
SOLVE PROBLEMSincluding
by applying
frequent
practice
their mathematics
to a variety
of with increasingly
REASON
MATHEMATICALLY
problems
over time, so that
routine andcomplex
non-routine
by following
line of enquiry,
pupils
developaconceptual
problems with increasing
conjecturing
understanding
the ability to
sophistication, including
breakingand relationships
and
generalisations,
and
apply
down problemsrecall
into aand
series
of knowledge rapidly
developing
and accurately.
simpler steps and persevering
in an argument,
justification or proof using
seeking solutions.
mathematical language.
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.
A Chinese takeaway
You need to find out what the
kites have each seen, and
Teachingensure
mathsthat
successfully
is like …
they’ve all
identified the key landmarks
Reasoning deepens connections
Reasoning deepens connections
Learning amounts to being able to discern certain aspects of
the phenomenon that one previously did not focus on or
which one took for granted, and simultaneously bring them
into ones focal awareness.
(Lo, Chik & Pang, 2006)
Comparison is considered the pre-condition to perceive the
structures, dependencies, and relationships that may lead to
mathematical abstraction.
(Sun, 2011)
A Chinese takeaway
Homework
With colleagues, choose a strand that runs across the curriculum,
and discuss how to secure your pupils’ conceptual understanding
and procedural fluency THROUGH reasoning about it.
Topic
Pupils
Year N
Year N+1
Year N+2
Year N+3
Year N+4
Content
The model / representation, discussion, questioning and
intelligent practice that will develop pupils’ …
… factual
knowledge
… procedural
fluency
… conceptual
understanding
Agree or challenge?
Bouquets to: Dan Abramson, Michael
Davies, Tony Gardiner, Debbie Morgan,
Matt Nixon, Maths Hub Leads, Team
NCETM, et al.
Brickbats to me!
[email protected]