Using key numeracy teaching
principles as the basis of leading
teaching improvement
Peter Sullivan
Numeracy keynote SA
Abstract
• Supporting improvement in numeracy teaching is
both demanding and complex. One way to
manage the complexity is to have explicit goals
for each step in the improvement process.
• After reviewing other similar lists, I identified six
principles that can form the basis, individually
and together, of improvement initiatives.
• Using the theme of the teaching of fractions, this
session will elaborate each of the principles.
Numeracy keynote SA
What is being recommended about
mathematics teaching?
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How does this connect to the AC?
Numeracy keynote SA
Describing the proficiencies
• Understanding
– (connecting, representing, identifying, describing,
interpreting, sorting, …)
• Fluency
– (calculating, recognising, choosing, recalling,
manipulating, …)
• Problem solving
– (applying, designing, planning, checking, imagining, …)
• Reasoning
– (explaining, justifying, comparing and contrasting,
inferring, deducing, proving, …)
Numeracy keynote SA
The (brand) new UK National
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
have conceptual understanding and are able to recall and
apply their knowledge rapidly and accurately to
problems
reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and
developing an argument, justification or proof using
mathematical language
can solve problems by applying their mathematics to a
variety of routine and nonroutine problems with
increasing sophistication, including breaking down
problems into a series of simpler steps and persevering
in seeking solutions.
Numeracy keynote SA
• https://www.education.gov.uk/schools/teachi
ngandlearning/curriculum/nationalcurriculum
2014/a00220610/draft-pos-ks4-englishmaths-science
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From Impactful practices
• Imagine a world where students, in every
mathematics classroom, are actively engaged
with worthwhile tasks that promote
mathematical understanding, problem solving
and reasoning.
Numeracy keynote SA
• Imagine classrooms where the interactions
among students, and with their teacher, are
focused on making sense of mathematics,
alternative approaches to solving problems,
and defending, confirming and verifying
possible solutions. These are thinking and
reasoning classrooms.
Numeracy keynote SA
An aside
• It is not a problem if we have told students
what to do
• It is not reasoning if students are reproducing
what we have told them
• It is not understanding unless students can
explain in their own words with their own
ideas
Numeracy keynote SA
What do you see as the most
critical aspect of being a powerful
learner of numeracy and literacy?
• Powerful learners connect ideas together, they
can compare and contrast concepts, and they
can transfer learning from one context to
another. They can devise their own solutions
to problems, and they can explain their
thinking to others.
Numeracy keynote SA
Two task examples that we will use as
the basis of the later discussion
Numeracy keynote SA
STRIPED RECTANGLE
If the dotted (blue) rectangle represents
2
3
what fraction is represented by the striped
(red) rectangle?
Work out the answer in two different ways.
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REPRESENTING A FRACTION
2
1
• If this represents 3 3 , draw what represents 1 2
(work this out two different ways)
Numeracy keynote SA
What is the point of the six key
principles ?
• We can all do these things better (although you
will find many of them affirming of your current
practice)
• Much advice is complex and hard to prioritise
• The principles can provide a focus to
collaborative discussions on improving teaching
• The principles can be the focus of observations
if you have the opportunity to be observed
teaching
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AVAILABLE TO DOWNLOAD FREE FROM
http://research.acer.edu.au/aer/13/ aer
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Key principle 1:
• Identify important ideas that underpin
the concepts you are seeking to teach,
and communicate to students that these
are the goals of your teaching, including
explaining how you hope they will learn
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Feedback - better when they know …
• Where am I going?
– “Your task is to …, in this way”
• How am I going?
– “the first part is what I was hoping to see,
but the second is not”
• Where to next?
– “knowing this will help you with …”
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In terms of learning intentions, we
know
• It is difficult to describe the purpose of lessons
and teachers benefit from discussions about
intentions
• The learning intention should
–
–
–
–
–
not restrict
nor tell
nor lower the ceiling
but provide focus for the students
and the teacher
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What would you say is the learning
intention for striped rectangle?
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This is taken from the lesson plan
• A fraction is a number.
• We can compare, add and multiply fractions
just like we do for numbers, even if the
calculation process is a little different.
• You will solve the problem for yourself and
explain your thinking
Numeracy keynote SA
goals
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Key principle 2:
• Build on what the students know, both
mathematically and experientially, including
creating and connecting students with stories
that both contextualise and establish a
rationale for the learning
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Part 1: Using data
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• It is as important to know what the students
know as it is to know what they do not
• Learning mathematics is not a hierarchy of
sequential steps on a ladder, but a network of
interconnected ideas
• Students benefit from work on tasks that are
beyond what they know
– Students at GP 2 can work on GP 4 tasks
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Helen has 24 red apples and 12 green
apples.
What fraction of the apples are green?
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Year 5
93%
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Year 5
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24%
Helen has 24 red apples and 12 green
apples. What fraction of the apples are
green?
• 55% of year 7 students
• 67% of year 9 students
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What does that tell you?
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Part 2: Connecting with “story”
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• A chameleon has a
tongue that is half as
long as its body ...
• … how long would your
tongue be if you were a
chameleon?
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Part 3: Creating experience
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goals
readiness
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Key Principle 3
• Engage students by utilising a variety of rich
and challenging tasks, that allow students
opportunities to make decisions, and which
use a variety of forms of representation
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Why challenge?
• Learning will be more robust if students connect
ideas together for themselves, and determine
their own strategies for solving problems, rather
than following instructions they have been given.
• Both connecting ideas together and formulating
their own strategies is more complex than other
approaches and is therefore more challenging.
• We need strategies to encourage students to
persist
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Related to those tasks above ..
• To what extent
– Are they challenging?
– Are they engaging?
– Do they allow student decision making
– Do they encourage different representations?
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What is 5 + 5 + 5 + 295 + 295 + 295 ?
Number of students that 1343
completed this question
Correct answers
413 (30.8%)
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Think about the question What is 5 + 5 + 5 + 295 +
295 + 295 ?
I prefer learning how to do questions like this
I prefer questions we work on in class to be
Much harder
About the
same
Much
easier
TOTAL
239
290
69
598
78
227
162
467
56
140
100
296
373
657
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331
1361
By myself
Working with other
students
By listening to the
teacher's
explanations first
TOTAL
Think about the question What is 5 + 5 + 5 + 295 +
295 + 295 ?
I prefer learning how to do questions like this
I prefer questions we work on in class to be
By myself
Working with other
students
By listening to the
teacher's
explanations first
TOTAL
Much harder
About the
same
Much
easier
TOTAL
239
118
290
103
69
21
598
242
78
15
227
64
162
13
467
92
56
17
140
36
100
18
296
71
373
657
150
203
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331
52
1361
405
Quotes from PISA in Focus 37
• When students believe that investing effort in
learning will make a difference, they score
significantly higher in mathematics.
• Teachers’ use of cognitive-activation strategies,
such as giving students problems that require
them to think for an extended time, presenting
problems for which there is no immediately
obvious way of arriving at a solution, and helping
students to learn from their mistakes, is
associated with students’ drive.
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goals
readiness
engage
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Key Principle 4:
• Interact with students while they engage in the
experiences, and specifically planning to
support students who need it, and challenge
those who are ready
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Enabling prompt
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IF YOU ARE STUCK
1
• If this represents 7 3 , draw what represents 2
(work this out two different ways)
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IF YOU ARE STUCK
If this represents 11 , draw what represents 5
(work this out two different ways)
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Extending prompt
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IF YOU HAVE FINISHED
4
• If this represents 8 5 , draw what represents 2.4
(work this out two different ways)
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goals
readiness
difference
engage
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Key Principle 5:
• Adopt pedagogies that foster communication, mutual
responsibilities, and encourage students to work in
small groups, and using reporting to the class by
students as a learning opportunity
Numeracy keynote SA
A revised lesson structure
• In this view, the sequence
– Launch (without telling)
– Explore (for themselves)
– Summarise (drawing on the
learning of the students)
Launch
Summarise
• … is cyclical and might happen more than
once in a lesson (or learning sequence)
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Explore
CONSOLIDATING THE LEARNING
1
1
• If this represents 5 2 , draw what represents 2 4
(work this out two different ways)
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readiness
goals
difference
engage
lesson
structure
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Key teaching idea 6
• Fluency is important, and it can be developed in two
ways
– by short everyday practice of mental calculation or
number manipulation
– by practice, reinforcement and prompting transfer
of learnt skills
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One aspect is transfer
• This connects to the consolidating task
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Another aspect is fluency
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3
4
+
7
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23
34
+
?
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13
?
+
35
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The unknowns are different
?
?
+
½
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?
?
?
+
+
?
?
0.2
+
?
+
+
1.7
?
+
?
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Think about the question What is 5 + 5 + 5 + 295 +
295 + 295 ? (row)
I prefer learning how to do questions like this
I prefer questions we work on in class to be
Much harder
By myself
239
39.97%
About the
same
290
48.49%
Working with other
students
78
16.7%
227
48.61%
162
34.69%
467
100%
By listening to the
teacher's explanations
first
56
18.92%
140
47.3%
100
33.78%
296
100%
373
27.41%
657
48.27%
331
24.32%
1361
100%
TOTAL
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Much easier
TOTAL
69
11.54%
598
100%
Think about the question What is 5 + 5 + 5 + 295 +
295 + 295 ? (column)
I prefer learning how to do questions like this
I prefer questions we work on in class to be
Much harder
By myself
239
64.08%
About the
same
290
44.14%
Working with other
students
78
20.91%
227
35.55%
162
48.94%
467
34.31%
By listening to the
teacher's explanations
first
56
15.01%
140
21.31%
100
30.21%
296
21.75%
TOTAL
373
100%
657
100%
331
100%
1361
100%
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Much easier
TOTAL
69
20.85%
598
43.94%
Think about the question What is 5 + 5 + 5 + 295 +
295 + 295 ? ( % of the overall 1361 responses)
I prefer learning how to do questions like this
I prefer questions we work on in class to be
By myself
Working with other
students
By listening to the
teacher's explanations
first
TOTAL
Much harder
About the
same
Much easier
TOTAL
239
17.56%
290
21.31%
69
5.07%
598
43.94%
78
5.73%
227
16.68%
162
11.90%
467
34.31%
56
4.11%
140
10.29%
100
7.35%
296
21.75%
373
27.41%
657
14.27%
331
24.32%
Numeracy keynote SA
Think about the question What is 5 + 5 + 5 + 295 +
295 + 295 ?
I prefer learning how to do questions like this
I prefer questions we work on in class to be
By myself
Working with other
students
By listening to the
teacher's
explanations first
TOTAL
Much harder
About the
same
Much easier
TOTAL
239 (100%)
118 (49.37%)
290 (100%)
103 (35.52%)
69 (100%)
21 (30.43%)
598 (100%)
242 (40.47%)
78 (100%)
15 (19.23%)
227 (100%)
64 (28.19%)
162 (100%)
13 (8.02%)
467 (100%)
92 (19.70%)
56 (100%)
17 (30.36)
140 (100%)
36 (25.71%)
100 (100%)
18 (18%)
296 (100%)
71 (24%)
373 (100%)
150 (40.21%)
657 (100%)
203 (30.90%)
331(100%)
52 (15.71%)
1361 (100%)
405 (29.76%)
Numeracy keynote SA