Transcript Slide 1

Developing specific planning
and pedagogies for improving
mathematics and numeracy
teaching
Peter Sullivan
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Program overview
• This program will build on the general strategies
for improving mathematics and numeracy
teaching suggested in AIZ numeracy programs,
and will develop specific approaches to
teaching selected mathematics concepts.
• We will use a particular aspect of the
curriculum to plan teaching sequences that are
engaging for students and which allow effective
differentiation, and will plan individual lessons.
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• By being specific we will extend the initial
exploration of the key principles for teaching
numeracy and mathematics, and examine what
these might be implemented in detail.
• We will also undertake an example of the Lesson
Study approach to allow detailed study of what
might be possible in mathematics teaching. This
will involve planning, implementation, and review
of a particular approach to teaching.
• This is a four day program. It is expected that
participants will attend all of the four days.
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The overall goal
• You will plan in detail a lesson sequence (in
which you are trying something different)
• You (or a colleague) will teach the lesson
sequence
• You will report back on what happened
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Day 1: Teaching mathematics for
curiosity and powerful learning
• This day will establish some principles of teaching that
will inform the planning of units. In particular these
include processes for differentiation, enquiry focused
teaching especially the proficiencies from the
Australian Curriculum, the importance of challenge and
high expectations, lessons structures including
grouping practices, and student motivation and
engagement. Collaborative teacher learning processes
will also be considered.
•
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Day 2: Planning mathematics for
curiosity and powerful learning
• This day will include goal setting, interpreting
curriculum statements, accessing and
adapting, resources, planning formats,
planning for assessment for learning, including
specifically designing assessments of students’
current knowledge for the units to be
developed.
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Day 3: Teacher planning
• This day will include opportunities for
reviewing initial assessments of student
learning, sharing of potential resources, and
time for the planning and recording of the
intended units of work.
•
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Day 4: Presentation of units of work
and reporting on their implementation
• This day is intended for all teachers involved in
the planning of the units to present to the
other schools, highlighting successes and
challenges, and particular the learning from
the process that can be generalised to future
team based planning. It is expected that
school leadership teams also attend these
presentations.
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• As well as being used in the developing school,
the units of work are intended to be shared with
others and to be illustrative of what is possible. It
is intended that the process for developing the
units be reflected in future planning process in
participating schools. We ask participating
schools to arrange the mathematics curriculum
for years 8 and 9 such that measurement topics
are planned to be taught in late term 2 or term 3.
•
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Today’s program
• Review of an approach to teaching building on
the Australian Curriculum
• Review of the six principles for teaching
mathematics
• Planning how we will work
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A task to get us started
I am concerned that my car is not
getting the best fuel economy.
I filled up my car on 27th April,
noting the odometer as being
2345 km. When I filled the car
next, I got this print out.
What is the fuel economy of my
car in L per 100 km?
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What would be the point of asking a
question like that?
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Engaging for boys
Ability to analyse information
Real life context, value for money
Ratio calculations
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What might make it difficult to ask
such a question in your school?
• The decimals, proportions
• Access to calculators (and trusting their
answers – rounding etc)
• Being exposed to more difficult numbers
• They give up
• Differences in readiness
• Reading, literacy level
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2/3
Draw this number line, and mark on it,
as accurately as possible, 0 and 2 ½
Explain your reasoning.
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What would be the point of asking a
question like that?
• Frctions are numbers
• Thinking about different ways to solve
problems
• Seeing fraction decimal relationships
• Value of numerator and denominator
• Proportions, number line
• Fractions and whole numbers
• Proving you are correct – being convincing
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What might make it difficult to ask
such a question in your school?
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Fractions
Why do I need to do this
Prior knowledge
Relating to number line
Have a go, record their work, explain their
thinking, listen to others
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Three content strands (nouns)
• Number and algebra
• Measurement and geometry
• Statistics and probability
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Looking at patterns and algebra
• Note that there is also a “linear and non-linear
relationships” section in these years as well
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Year 6 “Patterns and algebra”
Continue and create sequences involving whole
numbers, fractions and decimals. Describe the
rule used to create the sequence
Explore the use of brackets and order of
operations to write number sentences
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Year 7 “Patterns and algebra”
• Introduce the concept of variables as a way of
representing numbers using letters
• Write algebraic expressions and evaluate them
by substituting a given value for each variable
• Extend and apply the laws and properties of
arithmetic to algebraic terms and expressions
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Year 8 “Patterns and algebra”
• Extend and apply the distributive law to the
expansion of algebraic expressions
• Factorise algebraic expressions by identifying
numerical factors
• Simplify algebraic expressions involving the
four operations
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Year 9 “Patterns and algebra”
Extend and apply the index laws to variables,
using positive integral indices and the zero
index
Apply the distributive law to the expansion of
algebraic expressions, including binomials,
and collect like terms where appropriate
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Using the content descriptions
• Get clear in your mind what you want the
students to learn
• Make your own decisions about how to help
them learn that content
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A meta analysis of many studies (Hattie,
2007)
• Most important teacher influenced factors
– Feedback
– Instructional quality
– Direct instruction
– Remediation
– Class environment
– Challenging goals
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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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So far there is not much difference
from what you are doing
• It is the proficiencies that are different
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In the past, we made the distinction
between
• Knowing how
– (instrumental understanding)
• Knowing why
– (relational understanding)
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The action words (proficiencies)
• Understanding
– knowing why, what, where, …
• Fluency
– knowing how, when, …
• Problem solving
– finding out how, when, …
• Reasoning
– finding out why, what, where, …
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In the Australian curriculum
• 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, …)
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The proficiencies – why do we change
from “working mathematically”?
• These actions are part of the curriculum, not
add ons
• Mathematics learning and assessment is more
than fluency
• Problem solving and reasoning are in, on and
for mathematics
• All four proficiencies are about learning
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Choosing tasks will be a key decisions
• If we are seeking fluency, then clear explanations
followed by practice will work
• If we are seeking understanding, then very clear
and interactive communication between teacher
and students and between students will be
necessary
• If we want to foster problem solving and
reasoning, then we need to use tasks with which
students can engage, which require them to make
decisions and explain their thinking
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Another fractions tasks
• First do the task
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If the blue rectangle represents 2/3,
what fraction is represented by the red rectangle?
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Examining a task in detail
• What does this task do?
• Where does it fit with the content
descriptions?
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A review of student working on the
task
• What do you see in this video?
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The shaded rectangle represents 3/4.
What is the value of the whole square?
Statistic
Number of responses to this question
Number of correct answers
Value
82
24 (29.3%)
Please note that the results above do not include students that did not provide a response at all. For this question this was 5 students
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Choosing a topic: Like terms
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Is your planning sequence something
like this?
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Identify the topic
Examine curriculum content statements
Use data to inform decisions on emphasis
Select, then sequence, appropriate activities
Identify the mathematical actions in which
you want students to engage
• …
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Using data to informing instruction
From 2009 NAPLAN
2(2x – 3) + 2 + ? = 7x – 4
• What term makes this equation true for all
values of x ?
• 15% (Victorians) correct
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• Race to 10:
–Start at 0, in turn add on either
1 or 2,
– first to 10 is the winner
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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The videos
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Task A
• Race to 5x + 5y
–Starting at 0, you can add, in turn, x, or
y, or x + y
–The person who says 5x + 5y is the
winner
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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What do you have to think about
when playing that game?
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Now play Race to 8x + 12
–Starting at 0, you can add, in turn, x, or
2x, 1, or 2
–The person who says 8x + 12 is the
winner
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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What do you have to think about
when playing that game?
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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• Task B
• Choose some terms from the cloud and
write some expression that equal
5a+9
2a
3
a
4a
3a
6
7
2
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Task C
I want you to play Race to 32a5
Starting at 1, you can multiply by a, or 2,
or 2a
The person who says 32a5 is the winner
But first …
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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These are different
a + a + a + a + a + a = 6a
6
a
a×a×a×a×a×a=
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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How we say these
x
x2
x3
x4
x5
x6
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Write down the answer
3a + a =
3
a ×a=
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Now play Race to 32a5
Starting at 1, you can multiply by a, or 2,
or 2a
The person who says 32a5 is the winner
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Now play Race to 32a5b5
Starting at 1, you can multiply by a, b,
ab, or 2
The person who says 32a5b5 is the
winner
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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What do you have to think about
when playing that game?
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Task D
• Choose some terms from the cloud and
write some expressions that equal
6
2a
3
3
a
a
4a
3a
2
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6a
3a2
Task E
3a + 3
a+3
+ 2a + 3
-2a - 3
????
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3a + 6
Make
up
your
own
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Task F
• What might be the missing terms?
4x + 3 = __ + __ + __
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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• The underline means that something is
missing. What might be the bits missing in the
following?
__ + __ + __ + __ = 5x – 5y + 3
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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• The underline means that something is
missing. What might be the bits missing in the
following?
3( a + __ ) - __ = __ a + __
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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• The underline means that something is
missing. What might be the bits missing in the
following?
__ ( a – 2c) = __ a + __ c
INVESTIGATING THE RELATIONSHIP BETWEEN TEACHER EXPECTATIONS, STUDENT PERSISTENCE AND THE LEARNING OF MATHEMATICS
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Focus on these tasks collectively
What can you say about the nature
of those tasks?
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• Building from putting the answers to developing the
questions
• Reversing the process
• Alignment of learning: same concept in different ways
• Tasks test depth of understanding
• We need to find ways of assessing student learning
• Learning the rules of maths through playing the game
• Develop a feeling of success, they can enter at their
level
• Developing their own strategies
• Learning without a pen in their hand
• Does it engage the boys and girls differently
• Do students see this as the real learning?
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Matching those tasks to the content
descriptions
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What can you say about the nature of
those sequence?
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Get everyone’s attention
Continuing the concept and adding additional complexity
The jumps were progressive
Variety
Mode of delivery, structure of lessons varies
Enjoyable (hopefully) – decisions are enjoyable as is success
Can be differentiated readily
The sequence allows all students to follow the same pathway
They can go back if they struggle
Games etc encourage checking appropriateness of the answer
Accountable to their peers
Opportunity to learn together (especially the explaining)
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What might make teaching that
sequence difficult in your school?
Timetable
Length of period
Making connections, including the previous
algebra experiences
Getting all kids involved
Grouping students
Absenteeism
Teacher buy in
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What “Proficiencies” do these
address?
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Connecting the descriptions and
proficiencies to six key strategies
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What is the point of these six key
strategies?
• 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
• They can provide a focus to collaborative
discussions on improving teaching
• They can be the focus of observations if you
have the opportunity to be observed teaching
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Improving teaching by thinking about
pedagogy
• The following principles are a synthesis of:
– Good, Grouws, and Ebmeier
– Productive pedagogies
– Principles of learning and teaching
– Hattie
– Clarke and Clarke
– Anthony and Walshaw
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Key teaching idea 1:
• Identify big 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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What would you say to the
students were the goals of the
Race to 5x + 5y game?
• Would you write the goal(s) on the board?
• What would you say to the students about
how you hope they would learn?
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goals
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What are the implications for our
lesson sequence?
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Set clear goals for the sequence
Set clear goals for the lessons
Decide how to inform students
Saying how they will learn
Identifying big ideas
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Key teaching idea 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 of this is using data
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Using data to informing instruction
From 2009 NAPLAN
2(2x – 3) + 2 + ? = 7x – 4
• What term makes this equation true for all
values of x ?
• 15% (Victorians) correct
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Part of this is creating experience
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• How did that sequence connect with students’
experience?
• Or
• How could that sequence have connected
with the students’ experience?
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Partly about DIY experience
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readiness
goals
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What are the implications for our
lesson sequence?
• Where are they at, NAPLAN, VELS judgments,
on demand, pre testing
• Formative assessment for learning
• Them finding out what they know and can
build on and what they need to learn
• Identify common misconceptions
• Assessment/evaluation of their learning
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Relating to experience
• Find out (or at least assume) what they are
interested in
• Create a rationale for the learning, meaningful
and relevant for them now
• Relate the topic to past and future topics
• Link to other studies
• Build experience
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Key teaching idea 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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How might those activities TOGETHER
contribute to learning?
Progression, increasing complexity
Going both ways,
Start with concrete, moving to abstract
Various activities within a lesson, appealing to
different styles (the light bulbs can come at
different times/rates)
Kids are less bored,
Rigorous
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goals
readiness
engage
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What are the implications for our
lesson sequence?
• Different learning styles, including using a
variety of types of experiences
• Decisions, choice, of pathway, destination, and
form fo representation, including
incorporating this into the assessment
• Don’t tell
• Explaining and justifying their thinking,
strategies
• At least some of the tasks should be difficult
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Key teaching idea 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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Focusing on the “expressions and
relationships” activity
• How might we engage students who could
experience difficulty with it?
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How might we engage students who
could experience difficulty with it?
• Mixed groups (and how to manage those
groups)
• Like groups
• Try it in a row first
• Have only two expression cards, and a subset
of the relationship cards
• Give students a role
• Have one with just numbers
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What are enabling prompts?
• Enabling prompts can involve slightly varying
an aspect of the task demand, such as
– the form of representation,
– the size of the numbers, or
– the number of steps,
so that a student experiencing difficulty, if
successful, can proceed with the original task.
• This approach can be contrasted with the more
common requirement that such students
– listen to additional explanations; or
– pursue goals substantially different from the rest of
the class.
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Factors contributing to difficulty
• It may not be clear which aspects may be contributing
to a particular student’s difficulty, but by anticipating
some of the factors, and preparing prompts that, for
example,
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–
–
–
reduce the required number of steps,
simplify the modes of representing results,
make the task more concrete, or
reduce the size of the numbers involved,
• the teacher can explore ways to give access to the task
without the students being directed towards a
particular solution strategy for the original task.
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How might we extend students who
finish quickly
• A harder one
• Create your own
• Assist strugglers (by asking questions)
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goals
readiness
difference
engage
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What are the implications for our
lesson sequence?
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Prepare to differentiate
Commitment to interact with students
Plan to interact
Place a limit on textbook use
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Key teaching idea 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
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readiness
goals
lesson
structure
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I watched a mathematics lesson
when I was in Japan
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First the teacher told a story about
tatami mats that emphasised the
notion of area as covering
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Then the teacher posed the task
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The students had a worksheet with
TWO copies of the question on it that
emphasised to the students it was
the method, not the answer, that was
the focus
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How many squares?
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… And that they were meant to go
beyond counting the squares
The students worked individually
but talked with each other while
working
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The teacher selected students
to share their work, giving them
advance notice, an A3 sheet,
and a pointer
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• Emphasises there ar many ways to come to a
solution
• The teacher embraced the student solutions
• Focus on students explaining
• Open to scrutiny
• Student a got to see how student b did theirs
• Prepares them to later study in that much
mathematics is about making choices of methods
• Practical organisation is well doen
• Visual cues
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Solving one problem 5 different ways
Orderly display of the solutions
Sharing and celebrating the solutions
Contextualising the task
Conversation as a whole class rather than in small groups
Student generated solutions
Open-middled
Easy entry, chance of making connections, including important
mathematical properties
• Teacher allowed the students to learn and create and work through the
challenge
• Focus was clear
• She knew he rstudents
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What do you see as the connections to
“curriculum”?
• The lesson was connected to
students’ experience
–Relevance, engagement, utility
• It addressed at least one “big idea”
of mathematics
–Power of knowledge, building
connections
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• The clear expectation is that
students learn from each other
–Culture, community, relationships
• The emphasis was on the process
not on the answer
–Quality of thinking, building capacity to
learn
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What are those big ideas?
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The Japanese have better words
• Hatsumon
– The initial problem
– Kizuki - what you want them to learn
• Kikanjyuski
– Individual or group work on the problem
– Kikan shido – thoughtful walking around the desks
• Nerige
– Carefully managed whole class discussion seeking the
students’ insights
• Matome
– Teacher summary of the key ideas
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readiness
goals
difference
engage
lesson
structure
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What are the implications for our
lesson sequence?
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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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After the equations task
• What task might you ask next?
• What might the next lesson look like?
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practice
readiness
goals
difference
engage
lesson
structure
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What are the implications for our
lesson sequence?
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practice
readiness
goals
Collaborative
teacher
learning
difference
engage
lesson
structure
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What are the implications for our
lesson sequence?
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