Transcript AiZ Action Research PowperPoint

Developing specific planning
and pedagogies for improving
mathematics and numeracy
teaching
Peter Sullivan
nmr day 1 program 2 2011
Program overview
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Today’s program
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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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Choosing a topic: Like terms
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Is your planning sequence something
like this?
•
•
•
•
•
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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Task 1
• Choose some terms from the cloud and write
an expression that is equivalent to
5a+9
2a
3
a
4a
3a
2
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6
7
Task 2
• Race to 10:
– Start at 0, in turn add on either 1 or 2, first to 10 is
the winner
• Race to 5x + 5y
– Start at 0, in turn add on either x, or y, or x + y,,
first to 5x + 5y is the winner
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A different issue:
How might you support students
experiencing difficulty?
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How might you extend students
who find an answer quickly?
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Task 3
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3a + 3
a+3
+3
+ 2a
3a + 6
-3
- 2a
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+ 2a + 3
-2a - 3
3a + 3
+ 2a + 3
-2a - 3
a+3
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3a + 6
3a + 3
+ 2a + 3
-2a - 3
a+3
????
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3a + 6
Task 4
• What might be the missing terms?
4x + 3 = __ + __ + __
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Task 5
• If x + 5 < 12
What might x be?
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Focus on these five tasks collectively
• What do you hope that students might learn?
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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 that 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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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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goals
readiness
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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?
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goals
readiness
engage
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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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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,
–
–
–
–
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
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goals
difference
readiness
engage
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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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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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Devise a lesson plan for ???
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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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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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practice
readiness
goals
Collaborative
teacher
learning
difference
engage
lesson
structure
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