Developing specific planning and pedagogies for improving mathematics and numeracy teaching Peter Sullivan nmr 2012 day 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

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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?

What would be the point of asking a question like that?

• 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?

• 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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• 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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• 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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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,

• Fluency – knowing how,

• Problem solving – finding out how,

• Reasoning – finding out why,

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what, where, …

when, …

when, …

what, where, …

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 ValueNumber of responses to this question 82Number of correct answers 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?

• 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 NAPLAN2(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


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What do you have to think about when playing that game?


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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


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• Task B• Choose some terms from the cloud and

write some expression that equal

5 a + 9

3a

4a

26

a

3

2a 7


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Task CI 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 =

a × a × a × a × a × a =

6a

a6

a to the power of 6


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How we say these

xx2

x3

x4

x5

x6


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Write down the answer

3a + a =a3 × a =


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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


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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


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Task D

• Choose some terms from the cloud and write some expressions that equal

6 a3

3a

4a

26a

a

3

2a 3a2

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3a + 3Task E

a + 3 3a + 6

+ 3

- 3

+ 2a + 3

+ 2a

- 2a

-2a - 3

????

????

????

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Make up your own


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Task F

• What might be the missing terms?

4x + 3 = __ + __ + __


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• The underline means that something is missing. What might be the bits missing in the following?__ + __ + __ + __ = 5x – 5y + 3


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• The underline means that something is missing. What might be the bits missing in the following?

3( a + __ ) - __ = __ a + __


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• The underline means that something is missing. What might be the bits missing in the following?

__ ( a – 2c) = __ a + __ c


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Focus on these tasks collectivelyWhat 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?

• 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?

TimetableLength of periodMaking connections, including the previous

algebra experiencesGetting all kids involvedGrouping studentsAbsenteeismTeacher 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

What are the implications for our lesson sequence?

• 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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• 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 NAPLAN2(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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goals readiness


• 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 complexityGoing both ways, Start with concrete, moving to abstractVarious 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


• 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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• 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

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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

• A harder one• Create your own• Assist strugglers (by asking questions)

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goals readiness

engagedifference


• Prepare to differentiate• Commitment to interact with students• Plan to interact• Place a limit on textbook use

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• 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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goals

lessonstructure

readiness

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

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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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goals

lessonstructure

readiness

engagedifference


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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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goals

lessonstructure

readiness

practice

engagedifference


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goals

lessonstructure

readiness

practice

engagedifference

Collaborative teacherlearning


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Documents

Developing specific planning and pedagogies for improving mathematics and numeracy teaching Peter Sullivan nmr 2012 day 1