6 principles

Finding a focus for ongoing teacher learning in mathematics: Six key

principles

Peter Sullivan

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Overview

• Some stories• Six principles

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A story from a Grade 3 lesson I taught at Bathurst Island

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Football scores

Saints 105Bombers 98

How much are the Saints winning by?

(Work out the answer in two different ways)

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Football scores

Saints 25Bombers 18

How much are the Saints winning by?

(Work out the answer in two different ways)

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From a grade 3 lesson in the Kimberleys

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How many ways to make $10

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A suggestion as part of a geometry unit written for MiTK

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• Have pile of cubes. Ask the students to:– Build something using 15 cubes. Describe what

you have built.– Build something which is 3 cubes high and 3 cubes

wide made with 15 cubes. Describe what you have built.

– Something like “make a tower of 3 cubes, put a yellow cube to the left, and blue cube to the north, …”

• Repeat, adapting to the level of success of the students.

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A lesson I plan to teach to a year 8 class next Tuesday at CMS (Alice Springs)

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ONE HECTARE PARK

• A conservation park is to be enclosed by a fence that has exactly 6 internal right angles. What might the park look like?

• The total area of the park is 1 hectare. What might be the perimeter of the park? (give two different answers)

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What are the common characteristics of those tasks?

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Six principles as the basis of sustainable improvement

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What is the point of these 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

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 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 much more difficult to describe the purpose of lessons than we think

• The learning intention should – not restrict – nor lower the ceiling – but provide focus to the students– and the teacher

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In terms of the subtraction lesson

• There are many ways to find the difference between two numbers

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In this lesson, I need you to

• show how you get your answers• keep trying even if it is difficult (it is meant to

be)• explain your thinking• listen to other students

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goals

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 more important to know what the students know than what they do not

• Learning mathematics is not a hierarchy of sequential steps on a ladder, but a network of interconnected ideas

• Students can work on tasks that are beyond what they know– Students at GP 2 can work on GP 4 tasks

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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

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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For students to learn, two sets of factors must align

• The first set of factors include that the:

–students have the requisite prior knowledge;–curriculum is relevant to them; –classroom tasks match their expectations;–classroom tasks help them make connections–pedagogies use their knowledge and experience; –assessment regimes measure their learning.

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The second set of factors relates to

• whether the students– are motivated to learn – see participation in schooling as creating

opportunities– are willing to persist– connect effort and success

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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.

• It is potentially productive if students are willing to take up such challenges.

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This connects to “mindsets”

• Dweck (2000) categorized students’ approaches in terms of whether they hold either growth mindset or fixed mindset

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Students with growth mindset:

• Believe they can get smarter by trying hard• Such students– tend to have a resilient response to failure;– remain focused on mastering skills and knowledge

even when challenged; – do not see failure as an indictment on themselves;

and – believe that effort leads to success.

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Students with fixed mindset:

• Believe they are as smart as they will even get• Such students– seek success but mainly on tasks with which they

are familiar; – avoid or give up quickly on challenging tasks; – derive their perception of ability from their

capacity to attract recognition.

Teachers can change mindsets• This connects to

– the things we affirm (effort, persistence, co-operation, learning from others, flexible thinking)

– the way we affirm • You did not give up even though you were stuck• You tried something different• You tried to find more than one answer

– the types of tasks we pose

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Getting started “zone of confusion”

“four before me” •representing what the task is asking in a different way such as drawing a cartoon or a diagram, rewriting the question …

•choosing a different approach to the task, which includes rereading the question, making a guess at the answer, working backwards …

•asking a peer for a hint on how to get started

•looking at the recent pages in the workbook or textbook for examples.

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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Related to those 4 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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goals readiness

engage

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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Basketball scores

Eels 18Carp 13

How much did the Eels win by?

(Work out the answer in two different ways)

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Basketball scores

Cats 8Dogs 3

How much did the Cats win by?

(Work out the answer in two different ways)

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Extending prompt

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Darts scores

Parrots 1005Galahs 988

How much did the Parrots win by?

(Work out the answer in two different ways)

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goals readiness

engagedifference

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

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A revised lesson structure

• In this view, the sequence– Launch (without telling)– Explore (for themselves)– Summarise (drawing on the learning of the students)

• … is cyclical and might happen more than once in a lesson (or learning sequence)

shepp secondary

Launch

ExploreSummarise

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Football scores

Saints 18Bombers 13

How much are the Saints winning by?

(Work out the answer in two different ways)

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goals

lessonstructure

readiness

engagedifference

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

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Another aspect is fluency