NT 2012 Six principles of effective mathematics teaching Peter Sullivan Monash University.

NT 2012

Six principles of effective mathematics teaching

Peter SullivanMonash University

NT 2012

First, let us consider this task sequence

• The following sequence seeks to introduce students to – the nature of volume of prisms and cylinders (area

of the end × length) – surface area of prisms (total area of the faces)– efficient methods of calculating volume and surface

area of rectangular prisms– ways in which surface area and volume are

different

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The tasks are only intended to be illustrative

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This is a rectangular prism made from cubes. What is the volume of this prism?

What is the surface area?

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A set of 36 cubes is arranged to form a rectangular prism.

What might the rectangular prism look like?What is the surface area of your prisms?

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• A rectangular prism is made from cubes. It has a surface area of 22 square units. What might the rectangular prism look like?

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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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How might the AC help?

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Year 6 Year 7Year 8

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Three content strands (nouns)

• Number and algebra• Measurement and geometry• Statistics and probability

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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

• Overall• what do these suggest are the overall goals, the big

ideas, the important focus, etc

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AC content descriptions: Using Units of measurement

• Year 6 – Connect volume and capacity and their units of measurement

• Year 7 – Calculate volumes of rectangular prisms

• Year 8 – Choose appropriate units of measurement for area and volume and

convert from one unit to another – Develop the formulas for volumes of rectangular and triangular

prisms and prisms in general. Use formulas to solve problems involving volume

• Year 9– Calculate the surface area and volume of cylinders and solve related

problems– Solve problems involving surface area and volume of right prisms

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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 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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How is this represented in the AC?

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What would you say to the students were the goals the “36

cubes” task?

• 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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What proficiencies are associated with the 36 cubes task?

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goals

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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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This has two parts.Partly this is using data

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What % of Year 5 Victorians can do this?

A rectangular paddock has a perimeter of 50 metres. Each long side has a length of 15 metres.

• What is the length of each short side?

metres

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56% of students could do this

A rectangular paddock has a perimeter of 50 metres. Each long side has a length of 15 metres.

• What is the length of each short side?

metres

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What % of Year 7 Victorians (no calculator) can do this?

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12 % of Year 7 Victorians (no calculator) can do this

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This tells us

• Around half of the year 5 students are ready for challenging tasks about perimeter• Very few year 7 students have a sense

of volume

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Partly it is about connecting with story

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1mm of rain on 1 sq m of roof is 1 L ofwater.Design a tank for this building that captures all of the rain that usually falls this month.

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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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• 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 SA = 22cm2 activity

• How might we engage students who could experience difficulty with it?

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How might we extend students who finish quickly

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goals readiness

engagedifference

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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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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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Why do I tell you about this lesson?

• 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 words for the parts of the lesson• 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

• Neriage– 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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7

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35

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x2 - 5x + 6

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9x2 - 49

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1411

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goals

lessonstructure

readiness

practice

engagedifference

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goals

lessonstructure

readiness

practice

engagedifference

Collaborative teacherlearning

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AVAILABLE TO DOWNLOAD FREE FROM

http://research.acer.edu.au/aer/13

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[email protected]

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Words are important …

• But why does the word

• Monosyllabic

• have 5 syllables

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• but why is

• abbreviation

• such a long word

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• but why is

• phonetic

• not spelled the way that it sounds

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• but why is

• nostalgia

• not what it used to be

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• but why is the person who invests your money called a BROKER

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• but why is

• There only one word for thesaurus

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And finally

• Why do they nail down the lid of a coffin?

NT 2012 Six principles of effective mathematics teaching Peter Sullivan Monash University.

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