Judy Anderson The University of Sydney [email protected]

Implementing the Austra

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Implementing the Australian Curriculum for Mathematics F to 10

Judy AndersonThe University of Sydney

[email protected]

Key messages …

1. Balance is important

2. Evaluate the types of questions and tasks used during mathematics lessons

3. Assessment, assessment, assessment!!!

4. Alignment between curriculum, teaching and assessment

Mathematics teaching should include opportunities for (Cockcroft, 1982):

exposition by the teacher;

discussion between teacher and pupils and between

pupils themselves;

appropriate practical work;

consolidation and practice of fundamental skills and

routines;

problem solving, including the application of

mathematics to everyday situations; and

investigational work.

Understanding

Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas.

Fluency Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily.

Which tasks would support these proficiencies?

Examine the types of questions and tasks you use during mathematics lessons.

Gould, 2006

Because three is a larger number than 2

Because four is a larger number than three

Because six is a larger number than 3

Because 5 & 6 are larger numbers than 2 & 3

Because 12 & 13 are larger numbers than 9 & 10

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Problem solving

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively.

Reasoning

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying, and generalising.

Which tasks would support these proficiencies?

Examine the types of questions and tasks you use during mathematics lessons.

Bloom’s Taxonomy1. Understand

2. Remember

3. Apply

4. Analyse

5. Evaluate

6. Create

Higher order thinking

Problem solving

Reasoning

Cognitive process

What learners need to do Action verbs

Remember Retrieve relevant information from long-term memory

Recognise, recall, define, describe, identify, list, match, reproduce, select, state

Understand Construct meaning from information and concepts

Paraphrase, interpret, give egs, classify, summarise, infer, compare, discuss, explain, rewrite

Apply Carry out a procedure or use a technique in a given situation.

Change, demonstrate, predict, relate, show how, solve, determine

Analyse Separate information into parts and determine how the parts relate to one another.

Analyse, compare, contrast, organise, distinguish, examine, illustrate, point out, relate, explain, differentiate, organise, attribute

Evaluate Make judgements based on criteria and/or standards.

Comment on, check, criticise, judge, critique, discriminate, justify, interpret, support

Create Put elements together to form a coherent whole, or recognise elements into a new pattern

Combine, design, plan, rearrange, reconstruct, rewrite, generate, produce

Thinkers Bills et al. (2004)

Give an example of … (another and another)

Open-ended

Explain or justify

Similarities and differences

Always, sometimes or never true

Odd-One-Out

Generalise

Hard and easy

Approaches to teaching problem solving …

The approach …The outcome

…

Teaching for problem solving - knowledge, skills and understanding (the mathematics)

Teaching about problem solving - heuristics and behaviours (the strategies and processes)

Teaching through problem solving - posing questions and investigations as key to learning new mathematics (beginning a unit of work with a problem the students cannot do yet)



…

Teaching for problem solving - knowledge, skills and understanding (the mathematics)

Problems at the end of the

chapter!





…

Teaching for problem solving - knowledge, skills and understanding

Problems at the end of the chapter!


Problems used to ‘practise’ strategies and checklists




…

Teaching for problem solving - knowledge, skills and understanding

Problems at the end of the chapter!


Problems used to ‘practise’ strategies and checklists


Some success but limited implementation

Successful problem solving requires

Skills and Attributes

General reasoningabilities

Deep mathematicalknowledge

Heuristicstrategies

Personal attributeseg confidence,

persistence,organisation

Communicationskills

Helpful beliefseg orientation to ask

questions

Abilities to workwith otherseffectively

Stacey, 2005

Which tasks or problems?

Types of problems???Open-endedRich tasksReal-world problemChallengeInvestigationInquiryProblem-basedReflective inquiry

Which tasks or problems?

Content specific questions requiring a range of levels of thinking

Area and Perimeter in Year 5/6

Which shape has the largest perimeter?

Please explain your thinking.

Design a new shape with 12 squares which has the longest possible perimeter.



Communicationskills

Heuristicstrategies

Which card is better value?

Please explain your thinking.



Communicationskills

Heuristicstrategies

NumberandAlgebra

1. Make up an equation where the answer is x = 2

2. Make up an equation where the answer is x = 3

3. Make up an equation where ….

Another idea:

Change one number in the equation

4 x – 3 = 9,

so that the answer is x = 2.

NumberandAlgebraDeep mathematical

knowledge


Communicationskills

Helpful beliefseg orientation to ask

questions


Number and Algebra

Explain the difference between particular pairs of algebraic expressions, such as and

Compare similarities and differences between sets of linear relationships, eg.

x 2

2x

y 3x, y 3x 2, y 3x 2

Number and Algebra: Fractions

Explain why is less than

Explain why

1

4

1

8

2

31

43

7



Communicationskills


Informal and Formal Proof

Constructive alignment(Biggs, 2004)

Curriculum

Instruction

Assessment

Planning for Implementation(including Problem Solving and Reasoning)

• Identify the topic (mathematical concepts)

• Examine curriculum content statements

• Use data to inform decisions on emphasis

• Select, then sequence, appropriate tasks/activities

• Identify the mathematical actions (proficiencies) in which you want students to engage

• Design assessment for ALL proficiencies

Favourite SourcesMCTP (Maths 300 through www.curriculum.edu.au)

Bills, C., Bills, L., Watson, A., & Mason, J. (2004). Thinkers. Derby, UK: ATM.

Downton, A., Knight, R., Clarke, D., & Lewis, G. (2006). Mathematics assessment for learning: Rich tasks and work samples. Fitzroy, Vic.: ACU National.

Lovitt, C., & Lowe, I. (1993). Chance and data. Melbourne: Curriculum Corporation.

Sullivan, P., & Lilburn, P. (2000). Open-ended maths activities. Melbourne, Vic: Oxford.

Swan, P. (2002). Maths investigations. Sydney: RIC Publications.

Resources:MCTP (Maths 300) – Curriculum

Corporation website http://www.curriculum.edu.au

ABS – http://www.abs.gov.au

NCTM – http://www.nctm.org

NRICH website – http://nrich.maths.org.uk/primary

Others???

Key messages …

1. Balance is important

2. Evaluate the types of questions and tasks used during mathematics lessons

3. Assessment, assessment, assessment!!!

4. Alignment between curriculum, teaching and assessment

Documents

Judy Anderson The University of Sydney [email protected]