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Implementing the Austra
lia
n
Implementing the Australian Curriculum for Mathematics F to 10
Judy AndersonThe University of Sydney
Judy.anderson@sydney.edu.au
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)
Approaches to teaching problem solving …
The approach …The outcome
…
Teaching for problem solving - knowledge, skills and understanding (the mathematics)
Problems at the end of the
chapter!
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)
Approaches to teaching problem solving …
The approach …The outcome
…
Teaching for problem solving - knowledge, skills and understanding
Problems at the end of the chapter!
Teaching about problem solving - heuristics and behaviours (the strategies and processes)
Problems used to ‘practise’ strategies and checklists
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)
Approaches to teaching problem solving …
The approach …The outcome
…
Teaching for problem solving - knowledge, skills and understanding
Problems at the end of the chapter!
Teaching about problem solving - heuristics and behaviours (the strategies and processes)
Problems used to ‘practise’ strategies and checklists
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)
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
Types of problems???Open-endedRich tasksReal-world problemChallengeInvestigationInquiryProblem-basedReflective inquiry
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.
Deep mathematicalknowledge
General reasoningabilities
Communicationskills
Heuristicstrategies
Which card is better value?
Please explain your thinking.
Deep mathematicalknowledge
General reasoningabilities
Communicationskills
Heuristicstrategies
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
General reasoningabilities
Communicationskills
Helpful beliefseg orientation to ask
questions
Abilities to workwith otherseffectively
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
Deep mathematicalknowledge
General reasoningabilities
Communicationskills
Abilities to workwith otherseffectively
Informal and Formal Proof
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???
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