Thinking, reasoning and working mathematically

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MATHEMATICS
Years 1 to 10

• Thinking, reasoning and working mathematically

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Purpose of presentation

• to define thinking, reasoning and working
  mathematically (t, r, w m)
• to describe how t, r, w m enhances mathematical
  learning
• to promote and support t, r, w m through
  investigations.

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Thinking, reasoning and working mathematically

• involves making decisions about what mathematical
  knowledge, procedures and strategies are to be
  used in particular situations
• incorporates communication skills and ways of
  thinking that are mathematical in nature
• is promoted through engagement in challenging
  mathematical investigations.

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Thinking, reasoning and working mathematically
also

• promotes higher-order thinking
• develops deep knowledge and understanding
• develops students confidence in their ability
  to do mathematics
• connects learning to the students real world.

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What is thinking mathematically?

• making meaningful connections with prior
  mathematical experiences and knowledge including
  strategies and procedures
• creating logical pathways to solutions
• identifying what mathematics needs to be known
  and what needs to be done to proceed with an
  investigation
• explaining mathematical ideas and workings.

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What is reasoning mathematically?

• deciding on the mathematical knowledge,
  procedures and strategies to use in a situation
• developing logical pathways to solutions
• reflecting on decisions and making appropriate
  changes to thinking
• making sense of the mathematics encountered
• engaging in mathematical conversations.

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What is working mathematically?

• sharing mathematical ideas
• challenging and defending mathematical thinking
  and reasoning
• solving problems
• using technologies appropriately to support
  mathematical working
• representing mathematical problems and solutions
  in different ways.

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How can t, r, w m be promoted?

• By providing learning opportunities that are
• relevant to the needs, interests and abilities of
  the students
• strongly connected to real-world situations
• based on an investigative approach a problem to
  be solved, a question to be answered, a
  significant task to be completed or an issue to
  be explored.

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Planning for investigations
Select learning outcomes on which to focus
Identify how and when reporting of student
progress will occur
Select strategies to promote consistency of
teacher judgments
Identify how and when judgments will be made
about students demonstrations of learning
Make explicit what students need to know and do
to demonstrate their learning
Identify how evidence of demonstrations of
learning will be gathered and recorded
Choose the context(s) for learning
Identify or design assessment opportunities
Select and sequence learning activities and
teaching strategies
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How do investigations promote t, r, w m?

• Sample investigations present the learning
  sequence in three phases
• identifying and describing
• understanding and applying
• communicating and justifying.
• Each phase promotes the development of thinking,
  reasoning and working mathematically.

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Identifying and describing
Phase 1

• Students
• identify the mathematics in the investigation
• describe the investigation in their own words
• describe the mathematics that may assist them in
  finding solutions
• identify and negotiate possible pathways through
  the investigation
• identify what they need to learn to progress.

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Sample questions to encourage t, r, w m in phase 1

• What mathematics can you see in this situation?
• Have you encountered a similar problem before?
• What mathematics do you already know that will
  help you?
• What procedures or strategies could you use to
  find a solution?
• What do you need to know more about to do this
  investigation?

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Understanding and applying
Phase 2

• Students
• acquire new understandings and knowledge
• select strategies and procedures to apply to the
  investigation
• represent problems using objects, pictures,
  symbols or mathematical models
• apply mathematical knowledge to proceed through
  the investigation
• generate possible solutions
• validate findings by observation, trial or
  experimentation.

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Sample questions to encourage t, r, w m in phase 2

• What types of experiments could you do to test
  your ideas?
• Can you see a pattern in the mathematics? How can
  you use the pattern to help you?
• What other procedures and strategies could you
  use?
• What else do you need to know to resolve the
  investigation?
• Is your solution close to your prediction? If
  not, why is it different?

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Communicating and justifying
Phase 3

• Students
• communicate their solutions or conclusions
• reflect on, and generalise about, their learning
• justify or debate conclusions referring to
  procedures and strategies used
• listen to the perceptions of others and challenge
  or support those ideas
• pose similar investigations or problems.

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Sample questions to encourage t, r, w m in phase 3

• What is the same and what is different about
  other students ideas?
• Will the knowledge, procedures and strategies
  that you used work in similar situations?
• What mathematics do you know now that you didnt
  know before?

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Teachers can support t, r, w m by

• guiding mathematical discussions
• providing opportunities for students to develop
  the knowledge, procedures and strategies required
  for mathematical investigations
• presenting challenges that require students to
  pose problems
• providing opportunities to reflect on new
  learning.

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The syllabus promotes t, r, w m by

• describing the valued attributes of a lifelong
  learner in terms of thinking, reasoning and
  working mathematically
• encouraging students to work through problems to
  be solved, questions to be answered, significant
  tasks to be completed or issues to be explored
• advocating the use of a learner-centred,
  investigative approach in a range of contexts
• emphasising the connections between topics and
  strands that are often required in dealing with
  mathematics in real-life situations.

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Materials to support thinking, reasoning and
working mathematically

• How to think, reason and work mathematically
  (poster)
• About thinking, reasoning and working
  mathematically (information paper)
• Prompting students to think, reason and work
  mathematically (paper)
• Thinking, reasoning and working mathematically in
  the classroom (paper)
• Papers described in the annotated bibliography in
  the Additional information section of the
  support materials

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Contact us

• Queensland Studies Authority
• PO Box 307
• Spring Hill
• Queensland 4004
• Australia
• Phone [phone]
• Fax [phone]
• Visit the QSA website at www.qsa.qld.edu.au