Thinking, reasoning and working mathematically

MATHEMATICS
KLA Years 1 to 10
Thinking, reasoning and
working mathematically
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.
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.
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.
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.
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.
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.
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.
Planning for investigations
Select learning
outcomes on which
to focus
Identify how and when reporting of
student progress will occur
Identify how and when
judgments will be made about
students’ demonstrations of
learning
Select strategies to promote
consistency of teacher
judgments
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
Identify or design
assessment
opportunities
Choose the context(s)
for learning
Select and
sequence learning
activities and
teaching strategies
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.
Phase 1
Identifying and
describing
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.
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?
Phase 2
Understanding and
applying
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.
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?
Phase 3
Communicating and
justifying
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.
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 didn’t know before?
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.
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.
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
Contact us
Queensland Studies Authority
PO Box 307
Spring Hill
Queensland 4004
Australia
Phone: +61 7 3864 0299
Fax: + 61 7 3221 2553
Visit the QSA website at
www.qsa.qld.edu.au