1. Thinking, reasoning and working mathematically
MATHEMATICS
Years 1 to 10
Thinking, reasoning and working mathematically

2. 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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3. 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.
For more information, refer to the paper About thinking, reasoning and working mathematically in the support materials and on the QSA website.

4. 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.
When students have the opportunity to think, reason and work mathematically, they come to understand how they learn mathematics and draw on what they know and can do to investigate new ideas and situations. Through engagement in investigations, they become self-directed, independent learners who collaborate with one another, and plan, organise, evaluate and manage their thinking and reasoning.
Students who are given opportunities to communicate mathematical ideas develop deeper deep understandings of the Mathematics key learning area.
Learning experiences that foster enjoyment of mathematics and develop students’ confidence in dealing with mathematical situations should be provided at all levels and across all strands. Such experiences promote positive dispositions towards mathematical learning.
Connecting mathematics learning to the real-world and students’ prior experiences enables them to see the purpose of their learning and its application in everyday situations.

5. 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.
Students are also thinking mathematically when they have opportunities to:
draw on knowledge, procedures and strategies from a wide variety of mathematical topics
approach problems from different mathematical perspectives
pose problems.

6. 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.
Students’ engagement with Mathematics in the classroom should help them develop confidence in their ability to reason and to justify their thinking. If students are to develop reasoning skills, they need the freedom to explore their mathematical ideas, to conjecture, take risks and test and evaluate their ideas in new situations.
Students also need opportunities to:
use models, facts, patterns and relationships to draw conclusions and explain their thinking
use patterns and relationships to analyse mathematical situations
justify solutions, procedures and strategies
use logical reasoning
make, refine and explore conjectures referring to evidence collected
use a variety of reasoning and proof techniques to confirm or disprove those conjectures.

7. 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.
Students are working mathematically when they:
plan and implement procedures to conduct mathematical investigations
communicate procedures and strategies using clear and precise mathematical language
determine and apply correct mathematical procedures and strategies
develop knowledge of mathematical facts
apply what has been learned to solve real problems
tackle non-routine problems systematically
develop their abilities to measure, estimate and make sensible use of calculators and computers
represent mathematical situations using objects, pictures, symbols or mathematical models until they find methods that enable them to proceed with an investigation
work productively and reflectively and take responsibility for their mathematical decisions.

8. 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.
Contexts for learning should reflect the specific needs, interests and abilities of students in the class for which the investigations or units of work are planned (learning styles, special needs, target groups, previous experiences and prior learnings).
Learning should have value and meaning beyond the classroom. The mathematics students encounter in school must connect to their real-world experiences and interests as these provide them with the knowledge on which to build.
Framing an investigation in terms of a problem to be solved, a question to be answered, a significant task to be completed or an issue to be resolved provides a focus for learning and defines the parameters for the work to be done.

9. 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
This planning model shows one approach to planning investigations using learning outcomes. It highlights the dynamic and cyclic nature of the planning process and the integrated nature of planning for learning, teaching, assessment and reporting.
The highlighted text in the planning model draws attention to those steps in the model that promote thinking, reasoning and working mathematically. When developing an investigation in mathematics, a context is chosen that is broad enough to provide the opportunity for core learning outcomes to be demonstrated and core content to be developed. The context is then framed in terms of a problem to be solved, a question to be answered, a significant task to be completed or an issue to be explored.
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

10. 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.
This investigative approach is similar to inquiry approaches in other key learning areas. Investigations in mathematics:
stimulate students’ curiosity
illustrate connections between Mathematics and the world outside the classroom
respond to and cater for the needs of individual students as investigations can be planned across levels
promote student ownership of Mathematics learning.

11. Identifying and describing
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.
For example, in a situation where students are given opportunities to investigate ways of collecting information about their local area for presentation to an international exchange student, they may identify and describe:
what is required in the investigation
what information would be relevant to a visitor
categories for collecting data
proposed data collection methods
where they would begin (e.g. planning the data collection process) and what information they need to proceed with the investigation (e.g. how to represent the local area as a map).

12. 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?
In each of the three phases of an investigation, teachers need to ask questions that promote, challenge and extend students’ thinking, reasoning and working mathematically.
For further examples of questions, refer to the paper Prompting students to think, reason and work mathematically in the support materials and on the QSA website.
Refer to the poster How to think, reason and work mathematically to guide students through the three phases of an investigation.

13. Understanding and applying
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.
Students think, reason and work mathematically as they develop deeper understandings of the knowledge, procedures and strategies required to proceed with an investigation. They combine the new knowledge, procedures and strategies with their existing knowledge to resolve the investigation. For example, in an investigation to design and build display boards, students are understanding and applying as they:
either independently or with assistance from the teacher, learn about the relationship between length, width and perimeter, research the cost of framing materials and develop computation strategies to calculate the most economical way of buying materials
select procedures that will assist them to find the cost of constructing the boards. They may choose to find the total cost of materials and divide by the number of boards made to establish the cost of one board or they may choose to find the cost of the individual items required for one board and then total the costs
construct a paper model to represent the actual size of a finished board as well as show the working used to calculate costs
apply knowledge of addition, subtraction, multiplication, division, measurement of length, relationships between length of sides and area and perimeter
generate conclusions about the quantities of materials required for members of a class to construct one board each, the cost and the potential profit.
In the development of mathematical thinking and reasoning, it is important that students are encouraged to check and verify the use of procedures and strategies in a range of familiar and unfamiliar situations.
Students develop skills to select the most appropriate procedures, strategies or models for a particular situation (e.g. the use of a measurement formula to calculate the amount of material required for framing display boards or the procedure for calculating the costs of materials).

14. 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?
For further examples of questions, refer to the paper Prompting students to think, reason and work mathematically in the support materials and on the QSA website.

15. Communicating and justifying
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.
Students:
are given the opportunity to share their conclusions and the knowledge, procedures and strategies they used to arrive at their conclusions. The communication could be written, oral or incorporate the use of technology. It could be addressed to audiences such as small groups of peers, whole-class groups, a teacher or members of the community
develop the habit of writing down their thoughts, results, conjectures, arguments, proofs, questions and opinions about the mathematics used. They learn that a mathematical generalisation is something that is always true
use their written algorithms, formulae, resources or explanations of their procedures and strategies as evidence to support their conclusions and challenge the conclusions of others
convince their classmates that a particular result is true or plausible by giving precise descriptions of good evidence including proofs, procedures and strategies used. They could also use their learning to support or challenge the ideas of others. As they proceed through their schooling, students’ understandings of what constitutes a ‘good’ proof should become more sophisticated
extend their problem-solving abilities by posing a problem similar to the one they have solved (e.g. How would using a triangular shape for display boards change the results of the investigation?).

16. 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?
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17. 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.
Teachers facilitate discussions by asking questions that require students to think mathematically rather than recall facts and by providing opportunities for students to discuss aspects of an investigation with their peers.
Focused teaching that occurs within the context of an investigation provides students with knowledge, procedures and strategies from which they may select as they progress through the investigation.
Developing students’ abilities to pose problems is a vital component in helping them to understand problem structures. Students may pose problems at different stages of an investigation. For example:
students may pose a problem as a way of approaching a task to be completed, such as ‘If an international student were visiting our school, how could I use mathematics to help me prepare them for the visit?’
when trying to establish a pathway through an investigation, a student may think of a related, more accessible problem or reword a question using more familiar language
after a conclusion has been reached a student may pose an alternative related problem, such as ‘Would the results be the same if I surveyed Year 12 students instead of Year 9 students?’
after an investigation has been completed, students may ask ‘In what other situations could I apply the knowledge, procedures and strategies that I used in this investigation?’
Reflecting on their learning encourages students to actively think about past or current learning in order to improve future performances and helps them become self-directed learners. Teachers need to explicitly encourage reflection as students are unlikely to engage in reflections independently.
For more information about the teacher’s role, refer to the About thinking, reasoning and working mathematically paper in the support materials and on the Queensland Studies Authority website.

18. 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.
These ideas can be found in the syllabus rationale and in the information about planning.
Support materials such as sample investigations illustrate how thinking, reasoning and working mathematically can be promoted in the classroom.

19. 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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20. Contact us Visit the QSA website at www.qsa.qld.edu.au
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