Catholic Schools Office
Diocese of Lismore
DIOCESE OF LISMORE K-12 MATHEMATICS GUIDELINES
Introduction
The purpose of these Guidelines is to ensure a consistency in the pedagogical approach to the quality
teaching and learning of Mathematics within the Lismore diocesan schools.
The Diocesan Mathematics Guidelines reflect current research and the understandings and purposes of
Mathematics as a Key Learning Area. They are underpinned by the Foundational Beliefs and Practices - The
Essential Framework and the Contemporary Learning Framework.
Rationale
Mathematics prepares learners for full participation in our society and has a significant personal and social
value for the wider community.
All learners have the entitlement to develop deep-level understanding and mathematical reasoning.
It provides opportunities to:
“Educate students to be active, thinking citizens, interpreting the
world mathematically, and using mathematics to help form their
predictions and decisions about personal and financial priorities.”
(ACARA 2010a, p-5)
Mathematics allows learners to think critically and creatively in
ways that equip them to employ abstraction and generalisation to
identify, describe and apply patterns and relationships.
“Mathematics has its own value and beauty and it is intended that students will appreciate the elegance and
power of mathematical thinking and experience mathematics as enjoyable”. (ACARA 2010a, p-5)
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Catholic Schools Office
Diocese of Lismore
NSW Mathematics K–10 Syllabus
Mathematics K-10 focuses on developing increasingly sophisticated and refined mathematical understanding,
fluency, logical reasoning, analytical thought and problem-solving skills. These capabilities enable learners to
respond to familiar and unfamiliar situations by employing strategies to make informed decisions and to solve
problems relevant to everyday life and future learning.
The NSW Mathematics K-10 syllabus provides students with knowledge, understanding and skills in Number
and Algebra, Measurement and Geometry, and Statistics and Probability.
There are two key features within the structure of the NSW Mathematics K-10 syllabus
Strands
The NSW Board of Studies Mathematics K-10 syllabus is organised into three content strands and one
process strand. The content becomes more complex from K-6 to 7-10. The content presented in a stage
represents the knowledge, understanding and skills that are to be achieved by a typical student by the end of
that stage. It is acknowledged that students learn at different rates and in different ways, so that there will be
students who have not achieved the outcomes for the previous stage/s before commencing the next stage of
schooling.
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Catholic Schools Office
Diocese of Lismore
Working Mathematically – The process strand
The five interrelated processes of Working Mathematically are central to all Mathematics learning. Students
understand and connect related mathematical concepts: choosing, applying and communicating approaches in
order to investigate and solve problems.
As an essential part of the learning process, the Working Mathematically strand provides students with the
opportunity to engage in genuine mathematical activities and develop the skills to become flexible and creative
users of mathematics.
The Language of Mathematics
The symbolic nature of mathematical language provides learners with a powerful, concise and precise means
of communication.
“The language and literacies of mathematics must be explicitly taught by all teachers of mathematics in
recognition that language can provide a formidable barrier to both the understanding of mathematics concepts
and to providing students access to assessment items aimed at eliciting mathematical understandings.”
(National Review of Numeracy, 2008)
Teachers have a significant role to play in explicit teaching to help all learners deal with the complexities of
language in mathematics. Understanding the nature of language used in mathematics classrooms enables
teachers to support learners to deal with potential difficulties related to mathematical language.
Mathematics and Numeracy
The Mathematics K-10 syllabus makes clear the relationship between the components of Mathematics and
other disciplines. Learners can begin to apply their mathematical knowledge, understanding and skills in a
broad range of contexts beyond the mathematical classroom in core subjects such as Science, Technology,
HSIE, Creative Arts, PDHPE, Languages and English. Opportunities for students to appreciate connections
between mathematical ideas and concepts from other KLA’s are embedded within NSW Mathematics K–10
Syllabus.
Assessment in Mathematics
Assessment is the process of identifying, gathering and interpreting information about students' learning. The
central purpose of assessment is to provide information on student achievement and progress and set the
direction for ongoing teaching and learning. This process is referred to as ‘assessment for learning’ and is
designed to enhance teaching and ultimately improve learning outcomes. Teachers use the information gathered
from ‘assessment of learning’ to summarise and report on student achievement.
Mathematics assessment should allow learners to demonstrate their thinking and ability to work
mathematically, to effectively communicate mathematical ideas, findings and mathematical modelling. The
breadth of expectation in mathematics assessments requires teachers to provide opportunities that enable
learners to demonstrate the full extent of their learning, including application, fluency, thinking, reasoning and
problem solving. Through quality teacher feedback assessment also enables learners to know what quality
mathematics learning is.
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Catholic Schools Office
Diocese of Lismore
Mathematics within the Context of the Contemporary Learning Framework
The Contemporary Learning Framework underpins the vision for learning within the Lismore Diocesan
Schools. In this respect, it is critical that direct connections with the Key Learning Area of Mathematics are
delineated.
Rich Curriculum
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provides deep knowledge and understanding of mathematical content by
developing key concepts with clear and explicit links to prior learning;
recognises that mathematical problems can be solved in a variety of ways
making links to real world contexts;
allows opportunities to articulate mathematical language and symbolism
through the use of metalanguage;
develops explicit mathematical language by participating in substantive
communication of Mathematical ideas and strategies with peers and
teachers;
provides opportunities for students to undertake assessment of learning for
learning through valid, reliable, authentic and differentiated assessment; and
articulates an explicit quality assessment criteria in a language students
understand.
Pedagogy
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utilises higher-order thinking skills by developing mental processes beyond
simple recall of facts, and the manipulation of information to evaluate
understandings;
acknowledges the range of students’ learning styles, prior experiences and
different cultural and social backgrounds;
integrates knowledge by involving meaningful connections with other key
learning areas in mathematical problems;
draws upon a range of multimodal digital and non-digital resources to support
the learner; and
builds opportunities for rich feedback into the teaching and learning cycle.
Engaging and
adaptive environments
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providing a quality mathematics learning environment clearly focused on
pedagogy where high and explicit expectations are articulated;
nurturing positive relationships between students and teachers;
establishing classroom procedures where concepts are explicitly taught,
articulated and continually monitored;
providing students with the structure needed to feel safe and supported and
to maximise learning opportunities;
providing time for consolidation of conceptual knowledge and strategies;
establishing a learning environment where students are provided with support
to feel confident, take risks, share contributions and support their peers in
solving problems; and
providing a learning environment that is rich in relevant stimulus that supports
the learner.
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Catholic Schools Office
Diocese of Lismore
A Culture of Learning
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encouraging enthusiastic and sustained engagement in solving mathematical
tasks;
utilising students’ background knowledge as a building block to form new
learning;
valuing the cultural knowledge of students in linking mathematics learning to
real-life contexts;
creating learning experiences inclusive of the cultural knowledge and social
backgrounds of students;
connecting mathematical problems to the world beyond the mathematics
classroom; and
providing opportunities for collaboration and communication within and
beyond the classroom.
Learning Community
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setting consistent high expectations for students by providing rich, openended tasks accessible to all students;
providing opportunities for students to exercise autonomy and initiative
through self-regulation to allow quality learning to take place;
encouraging students to exercise control over the direction of their learning
through choice of strategies and problems; and
seeking opportunities for the school community to work together to support
and enhance learning opportunities.
Leadership for learning
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A continual
focus on
leadership for
learning
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making available professional learning leading to enhanced learning
outcomes;
promoting professional learning informed by professional standards specific
to the discipline of mathematics;
supporting the development of explicit and articulated learning that
underpins classroom teaching and learning; and
developing strong leadership within the school community to build teacher
capacity.
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Catholic Schools Office
Diocese of Lismore
SIX KEY PRINCIPLES FOR EFFECTIVE TEACHING
OF MATHEMATICS
The following six principles for effective teaching of mathematics
draw upon a synthesis of research recommendations. As a group,
they underpin critical aspects of pedagogy that inform teacher
learning and classroom practice.
Principle 1: Articulating Goals
Identify key ideas that underpin the concepts you are seeking to teach, communicate to the students
that these are the goals of the teaching and explain to them what you hope they will learn.
Principle 2: Making Connections
Build on what students know, mathematically and experientially, including creating and connecting
students with stories that both contextualise and establish a rationale for the learning.
Principle 3: Fostering Engagement
Engage students by utilising a variety of rich and challenging tasks that allow them time and
opportunities to make decisions regarding ways of demonstrating their learning.
Principle 4: Differentiating Challenges
Interact with students while they engage in the experiences; encourage students to interact with each
other, including asking and answering questions and specifically plan to both support and challenge
students at their particular stage of learning.
Principle 5: Structuring Lessons
Adopt pedagogies that foster communication and devolve individual and group responsibilities; use
students’ reports to the class as learning opportunities with teacher summaries of key ideas.
Principle 6: Promoting Fluency and Transfer
Flexibility and fluency is important and it can be developed in two ways: 1) by short everyday practice
of mental processes and 2) the practice, reinforcement and transfer of learnt skills.
Sullivan Peter, Australian Education Review, Teaching Mathematics: Using research-informed
strategies pg. 24-30.
Critical Aspects of Mathematics
To ensure a common language and understanding of the critical aspects of a mathematics session (K6) and lesson guide (7-12), the following diocesan planners have been developed. They indicate the
sequential and differentiated nature of learning and teaching to ensure maximum opportunity for the
development of conceptual knowledge and understanding. These planners reflect the language and
practices outlined in the Contemporary Learning Framework and promote quality Mathematical learning
and teaching across the diocese.
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Catholic Schools Office
Diocese of Lismore
K-6 Mathematics Sequence
Mathematical
Reflection and
Review
Learning
Concept
Development
Comprehension
Introduction
Development
of Number
Sense
Links to CLF
Mental
Computation and
Procedural
Fluency
The development of mental computation and procedural fluency of known
facts and strategies. Number Sense is the ability to understand and use
numbers flexibly. The emphasis should be on the areas of Whole Number
and Patterns & Algebra.
Rigorous
Metalanguage
Background
Review
Identify the focus. What do the students know about this focus area?
Include the use of higher order questions.
Critically Engaging
Build Upon Knowledge
Higher Order thinking
Learning Goal
WILF (What I’m Looking For)
WALT (We are Learning to…)
Explicitly outlining to the students the articulated learning goal/s of the
lesson and the aims/expectations from the Syllabus in the language
students can understand.
Explicit and
Articulated Learning
Goals
Set High Expectations
Metalanguage
Focus on a Question
connected to the main
focus of the lesson
Teachers should encourage students to think about the question deeply,
identify distractors, common misconceptions, identify the calculation
pathway to all options, change the question slightly to give a different
answer, write a similar question and change the numbers/words in the
question. Questions should be presented in a variety of ways fostering the
development of metalanguage.
Exploration
Shared Critical Thinking
Higher Order thinking
Metalanguage
The explicit modelling, demonstrating using hands-on materials, checking
for understanding, questioning, practising together, explaining the
terminology and vocabulary and providing opportunities to the students to
use the terminology and language. Feedback from previous related
sessions should be utilised.
Planned
Based on
Assessment of
Learning for Learning
Metalanguage
Rich Feedback
Guided Teaching
Purposeful, meaningful and relevant activities are selected generally at
three levels, e.g. for students who have:
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a basic knowledge and understanding of the content and have
achieved a limited level of competence in the processes and skills.
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a sound knowledge and understanding of the main areas of content
and have achieved an adequate level of competence in the
processes and skills
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a thorough knowledge and understanding of the content and a high
level of competence in the processes and skills.
Activities can be completely different or variations of the one activity
(differentiation). Activities are not rotated, but repeated each day by the
selected groups while this key idea remains the focus. Groups can be
pairs or larger and any number of groups can work on the same activity.
Collaboration and
communication
Personalised
Multimodal
Ensures inclusivity
Rich Feedback
Connection to Real
world
Independent
Personalised learning activities to suit the different ability levels of the
range of students. Short, sharp focused activity. A few well-chosen tasks
rather than a whole page of activities that can be utilised as an assessment
opportunity.
Flexible and
Negotiated
Quality Assessment
Rich Feedback
Talking About
Learning
(WILF/WALT)
Students reflect on their learning and achievement of outcomes through:
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timely feedback from teachers and peers
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writing/drawing a reflective journal outlining their learning/new
knowledge. (Teacher/Aide may also scribe for younger students.)
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discussing, questioning and clarifying key ideas.
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recounting to the whole class the activity they were engaged in.
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writing an explanation of strategies used/new knowledge to
parents/carers.
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using a graphic organiser to demonstrate thinking.
Collaboration
Evaluated
Rich Feedback
Metalanguage
Main Focus
+ Vocabulary
Modelled/Shared
Teaching
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Catholic Schools Office
Diocese of Lismore
7-12 Mathematics Lesson Guide
Concept
Development
Introduction
Links to CLF
Background
Review
Learning Goal
Main Focus
+ Vocabulary
Modelled/Shared
Teaching
Learning
Guided Teaching
Independent
Consolidation
and Deliberate
Practice
Reflection and Review
Immediate engagement to embed learning by reviewing prior learning.
Rich feedback within 24 hours is critical to inform the Teaching and
Learning cycle. The aim is to identify concepts, skills and strategies that
need reinforcement or consolidation. Focus on identifying students’ prior
knowledge as basis for curriculum differentiation.
Rich feedback
Background knowledge
Assessment for
learning
Start lesson with the “End in Mind”1. Explicitly communicate learning goals
to the students with emphasis on high expectations. Clearly outline to the
students the outcome/s of the lesson and the aims/expectations from the
Syllabus in language that students can understand. Target questions to
underpin the lesson and address key issues noted from the students’
discussion around prior learning.
Communication of
Mathematical ideas
and strategies
Explicit and
articulated learning
goals
High expectations
Explicit teaching (teacher or student modelling) to demonstrate the
concepts and strategies that are critical to achieve the lesson outcomes.
Demonstrate using appropriate pedagogy including, using hands-on
materials, checking for understanding, questioning, introducing and
explaining the terminology and vocabulary that will be used. The use of
appropriate representations and mathematical language are key elements
during this modelled /shared teaching time.
Utilise higher order
thinking skills
Integrating knowledge
Multimodal resources
Metalanguage
A collaborative approach allows students to discuss, explore and engage
in the set tasks. The tasks should be differentiated as a result of preassessment including the discussions in the Background Review.
Activities are selected generally at three levels. For students who have:

a basic knowledge and understanding of the content and have
achieved a limited level of competence in the processes and skills.

a sound knowledge and understanding of the main areas of content
and have achieved an adequate level of competence in the
processes and skills

a thorough knowledge and understanding of the content and a high
level of competence in the processes and skills.
Collaboration
Engagement
Ensure inclusivity
Independent and personalised activities to suit the different ability levels of
the range of students. Short, sharp focused activity that can be utilised as
an assessment opportunity. Provide meaningful and timely feedback.
Deep knowledge and
understanding
Rich Feedback
Opportunities are given to allow students to master the goal. Reengagement within a 24 hour period is crucial to maximise opportunities to
make concepts, strategies and procedures secure for each student.
Rigorous
Students to summarise and wrap up based on 3 criteria2:
1. Do students understand the problem?
2. Have they got a strategy to solve it?
3. Are they able to describe how they reached the solution?
Students may display their thinking by using a graphic organiser or in
written form, recording their experiences .Through this, students reflect and
evaluate their achievement of outcomes.
Teachers provide formative feedback.
1
Harvey Silver ( Author “ The Thoughtful Classroom”) – Backward Design
2
Anne Davies (Author “Knowing What Counts”) – What Counts in Solving a Maths Problem
Personalised
Self-regulation
Positive and
supportive
Reflection
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Catholic Schools Office
Diocese of Lismore
References
AAMT (2006) Standards for Excellence in Teaching Mathematics in Australian Schools.
AAMT (2008) School mathematics for the twenty-first century: Some key influences
<http://www.aamt.edu.au/content/download/8004/102828/file/21C_inf.pdf> Downloaded July 2012.
ACARA Australian Curriculum, Assessment and Reporting Authority. Mathematics curriculum (2012)
Downloaded July 2012. <http://www.australiancurriculum.edu.au/Mathematics/Rationale>.
Catholic Schools Office, Diocese of Lismore, 2012. Contemporary Learning Framework.
MCEETYA (2008) Melbourne Declaration on Educational Goals for Young Australians.
<http://www.mceetya.edu.au/mceetya/melbourne_declaration,25979.html> Downloaded July 2012.
NSW Board of Studies (2012) Mathematics syllabus K–10.
Pegg, J, Lynch, T and Panizzon, D. Exceptional Outcomes in Mathematics Education. Post Pressed,
Queensland 2007.
Stanley, G. et al. (2008). Numeracy Review Report. Canberra: DEEWR
<http://www.coag.gov.au/reports/docs/national_numeracy_review.pdf> Downloaded July 2012.
Sullivan, Peter. Six Key Principles for Effective Teaching of Mathematics, Australian Education Review
Teaching Mathematics: Using Research-informed strategies, pages 24-30.
The Australian Association of Mathematics Teachers. 2008. Position paper The practice of assessing
mathematics learning.
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