1. Transforming professional practice: Teaching numeracy across the curriculum
This is a research and development project that helped teachers plan and implement numeracy strategies across the school curriculum.
Merrilyn Goos

2. Numeracy means … different things to different people
PISA definition of mathematical literacy:
An individual’s capacity to identify and understand the role mathematics plays in the world, to make well-founded judgments, and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen.
An Australian definition of numeracy (AAMT, 1997):
To be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for participation in community and civic life.
Numeracy is a word used in some countries but elsewhere it is more common to speak of quantitative literacy or mathematical literacy. Internationally the best known definition of what we in Australia call numeracy is the PISA version you see here.
In Australia, we have come to accept this definition (appears in national numeracy policy documents).

3. Distinguishing numeracy from mathematics
Mathematics climbs the ladder of abstraction to see, from sufficient height, common patterns in seemingly different things. Abstraction is what gives mathematics its power; it is what enables methods derived from one context to be applied in others. But abstraction is not the focus of numeracy. Instead, numeracy clings to specifics, marshalling all relevant aspects of setting and context to reach conclusions.
Steen, 2001
In fact it’s context that distinguishes numeracy from mathematics. Steen offered this elegant comment on the relationship between mathematics and numeracy.

4. Numeracy is an across the curriculum commitment
To enable students to become numerate, teachers must encourage them to see and use mathematics in everything they do … Fortunately, because numeracy is ubiquitous, opportunities abound to teach it throughout the curriculum.
(Steen, 2001, p. 18)
Steen went further to say that numeracy should be taught throughout the school curriculum.

5. Numeracy is an across the curriculum commitment
Australian National Numeracy Review (2008) recommended:
That all systems and schools recognise that, while mathematics can be taught in the context of mathematics lessons, the development of numeracy requires experience in the use of mathematics beyond the mathematics classroom, and hence requires an across the curriculum commitment. (p. 7)
The key idea in this project is that numeracy needs to be developed in all subjects, not just in mathematics, because every curriculum area has its own distinctive numeracy demands.

6. Numeracy in the Australian Curriculum
Numeracy is one of 7 General Capabilities to be developed and used by students across all learning areas, in co-curricular programs, and in their lives outside school. In the Australian Curriculum, students become numerate as they develop the knowledge and skills to use mathematics confidently across all learning areas at school and in their lives more broadly. Numeracy involves students in recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully.
The view of numeracy as a capability every student should possess is endorsed by educational policy makers and curriculum authorities.

7. An Australian definition of numeracy
To be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for participation in community and civic life.
(Australian Association of Mathematics Teachers, 1997)
Earlier I referred to this Australian definition of numeracy that was formulated in 1997 and adopted in the government’s numeracy policy.
Is this description good enough for the 21st century?
I’d argue that we need a broader description of numeracy for new times that captures the rapidly evolving nature of knowledge, of work, of technology, of social structures, and the changing characteristics of the learners in our schools. And I’d also argue that any description of numeracy is useless unless it makes sense to teachers and is able to be implemented by teachers.

8. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Contexts
I’ve tried to represent the dynamic and multi faceted nature of numeracy in this model. I originally designed it as a 2 dimensional representation, but people quickly pointed out that this is the net of a tetrahedron, so I call it my tetrahedral model for numeracy.
Mathematical Knowledge

9. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Contexts
Problem Solving
Estimation
Concepts
Skills
You still need mathematical knowledge to be numerate! This includes concepts, skills, and problem solving strategies, as well as the ability to use sensible estimations.
Mathematical Knowledge

10. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Contexts
Problem Solving
Estimation
Concepts
Skills
A numerate person also has positive dispositions - a willingness and confidence to engage with tasks - independently and in collaboration with others - and apply their mathematical knowledge in flexible and adaptable ways.
Mathematical Knowledge

11. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Representational
Physical
Digital
Contexts
Problem Solving
Estimation
Concepts
Skills
Numerate practice often involves using tools. These include:
representational tools like ready reckoners and charts and tables that might be used in a manufacturing context,
physical tools like mathematical drawing or measuring instruments and the work related tools of a trade or profession
digital tools - technology.
Mathematical Knowledge

12. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Representational
Physical
Digital
Contexts
Problem Solving
Estimation
Concepts
Skills
Numeracy is about using mathematics to act in and on the world. People need to be numerate in a range of contexts.
Mathematical Knowledge

13. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Representational
Physical
Digital
Contexts
Personal and Social
Problem Solving
Estimation
Concepts
Skills
A numerate person can organise their personal finances, for example in relation to credit card spending and mobile phone use.
They manage their personal health by making decisions about their eating and exercise habits.
They engage in leisure activities that require numeracy knowledge, such as travel, sport, perhaps gambling.
Mathematical Knowledge

14. Mathematical Knowledge
21st century numeracy
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Representational
Physical
Digital
Contexts
Personal and Social
Problem Solving
Estimation
Concepts
Skills
Work
All kinds of occupations require numeracy. Many examples of work-related numeracy are very specific to the particular work context, and often the mathematics used is either invisible to the user or is used in very different ways from how mathematics is taught at school.
Mathematical Knowledge

15. Mathematical Knowledge
21st century numeracy
Citizenship
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Representational
Physical
Digital
Contexts
Personal and Social
Problem Solving
Estimation
Concepts
Skills
Work
Informed and critical citizens are numerate citizens. Almost every public issue depends on data, projections, and the kind of systematic thinking that’s at the heart of numeracy.
Mathematical Knowledge

16. Mathematical Knowledge
21st century numeracy
Citizenship
Dispositions
Tools
Confidence
Flexibility
Initiative
Risk
Representational
Physical
Digital
Contexts
Personal and Social
Problem Solving
Estimation
Concepts
Skills
Work
Model is grounded in a critical orientation because numerate people need to evaluate the reasonableness of methods used and results obtained, and understand how mathematics can be used appropriately and inappropriately to analyse situations and draw conclusions, and to manipulate, persuade, and shape opinions about social and political issues.
Mathematical Knowledge
Critical Orientation

17. Research study
A one year action research study that investigated approaches to help teachers plan and implement numeracy strategies across all curriculum areas in Grades 6-9
20 teachers in 10 schools – primary school teachers, secondary mathematics teachers, secondary non-mathematics teachers
To what extent did teachers’ classroom practice change over time as they engaged with the numeracy model?
How effective was the professional development approach in building teachers’ confidence in numeracy teaching?

18. Research design Time Activity Data sources March
Teacher meeting (1 day): introduce numeracy model; try out numeracy teaching strategies and tasks; plan for implementation
Surveys
numeracy teaching confidence
understanding of numeracy
June
School visits (1 day per school): observe and evaluate implementation
Lesson observations, interviews, teaching materials
August
Teacher meeting (1 day): evaluate implementation; share teaching resources and strategies; plan for implementation
October
November
Teacher meeting (1 day): evaluate implementation; reflect on professional learning
Map trajectory through the numeracy model

19. Mathematical Knowledge
Changes in classroom practice: Teachers’ trajectories through the numeracy model
Dispositions
Tools
Contexts
We asked teachers to annotate a copy of the model to show what their main concern was when they joined the project, and how they believed they successively explored and incorporated other elements of the model into their planning and teaching.
Most teachers started with a concern for improving their students’ dispositions or their mathematical knowledge, and then moved from D to K or K to D.
Some teachers linked use of tools with developing mathematical knowledge.
Teachers then moved from dispositions to contexts or [knowledge-tools] to contexts.
Only four teachers indicated that they considered critical orientation – and this was their end point.
Mathematical Knowledge
Critical Orientation

20. Changes in classroom practice: Case study
First school visit: Investigating newspapers
What is the percentage of different forms of “news” in a newspaper? (e.g., sport, local news, world news, feature articles, weather)
Karen: in her second year of teaching. Taught Grads 8-10 science, English, mathematics, society and environment
A Grade 8 English class.
Students were to determine the percentage of different forms of “news” in a newspaper; that is, the component elements of a newspaper in terms of types of reports (sports, local news, world news, special interest, weather) and to represent this data using a graph. This would be the basis for students developing their own newspapers and thinking about the balance of material in a newspaper.
Karen was unhappy with this initial unit because she felt the balance between the English curriculum content and the numeracy elements was inappropriate.

21. Changes in classroom practice: Case study
Year 8 Society and Environment: Building an Expressway
What is the best route for a new expressway to be built between Whyalla and Adelaide?
Students provided with maps (1cm square grid) and grid references for start and end points.
Route must only cross one river.
Route must cross Gawler-One Tree Hill Road then rejoin Main North Road.
Bends must be equal to or greater than 110 degrees.
For the second action research cycle she drew on her experience as the winner of an Industry Award, where she spent two weeks observing how mathematics and science were used in a civil engineering company. She noted the type of projects the company was engaged in, for example, building bridges, making roads and studying environmental impact of projects. She also noticed the way mathematics was embedded in the work life of the company.
In Karen’s second numeracy unit, students were to find the best route for a new expressway to be built between their town and the State capital city, about 100 kilometers away.
Must not pass nearer than one grid square to a quarry.
Residents to be compensated at $150,000 per grid square for any land resumed.
Crossing Uleybury vineyard to be kept to a minimum because compensation to residents is $500,000 per grid square.
DESIGN FOR MINIMUM COST.
USE A LENGTH OF WOOLLEN YARN TO MARK OUT THE ROUTE.

22. Building an expressway
Dispositions
Tools
Confidence
Flexibility
Authenticity
Protractors
Rulers
Maps
Contexts
Engineering
NIMBY
Measurement
Estimation
Number
Make decisions about cost vs most direct route
Mathematical Knowledge
Critical Orientation

23. Changes in classroom practice: Case study
“I felt that my involvement in the project has changed who I am, both professionally and personally.”
During the initial project meeting, where the model was described for what it was to be numerate, exemplar activities were provided that helped me with knowing about numeracy. Returning to school and trying out initial ideas was part of me doing in relation to numeracy. Eventually, though, the continued interaction of my developing knowing and doing led to my present state where my approach to teaching numeracy had become part of my being.

24. Changes in numeracy teaching confidence
A survey was designed, based on Queensland’s Numeracy Standards for Graduates of Pre-Service Teacher Education Programs.
The Numeracy Standards address three domains:
Professional knowledge: knowledge of students, of numeracy, of students’ numeracy learning
Professional attributes: personal attributes, personal professional development, community responsibility
Professional practice: learning environment, planning, teaching, assessment
32 survey items, How confident are you that you can …
Likert scale 1-5
Items referred to numeracy teaching standards developed in Qld (professional knowledge, professional attributes, professional practice)
Substantial rise in confidence for 16/32 items (shift in total scores equivalent to half the group changing their confidence level by 1 point on Likert scale).

25. Changes in numeracy teaching confidence
Pre
High
Low
Post
Recognising the diversity of students’ numeracy learning needs.
Understanding the nature of numeracy and its relevance to all curriculum areas.
Understanding how students learn mathematics.
Planning for numeracy learning and assessment.
Using a range of effective numeracy teaching and assessment strategies.
Using multiple representations and digital technologies to enhance students’ numeracy learning.
Fostering risk taking and critical inquiry in numeracy learning.
Catering for the diversity of students’ numeracy learning needs.
At the start of the project, teachers felt confident they possessed the personal attributes and commitment to professional learning required for numeracy teaching. They also had confidence in some aspects of their professional knowledge (knowledge of the diversity of students’ numeracy needs, of the pervasive nature of numeracy, of numeracy learning opportunities across the curriculum).
However, they lacked confidence in their ability to establish an appropriate numeracy learning environment, plan for numeracy learning, and demonstrate effective numeracy teaching and assessment strategies.
By the end of the project, these teachers felt confident in almost every aspect of numeracy teaching, apart from their ability to foster risk taking and critical inquiry in numeracy learning and to cater for the diversity of mathematical abilities and numeracy needs of learners

26. Conclusions
We found encouraging evidence of teacher change and development.
Teaching in context is difficult – mathematical knowledge may not transfer easily from one context to another.
Recognising numeracy opportunities as they arise during a lesson is challenging for teachers
The numeracy model can be used by teachers for curriculum planning and by researchers to track teacher development.

27. Transforming professional practice: Teaching numeracy across the curriculum
Merrilyn Goos