Along with my colleague, Cath Pearn, we were honoured to be the first presenters to work with a cohort of teachers at the new state-of-the-art Orbis facility. It was great to have the opportunity to work with some of the many committed reception to year 3 educators in South Australia. The aim of the 8-day program we are facilitating is to support teachers to develop children’s confidence in understanding and using mathematical concepts in the early years of schooling. The program sets out to support teachers to enhance their teaching expertise by building pedagogical content knowledge and develop their capacity to design numeracy learning that engages all learners. So while the program has sound aims, we would like to use this blog to share some details about what it means for educators to develop content pedagogical understanding and to offer some guidance on this critical area of teaching and learning in mathematics.
Why focus on the maths proficiency strands?
Presently across the world, mathematics curricula are now requiring teachers to teach and assess problem solving, reasoning and critical thinking (ACARA, 2018; CCSS, 2010). As such we have curriculum that advocates that it is no longer enough to require students to ‘do mathematics’. Rather, there is an increasing demand that students should be able to demonstrate an ability to think, behave and communicate mathematically (Boaler, 2008). In the US, the common core standards for mathematics include Standards for Mathematical Practice. These describe varieties of expertise that mathematics educators at all levels should seek to develop in their students (CCSS, 2010). These practices are based on what are considered essential ‘processes and proficiencies’ with longstanding importance in mathematics education. They articulate the level to which students should be able to make sense of, persevere with, reason, argue and critique, model and choose the appropriate tools and strategies when engaging in mathematical activity. In the Australian mathematics curriculum (ACARA, 2018) our version of the Standards for Mathematical Practice takes the form of the ‘proficiency strands’. The proficiency strands focus on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. These proficiencies enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.
In Shape of the Curriculum: Mathematics (2009) developed by the National Curriculum Board in Australia, the proficiency strands were described as ’the ”how” of the way content is explored or developed i.e. the thinking and doing of mathematics’ (p.7). That is, the proficiency strands describe the actions in which students can engage when learning and using mathematical content. There is no doubt the proficiency strands are an important aspect of our mathematics curriculum in Australia. They should guide the way we ask students to interact with the content, which obviously has significant implications for what classroom instruction should look like. With the international shift towards adopting curricula that emphasises the role of critical thinking and communication of mathematical ideas, an instructional shift may also be required (Symons & Dunn, 2019).
With this in mind, we offer some ideas to support teachers to enact the proficiency strands for mathematics in their classrooms.
1. Reasoning: justify their answers and critique the reasoning of others
Enacting classroom discourse and discussion as a practice to have students ‘justify their answers and critique the reasoning of others’, is one approach to enhance mathematical reasoning. Classroom discourse can be used to cultivate students’ ability to engage in productive talk in which they deduce appropriately, infer appropriately, arrive at conclusions, generalise and articulate relationships amongst concepts. Explicitly teaching the language of mathematics and how to incorporate this into mathematical justification is an important element of any classroom. Dr. Duncan Symons and I explored some guidelines for establishing productive classroom talk in a recent article for Australian Primary Mathematics Classroom.
2. Problem solving: make sense of problems and persevere in solving them
In the Australian Curriculum students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable (ACARA, 2019). This might look like a student explaining to themselves the meaning of a problem and looking for entry points to its solution. They make conjectures about the meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They would monitor and evaluate their progress and change course if necessary. Using the South Australian numeracy guidebooks for guidance on setting up positive classroom norms (PDF 267KB) and cultivating ‘productive struggle’ is a great place to begin conversations about setting a climate for mathematical problem solving.
3. Understanding: use multiple representations to develop understanding
To develop conceptual understanding Reys et al (2012) advocates we need to move learning through concrete, pictorial and abstract representations. Using multiple representations that are connected develops mathematical understanding. How long a representation should be used is dependent on both the student and mathematics content. Generally, students are rushed (or dragged) too quickly through first-hand experiences with models and then confronted with symbolic (abstract) representations at the expense of understanding (Reys et al, 2012).
4. Fluency: students are reasonably fast, accurate and flexible
Consider what it means to be fluent in a language. Within this context it usually means there is flow in the way you are able to speak. Being fluent in maths is similar. Being reasonably fast as well as accurate is a key characteristic. The word ’fluent’ is used to mean the ability to use certain facts and procedures with enough competence that using them does not slow down or derail the problem solver as he or she works on more complex problems. Using instructional routines like number talks and numbers strings can be effective ways to develop accurate and flexible use of number knowledge.
So while the Australian Curriculum enables students across the country to have access to consistent content in mathematics, it also gives us the opportunity to a consistent approach to mathematics pedagogy. As outlined, a good starting point for schools to discuss pedagogical content knowledge is to explore practices that enact the proficiency strands and test the effectiveness of these practices in your context.
Looking forward to seeing you at Orbis.
Dr Ryan Dunn
The University of Melbourne