Mathematics is a discipline that is central to all national and state curriculum models. It is one of the core subjects studied by students across the world, and an understanding of mathematics is considered an essential skill for success beyond school. Mathematics is broadly acknowledged as playing an essential role in our lives and its importance in modern times is amplified rather than diminished by our ever-increasing reliance on technology. As is the norm across education, questions are being debated about the nature of mathematical learning and how we might ensure our students leave school with the skills, knowledge and dispositions they need. What does it mean to be a mathematical thinker? What mathematical skills and knowledge will our students require in the lives they are likely to lead? How do we increase mathematical confidence? Which pedagogies are most appropriate if our goal is mathematical understanding? What place do calculation and basic skills occupy? How do we increase participation rates of all students in advanced mathematics courses, particularly groups which are currently underrepresented?
The response to this is a growing body of research that describes an emerging approach to mathematics. This contemporary research is reflected in the designs of the modern mathematical curriculum. In Australia both the Australian Curriculum and the New South Wales curriculum place emphasis on the ‘doing’ of mathematics.
In Mathematics, the key ideas are the proficiency strands of understanding, fluency, problem-solving and reasoning. The proficiency strands describe the actions in which students can engage when learning and using the content. - Australian Curriculum
The proficiencies reinforce the significance of working mathematically within the content and describe how the content is explored or developed. - Australian Curriculum
Students develop understanding and fluency in mathematics through inquiry, exploring and connecting mathematical concepts, choosing and applying problem-solving skills and mathematical techniques, communication and reasoning. As an essential part of the learning process, Working Mathematically provides students with the opportunity to engage in genuine mathematical activity and develop the skills to become flexible and creative users of mathematics. - New South Wales Educational Standards Authority (NESA)
The five components (communicating, problem solving, reasoning, understanding & fluency) of Working Mathematically describe how content is explored or developed − that is, the thinking and doing of mathematics. - (NESA)
This growing focus on mathematical thinking, reasoning, problem solving and understanding is plainly evident and in response to this teachers are exploring new pedagogical models. The shift away from teaching for the rote memorisation of prescribed methods requires teachers to rethink their approach to the discipline. With this new pedagogy comes a need to understand the processes of mathematical thinking in ways not previously required. When we require our students to be able to reason and problem-solve through unique challenges we also require our teachers to have an understanding of the mathematical moves that their learners are likely to call upon. This is challenging if the teachers only mathematical learning has occurred in classrooms with a strong focus on accurate replication of set methods. A teacher may know that the method for solving a question that requires division with fractions involves inverting one fraction and multiplying, but may not know why this method works.
According to the Australian Curriculum "Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations”, and as the curriculum’s writer, Peter Sullivan shares 'Students are not solving problems if they have been told what to do’. Peter Sullivan adds that 'students are not reasoning if they are merely repeating an argument developed by someone else – the reasoning needs to be their own’. This presents a real challenge to the average primary teacher and requires a shift not only pedagogy but access to professional development that addresses potential gaps in their mathematical understanding. Even to those who may be described as experts might have achieved this status through their mastery of ‘mathematical tricks’ or ‘shortcuts’ that allow a solution to be quickly determined but obfuscate why and how the method works.
There are also significant questions to be addressed here about the philosophical beliefs which underlie our approach to mathematics. If we believe that mastery of fundamental basic knowledge is required before students have opportunities to engage in problem-solving we will prioritise this learning above all else. If we require students to demonstrate accurate and rapid recall of number facts and in particular times tables we will dedicate time and energy to achieving this capacity and in doing so send strong messages to our students about what mathematical knowledge is and what the doing of mathematics involves. When we do introduce our students to mathematical thinking, understanding, reasoning and problem-solving, they are likely to approach it with a mindset that limits their capacity to learn. In place of a willingness to embrace challenges and engage with uncertainty, they are likely to come to mathematics with a belief that speed and the accurate application of a prescribed method is what the subject is all about. When we only experience problem-solving as an extension task at the end of a unit of learning it should not be surprising that student and teacher view it as a black-art that exists slightly beyond what they know or even what they might want to know.
Learning mathematics creates opportunities for and enriches the lives of all Australians. . . The curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. - Australian Curriculum
The study of mathematics provides opportunities for students to appreciate the elegance and power of mathematical reasoning and to apply mathematical understanding creatively and efficiently. The study of the subject enables students to develop a positive self-concept as learners of mathematics, obtain enjoyment from mathematics, and become self-motivated learners through inquiry and active participation in challenging and engaging experiences. - NESA
It is enlightening to read the rationale and aims of these two curriculum documents. They describe mathematics as a beautiful and empowering discipline full of opportunities for creative and original thinking that leads to generalisations and abstractions. It is a discipline founded on essential dispositions such as problem-solving, communicating, understanding, representing and reasoning. It is perhaps a shame that the online presentation of both curriculums make it easy for the aim and rationale to be skipped over by teachers looking for the content they are to teach. The thinking behind these curriculums is not unique to Australia. A quick look at the Common Core Standards from the United States of America reveals a similar emphasis in the identification of essential practices.
"These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections.” Common Core State Standards Initiative
As is the case in Australia, students in America explore content areas such as geometry, number, measurement and data through their engagement with and application of these proficiencies. Mathematical thinking in all cases is built on rather than extending by approaches to mathematical thinking, reasoning and problem solving.
What is now required is a conversation around how we make this thinking accessible to all who are charged with teaching mathematics, particularly those charged with introducing students to the discipline and those whose experience with mathematical thinking might be limited.
By Nigel Coutts