As we await the release of a new Australian curriculum for mathematics, debates about its contents are developing. As is typical with educational debates, the issues are often painted in binary terms: traditional vs progressive, explicit teaching vs problem solving, content vs skills, procedural vs conceptual knowledge. In mathematics education, these debates have existed for some time, pitting supporters of explicit teaching of clearly defined content against those who advocate for more opportunities for mathematical problem solving and reasoning. However, in order to produce mathematically-able citizens at one end of the spectrum, and at the other, the mathematicians of the future, we need both. We need students with extensive mathematics knowledge and skills, who can also think flexibly and creatively with that knowledge to be confident problem solvers.
So what do teaching approaches have to do with the new Australian curriculum? A curriculum should outline what is to be taught alongside its purpose and intent. In addition to outlining the mathematics content for each year level, the current Australian curriculum for mathematics includes four proficiency strand: understanding, fluency, problem-solving and reasoning. The strands describe “how content is explored or developed; that is, the thinking and doing of mathematics.” These proficiencies are integrated with the content to ensure that students can work adaptively with mathematics content rather than just be competent users of set procedures. The world that students will inhabit in the future will require them to have strategies to solve complex problems and apply their mathematical content knowledge to unfamiliar situations. Taking on these challenges will require them to be confident and experienced problem solvers. Knowing how to apply mathematical procedures and algorithms only to familiar situations is not sufficient.
Criticisms of learning approaches that enable students to problem solve or to reason mathematically often assume that teachers are trying to teach generic skills in the absence of content. This is not the intent of the Australian mathematics curriculum, where it is expected that the proficiencies are integrated with the mathematics content. Students need sound content knowledge in order to draw on that knowledge to solve problems. They cannot think critically without sound underlying knowledge to think critically about. In mathematics they need both procedural and conceptual knowledge, which develop iteratively alongside one another. Teachers have a role to play in providing quality explicit instruction which takes into account cognitive load theory but students also need time to explore and compare alternative problem solutions and to communicate their mathematical reasoning.
We know that mathematics is a polarising subject. Most people have strong views about their attitudes towards the subject and their capacity to do mathematics. Such views were generally formed as a result of their schooling. Unlike other school subjects, mathematics induces widespread anxiety and negative feelings, often attributed to traditional teaching and assessment practices. These negative attitudes cause many learners to opt out of mathematics in the senior years of schooling and to disengage with the subject from an even earlier age. The TIMSS data tells us that more positive attitudes, in terms of valuing, liking and confidence with mathematics are related to higher achievement. So teaching approaches that support the development of positive attitudes are vitally important so that students don’t start to consider themselves as a ‘non-maths person’, losing interest in mathematics early in secondary school. Traditional explicit teaching, while appropriate some of the time during a teaching sequence, if overused, can alienate some learners, leading to disengagement and a reduction in student confidence leading to a decrease in enrolments in mathematics in senior years of schooling and beyond.
The new Australian Curriculum is an opportunity to create a curriculum that will enable students to develop a deep understanding of and appreciation for mathematics. The curriculum should provide guidance for teachers so that they can teach students in ways that develop understanding and competence but also develop an interest in the subject and a desire to learn more. We want our students to finish school with positive attitudes to mathematics and confidence to use mathematics within their personal and working lives. This will only happen if teachers provide a balanced approach to learning mathematics, with time for students to learn from explicit instruction, but also to apply their creativity and knowledge to unfamiliar situations, building confidence in themselves as learners, preparing them for whatever the future holds.
Professor Kathryn Holmes is the director of the Centre for Educational Research in the School of Education at Western Sydney University. She is also the asssociate dean, research, School of Education
Associate Professor Catherine Attard is the deputy director of the Centre for Educational Research in the School of Education, Western Sydney University. She is also president, Mathematics Education Research Group of Australasia
Tomorrow Emeritus Professor of Educational Psychology at UNSW John Sweller adds to the debate.