Working mathematically sits at the heart of the New South Wales syllabus just as the proficiencies sit at the heart of the Australian Curriculum: Mathematics. As Sullivan explains, ‘the use of the proficiencies can be described as the verbs of the curriculum that go with the nouns that are the content’ (Sullivan, https://youtu.be/bkDqWrf_WVY). The curriculum and syllabus are designed to integrate both the nouns and the verbs, embedding content within the behaviours required to become numerate.
Whilst the components of Working Mathematically are often discussed separately (often for ease of communication) they are inextricable connected.
- Communicating refers to the ability to describe mathematical situations using language, symbols, diagrams, enactment, materials, etc. so that it makes sense to others and helps clarify understanding. It involves both the expressive and receptive components of communicating;
- Problem solving refers to ‘the ability to formulate, represent and solve mathematical problems’ (Sullivan, 2007, p. 7). It requires students to work with tasks where the answer is not immediately evident to them. As such, solving problems requires an acceptance of uncertainty and discomfort as students from a place of not knowing to being able to offer a reasonable response;
- Reasoning is essential to mathematics and as Sullivan argues, ‘it is what doing mathematics is all about’ (Sullivan, https://youtu.be/bkDqWrf_WVY). Described as the ‘capacity for logical thought, reflection, explanation and justification’ (Sullivan, 2008, p.7), reasoning is also considered ‘the 'glue' which helps mathematics makes sense’ (NRich, https://nrich.maths.org/10990). Reasoning involves actions such as sorting, comparing, contrasting, explaining, generalising, convincing, forming, testing, validating and justifying;
- Understanding refers to conceptual understanding and ‘includes the comprehension of mathematical concepts, operations and relations’ (Sullivan, 2008, p.6); and
- Fluency is generally considered to involve flexibility, efficiency and accuracy in the application of mathematical skills and knowledge.
An additional component of numeracy and working mathematically is a productive disposition. Described as ‘a habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy (Watson and Sullivan, 2008, p.), positive dispositions can be fostered (Sullivan 2008). In general, our mindset can be fixed or focussed on growth (Dweck, 2006). Mindsets, like the brain, are malleable and it is possible for a person to have a fixed mindset in one context yet a growth mindset in another. Under the right conditions, everybody’s mindset can be nurtured and when they are, achievement in mathematics has been shown to improve (Boaler, 2016).