This week I spent three days in Brisbane attending the Australian Association of Mathematics Teachers’ national conference. The theme of the conference was “Why Maths?” and along with 500 other mathematicians, we looked to find inspiring answers to this provocative question beyond the classroom. Here are my key takeaways from this event.

“How to think brilliantly and creatively in maths” was the title of the opening keynote from James Tanton. James is a research mathematician deeply interested in bridging the gap between the mathematics experienced by school students and the creative mathematics practised and explored by mathematicians. Having been a college professor for a decade, James realised that high school mathematics was where he could have the most significant impact. Today he is the Mathematician in Residence at the Mathematical Association of America in Washington D.C. James offered the following strategies for being brilliant at mathematics.

Do Something - replace the fear of not knowing how to start by taking some action, evaluate the results and adjust your strategy accordingly.

Use Visualisations - rather than relegating visualisations to our early years of mathematical learning, we should embrace the approach as a life-long path to brilliantly thinking. Visualisations not only help us to solve problems, they also help us to build deep-understandings and see patterns and connections which otherwise may remain invisible

Work Hard to avoid working hard - one of the best ways to be truly brilliant at mathematics is to look for an easier way to solve a problem. Simple, elegant and beautiful solutions should be our goal. If our thinking relies upon complex methods, maybe we don’t truly understand what we are doing.

Seek the story behind the topic at hand - Thinking in stories is a powerful strategy to understand what is truly going on. When we start to tell the story behind the mathematics, instead of merely looking at the numbers and symbols, we allow ourselves to build a more complete understanding

Got haze - Walk into hazy thinking - There will be times when a confusing haze confronts even the brilliant mathematical thinker. The path forward is unclear, and you will need to confront the unknown. Brilliant mathematicians are comfortable with admitting what they don’t know, but they don’t retreat. Instead they extend their thinking into the haze by building on what they do know.

James is the mathematician behind Exploding Dots, a strategy for visualising and understanding many mathematical concepts, especially in the area of number. Where other methods for explaining place value, the four operations and algebra obfuscate the essential concepts at play, the Exploding Dots method makes this transparent and allows students to develop a true understanding of the mathematics. In a fast-paced fifty minute workshop, James explained the fundamentals of Exploding Dots. For those wanting to explore the possibilities of using Exploding Dots with your class, these websites are the best place to start. G’Day Maths - Exploding Dots

In the image below, we see how Exploding Dots are used to visualise counting with Base 2. The aim is to demonstrate how as the number of dots increase, we show the larger quantity by using the next spot in our place value table. James refers to this as a two into one machine as it takes two dots in any square, explodes them and replaces them with a square in the next one to the left. This sequence shows a single dot in the one column. Then two dots in the one column which in the next image explode and are replaced by a single dot in the twos column. The final image shows what happens when we add a fourth dot. The two dots in the ones column explode, and one dot is added to the twos column. As there are now two dots in the twos column they explode and one dot appears in the fours column. This principle can be extended to any base value and can be used to demonstrate the four operations and algebra.

The value of mathematical reasoning quickly emerged as a common theme for the conference. By asking students to explain their thinking, validate their solutions, test their proofs, solve problems in multiple ways and reframe questions, teachers are including opportunities for students to develop mathematical reasoning. By valuing mathematical reasoning, teachers require their students to think like mathematicians.

Tingalpa State School is supporting mathematical reasoning in many ways. They have created a culture that is tolerant of mistakes and understand that much can be learned by reflecting on the mistakes we make. Their students engage in mathematical reflections through the use of Maths Journals supported by a metalanguage for mathematical understandings. Teachers and students utilise a common set of prompts for mathematical reasoning such as “Convince me...?”, “What stays the same and what changes?” and “Is it just sometimes true, or always true?”. The use of these prompts ensures that the students are thinking throughout their maths lessons and that passive absorption of mathematical methods is banished. These prompts are used alongside visible thinking routines within classrooms which value thinking. This use of a common language for learning, coupled with routines for thinking maximises learning opportunities over time as students engage with new concepts and strategies in a familiar learning environment.

Dr Toh Tin Lam of Singapore’s National Institute of Education shared strategies for developing mathematical investigations. The emphasis once again was on the use of strategies which would require thinking and in particular mathematical reasoning. A mathematical investigation is described by Dr Lam as a task that is open-ended and where the goals are ill-defined. A problem-solving task, by contrast, most likely has a solution and while there may be multiple ways to solve it, the set of possible strategies one might deploy and the mathematical concepts involved are likely to be limited.

One particularly useful strategy for designing a mathematical investigation described by Dr Lam is to explore the common mistakes made by students. He used the example of a common mistake seen when students are investigating fractions. A student may think that the strategy used to simplify the fraction as shown in the example below is valid, particularly as in this example, it results in a correct answer. When students are invited to investigate this further and to test this solution in multiple instances, they begin to understand where they went wrong. Encouraging students to investigate their errors and find an explanation for where their thinking goes wrong seems like a much more effective strategy than returning work full of red crosses and hoping the students correct their strategy before the next assessment.

Libby Foley shared her experience of working in remote regions of Far North Queensland. She reminded us of the importance of building positive, supportive relationships with our students and especially those of Aboriginal or Torres Strait Islander descent. The strong message here is that our pedagogy must always be aligned with the context in which it is practised. Foley’s deep respect for the culture of her students and the community in which her teaching is situated is impressive. There is a range that our faith in Western epistemology as the one path to truth and understand can blind us to the cultural bias of such a view. The wisdom and knowledge of our Indigenous Australians, founded on over 65,000 years of living in harmony with the land cannot be ignored and should not be diminished by cultural elitism.

Cathy Foley, the Chief Scientist for Commonwealth Scientific and Industrial Research Organisation (CSIRO), offered a strong case for rethinking the capacities we focus on in education. The CSIRO has an impressive record in science, and its contribution to our collective understanding should not be overlooked. Foley outlined the work of the CSIRO in response to Australia’s greatest challenges, which could also serve as a starting point for exciting investigations in schools:

Resilient and Valuable Environments: Enhancing the resilience, sustainable use and value of our environments, including by mitigating and adapting the impacts of climate and global change.

Food security and quality: Achieve sustainable regional food security and grow Australia’s share of premium AgriFood markets.

Health and Wellbeing: Help enhance health for all through preventative, personalised, biomedical and digital health services.

Future Industries: Help create Australia’s future industries and jobs by collaborating to boost innovation performance and STEM skills.

Sustainable Energy and Resources: Build regional energy and resource security and our competitiveness while lowering emissions.

A secure Australia and region: Help safeguard Australia from risks (war, terrorism, regional instability, pandemics, biosecurity, disasters and cyber attacks).

In response to these the CSIRO is evolving eight future science platforms each as exciting as the next and all demanding mathematics:

Active Integrated Matter - Creating Industry 5.0

Deep Earth Imaging - Unlocking our resource potential

Digiscape - Digital solutions for the land

Environomics - Environmental genomics to care for biodiversity

Hydrogen Energy Systems - Next generation energy industry

Precision Health - Integrated and tailored health solutions

Probing Biosystems - Innovative medical devices and diagnostic technologies

Synthetic Biology - Artificial engineering of biological systems

This is but a small taste of al the ideas shared over the three days. As is so often the case much of the best learning came from conversations with other educators along the way. What was clear is that mathematics education in Australia is in good hands. We are fortunate to have many teachers with a genuine passion for mathematics who believe that all learners can be successful in their learning. It is also clear that mathematics has a vital role to play in our collective futures. And, Why Maths?, there were many answers shared but perhaps Dr Cathy Foley offered the definitive answer, because it will help us solve the challenges of today and the future.

By Nigel Coutts