What maths teaching could and should learn from cognitive science

In the arena of educational policy, where decisions shape the future of our students, the plight of mathematics education in New Zealand has become a drawn-out match of political football.

As head of the mathematics education unit in the mathematics department at the University of Auckland, I’ve spent the last three years dissecting the factors contributing to the steady decline in mathematical proficiency across our nation’s schools. What I’ve observed is a troubling dissonance between the prevailing educational theories that have shaped policies for decades and the empirically grounded insights offered by cognitive science. The widening disconnect at the academic level has led to what I perceive as an undeclared paradigm war.

In New Zealand, the educational landscape has a multitude of approaches, each rooted in various educational philosophies and theories – project-based learning, inquiry-based learning, and traditional teacher-led instruction to name a few.

As a mother of three children navigating the New Zealand school system, I’ve witnessed the diverse methods used in their classrooms. My children have shown a distinct lack of enthusiasm for the project-based learning environment embraced by their intermediate school. Having spent two years immersed in this approach, they felt they haven’t progressed beyond the primary school curriculum. The teacher’s role seems confined to that of a facilitator – a telltale sign of adherence to a particular educational philosophy. My son recounted his frustrations: “Our teacher doesn’t offer much explanation; she simply assigns problems for us to solve in groups, and every third lesson, she checks in to see if we have any questions.”

While these approaches reflect differing perspectives on education, some of them neglect to fully integrate the findings of cognitive science, which offer invaluable insights into how we learn and retain information.

One damaging consequence of this paradigm war is the false dichotomy between ‘rote learning’ and ‘learning with understanding’. Mathematics education in New Zealand, as well as in other English-speaking nations, has fallen prey to a misguided notion: that prioritising the memorisation of key mathematical facts, such as multiplication tables, is detrimental to students’ comprehension and enthusiasm for the subject. This belief rests on the misguided notion that long-term memory plays a minimal role in fostering understanding. In reality, the information stored in our long-term memory is instrumental in facilitating effective thinking and learning.

When encountering and processing new information, the thinking process occurs primarily in the working memory. However, the capacity of the working memory is limited in terms of the number of elements it can handle simultaneously and the duration of this processing. We can only process a finite amount of novel information at any given time within our working memory. Exceeding this limit leads to cognitive overload, hindering the ability to absorb and comprehend new material.

These limitations essentially vanish when the working memory processes information that is easily retrieved from long-term memory. Well-organised information that can be easily accessed (without thinking twice) from long-term memory means the working memory operates without apparent constraints. The more we remember, the easier it is for us to think and learn. Neither Google nor ChatGPT are substitutes for long-term memory; the novel information found on Google is subject to the limitations of our working memory – it adds to the count of the elements to be processed. Thus, the metaphor of an external memory stick expanding memory reserves fails to accurately depict the intricacies of human cognitive architecture.

Another detrimental fallout of the paradigm wars is the vilification of the teacher-led classroom model. Teacher-led instruction is often said to stifle students’ innate curiosity, extinguish their interest, and obstruct conceptual comprehension. This conflicts with the science. A well-crafted explanation of a mathematical concept or procedure by a proficient educator serves to channel learners’ attention towards incoming information, fostering active cognitive engagement crucial for selecting pertinent information for further processing in the working memory. This, in turn, enhances our capacity to comprehend new mathematical concepts.

Scientific research has clearly demonstrated the efficacy of explicit instruction when students are introduced to novel concepts and procedures. Teacher-led instruction involves the provision of clear explanations, demonstration of examples, and immediate corrective feedback during initial practice attempts.

Subsequently, as proficiency increases to a satisfactory level of accuracy, students should transition to a phase characterised by extensive practice – a phase marked by repetition and reinforcement. It is through copious and meticulous practice that students achieve the requisite speed of execution, paving the way for the development of automaticity – the ability to do things without thinking too hard, such as riding a bike or driving a car.

The speed is not merely ‘nice to have’, it is a crucial catalyst for mathematical generalisation. Without the attainment of automaticity (mathematical fluency), the ability to generalise becomes arduous. After all, the capacity to generalise is intricately linked with what is commonly perceived as conceptual understanding – the oft-cited pinnacle of learning objectives.

There is much more that we have learned about mathematical cognition (for further reading I’d recommend An Introduction to Mathematical Cognition) and the most effective ways to develop mathematical thinking based on decades of research. I hope policymakers will take a break from the ongoing political football match, which seems to be progressing without on-field referees, who are left yelling from the sidelines, often completely ignored.

I urge them to instead prioritise the science of learning over political expediency. It is imperative that we heed the insights provided by research to inform policy and shape recommendations for a revitalised mathematics curriculum and a common practice model for mathematics teaching and learning.