Expert opinion from University of Auckland - Waipapa Taumata Rau
Opinion: 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.
Good analysis of issues facing students. Teacher led instruction was discarded when low wages meant it was difficult to recruit teachers capable of leading instruction.
Bullseye! After a lifetime in science education (and having seen so many educational fads come and go) it is refreshing to see such an honest, evidence-based article. There are so many issues that impact on the effectiveness or otherwise of formal education that it’s hard to untangle them all, but thank you for this very incisive critique.
Beautifully written – my highlights being, “undeclared paradigm war” and “automaticity (- the ability to do things without thinking too hard)”
Agree totally
We no longer teach Mathematics. We facilitate “standards”
The discussion fits with Daniel Kahnman’s distinction between System 1 (jumping immediately to a conclusion) and System 2 (pondering) distinction. Very often, in mathematical as in other mental challenges, what is required will be a mixture of the two. One wants to be able to handle as much as possible of any challenging problem in a System 1 way, so that the System 2 thinking can focus on the hard parts. Rote learning, as with multiplication tables, is an important contributor to making key parts of responses to mathematical challenges automatic. Problems arise, of course, when wrong ideas find their way into System 1 — what it has to say has always to be open to review and challenge.
Thank you, Dr Evans! It’s been enough to make me weep with frustration to witness my children subjected to teaching that defies basic common sense in the name of the latest theories, and from teachers who benefited as children from the exact same skills, knowledge, and teaching they are now discrediting. My (very clever) eldest son’s last primary teacher’s parting shot to him was, “learn your times tables”. Really? Don’t get me started on handwriting and writing in general…I noted with grim pleasure Dr Evans’ explanation of automaticity and the need for practice. It follows that it’s not sufficient to teach children how to link their letters with a pen then switch entirely to one-fingered input on a device. Again, common sense–and supported by the science–but that’s too late for my guys, who couldn’t handwrite their way out of a paper bag with the end open. To further Dr Evans’ metaphor, parents have also been on the sidelines of this football match also, shouting and being completely ignored.
this is the explanation we have needed for some time. Excellent piece.
Great: “The more we remember, the easier it is for us to think and learn.” When it comes to multiplication automaticity, we must remember that understanding and memorization are interconnected, with memory connections formed through understanding and strategies serving as anchors for storing facts.
A key challenge is efficiently practicing multiplication automaticity across an entire class, without stress, using research-based methods such as individual practice, spaced repetition, and active recall. In response, I developed Fluent Math – Multiplication, a free tool designed to help teachers facilitate effective automaticity practice in the classroom.
Striving to achieve automaticity, to the greatest extent possible for each individual, is an inspiring goal.