Of all the subjects that our students engage in, mathematics is the one most requiring an injection of learner agency. What is it about mathematics that engenders it to modes of teaching that are so heavily teacher-directed? How might this change if we seek to understand the place that learner agency plays in producing learners who will emerge from our classrooms with a love of mathematics and a deep understanding of its beauty?
Contemplating the effects of traditional mathematics in “A Mathematicians Lament”, Paul Lockhart wrote:
“If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done - I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.” (Lockhart. 2009 p.2)
The dominant mode of delivery for mathematical education is well-rooted in the history of education. It sends a clear message to learners about the nature of learning in the discipline. In these classrooms, the role of the teacher is to demonstrate the methods that students will then follow. The teacher asks the questions, and the students apply the given method and, if they accurately follow the steps as described, arrive at the required solution. Students learn that mathematics is about the transfer and absorption of a body of knowledge. The roles of the teacher and the student in this model are clear, and it is the lack of any compulsion towards an emotional or personal connection with the content that leads to the “senseless, soul-crushing” model described by Lockhart.
These traditional methods of teaching maths have more in common with how we programme a computer than what we might do if we wanted to encourage learner agency. The product of most computational thinking, an algorithm is in essence just a step by step list of instructions that a machine can follow. We shouldn’t be overly surprised then when our students consider mathematics as a discipline without soul. There are few, if any, opportunities for creativity or critical thinking and no place for personal expression or agency. How then might we shift this model, and what role does learner agency play in a reimagined mathematics?
Learner Agency is the most transformative ingredient we might add to our math classrooms. Learning is best achieved when it is driven by the learner. When the learner owns the process and when their success in the learning endeavour results from the strategic actions that they take, ‘learner flow’ becomes possible. When critical decisions are made for the learner, when the learning requires that they merely follow directions, when learning happens to you rather than because of you, engagement declines.
“By giving students a voice you send the message that their ideas and thinking are relevant to the learning that takes place and they begin to naturally take agency over their learning if we hand them the baton.” (Jennifer LaTarte cited in Ritchhart and Church. 2020 p. 19)
Our children will need a sense of agency empowered by capacities required to activate or perform their intentions (Clapp, et al. 2017). They must become creative problem finders through learning opportunities that allow them to “sense that there is a puzzle somewhere or a task to be accomplished” (Csikszentmihalyi. 2013 p. 95) and respond strategically, creatively and collaboratively towards solutions devised with empathy and a long-term view of impacts and real-costs (Kelley. 2013). Our students must be shown the value of acquiring deep understandings through weaving ideas together, going beyond information and figuring things out (Ritchhart. 2015).
Such learning requires a reimagining of what the discipline of mathematics is about. Instead of teaching our students how to do mathematics, our goal should be to teach them and empower them to be mathematicians. Such an approach requires an understanding of what it is that mathematicians do and what mathematics is.
“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” (Hardy. 1940 p. 13)
With such a description of what a mathematician does in mind, our approach to the discipline should begin to change. Our students require the capacity to become makers of patterns derived from the mathematical ideas they explore and uncover. Just as a poet or painter expresses their connection with the world, the mathematician finds their voice in the patterns they explore through their mathematical inquiries. Sadly, our current model of mathematics education borrows little from the field of art education, going no further than elementary art appreciation. Our students learn to look at mathematics and copy mathematics without understanding why and with no compulsion to be mathematicians.
“By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.” (Lockhart. 2009)
This reimagining of mathematics begins when we reorient our approach to teaching and learning. Instead of dry lessons that strictly adhere to an “I do, We do, You do” framework, mathematics education can be liberated by first allowing students to explore rich contexts with the potential for connectedness and meaning. “Exploring meaningful and important concepts that are connected to the world often means students want to take action. Providing opportunities and structures for them to do so encourages students’ agency and power while making the learning relevant.” (Ritchhart. 2015 p.8) We can do this by flipping the order of our planned curriculum. Instead of ending our learning journey with a rich problem, this should be the place where a mathematical inquiry begins. When we start our learning with a problem, when that problem matters to the learner and invites them to seek a solution, we begin to invite learner agency into our classrooms.
“agency”, is the ability to make choices and direct activity based on one’s own resourcefulness and enterprise. This entails thinking about the world not as something that unfolds separate and apart from us but as a field of action that we can potentially direct and influence.” (Ritchhart, 2015 p.77)
Such a definition of agency requires our students to be self-navigating learners who routinely make choices about the directions that their mathematical inquiries will take. Such a model will invoke situations where our learners struggle to find a path towards a viable solution. They will make mistakes and fail. At these times, we have a choice to make, do we allow them to engage in productive struggle, or do we rescue them? The challenge is to trust in our ability to spot the difference between a student engaged in productive struggle and a student who lacks the resources to move their learning forward. Productive struggle is best met by supportive mentoring towards sustained effort and reflective practices. The student who is overwhelmed and lacks the necessary knowledge for a step in the right direction is likely to be open to the introduction of a new tool. The trick is to offer a new tool, not a lifeline. To leave the challenge in place while providing a way forward.
While the prospect of empowering mathematical agency may seem daunting, it does not need to be. Including time for a Number Talk within each mathematics lesson is a simple step in this direction. Doing so sends a powerful message to students about the nature of mathematics. When we emphasise mathematical reasoning, as the curriculum invites us to, we create rich opportunities for student thinking. Mathematical reasoning is defined by the writers of the Australian Curriculum as follows:
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, deduce and justify strategies used and conclusions reached, adapt the known to the unknown, transfer learning from one context to another, prove that something is true or false, and compare and contrast related ideas and explain their choices. (ACARA. 2010)
Source: ACARA
In this extract from the Australian curriculum, we see the seeds of learner agency tied closely to the act of being a mathematician. It is critical to note here that it is our students who are developing their capacity for logical thought as they reason mathematically and as they explain, deduce and justify their thinking and adapt to the known and unknown. Such an outcome will not be achieved by pedagogies that privilege rote learning. If our students are to engage with agency in mathematical reasoning, they must encounter learning opportunities that require this within a classroom culture that values thinking. Our students need to routinely engage with learning that requires thoughtful action and justification of chosen methods. This is likely to occur in classrooms where student thinking is named, noticed and celebrated. Teachers who use thinking routines as scaffolds for low-floor, high-ceiling, wide-walled (open-ended) tasks are more likely to foster learner agency. In this, we are describing a task that is accessible by all learners (low-floor), has scope for enrichment and extension (high-ceilings) and has multiple possible solutions and methods of engagement (wide-walls). Such rich tasks, when linked to learning that is likely to matter in the lives our learners are likely to live, are highly engaging and naturally supportive of learner agency.
When we combine opportunities for our students to explore meaningful and important concepts connected to their world, when we encourage them to embrace productive struggle and seek out the patterns and beauty of mathematics, we create a space in which learner agency is valued. When we do these things, and in doing so, invite agency into our math classrooms, we allow mathematics and mathematicians to thrive.
By Nigel Coutts
Originally published in Connect Number 248 May 2021
Connect is an independent practice journal, published bimonthly since 1979!
It aims to:
document student participation approaches and initiatives
support reflective practices;
develop and share resources.
You can access all past issues of Connect HERE
Clapp, E., Ross, J., Oxman Ryan, J. & Tishman, S. (2017) Maker-centered learning: empowering young people to shape their worlds. San Francisco, Josey Bass.
Csikszentmihalyi, M. (2013) Creativity: Flow and the psychology of discovery and invention, New York, Harper Perennial
Hardy, G. H. (1940) A mathematician’s apology. University of Alberta Mathematical Sciences Society
Kelley, D. (2013) Creative Confidence: Unleashing the Creative Potential Within Us All”, London, Harper Collins.
Lockhart, P (2009) A Mathematician’s Lament: How school cheats us out of our most fascinating and imaginative art form. Bellevue Literary Press
Ritchhart, R. (2015) Creating cultures of thinking: The eight forces we must truly master to transform our schools”, San Francisco, Josey-Bass.
Ritchhart, Ron, Church, Mark. (2020) The Power of Making Thinking Visible: Practices to engage and empower all learners. New Jersey; Josey Bass