Mathematics education in Australia, much like in other countries, is in the grip of a conflict between two types of pedagogies: explicit teaching and inquiry-based learning. Unhelpfully sensationalised as ‘maths wars’ by the media (e.g. here and here), this perceived divide only serves to harm rather than help students and the teaching profession.
In an earlier blog post on various approaches to problem solving, I asked: ‘How do we move on from what seems to be an ever-growing divide?’ It turns out that the way forward might involve working together in seemingly unconventional ways.
‘Adversarial collaboration’ is an approach pioneered by Daniel Kahneman, recipient of the Nobel Prize in Economics and acclaimed author of the bestseller ‘Thinking, Fast and Slow’. Kahneman’s idea is that individuals with opposing views purposefully engage in a structured process to challenge each other’s positions through respectful exchanges designed to advance knowledge in a specific field.
In their 2015 paper, mathematics education researchers Charles Munter, Mary Kay Stein, and Margaret S Smith adopted the spirit of adversarial collaboration to advance the debate around two main models of mathematics instruction: ‘dialogic’ and ‘direct’. These labels were chosen by the authors in an attempt to characterise the two ‘sides’ of the debate. (Alternative, and often more value-laden, terms exist.) ‘Dialogic’ instruction is meant to capture student-centred approaches such as reform, discovery, inquiry, and constructivist. ‘Direct’ instruction is meant to imply teacher-led explanations, often termed explicit teaching or back to basics. These two models are often positioned as extremes of a continuum which, although oversimplified, provides a starting point for the kind of discussion that Kahneman proposes.
Munter, Stein and Smith hosted a series of semi-structured discussions among nationally recognised experts in the US (mathematicians, educators, psychologists, and learning scientists) with opposing views regarding mathematics instruction. Representatives of different perspectives were invited to outline their stance with respect to what it means to know mathematics, how students learn mathematics, and how mathematics should be taught. They then met with the opposing side to come to agreement on exactly how they disagreed on issues of mathematics knowledge, learning, and teaching.
The goal was not consensus. Instead, the intent was to convert disagreement into something productive (my emphasis) through a good-faith effort at understanding each other’s view, identifying areas of commonality, and ‘surfac[ing] and highlight[ing] the underlying sources (rationales, perspectives, theories, and priorities) that give rise to disagreement’ so as to add ‘clarity and depth to the debate’.
The discussions served to provide clear descriptions of direct and dialogic instruction, and the points of similarities and difference. I’ll include the first two verbatim so as to accurately represent the models. I’ll summarise the others, drawing heavily on the original text. I highly recommend that you read the details yourself in the paper, which can be accessed for free here.
Describing direct and dialogic instruction
“In the direct instruction model, pedagogy consists of describing an objective, articulating motivating reasons for achieving the objective and connections to previous topics; presenting requisite concepts (if they have not been presented previously); demonstrating how to complete the target problem type; and providing scaffolded phases of guided and independent practice, accompanied by corrective feedback. Across these phases, lessons should be made engaging, which can be accomplished through keeping a brisk instructional pace, inviting group unison responses to questions, encouraging student motivation by supporting them in experiencing success, and providing focused praise.”
“In the dialogic model, across a series of lessons, students must have opportunities to (a) wrestle with big ideas, without teachers interfering prematurely, (b) put forth claims and justify them as well as listening to and critiquing claims of others, and (c) engage in carefully designed, deliberate practice. This requires teachers, first, to engage students in two main types of tasks—tasks that introduce students to new ideas and deepen their understanding of concepts, and tasks that help them become more competent with what they already know; second, to orchestrate discussions that make mathematical ideas available to all students and steer collective understandings toward the mathematical goal of the lesson; third, to introduce tools and representations that have longevity (i.e., can be used repeatedly over time for different, but likely related, purposes, as students’ understanding grows); and, finally, to sequence classroom activities in a way that consistently positions students as autonomous learners and users of mathematics.”
From my perspective, these seem to be accurate and impartial descriptions. What are your thoughts?
Similarities and differences
The authors found several similarities in discussions of the two models:
both value conceptual understanding and procedural fluency and view them as being developed together
both emphasise tasks that are carefully designed, purposefully sequenced, and mathematically rigorous
both recommend closely monitoring students’ reasoning
both advise regular opportunities for practice
noting that some of these points are typically more associated with one model of instruction than the other.
The authors identified nine key areas of difference between the two models (unpacked in detail in the paper):
The importance and role of talk
The importance and role of group work
The sequencing of mathematical topics
The nature and ordering of mathematical instructional tasks
The nature, timing, source, and purpose of feedback
The emphasis on creativity (i.e. authoring one’s own learning; mathematising subject matter from reality)
The purpose of diagnosing student thinking
The introduction and role of definitions
The nature and role of representations
Our individual views on what it means to know and learn mathematics have a direct connection to how we teach. (I discussed this in relation to teacher listening here.) An interesting clarifying exercise is to ask yourself what each of these dimensions means to you, and the relative value or emphasis you place on each.
The authors point to multiple possible sources of differing perspectives. They elaborate the first four in the paper, and refer to the last three in the notes:
Content — which mathematical ideas should be taught and how they should be represented
Epistemology — definitions of mathematical knowledge or proficiency, and whether constructs such as student identity or mathematical authority should be included in the goals of school mathematics
Learning — alignment with different theories of how children learn, and whether those are driven by cognitive, behavioural, participationist or constructivist approaches
Pedagogy — commitments to different kinds of instruction, following on from beliefs about what should be learned, and how it is learned
Evidence — differences in what constitutes evidence for either approach to instruction
Purpose of mathematics education — e.g. social mobility or social efficiency
Equity — who has opportunities for success in each model of instruction
Some final thoughts
I believe that everyone, irrespective of which ‘side’ they are on, is working with good intentions and with the best outcomes in mind for their students. We just have differing views of what those outcomes might be and how to get there. This article has been incredibly useful in sharpening my thoughts around the sources of those differing opinions and how they might manifest in our teaching.
Munter, Stein and Smith encourage us ‘to press each other to identify exactly which of the facets of a debate have motivated one’s concern’ and ‘to hold each other accountable for articulating rationales and, perhaps most importantly, citing evidence.’ In this way, they have given us a useful way forward in turning disagreement into something productive and, perhaps, bridging the divide.
References
Munter, C., Stein, M. K., & Smith, M. S. (2015). Dialogic and Direct Instruction: Two Distinct Models of Mathematics Instruction and the Debate(s) Surrounding Them. Teachers College Record, 117(11), 1–32.
