{"id":4224,"date":"2024-12-30T22:49:29","date_gmt":"2024-12-31T04:49:29","guid":{"rendered":"https:\/\/www.mctmmathbits.org\/?p=4224"},"modified":"2024-12-30T22:49:29","modified_gmt":"2024-12-31T04:49:29","slug":"adversarial-collaboration-the-way-to-bridge-the-great-maths-divide","status":"publish","type":"post","link":"https:\/\/www.mctmmathbits.org\/?p=4224","title":{"rendered":"Adversarial collaboration: the way to bridge the \u2018great maths divide\u2019?"},"content":{"rendered":"

The following is reprinted here with the permission of the author, Amie Albrecht<\/strong><\/em><\/p>\n

Read more of Professor Albrecht’s work at\u00a0amiealbrecht.com<\/a><\/strong><\/em><\/p>\n

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 sensationalized as \u2018maths wars\u2019 by the media (e.g. here<\/a>\u00a0and\u00a0here<\/a>), this perceived divide only serves to harm rather than help students and the teaching profession.<\/p>\n

In an earlier\u00a0blog post<\/a>\u00a0on various approaches to problem solving, I asked: \u2018How do we move on from what seems to be an ever-growing divide?\u2019 It turns out that the way forward might involve working together in seemingly unconventional ways.<\/p>\n

\u2018Adversarial collaboration\u2019 is an approach pioneered by Daniel Kahneman, recipient of the Nobel Prize in Economics and acclaimed author of the bestseller \u2018Thinking, Fast and Slow\u2019. Kahneman\u2019s idea is that individuals with opposing views purposefully engage in a structured process to challenge each other\u2019s positions through respectful exchanges designed to advance knowledge in a specific field.<\/p>\n

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: \u2018dialogic\u2019 and \u2018direct\u2019. These labels were chosen by the authors in an attempt to characterise the two \u2018sides\u2019 of the debate. (Alternative, and often more value-laden, terms exist.) \u2018Dialogic\u2019 instruction is meant to capture student-centred approaches such as reform, discovery, inquiry, and constructivist. \u2018Direct\u2019 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.<\/p>\n

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.<\/p>\n

The goal was not consensus. Instead, the intent was to\u00a0convert disagreement into something productive<\/em>\u00a0(my emphasis) through a good-faith effort at understanding each other\u2019s view, identifying areas of commonality, and \u2018surfac[ing] and highlight[ing] the underlying sources (rationales, perspectives, theories, and priorities) that give rise to disagreement\u2019 so as to add \u2018clarity and depth to the debate\u2019.<\/p>\n

The discussions served to provide clear descriptions of direct and dialogic instruction, and the points of similarities and difference. I\u2019ll include the first two verbatim so as to accurately represent the models. I\u2019ll 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\u00a0here<\/a>.<\/p>\n

Describing direct and dialogic instruction<\/h3>\n

\u201cIn\u00a0the direct instruction model<\/strong>, 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.\u201d<\/em><\/p>\n

\u201cIn\u00a0the dialogic model<\/strong>, 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\u2014tasks 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\u2019 understanding grows); and, finally, to sequence classroom activities in a way that consistently positions students as autonomous learners and users of mathematics.\u201d<\/em><\/p>\n

From my perspective, these seem to be accurate and impartial descriptions. What are your thoughts?<\/p>\n

Similarities and differences<\/h3>\n

The authors found several similarities in discussions of the two models:<\/p>\n