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were the least effective practices for occasioning and necessitating thinking. These practices mark a significant evolution of teaching from the institu tionally normative practices that permeate and have permeated education for the last 150 years. The results of this work have been implemented in hun dreds of face - to - face classrooms around the world with incredible volition for engendering student thinking and transforming teachers ’ experiences of their students ’ abilities. And then came COVID - 19. In an attempt to con tain this nefarious and cunning virus, countries all over the world moved education from the familiar face - to - face classrooms to online platforms. And millions of teachers were required to transform their practices seemingly overnight. Suddenly, words like synchronous and asynchronous became part of our daily vocabulary as we tried to cope with this new reality. Some teachers met this chal lenge by mapping their practices to a new medium of instruction—swapping the whiteboard for a doc ument camera and the raised hand for the chat box. Other teachers met this challenge by evolving their teaching and exploring the affordances of online collaboration, simulations, and discussion plat forms. Regardless of the nature of the change, how ever, all teachers were now operating outside of the constraints of the institutionally normative practic es that were anchored to the bricks and mortar in stitution of school. Even teachers enacting all, or a subset, of the 14 thinking practices had to evolve their already evolved practice to meet this new challenge. In this article, we detail some of the lessons learned in adapting these thinking practices to rise to the challenge of teaching online. We look specifically at one of the general BTC practices – the types of tasks we use. This practice was in most need of ad aptation for an online setting because of its orienta tion around supporting collaboration among learn ers, and the adaptations based on the currently available technological tools we have at our dispos al. In what follows, we present this practice in the context of online teaching and learning and explore how the shift in the medium of instruction necessi

tated, or not, shifts in the execution of the practice.

The Types of Tasks we Use

If we want our students to think, we need to give them something to think about—something that will not only require thinking but will also encour age thinking. In mathematics, this comes in the form of a task, or more specifically, a thinking task . When it comes to talking about tasks that get stu dents to think in mathematics, the best place to start is with problem solving. From Pólya's (1945) How to Solve It to the NCTM Principles and Standards (2000), the literature is replete with the benefits of having mathematics students engage in problem solving. Although there are arguments about the exact processes involved and the exact competen cies required, there is universal agreement that problem solving is what we do when we do not know what to do. That is, problem solving is not the precise application of a known procedure. It is not the implementation of a taught algorithm. And it is not the smooth execution of a formula. Prob lem solving is a messy, non - linear, and idiosyncrat ic process. Students will get stuck. They will exper iment, guess and check, try and fail, and they will apply their knowledge in novel ways in order to get unstuck. In short, they will think . Tasks that invoke this sort of behaviour are often called non - routine tasks because they require stu dents to invoke their knowledge in ways that have not been routinized. Once routinization happens, students are mimicking rather than thinking —or as Lithner (2008) calls it, being imitative rather than creative. These tasks are also referred to as rich tasks in that they require students to draw on a rich diversity of mathematical knowledge. Taken to gether, thinking tasks are problem solving tasks that require students to use their diverse mathemat ical knowledge in innovative and creative ways.

Towards an Online Thinking Classroom

Even in online settings, thinking tasks should still aim to prompt students to think rather than to mim ic , to use their diverse mathematical knowledge in

Virginia Mathematics Teacher vol. 47, no. 1

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