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Virginia Council of Teachers of Mathematics | www.vctm.org

VIRGINIA MATHEMATICS

TEACHER

Vol. 47 No. 1

Teaching Mathematics During the COVID-19 Pandemic!

Virginia Mathematics Teacher vol. 47, no. 1

Editorial Staff

Dr. Agida Manizade Editor - in - Chief Radford University vmt@radford.edu

Dr. Jean Mistele Associate Editor Radford University

Mr. Alexander Burnley Assistant Editor Radford University

Ms. Grace Chaffin Assistant Editor Radford University

Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050 Blacksburg, Virginia 24060

Virginia Council of Teachers of Mathematics

Many Thanks to our Reviewers for 47(1)

John A. Adam, Old Dominion University Michael Bosse, Appalachian State University Hunter Dickenson, Atlee High School Anthony Dove, Radford University Joe Garofalo, University of Virginia Christopher Hildenbrand, Radford University Jamey Lovin, Virginia Commonwealth University Harold Mick, Virginia Tech

President: Lynn Reed

Past President: Pam Bailey

Secretary: Kim Bender; Historian: Beth Williams

Jean Mistele, Radford University Alexander Moore, Virginia Tech Laura Moss, Radford University Anderson Norton, Virginia Tech

Treasurer: Virginia Lewis

Webmaster: Ian Shenk

Tracy Proffitt, R.S. Payne Elementary Tammy Sanford, Staunton City Schools Toni Sorrell, Longwood University Skip Tyler, Henrico County Schools

NCTM Representative: Skip Tyler

Elementary Representatives: Tracy Proffitt and Tammy Sanford

Special thanks to all reviewers! We truly appreciate your time and service. The

Middle School Representatives: Lisa LoConte - Allen

high - quality journal is only possible because of your dedication and hard work.

Complete VMT Editorial Staff

Secondary Representatives: Reagan Davis and Timothy Barnes

Agida Manizade, Radford University Jean Mistele, Radford University Alexander Burnley, Radford University Grace Chaffin, Radford University Alexander Moore, Virginia Tech Eric Choate, Radford University David Shoenthal, Longwood University Betti Kreye, Virginia Tech

Math Specialist Representative: Allison DePiro

Two - Year College: Doniray Brusaferro

Four - Year College: Darryl Corey

VMT Editor in Chief: Dr. Agida Manizade

VMT Associate Editor: Dr. Jean Mistele

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Table of Contents:

Note from the Editors ............................................ 5

Message from the President .................................. 7

Building Thinking Classrooms Online: A Closer Look at the Types of Tasks We Use ....................... 8 COVID - 19 and the Chain Rule are Linked: Understanding R 0 through Calculus .................... 15 Technology Review: Teaching Online During COVID - 19 ............................................................ 21 Exploring Linear Functions Using Arrow Diagrams …. ......................................................... 24 Math Jokes ........................................................... 20

Affiliate Information ............................................ 30

Note from the VDOE ........................................... 31

Busting Blockbusters ........................................... 33

Shapes in the Octagon Wreath ............................ 34

Meet Your Representatives .................................. 39

Every Equation Tells a Story: Waves on Water .. 40

Call for Manuscripts ............................................ 47

HEXA Challenge ................................................. 48

NASA Resources .................................................. 50

Upcoming Mathematics Competitions ................. 51

Good Reads ......................................................... 52

Membership Information ..................................... 56

Teaching Place Value in Elementary Mathematics to First Grade During a Global Pandemic: Reimagining COVID ......................... 57

Grant and Scholarship Opportunities ................. 68

Key to the 46(2) Puzzlemaker .............................. 69

Solutions to 46(2) HEXA Challenge .................... 70

Puzzlemaker ......................................................... 75

Conferences of Interest ........................................ 76

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Teaching Mathematics During the COVID - 19 Pandemic: Note from the Editors

We are approaching the one - year anniversary when the K - 12 schools, universities in Virginia, and the rest of the world began to close their doors due to the Covid - 19 pandemic. Teachers scrambled to shift their face - to - face teaching to an online envi ronment and, later, to a hybrid teaching model. Words like Zooming, synchronous, asynchronous, and hybrid to name a few became common over the last year . The change was swift, and teachers at all grade levels, K - 16+, were thrust into a new world of teaching. To keep students ’ engagement in this new digital environment, mathematics teachers had to face a challenge of limited digital interactive mathematics content appropriate for their classes. Developing all of the digital content for their classrooms from scratch was time consuming and, in some cases, impossible. The many venues on the internet such as blogs, YouTube, webinars, and mathematics apps switched to support teachers in their heroic effort to sustain student learning. On another hand, common pedagogical practices also had to change. For ex ample, classroom management, student engage ment, parent communication practices in an online school looked very different from what was once used in a face - to - face class. Teachers now held du

al roles as learners of new teaching practices and pedagogy and doers of these new teaching practic es all at the same time. Overnight, all teachers were thrust into a massive professional development ex perience that exponentially increased their technol ogy skills and newfound online teaching skills. In this issue, we share how teachers used their new teaching and technology skills during the pandem ic, the ways that technology augmented student learning, the ways that changed how we maintained a sense of community with students and parents, and the way mathematics modeling helps us under stand the meaning of pandemic and herd immunity from a mathematics perspective. We open the issue with an article by an internation ally renowned mathematics educators, Peter Liljedahl from Simon Fraser University in Canada and his co - author Judy Larsen from the University of Fraser Valley in Canada. In their article, “ Building Thinking Classrooms Online: A Closer Look at The Types of Tasks we Use , ” the authors ground their article on Dr. Liljedahl ’ s framework that structures classroom teaching to foster student thinking. The authors explain how shifting teaching

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practices from face - to - face to online can be done successfully to continue to build thinking class rooms, where students are engaged in mathematical thinking. The article “ COVID - 19 And The Chain Rule Are Linked: Understanding R 0 Through Calculus , ” ex plains the meaning of R 0 , its value, and its relation ship to herd immunity. The author takes the reader through the modeling process using mathematics appropriate for students enrolled in a calculus course in high school, a community college, or a university. The students see how the Chain Rule is used in a real - world situation, and the related graphs of the functions demonstrate how the math ematics modeling unfolds. “ Teaching Place Value in Elementary Mathematics to First Grade During a Global Pandemic – Reimagining COVID, ” the author shares another way to interpret COVID: C – caring for children and the community; O – once daily lesson; V – vi vacious teaching that is active, lively and animated; I – independent at - home learning; and D – direct instruction using asynchronous videos. This article shared how teaching place value to first grade stu dents can maintain rigor while remaining hands - on and engaging. The Technology Review section introduces us to MathBot, a virtual manipulative that has many ap plications. In this section, we learn about its strengths and limitations. In the NASA section, we share links to ways that NASA is studying and exploring the COVID - 19 pandemic from space. Students seeing the pandem ic from a new and global perspective reveals the pandemic as a shared experience for all humankind around the world. For example, students see the changes in the heat patterns ’ levels and the nitrogen dioxide levels. The question arises as to the role that weather patterns may have played in COVID ’ s spread. The article, “ Exploring Linear Functions Using Ar row Diagrams, ” explains how linear functions can be explored using a visual representation, the arrow

diagram among the author ’ s preservice and in service teachers. The inquiry activity uses a graph ical representation, the arrow diagram, which uses ordered pairs of inputs and outputs. This represen tation shows the relationship between linear func tions and their arrow diagrams, which is a novel approach to learning about linear functions. Stu dents engage in conjecturing and reasoning as they seek patterns in the arrow diagrams. Paper folding is an engaging activity for students in grades K - 3. In the article, “ Shapes in the Octagon Wreath, ” students learn geometry. They identify shapes as they fold the paper, or they identify lines of symmetry or classify triangles while engaged in the paper folding activity. A truly hands - on activi ty, which can be used in the online classroom. For younger students, the teacher asks them geometry questions during the folding process. Older stu dents may be able to create their own octagon when supplies are provided to them. The exploration of patterns continues when explor ing waves on water in the article, “ Every Equation Tells A Story: Waves On Water. ” The author ex plains the mathematical structure of a wave and how it differs when in shallow water or deep water and how the waves also differ based on their length. The mathematics in this article is suitable for high school students enrolled in a trigonometry course. The appendix includes other types of waves, refraction or tides, the types of wave creat ed by speedboats and ships, and the hyperbola cre ated when circular waves intersect. This Special Issue, Teaching Mathematics During the COVID - 19 Pandemic , highlights the innovative ways that teachers responded to the pandemic— ways that continued to keep students engaged, pro mote the conceptual learning of mathematics, and emphasized students thinking in the classroom. We applaud all of the teachers across Virginia for their hard work during an extraordinary time in world history. We hope you enjoyed this Special Issue as much as we enjoyed working on it!

– Dr. Jean Mistele and Dr. Agida Manizade

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Message from the President Lynn Foshee Reed

Hello VCTM Colleagues!

Shelly Pine for the professional community events and to Theresa Wills for the virtual VCTM Confer ence in March. Another suggestion, which may seem counterintui tive, is to “ engage ” your brain in something differ ent. Maybe attend a webinar on the Virginia Math ematics Pathways Initiative? VCTM has many of our members working very hard to envision, as well as provide feedback on, the framework. Or check out a virtual national or regional conference such as the Teaching Contemporary Mathematics conference in February? Watch a Numberphile video? Make math - themed arts/crafts? Read bite sized math - morsels in Plus online magazine or the Global Math Project website? And of course, this special issue on "Teaching Mathematics During a Pandemic" offers a look at how teachers have re sponded and adapted to teaching during COVID 19. Whatever you choose, whether a 5 - minute or 5 hour endeavor, let it bring you energy or peace or strength, as you need. And then, share with a friend or colleague! What I want for every one of you as we move into 2021 is to be filled with light and to reflect it back to your friends and family, colleagues and students. Please consider attending one or more of upcoming virtual VCTM events to share your successes and struggles of the past year. We are a community of strong, resourceful, compassionate, and resilient educators. Thank you for the hard work you do every single day!

My family knows that I am NOT a winter person – it ’ s not really the cold, but the DARK that gets me down. But the winter solstice marks the minimum number of daylight hours here in Virginia, and my mood will get a boost from knowing that every day through the rest of the school year will be “ lighter ” than the one before. What a wonderful, positive rate of change (sorry, the calculus teacher in me just couldn ’ t resist)! In December 2012, I traveled to Antarctica as part of my Einstein Fellowship. One of the things that I loved about that trip was the 24 - hour daylight that surrounded me – it was completely energizing. However, the curiosity and joy which came with witnessing and learning about the wonders of such a beautiful place undeniably “ lightened ” me as well. The announcement of ef fective vaccines coming in 2021 is also the kind of news that provides reassurance that things will get better.

Yet … I am bone - tired. I ’ ll bet you are, too. How can we continue to serve our students in the best way that we can, given the extraordinary circum stances of teaching and

learning during a pandemic? How do we find time for family - time and self - care? I suggest connecting to other teachers for support and to serve as a sounding - board. VCTM will be offering many op portunities to connect in 2021. I am grateful to

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Building Thinking Classrooms Online: A Closer Look at The Types of Tasks we Use

Peter Liljedahl and Judy Larsen

Many activities of teaching, and thus also activities of learning, are dictated by what can be referred to as institutional norms (Liu & Liljedahl, 2012). These are practices that transcend classroom norms (Cobb, Wood, & Yackel, 1991; Yackel & Cobb, 1996), and crystallize into patterns that occur in classrooms. Although what is taught in classrooms has evolved greatly over the course of the last 150 years, institutional norms that were laid down at the dawn of public education still dictate much of how teaching looks in classrooms today. In visits to 40 different K - 12 mathematics classrooms in 40 different schools, Liljedahl (2016, 2020) observed that in a typical lesson, there was very little oppor tunity, and even less need, for students to do much thinking. He posited that in order for this reality to change—in order to get more students thinking and thinking for longer—a radical departure from the institutional norms would be needed. Thus was born the Building Thinking Classrooms (BTC) pro

ject which, for over 15 years, sought to empirically emerge and test pedagogical practices that not only afford opportunities to think, but necessitate think ing. This research involved more than 400 classroom teachers implementing hundreds of two - week mi cro - experiments. Each micro - experiment sought to measure the degree to which a specific practice im pacted the amount of thinking in the classroom. Emerging out this research are 14 teaching practic es that have been proven to produce more thinking in the classroom than the institutionally normative practices they sought to replace, but also more thinking than any of the other hundreds of practices experimented with (Liljedahl, 2020). In many cas es, these practices are radical departures from the norm. In some cases, less so. But in all cases, it was shown that the institutionally normative prac tices laid down at the dawn of public education

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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

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innovative and creative ways, and to engender a culture of collaborative thinking in the classroom. However, online settings offer certain constraints that need to be considered. Since thinking tasks involve communicating and representing ideas about a problem, the most prominent aspect for de termining which tasks are appropriate for an online setting has to do with the modes of communication it requires from students as well as their technolog ical competencies . In face - to - face BTC environments, students have ubiquitous access to a variety of proximally located shared workspaces which, together, allow them to employ a variety of modes of communication. With little to no effort, students in these face - to - face set tings naturally employed a combination of commu nicative tactics including gestures, verbal utteranc es, diagrammatic presentations, and various written notations for communicating their mathematical ideas around a thinking task. However, these taken for - granted affordances are hindered in an online space. To communicate ideas in an online setting, students are not only limited by the technological tools they have at their disposal (e.g., cameras, microphones, writing inputs, etc.), but also by their technological competencies and willingness to engage while con necting remotely (Roddy et al., 2017). The effec tiveness of a thinking task is therefore intimately interwoven with the modes of communication it requires of students and their ability to express their ideas through technological means. As such, choosing thinking tasks for online collaborative settings and identifying ways to use them with stu dents in such settings needs to take into account not only the nature of the task itself, but also the stu dents ’ technological competencies. For example, if the state of students ’ technological competencies are unclear, then it is best to begin with tasks that invoke verbal or gestural modes of communication without the need for notation heavy or diagrammatic output. Such tasks may in vite students to visualize, explain, predict, conjec ture, justify, or verify (Van de Walle et al., 2015). In a synchronous environment, students may be

collaborating on the task in breakout groups using a microphone and camera, while in an asynchronous setting, they may be using basic text functions in a discussion board dedicated to collaborating about the thinking task. Either way, the thinking task needs to allow for multiple ways of thinking and be focused on emphasizing discussion of various viewpoints rather than on solving to get a single solution. An example of a thinking task that achieves this and that does not require students to have high technological competencies for commu nication of ideas may be introduced as follows: A bicycle crosses a freshly painted road line, and the smears on the bike ’ s tires from the road line continue to make imprints on the pavement as the bike continues down the road. What do these imprints look like? (Mason, Burton, & Stacey, 2011) Given a limited technological space that does not offer written or diagrammatic output, but that simp ly relies on communication through a microphone and possibly a camera, this task then becomes heavily reliant on students ’ ability to visualize pos sible solutions. The limitation of technology here provides an opportunity for emphasizing visualiza tion, followed by verbal and possibly gestural com munication. In using this task in multiple instances with different groups of students, we observed that students debating heatedly about their predictions, often using gestures through their video cameras while explaining their ideas verbally through their microphones. Even when we provided students with a collaborative writing surface, very little was drawn on it and the primary mode of communica tion was verbal as well as gestural. The nature of this task lends itself to verbal communication, and therefore, by using it we may harness the potential ities of the online setting rather than being limited by its constraints. While verbal and gestural communication about thinking tasks offers a plethora of possibilities for both non - routine and curricular thinking tasks, the need for written and diagrammatic communication is bound to arise. In such cases, another tool is needed for students to use for communicating their

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ideas and the thinking task selected needs to be compatible with the capabilities of both the tech nology and students ’ abilities to use the technolo gy. One such tool is Google Jamboard, a collabora tive online whiteboard that allows users to type and draw synchronously in real time from many devic es on one shared workspace that is accessible via hyperlink. It acts similarly as a Google Doc but al lows for more varied forms of input. Although it offers advanced tools, students tend to use only the most basic of its functionality. For instance, one student may only want to use the typing option since it may be easier to use with a desktop com puter, while another may be more comfortable with the drawing feature that is more natural to use with a mobile device or touch screen computer. Taken together, the technological tools available to stu dents that they will willingly use to communicate their mathematical thinking should be taken into account when selecting and devising appropriate thinking tasks. This is because the technology can be a hindrance or a catalyst for thinking about the task depending on the student ’ s technological com petencies. As such, it is best to hold the assumption that students will have a range of technological competencies and that as long as the task does not require overly complex notation or diagramming to communicate its basic principles, students should be able to adequately communicate their thinking. To illustrate what such tasks could look like, we offer a few examples of thinking tasks that have been introduced in a Zoom environment along with samples of the kind of written work students pro duced when working on these tasks in breakout groups with a Jamboard. The first of these tasks is a number puzzle that could be introduced as fol lows: Below is a list of five answers. These are the answers to five arithmetic expressions each consisting of two numbers and an operation.

__ __ = 3

__ __ = 2

Using each of the numbers from 1 to 10 exact ly once, and each of the operations (+ - ×÷) at least once, find what the five expressions are such that the answers are 17, 2, 21, 3, 2. When students solve this problem for the first set of answers, they are then given a more difficult one, etc. Below in Figure 1 is a sample student work on this task on a Jamboard.

Figure 1: Example of student work on a Jamboard in a breakout room

It may be observed in the above student work that the group of students used a combination of drawn inputs and typed inputs to communicate their think ing. This was based on what each student was most comfortable using to make their contributions. No tably, the complexity of notation required in this task was not overly obstructive of the communica tion among group members. It is interesting to contrast this work with what stu dent work looked like for this same thinking task in a face - to - face thinking classroom as seen in Figure 2. As may be observed, the main difference in the face - to - face student work is that there were more small workings on the side (see the far - right of Fig ure 2) that are not present in the online setting. This may be because students in an online setting may have been using their own paper to help them work through a problem and only wrote what they identi fied as correct on the digital workspace. Alterna tively, they may also have been discussing their ideas verbally rather than by writing them down in

__ __ = 17

__ __ = 2

__ __ = 21

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the cheques (see Figure 3) rather than crossing them out in order to try their attempts at beating the tax collector. This makes it an object - oriented me dium rather than simply a writing medium , offering a form of interaction for students not automatically available on a whiteboard in a face - to - face thinking classroom. However, the disadvantage was that students be came less inclined to take note of the order of each move they made, nor of the brainstorming of fac tors available for each number. As may be seen in Figure 4, a whiteboard as a writing medium for this same thinking task engendered rather different trac es of mathematical thinking, which perhaps oc curred only verbally in the online setting.

Figure 2: Example of student work on a thinking task in a face - to - face BTC setting

the shared workspace. This illustrates the shift in modes of communication as determined by the type of workspace used.

Another example of a task used in a Jamboard is as follows:

Start with a collection of paychecks, from $1 to $12. You can choose any paycheck to keep. Once you choose, the tax collector gets all paychecks remaining that are factors of the number you chose. The tax collector must re ceive payment after every move. If you have no moves that give the tax collector a paycheck, then the game is over and the tax collector gets all the remaining paychecks. The goal is to beat the tax collector.

Figure 4: Example of student work on a whiteboard in a face - to - face BTC setting

Another example of a thinking task used in a Jam board is one that relies mostly on students drawing to communicate their thinking, and may be intro duced as follows: You have a triangle of numbered disks as shown on the left of Figure 5, and we want to move the disks so that this triangle transforms into what is shown on the right of Figure 5. What is the least number of disks that need to be moved to achieve this transformation? Can you predict the number of disks you will need to move for larger triangles? In this problem, students were primarily communi cating through drawings in a way that would have been difficult to communicate about strictly verbal ly or textually. Therefore, a collaborative space

This problem in particular was advantaged by the online medium because it allowed students to drag

Figure 3: Example of student work on a Jamboard with an object - oriented setup

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familiar shared workspace is available, then we can ask students to draw graphs of functions. If the col laborative medium does not have a shared work space, however, we would need to pivot these questions from drawing graphs to asking questions that may be discussed verbally about a provided graph— what do you think would happen to this graph if we doubled the leading coefficient? As such, many of the same strategies of designing thinking tasks in face - to - face thinking classroom settings are therefore transferrable to the online set ting, but an enhanced attunement to the collabora tive medium needs to be considered. The BTC framework evolved from a systematic opposition and reformation of the normative struc tures that permeate classrooms around the world, towards a toolkit of strategies that together guaran tee a thriving classroom environment brimming with student thinking, and in turn, learning. The framework evolved from over 15 years of research in face - to - face classrooms. Since the COVID - 19 lockdown, however, a new evolution has taken place. One that is more sudden and even more globally challenging of various kinds of normative structures. The sudden onset of its incredibly trans formative implications shook our educational sys tems into an unanticipated and uncontrollable reor ganization where everything that was known, now became questioned. Without the comfort of regular face - to - face interactions with students and with shifting landscape of all aspects of society, no longer could the same rules and norms continue. The adaptations were born out of the protocols as sociated with the COVID - 19 lockdown, which caused many teachers to have to reorganize their ways of teaching into a mix of asynchronous and synchronous online learning approaches, at times paired with some limited face - to - face components with various degrees of physical distancing proto cols. For those who had grown accustomed to teaching in a BTC environment, it created an ur gency to find ways to transfer what BTC offered them into their respective lockdown contexts, the Conclusion

Figure 5: Image used to introduce the triangular disks task

with an option for synchronous collaborative draw ing, offering a diagrammatic medium , advantages the thinking space for students for such tasks. Stu dent work completed on Jamboard for this task is shown in Figure 6.

Figure 6: Example of diagrammatic student work on Jam board in breakout group

As may be seen, all the inputs were made with a drawing tool and the drawing required was simple enough to create with a mouse if no touch screen was available. However, it is possible that students uncomfortable with drawing digitally may find it easier to work on their own papers, which could make it difficult for collaboration to occur if they choose not to use their cameras to show each other their ideas. While the above - mentioned thinking tasks are not directly linked to curriculum (even though they may be mapped to curricular outcomes), they em phasize mathematical competencies such as reason ing, describing, formulating, predicting, justifying, and verifying. Just as in a face - to - face thinking classroom, tasks that involve curricular outcomes may be more directly introduced as thinking tasks after a thinking culture is established. When mak ing this transition, however, we need to be likewise aware of both the students ’ technological compe tencies and the affordances and limitations of our collaborative medium. Take for example identify ing tasks for the topic of graphing functions. If a

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most drastic of which was a strictly online context, which is the one we have chosen to address in this article. The transition from face - to - face to online teaching necessitated shifts in some of the thinking class room practices. We have discussed one such prac tice—the types of tasks we use. Our exploration of the adaptations needed for this practice to be effec tive in the online setting indicates a need for con sidering the modes of communication required of students and their technological competence with the tools being used for this communication. In particular, tasks may be designed to emphasize ver bal, gestural, written, object - oriented, or diagram matic modes of communication depending on the technological capacity in the learning environment. We conclude that engendering a culture of collabo rative thinking in an online classroom setting re mains possible even if there are some technological limitations. In the end, the COVID - 19 lockdown has forced upon us a new perspective on which practices continue to work, why they work, and what they offer in the synergy of the overall BTC framework. By creating a breach between the fa miliar face - to - face context and a new online con text, the aspects that make the BTC framework so effective have been illuminated. And, by consider ing these aspects, appropriate modifications have become evident for alternative settings. While there are evidently limitations, the essence of a thinking classroom can live on in novel contexts. Cobb, P., Wood, T., & Yackel, E. (1991). Analo gies from the philosophy and sociology of science for understanding classroom life. Science Education, 75 (1), 23 - 44. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in math ematics Education, 27 (4), 458 - 477. Liljedahl, P. (2020). Building thinking classrooms in mathematics (Grades K - 12): 14 teaching practices for enhancing learning. Thousand Oaks, CA: Corwin Press Inc. References

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (Eds.), Posing and Solving Mathematical Prob lems: Advances and New Perspectives (pp. 361 - 386). New York, NY: Springer. Liu, M. & Liljedahl, P. (2012). ‘ Not normal ’ class room norms. In T.Y. Tso (Ed.), Proceed ings of the 36th Conference of the Interna tional Group for the Psychology of Mathe matics Education , Vol. 4(pp. 300). Taipei, Taiwan. Lithner, J. (2008). A research framework for crea tive and imitative reasoning. Educational Studies in Mathematics, 67 (3), 255 - 276. Mason, J., Burton, L., & Stacey, K. (2011). Think ing mathematically (2nd ed.). Pearson Higher Ed. National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. NCTM, Reston, Va. Pólya, G. (1945). How to solve It. Princeton, NJ: Princeton University. Roddy et al. (2017, November 21). Applying best

practice online learning, teaching, and sup port to intensive online environments: An integrative review , Frontiers in Educa tion. https://www.frontiersin.org/ articles/10.3389/feduc.2017.00059/full

Van de Walle, J., Karp, K., Bay - Williams, J., McGarvey, L., & Folk, S. (2015). Elemen tary and middle school mathematics: Teaching developmentally (4th ed.). Pear son.

Dr. Peter Liljedahl Professor Simon Fraser University liljedahl@sfu.ca

Dr. Judy Larsen Associate Professor University of the Fraser Valley Judy.Larsen@ufv.ca

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COVID - 19 and the Chain Rule are Linked: Understanding R o through Calculus Eric P. Choate

• If R 0 < 1, on average an infected person re covers before infecting another person. New people still become infected, but recoveries outpace new infections so that transmission chains are easily broken, and the outbreak dies out before spreading rapidly. • If R 0 = 1, the outbreak neither grows nor de clines, and the number of infected people re mains a stable proportion of the population. This kind of disease is called endemic . • If R 0 > 1, the outbreak exhibits sustained growth, given a sufficient reservoir of people with no immunity to it. This is considered the definition of an epidemic .

Introduction

If any aspect of modeling an epidemic appears in the popular media, such as the first diagnosis of a deadly virus in the United States (Doucleff, 2014) or the 2011 film Contagion (Soderbergh, 2011) or news articles fact - checking Contagion (Kritz, 2020), it is the reproduction number R 0 . For an au dience that does not know calculus, R 0 has a practi cal definition and an intuitional effect: the average number of new infections that can be traced to a single infectious person. That is, if a college stu dent returns from spring break with a new disease with R 0 = 3, he infects his roommate, his chemistry lab partner, and his math professor when he makes up the test he missed leaving early for spring break. Each of these three in turn infect three more peo ple, for a total of nine new cases at this stage of the outbreak. These nine lead to 27 new cases in the third generation, and so the outbreak begins to grow exponentially.

Estimates of R 0 for COVID - 19 range from 2 to 4 (CDC, 2021).

In this article, we define R 0 in a different way as a parameter—and the only parameter—in a mathe matical model for an outbreak of a disease like COVID - 19, and then we connect the above effects of its value with causes in the model ’ s equations, with an eye toward how it could be useful to an

Given this loose definition, we see three different modes of outbreak behavior:

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Figure 1: A diagram of the two processes—infection and recovery—by which people move from one compartment to another.

• Recovered R (t): People who were infected but are no longer at risk of infecting others.

instructor of high school calculus. The model tracks the numbers of people who have not been infected, who have an active infection, and who have recovered and have immunity, and since it draws connections between the size of these groups and the rates of change of these groups, the model naturally takes the form of a system of differential equations. Students in a high school calculus class (and possibly their teachers) may not have studied differential equations, and so we will not attempt to solve the system. Instead, our focus here is on how the ideas of rates of change and proportionality can be used to create a model, and even though we may not know how to solve that model, we are still able to gain important qualitative information about how that unknown solution behaves by applying our knowledge of the concept of critical values. The definition of R 0 arises in the process of nondi mensionalizing the model, which involves an appli cation of the Chain Rule. A common framework for modeling a disease out break is a “ compartment model. ” Brief explana tions of this approach and other more complicated models may be found in (Abou - Ismail, 2020; Bertozzi, et al., 2020; & Hethcote, 2000). In the most common compartment model—the Suscepti ble - Infectious - Recovered (SIR) model—we let N be the fixed total population of a community. This could be a country, but it could also be a smaller community such as a rural town, a school, an as sisted living community, or a prison. At time t , the people of the community are split among three groups or “ compartments ”: The Susceptible - Infectious - Recovered Model • Susceptible S (t): People who do not have the disease and have no immunity to it. • Infectious I (t): People who have an active in fection that can spread to the susceptible.

Some modelers call R “ resistant ” because it is as sumed the previous infection provides immunity, and others called it “ removed ” because it also in cludes people who have died from the disease. Since these three groups account for the entire pop ulation, S (t) + I (t) + R (t) = N. More complicated models have more compartments and more con necting pathways. We build up the model as a system of differential equations for these unknown functions by connect ing the two aspects of the disease cycle we consid er—recovery and infection—with the rates of change of the compartments. A diagram of these two pathways is shown in Figure 1. People move from the infectious compartment I to the recovered compartment R when they are no longer able to infect others. Thus, the rate of change of R with respect to time is directly propor tional to the total number of infected at a given time. That is, the more infections there are, the more people who recover during a given time peri od. This gives us the equation: The proportionality constant γ is known as the re covery rate , and it is the reciprocal of the recovery time. That is, if it takes on average 10 days to stop being infectious, γ = 0.1 day - 1 . This term γ I appears twice in the model. In the righthand side of (1), it represents the number of people per day entering the recovered compart ment, but these people simultaneously leave the infectious compartment, and so the equation for the rate of change of I must include – γ I . However, that

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equation must also include positive term to account for the newly infected.

The values of β and γ do depend on biological as pects of the particular disease pathogen, but they also depend on properties of the population prior to the outbreak and the response of public health offi cials. If the underlying health of a community is poor, γ could be lower than for a generally healthy community facing an outbreak of the same disease. Conversely, effective treatments and a robust health care system could increase γ. The transmis sion rate β is affected by whether or not the patho gen can be aerosolized and how long it survives on surfaces, but it could be reduced through preventa tive measures such as mask wearing that would re duce the likelihood that a single encounter would result in a new infection or effective social distanc ing, which would reduce the number of contacts. As derived, the model (1) - (3) is governed by three parameters, β, γ, and N . However, we can reduce this model to only one parameter without losing any aspects of the behavior of the system though a process called nondimensionalization . We now ex plain this process and then analyze the behavior of the solution directly from the equations without actually solving the system. The first step is to switch our unknown functions from the total people in each compartment to the proportion of the total population in each compart ment. That is, we rescale the unknowns in the sys tem by the total population as: Nondimensionalization and R 0 Below, we will see how this eliminates N as ex plicit parameter without effectively changing the model. Also, since of these new variables represent fractions of the population and everyone in the population is in one of these groups, we have s + i + r = 1. The key to the second step lies in carefully choos ing the problem ’ s timescale. There is no inherent link between the disease outbreak and how fast the earth turns on its axis. Therefore, instead of meas-

Infection is more complicated to model. Recovery of an individual is unaffected by the total number of infectious or recovered people. The growth of new infections, however, is driven by interactions between the infectious and the susceptible. The more infectious people there are, the more people who are able to someone else, and so it is clear that the rate of change of I is proportional to I . Howev er, no matter how many people are infectious, new infections may only occur if there remain suscepti ble people to catch the disease, and so the more susceptible people there are, the more who will be come infected. Thus, the rate of new infections must be also directly proportional to S . Together, these mean the new infection term transferring sus ceptible people to infectious must be proportional to the product IS . From another point of view, the rate of change of S must be proportional to S , but with a proportionali ty constant that actually depends on I . This “ constant ” consists of two factors. The transmis sion rate or contact rate β measures the number of contacts with another person per unit of time that were close enough to result in a new infection if the contact had been with an infectious person. The other factor in the proportionality is therefore not truly the number of infectious people I , but instead it is the fraction of the total population that is infec tious, or I / N . This is important because it represents the likelihood that the susceptible person ’ s contact was infectious. Thus, β ( IS / N ) represents people leaving the susceptible population and would be negative in the dS/dt equation, but it would be posi tive in the dI/dt equation as these are the people entering the infectious compartment. Therefore, we have the other two equations to complete our sys tem:

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uring time in days, we should pick a timescale that is defined by the problem itself: the recovery time γ - 1 , or the reciprocal of the recovery rate. This al lows us to define a new variable τ = γ t , which plays the role of time but is a pure number with no units because γ is measured in day - 1 and the dimensional time t is measured in days. We then treat of our new unknown proportions s ( τ), i ( τ), and r ( τ) as functions of this new nondimensional time. Now, we focus on dR / dt in Eq. (1). It has two vari ables that we want to replace, R and t . From Eq. (4), it is simple to replace R with Nr . The compli cating factor, however, is replacing differentiation with respect to t with differentiation with respect to τ . The answer is straight out of introductory calcu lus—a careful application of the Chain Rule. If we view our definition of the nondimensional time as τ = yt being a function of t , we may differentiate this function to get d τ/ dt = γ . Also, if we think of r as a function of τ with τ being a function of t , then dif ferentiation of R with respect to t is actually now the derivative of a composition of functions using the Chain Rule:

While β and γ both still appear in these equations, they only appear in their ratio β/γ , which is dimen sionless because it is a measure of the rate of con tacts per day divided by the number of recoveries per day. That is, it is exactly how we described the reproduction number R 0 above—the average new infections per resolution of a single infection. Therefore, we define the nondimensional parameter R 0 as β/γ , and it becomes the only parameter in the system (5) - (7). We can now see how the earlier intuitive effects of R 0 being less than, equal to, or greater than 1 can be predicted from this model and with greater in sight. From Eq. (7), dr / d τ = R 0 is - i = ( R 0 s - 1) i . Since s is the proportion of the population that is susceptible, 0 ≤ s ( τ ) ≤ 1. The importance of R 0 < 1 is now clear: R 0 s ( τ) - 1 < 0 for all τ so that dr / d τ is always negative. Therefore, if R 0 < 1, the infectious proportion i ( τ ) is decreasing, and the outbreak dies out without growing. The endemic case with R 0 = 1 also has this property, but its decay rate will be slow enough that it may not be observed. Now let R 0 > 1. The initial susceptible proportion s (0) is close to 1 if the disease is new to the commu nity, and so we would expect s (0) > (1/ R 0 ) if the disease is new to the community. This means that di/d τ = ( R 0 s - 1) i is initially positive, indicating a rapid initial growth phase. However, from Eq. (6), di/d τ is negative, and so s decreases, which in turn slows the growth rate of i because ( R 0 s - 1) be comes smaller. Eventually, s will pass through a special value we will denote as s * = (1/ R 0 ), which makes di/d τ = 0 . As s continues to get smaller after that, di/d τ becomes negative and the infectious pro portion begins to decrease. Therefore, the infec tious proportion changes from increasing to de creasing, and the First Derivative Test says that i passes through its maximum value. The outbreak then begins to recede because it becomes increas- Herd Immunity

When we plug this into Eq. (1) and replace I with Ni , we see that both sides are multiplied by N γ . Di viding both sides by this eliminates all parameters from the new equation for the proportions.

Similarly, we must now replace the derivatives in Eqs. (2) and (3), but here after we divide by γ , the parameters do not all go away. We are left with:

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