If I won the lottery, I would…

Chernoff, E. (2019). If I won the lottery, I would…. En J. M. Contreras, M. M. Gea, M. M. López-Martín y E. Molina-Portillo (Eds.), Actas del Tercer Congreso Internacional Virtual de Educación Estadística. Disponible en www.ugr.es/local/fqm126/civeest.html If I won the lottery, I would… Si ganase la lotería, yo… Egan J. Chernoff University of Saskatchewan, Canada Abstract This article is an exploration of the work-related consequences of me winning the lottery. As detailed, depending on which lottery I hypothetically won would dictate whether or not I quit my job. In the one scenario where I do quite my day job, I imagine a scenario where I look back at an interrupted career in probability (and statistics) education. Topics pondered include: gambling education, seminal articles, the changing nature of publication and conference travel, the old guard of stochastics education, missed opportunities (e.g., sports analytics education and consequential probability) and the grand narrative of school mathematics. Keywords: gambling; conference travel; consequential probability; The Math Myth; sports analytics education. Resumen Este artículo explora las consecuencias relativas al trabajo si ganase la lotería. Como se detalla, dependiendo de qué lotería gane hipotéticamente, dependerá de si dejo o no mi trabajo. En un escenario en que deje mi trabajo diario, imagino un escenario en que miro una carrera ininterrumpida en educación en probabilidad (y estadística). Los temas posibles incluyen: educación para el juego, artículos seminales, naturaleza cambiante de las publicación y viaje a conferencias, la vieja guardia de la educación estocástica, oportunidades perdidas (e.g., educación analítica deportiva y probabilidades consecuentes) y la gran narrativa de la matemática escolar. Palabras clave: juego, viaje a conferencias, probabilidad consecuente, el mito matemático, educación analítica deportiva. 1. Introducción This paper is a thought experiment, based on a simple question: “If I won the lottery, I would…” For those of you not familiar with the Canadian lottery landscape, there are three main lotteries. Of the big three, that is, Lotto 6/49, the provincial versions of Lotto 6/49 (e.g., BC 49, Western 6/49, Atlantic 49 and others) and Lotto Max (Akin to the provincial versions of Lotto 6/49, there are two provincial versions of Lotto Max, known as Québec Max and Western Max). Lotto Max is the lottery with the largest jackpot. Held every Friday, for a mere $5, Canadians have the opportunity to win, at a minimum, $10 million, and, at a maximum, $60 million. On a related note, the odds of winning the Lotto Max Main Jackpot are not great. Those who buy a ticket must choose seven numbers from a field of 49; and, as such, the odds of winning Lotto Max are 1:85900584. The focus, here, on Lotto Max is purposeful because it is integral to the thought experiment. Please don’t get me wrong, yes, it would be nice to win any of the big three Canadian lotteries. Take Lotto 6/49, for example: the odds of winning, at 1:13983816, are much better than Lotto Max; there are two draws per week for Lotto 6/49, as opposed to once a week for Lotto Max; and, it only cost $3 to play. And while, yes, the largest single 2 If I won the lottery, I would… jackpot in Canadian lottery history was a Lotto 6/49 jackpot of approximately $64 million, the average jackpot is just shy of only $10 million and the minimum jackpot is a paltry $5 million. Initially, it might sound absurd for me to using words like “only” and “paltry” when discussing $10 and $5 million. But, to be honest, I am not going to quit my job if I won $5 million playing Lotto 6/49, which brings me back to the thought experiment. Lotto 6/49 If I won the lottery, I would… If I won the minimum Lotto 6/49 jackpot, recall: $5 million, I would probably put the money into some high interest savings account and, yes, live extremely comfortably for the rest of my days. Sure, I would quickly pay off the mortgage I have on my house. I would get Kristen (my wife) to retire as soon as possible, of course. I’d probably buy a new car; nothing fancy, just new (e.g., a 2019 Subaru Forester). Make sizeable donations to amazing animal charities (e.g., The David Sheldrick Wildlife Trust). And, I would have all the latest and greatest expensive technology (e.g., iPad Pro, etc.). I would not, however, quit my job here at the University of Saskatchewan. Sure, I would probably look into whether or not I could buy myself out of my teaching duties — this way I could focus all of my time and efforts on reading and writing about the teaching and learning of probability. If this wasn’t possible then I would look into whether or not I could fund my own endowed chair or professorship. This would run, I’m assuming, into some ethical issues. Deterred, I would probably just accept my regular assignment to duties and my day to day activities wouldn’t look all that different from what they look like today. In this particular scenario, I see myself continuing to conduct research in the field of probability education. (I think you see where this is going.) Lotto Max If I won the lottery, I would… Let’s say, rather than winning the jackpot minimum for Lotto 6/49, that I win the Lotto Max maximum jackpot, that is, $60000000. Sixty million dollars! Sure, the bulk of the money would still go into savings and investments. We would make even more sizeable donations to amazing animal charities. I would get Kristen to retire the very next day. We’d probably have a super-fancy car (like a Volvo or a Lexus). And, of course, we would buy one of those oceanfront houses just below the endowment lands of the University of British Columbia in Vancouver, British Columbia. We would also have an oceanfront cabin on one of the Gulf Islands that we would visit frequently. Setting our life back up on the West Coast of Canada would be possible because, of course, I too have retired in this scenario. Not only am I no longer working at the University of Saskatchewan, upon winning $60 million, but I would also be done with any and all current or future investigations in the field of probability education. Done! 2. Saying goodbye to probability (and statistics) education I understand that the passage from researcher in the field of probability education to lottery winner would not take place overnight. There would be a number of loose ends that would need to be dealt with. Case in point, I have a few conferences already on the books for this year and next. I am reviewing a number of manuscripts for various mathematics education journals. And let’s not forget about all the emails that keep Egan J. Chernoff 3 incessantly showing up in my inbox. Slowly, though, over time, my connection to the world of probability education would slip into the background. I like to tell myself that I would definitely keep up with new articles as they were published. (Although I would have given up my university access to the articles, my new found wealth would probably allow me to pay the exorbitant prices to access the articles stuck behind paywalls.) It is more likely, though, that I would deep dive into new passion projects that arose from my new station in life. I’m not sure exactly how long it would take, but I do see a lottery based scenario where I am became completely cut off from the world of probability education for decades. There’s this image that is burned into my brain. After all, even with $60 million burning a hole in my pocket, I won’t be able to avoid Mother Nature and Father Time. I picture myself much, much older than I am today. My hair has turned completely white, I’ve keep most of it, and even shed a few pounds. I see myself sitting in a nice wooden rocking chair with a red, plaid blanket over my lap. I’m not reading. I’m not listening to the radio or a podcast. I just am looking out at the ocean from the top floor of my home. The wood that surrounds me is lit by sunlight. It’s just me and my dog. And, as I’m sitting there, I’m thinking. More accurately, I’m thinking back. Having shut down a career in probability education at a relatively young age, it’s conceivable that one would start wondering about what might have been. After all, some have argued that the field was just coming into its own. There would probably be time spent projecting how many articles in refereed journals would have published if it weren’t for winning the lottery. There would also be thoughts about whether or not the career path given up would have ever led to a keynotes presentation at a major mathematics education conference. Thankfully, these vain initial thoughts about life in the field would give way, eventually, to thoughts about the field, that is, thoughts about research in the field of probability education. The remainder of this article, then, is the wonderings of an imagined, future version of myself. An older me who is looking back on the field of probability education research, the field that I completely abandoned for decades, because I had won the lottery. Stated in more confusing terms, in what follows, I present a personal look back at a quizzical look forward at the field of probability and statistics (mostly probability) education. Given that prominent researchers, especially in publications important to the field (as I best remember it), utilized (wish) lists, I too have made a list, albeit in no particular order. Let’s begin. 3. Gambling education Based on my unique circumstances — having won the lottery, that is — one my first thoughts will lean towards the connection between the lottery and probability education. I’ll have a quick thought reminding myself to finally read The Improbability Principle and then I’ll begin by wondering what would have happened had I won the lottery but not quit my job as an Associate Professor at the University of Saskatchewan. Obviously, my lesson for future math teachers on winning the lottery would need to be rewritten. At the time I left, I began my lesson telling my students that I have a full proof plan for us, as a class, to win the next drawing of the Lotto Max. I proceed to tell them that we begin by pooling all our money together so that we can buy as many tickets as possible. I tell them that we can take out loans, sell cars, etc. so that we can amass the most amount of money that we can. Pull out all the stops! In a room of 80 people, after all, we should be able to put enough cash together. After we work out the Lottery 4 If I won the lottery, I would… calculation we find, surprisingly, that we are approximately $85900584 short. Currently, the lesson gets a decent laugh and, I believe, the odds of winning the lottery hit home for some students. However, if I’m standing at the front of the room with $60 million hanging out of my pockets, the laugh just wouldn’t be the same. Thinking about the lottery in this manner, my next thoughts would lead to gambling, in general. Even before I left the game, I’ve always been perplexed about the lack of gambling education that was provided in schools here in Canada. We go to extreme lengths to teach future citizens about a number of different subjects, including health and physical fitness, history (Canadian and otherwise), literacy and many more, to better inform their future life. However, we do not dig into the perils of gambling. Instead, Canadians have to find out about gambling the hard way through the school of hard knocks. Perhaps playing the lottery could act as a gateway investigation into gambling education. Speaking of gateways, I would wonder as to whether there were pious reasons for not teaching gambling in school and how much society had or has not changed around gambling. Thoughts, though, would lead back to probability education. As I’m sitting in my room, looking out over the ocean, I would rack my brain trying to figure out who from the field of probability education would be best equipped to have had a good run of research and become the face of gambling education. First off, this person would have to be familiar with much of the research in the field of probability education. They would also have to be extremely well versed in psychological research, as well. These two key factors would provide me with an aside where I wonder about whether the connection between psychology and probability education has strengthened or waned since I left the profession. After that brief aside, I would, once again, wonder about developments that have taken place in the outside world over the years. Suspecting, here and now, that the lottery and gambling and related activities have moved almost if not entirely online, I would, when looking back, add a third factor about the researcher being well versed in the role of technology in teaching and learning probability and statistics and well as in gambling. After going through a list of the usual suspects, my thoughts about the lottery, which acted as a gateway back into the field of probability education, would move on to other subjects. 4. The changing face of publications I’m not going to deny it, I will definitely spend some of my time thinking about certain projects that I did complete before winning the lottery. Sure, there will be some time spent wondering which and to what extent certain publications were referenced, if at all, during the intervening years. Knowing full well that if something I did had become referenced all the time that I would have been contacted by somebody at some point, and having not been contacted by anybody and any point, thoughts would quickly move to the changing face of publications in probability education. Everything was in transition at the time of my departure. Libraries, yes, still had row after row after row of books. Libraries also had many rows of journals. At the same time, though, the majority of the students were accessing these materials with the electronic devices that they carried around with them all the time. Should option number one be: First, get to the library; second, find the location of the material that you are looking for with a reference device; third, sojourn the library looking for the material; find the material; make sure that you are interested in the material by reading some or all of it; lastly, capture the material for use at a later date. And should option number two be: an article just appeared on my device in my hands. There is no doubt that people Egan J. Chernoff 5 will opt for option number two. Also at the time of my departure, a weird shift was taking place: ease of access was starting to dictate the publications success. As mentioned, I had noticed a shift when I was leaving. The references that I was finding at the end of their papers had begun to change. The bulk of the references were no longer to articles that were found in paywalled journals from long ago or those just recently published. Instead, the bulk of the references were to online articles that were freely accessible. Having tested this scenario out, I would go through reference list and, sure enough, they were full of articles that just a few clicks away on my most portable of portable devices. Advances in technology, clearly, had not slowed down in the subsequent years since my testing of the reference shift, which got me thinking back to the physical nature of certain projects that I had completed. Around the time I left, they said vinyl was dead; they were wrong. They also said that the physical manifestation of the book was on the way out, as well; they, too, were wrong. Years on from when I left now, though, the advances with reading books and articles on digital devices has put to shame the experience that was taking place. Sure, people are still reading books decades later from when I left, but the physical book has become like vinyl, that is, books have become the purview of nostalgic diehards. The book reader became the kind of person you see in a special interest story on the news. The other current trend, the ever surging dominance of audiobooks and podcasts will also have a large impact on who, if anyone, is still holding onto that big, yellow bound, paperweight known as Probabilistic Thinking: Presenting Plural Perspectives. I remember the day when my editor copies of Probabilistic Thinking arrived in the mail at my office. People in my office joked about lifting the box with my legs and not my back (it was that heavy). Given a number of copies, they took up a lot of space. And I have to admit, there was something satisfying about the physical nature of the book, especially the thickness. And it made quite a thud when I casually tossed it on my desk after looking it over. I also, that day, got a copy of the .pdf version of my book. It just wasn’t the same. There was no sense of thickness to the digital file that sat on my computer desktop. I would have to open the file and scroll and scroll and scroll but even then something was lost in the experience. The digital file, another potential option to measure the book’s heft, was also much smaller than expected. In fact, given the email limitations of the day, the book could easily be attached to any email that I wanted to send and, just like that, a copy of the book could be sent to the other side of the world. Getting very nostalgic at this point, I’ll probably get up from my chair to go look for my one copy of the big yellow book. Displayed, not necessarily prominently, but definitely where it could be seen, I’ll blow off all the dust that has accumulated on the cover before I crack it back open and start to flip through the pages. 5. The seminal article As I flip through the pages of the big yellow book, I’ll wonder which article (if any), and to what extent, became most referenced by those in the field of probability education. My thoughts would extend, naturally, to wondering about which article, of all the articles, became the seminal article in probability education. Sure, the seminal article may have been published in the period of time since I left probability education, which would prompt me to look things up and see what the big article that I missed was all about. If the seminal article in probability education was published after I left then I know that I would read and, to the best of my ability, try to 6 If I won the lottery, I would… make sense of the new philosophical interpretation that was presented or how quantum computing played a role in the teaching and learning of probability or whatever made it “the” article. Probably not being able to fully comprehend the piece, I would do my best to dip my toe back into the water. However, not wanting to look up all the other, new, unfamiliar references supporting this seminal piece, I would just succumb to not being able to fully comprehend the importance of the work. The other scenario, though, is the one I would be able to sink my teeth into a bit more. Perhaps the seminal article in probability education had already been published at the time of my lottery-based leaving of the field of mathematics education. Thinking of all the usual suspects would also get me thinking about all of the different ways to identify an article as seminal. Biased because of my work, I would begin by thinking about the beginning, not the beginning-beginning, but to the Tversky and Kahneman’s article published in the journal Science that had such a large impact on the field of probability education. Seminal, sure, but not really an article in the field of probability education. I would then pour through all the different major contributions housed in articles and wonder whether and to what extent this research permeated not only probability education but also the field of mathematics education. To do this I would start to look up research syntheses. Engrossed in research syntheses, and recognizing the varied definitions of seminal, I would go through all of the articles that I had read well getting acquainted with the field. In doing so, I would recall Shaughnessy’s article in the first handbook of the National Council of Teachers of Mathematics. There is no doubt, the article is well written, well organized and well referenced; however, there is something intangible about the timing of the article, as well. Perhaps related to the age of the field of mathematics, the age of the field of probability education, what was happening in related fields such as psychology, and other factors I would ponder the seminal sense of the article in an Iliad and Odyssey fashion. Digging through these syntheses would draw my attention to the fact that at one point, before my leaving, my name started to pop up in a few handbooks. Not being able to resist, I would start to wonder about what happened to the research I, personally, was focused on just before I left. 6. Consequential probability Looking back on the my writing, right at the time of my lottery win, there was one idea that I was floating around that never came to fruition after the insanity of winning the lottery started to set in. If there was one piece that I didn’t finish, one that I wish had, it was the piece on, what I was going to call, Consequential Probability. Not necessarily a theoretical or philosophical interpretation of probability, consequential probability was the type of probability that I wish students were learning in classrooms. Given that it was that this one piece was the one itch that I never got to scratch, I still had a copy of the beginning of the paper with me that sat in my desk for all these years. Here’s an excerpt: Imagine, if you will, a standard deck of 52 playing cards. Consider, now, the following two scenarios. Scenario one: a card is drawn from the deck; the card is replaced; then a second card is drawn from the deck. What is the probability that the second card drawn is a king? Spoiler alert! The probability, in this particular scenario, scenario one, to nobody’s surprise, is 4/52 or, if you like, 1/13. Scenario two: a card is drawn from the deck; the card is not replaced, but, rather, is placed (faced down) beside the deck; then a second card is drawn from the deck. What is the probability that the second card drawn is a king? Well, the probability that the second card drawn is a King, denoted P(K2), is 4/52. Egan J. Chernoff 7 We see, then, that P(K2) is 4/52 in scenario one and P(K2) is also 4/52 in scenario two. In other words, for this particular problem, the probability that the second card drawn is a King is the same whether the first card is replaced, as in scenario one, or not replaced, as in scenario two. We denote this the The Replaced Equals Not Replaced Problem or, more succinctly, the R=NR Problem. Based on our experiences with the R=NR Problem, to some, the answer is intuitive; but, to others, also known as the vast majority, the answer is counterintuitive. Primarily, the counterintuitive nature of the R=NR Problem stems from the probability being the same whether the card is replaced or not, which is anathema to the secondary intuition that is developed when the teaching and learning of probability hinges largely upon the overarching bifurcation of “with replacement” or “without replacement” leading to different probabilities. Cementing this counterintuitive nature, the answer to the R=NR Problem remains the same as the problem is extended. We ask that you consider, now, a third scenario to the R=NR Problem. In scenario three: a card is drawn from a standard deck of cards; the card is not replaced, but, rather, is placed faced down beside the deck; a second card is drawn from the deck, which is also not replaced, but, rather, placed face down beside first card that was not replaced; then a third card is drawn from the deck. What is the probability that the third card drawn is a king? That’s right, P(K3)=4/52. Our final extension to the R=NR Problem, which we call scenario four, extends the problem in a similar fashion to the extension from scenario two to scenario three. In scenario four: a card is drawn from a standard deck of cards; the card is not replaced, but, rather, is placed faced down beside the deck; a second card is drawn from the deck, which is also not replaced, but, rather, placed face down beside first card that was not replaced; a third card is drawn from the deck, which, again, is not replaced, but placed face down beside the two cards that were not replaced; then a fourth card is drawn from the deck. What is the probability that the fourth card drawn is a king? Once again, the answer is 4/52, that is, P(K4)=4/52. As demonstrated, the solution to all four of our scenarios to the R=NR Problem is the same: 4/52. As such, our following general discussion of the solution, while specific to scenario three, applies to both scenario two and scenario four. The probability that the third card drawn is a King, in scenario three, is predicated on the first and second card, the ones that are not replaced, being placed face down. Worthy of note, whether the card was actually placed “beside the deck” is, for all intents and purposes, irrelevant. In other words, the cards that are placed faced down could, if one so chooses, be placed on the floor, on an adjacent table, at the back of the room, in another room, on a wall or wherever one sees fit. Wherever the first and second cards are placed, however, it is crucial that the cards are placed faced down. In addition to the card remaining face down, it is also of vital, vital importance that, in the act of placing the cards face down beside the deck, one does not get a peek at the card… It’s at this point that the article delves into the Men in Black movie franchise and what a neuralzyer is and what it does, which is key to erasing the memories of anyone that peeks at any of the cards because, of course, the information obtained changes the probability that the third card drawn is a King from 4/52. Anyways, like I said, interesting idea and it’s the one that, had I not won the lottery, I’d have wished got out there for all to read. I am pleased to say though, looking back, I did enjoy chatting about this problem with Sir David Spiegelhalter over beer and chicken wings in Arizona, USA before I left the profession. Speaking of those that might leave the profession, I got to thinking about (and said respectfully) the old guard of stochastics education. 7. The old guard Now old myself, I would definitely spend some time thinking about a unique situation in the field of probability education, and statistics education, for that matter, that was taking place just at the time of my leaving the field. Essentially, and again this is said with all due respect, the majority of the old guard was at or near retirement. All of the 8 If I won the lottery, I would… major players in the field, at the time of my leaving, were getting older and contemplating retirement. This would lead me towards two thoughts in particular. My first thought about the who’s who of probability and statistics education all leaving the profession in a rather short time span led me to wonder about the people that replaced them. I would spend time wondering about the next who’s who of stochastics education. Interested in their academic lineage, I would look to see if any of the new guard had worked with the old guard. I would also be interested in any new players on the scene who didn’t necessarily work with the old guard yet were able to establish themselves in the field. Interest would also lie in the work that they were conducting. Essentially, there would time dedicated in my pondering to who was standing on the shoulders of which giants and whether or not they had personal connections in any manner. Less familiar with the new who’s who and more familiar with the old who’s who, I would wonder about one other thing. For a group of people that are well aware of various cognitive biases, I would wonder if the group, as a whole, fell prey to creeping normality and whether they had organized one last hurrah. In other words, time, for me, would be spent looking for that one final collaboration, before the who’s who all began to retire. This collaboration that I was seeking would house some work from all the major players in probability and statistics education, before it was too late. Given the timing of everything, I would first begin my search by looking at the different book series in mathematics education. Familiar with the Advance in Mathematics Education Series, I would fist look there to see if, perhaps, a volume where the old guard put together a project that not only helped encapsulate the hard work and efforts of a pivotal generation in the field; but, also if they had maybe looked forward to let the new guard know about the things that interested them even though they would not necessarily be the ones conducting the investigations. If not found in that particular series, I would to The Mathematics Education Library series to see if the material was housed there. If I did not see the project in either of those series I would, I think, honestly get a tad excited because that meant that, perhaps, something I’ve been advocating for some time had perhaps come to fruition. Excitedly, I’d start telling my computer to look for and report back on any and all ICMI Study Series that had been conducted since my big lottery win. As I listened and watched my computer show me all the different books that resulted, I would hold out hope that the band got together one last time and produced an ICMI Study on probability education. Yes, in 2011, a statistics education study was completed, but I would still be holding out hope for a probability education analog. Given the timing that was discussed, that is, the leaving of all the members of the old guard, it would seem like a perfect venue for one last hurrah. Thinking even bigger, my thoughts would then tend to an ICMI Study on Stochastics where not only the who’s who of probability education but also the who’s who of statistics education all got together, including some of the newer generation, to put together a seminal book that would, generations and generations from now, be referenced over and over. Not finding the book, would lead to thoughts about whether such a project was proposed, who was in the running for editing such a project, and the reasons as to why such a project never really came to fruition. The timing was perfect! Alas, my disappointment would not last too, too long because, after all, there were many other topics that garnered my attention — like conferences. Egan J. Chernoff 9 8. Conference activity As I looked back to my time in the game, I know that I would fondly remember my memories from various mathematics education conferences. These memories, of course, are lovely mixture of both personal and professional experiences. My very first major conference was important for my career, and as a kind who grew in the smallish town of Kamloops, British Columbia, seeing Prague was a big moment, personally. I do remember, however, the lack of talks dedicated to probability and statistics education in Prague. Similar memories existed for certain North American conferences. Sure, there was a dedicated group discussing probability and statistics education at the North American Chapter of the International Group for the Psychology of Mathematics Education, but as that group fizzled out so did my attendance at those conferences. At the same time, though, I was making my way to and getting involved in the dedicated Topic Study Groups of the International Congress on Mathematical Education and Topics at the International Conference on Teaching Statistics. These conferences were different. Fondly looking back at the conferences, recalling memories of meeting major names in the fields of probability and statistics education (and just how polite everyone was), me and the computer would spend timing sifting through all the subsequent conferences that I had missed. All the great places that I did not get to travel. I mean, sure, with my lottery winnings I had the ability and did travel rather extensively once away from the field. But, the fun part of attending conferences, is attending places that you may not have on your list but went to anyways. Sure, for example, I had spent some time in Phoenix during the cold Canadian winters but heading over to Flagstaff was a direct result of a conference. Scrolling through all the conference proceedings, particularly checking out the names and the titles of the keynotes for these big conferences, I would start to ponder about how the conference scene had changed during my time away from the field. As a person concerned about what is happening to the planet, I am in the unique position of being able to afford to offset all my jet setting around the world. But, I’m lucky, I won the lottery. I wonder how the rest of society saw the continuous flying of researchers to different corners of the world to talk, time and again, to the same handful of individuals who were in the same room from all over the globe. Naturally, this view point is not just restricted to those in probability and statistics education, all academics would fall under this criticism. I suppose the viewpoint of everyone’s respective university would dictate to what extent travel was either supported or not supported. There is also the individual, though. Intrigued by the possibility, I would begin looking for a person, anyone, who had eschewed the traditional conference scene for environmental reasons. Whether or not I found such a person, I would be interested in how their curriculum vitae might differ from others. While suspecting that they chose to just write more articles in refereed journals, maybe not being part of the scene would impact their impact their vitae in ways I had not imagined. If and how such a person did “attend” conferences would also be of great interest. The technology, at the time of my leaving, was there for conferencing without travel. Look, the technology at the time was not great. Arguably, the technology was in its infancy. Certain programs, for example, Zoom, were much better than the more mainstream programs at the time, such as Skype and FaceTime, but they were also