Math Students Hunt For Errors in False Proofs!

Posted on June 27, 2020 by Error Not Found

Communicating mathematics is a crucial part of a developing mathematician’s career. Really, any mathematician’s career. In the classroom, with peers, and at conferences, math students organize their learning and research in order to effectively question and convey concepts that require significant math background. Of course, mastery of the many levels of communication spans everything from talking through word problems with curious elementary schoolers to defending one’s thesis.

In the spirit of celebrating the importance of effective communication, we decided to play a game with some PhD students at the University of Michigan! To test their math communication skills, we selected several “proofs” from around the internet (thanks, reddit) which have subtle errors leading to an ultimately false conclusion. For example, many math students have seen at one point a “proof” that 1 = 0. The volunteers then had to spot the error(s) in the reasoning and do their best to explain it to a broad audience.

Note: only proofs that appeal to a wide audience were selected so that more students can enjoy. There are certainly examples of error spotting in more “high tech” math (see here).

It’s fun to try it yourself! Pause the video before each section and see if you can spot the error(s).

 

Disclaimer: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

Comments Guidelines: The AMS encourages your comments, and hopes you will join the discussions. We re- view comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in AMS, Arts & Math, Grad School, Interview, Math, Math Education, Math Games, Math in Pop Culture, Math Teaching, Mathematicians, Mathematics in Society, Mathematics Online, puzzles, Teaching, Technology & Math, Uncategorized | Comments Off on Math Students Hunt For Errors in False Proofs!

Let’s Take Responsibility For Our Math

Posted on June 27, 2020 by Error Not Found

In an open letter to the AMS Notices, a collaboration of prominent mathematicians and other stakeholders insist that mathematicians and universities suspend any relationship with law enforcement (see here). Their reasoning, as the letter makes clear, is that the tools mathematicians have developed for law enforcement have exacerbated harm and furthered discrimination of historically subjugated communities in the United States. Given the current national conversation concerning the treatment of Black Americans spurred by the murder of George Floyd and others, it’s only natural that these letter writers would re-evaluate mathematics’ relationship to law enforcement.

For some, it may initially seem far-fetched that mathematics, often lauded as a pure and objective discipline, might play a role in divisiveness and even harm. Surely, they argue, mathematicians don’t need to revisit our pre-sheafs and morphisms for fear that we’ve somehow played a role in societal destruction. Well, yes and no. The line between pure and applied mathematics is blurry at best. A theorem in category theory might have implications in optimization and control. An analyst might dabble in applied probability and machine learning. Even abstract mathematicians might relate their work to important scientific motifs in other fields when applying for a grant. Simply put, mathematics does not exist in a vacuum and as its students, we are all partially responsible for its use.

The letter writers focus on PredPol, a clear example of advanced mathematics directly impacting society. PredPol, “the predictive policing company,” boasts that visitors can “join 1,000s of other Law Enforcement and Security Professionals” in using its service (1). Indeed, in 2019, PredPol algorithms were in use by more than fifty police departments (2, 3). In addition to PredPol, there are many companies (Palantir and Third Eye Labs, for instance) that serve as a bridge between mathematics and its applications in policing. Unfortunately, these companies often treat the algorithms and AI they employ as a black box for profit, just as law enforcement treat them as a black box for arrests. As one police Captain succinctly put it, “It’s PredPol, and it’s going to reduce crime” (3).

So, what exactly is the failure of these predictive policing algorithms? As one team of researchers puts it, such algorithms have been “empirically shown to be susceptible to runaway feedback loops, where police are repeatedly sent back to the same neighborhoods regardless of the true crime rate” (4). First, over-policed areas lead to over-reported crime. For example, the Stanford Open Policing Project elucidates a significant disparity in traffic stops for Black Americans compared to non-Black Americans (5). Moreover, activists often point out the long history of racist policing policy that further skews the frequency of incidents involving Black Americans (6, 7). This creates biased crime statistics. Companies then use these biased crime statistics to train reinforcement learning algorithms or to measure the “accuracy” of their crime models. Law enforcement then purchases these flawed models and uses them to inform their policing practices.

A natural rejoinder from advocates of predictive policing is to acknowledge the bias and potential harm, but to argue that the existence of bias only reasserts the necessity for continued innovation of the algorithms in play. As any applied mathematician will say, however, every model is an approximation of reality. Since policing deals with life-altering situations, are we really comfortable with the error of that approximation being human life? I am not. And if the reality we attempt to approximate is structurally racist, is it ethical to build models which reflect that structural inequity?

Moreover, we should remain wary of arguments which allege that predictive policing algorithms only require further refinement. From a purely mathematical perspective, a problem that optimizes for one outcome is an interesting publication. A problem that optimizes for many outcomes is a field of research. A claim that one can eventually solve issues with predictive policing through mathematical research seems grandiose at best. After all, the implementation of such tools relies on the discretion of law enforcement in the first place. And much like a game of telephone, the intent of the original mathematicians involved in creating predictive models inevitably becomes obscured through company adaptation and police implementation.

Unsurprisingly, the misuse of mathematics goes beyond the current predictive policing debate. Internationally, different law enforcement agencies have faced censure for their use of flawed facial recognition software (8, 9, 10). Besides the obvious privacy concerns about facial recognition technology, activists have raised the argument that facial recognition far more often misidentifies darker complexions (8). While predictive policing has been the genesis for productive calls to action within mathematics, there are clearly other ethical concerns which require continued attention.

Mathematics, powerful as it may be, has never been a panacea for society’s ills. So as mathematicians, what can we do? Ideologically, we must humbly accept that the conversation about predictive policing requires a diverse coalition of experts in order to avoid perpetuating harm. In doing so, we acknowledge the limitation of our mathematical expertise, allowing ourselves to learn about issues outside our field. Crucially, we must sign our support for the petition to suspend cooperation with law enforcement, and work to see its goals are realized within our home institutions. In the future, we must ensure we engage only with responsible companies and organizations. In pursuing the question of what companies are “responsible,” we must be sure to solicit the opinions of a diverse array of colleagues and peers, in addition to doing our own research. More broadly, we must support initiatives to diversify our science in our classrooms, at our universities, and nationally.

If you have suggestions for additional actionable ways to address these challenges, please leave a comment.

Edit: Since the writing of this article, the Association for Women in Mathematics has also written an extensively sourced petition (see here) and I encourage people to lend their support. Also, thank you to Michael Breen for providing this New York Times article, which is a humanizing case study detailing the actual harm that results from the misuse and inaccuracy of facial recognition software.

(1) Predpol

(2) Predictive Policing Using AI Tested by Bay Area Cops

(3) Predictive Policing Lacks Accuracy Tests

(4) Runaway Feedback Loops in Predictive Policing

(5) Open Policing Stanford

(6) Black Lives Matters Police Departments have Long History Racism

(7) 13th

(8) Federal Study Confirms Racial Bias Many Facial Recognition Systems Casts Doubt Their Expanding Use

(9) How China Is Using Facial Recognition Technology

(10) The Global Expansion of AI Surveillance

 

Disclaimer: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

Comments Guidelines: The AMS encourages your comments, and hopes you will join the discussions. We re- view comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in Diversity, General, Grad School, Jobs, Math in Pop Culture, Mathematics in Society, News, Social Justice | Comments Off on Let’s Take Responsibility For Our Math

Lockdownward mobility

Posted on May 15, 2020 by Aram Bingham
These days, I feel like I need to think actively and creatively on ways to spend less time in front of a screen. So if you don’t have the wherewithal to make it through another piece of commentary about what it’s like to live through this pandemic, I feel you. But the blog’s been quiet recently, and I thought it might be alright to check in on a few things.

Congratulations!

A lot of us just made it through the weirdest semester of teaching and coursework we will (hopefully) ever have to deal with. Every semester is an achievement, and this one — for all its frustration, heartbreak and awkwardness — counts too. Personally, I felt fortunate to have the thread of continuity with students, classmates and professors as we all adapted, and I was heartened by the numerous acts of consideration, cooperation and kindness that combined to make this thing work.

Unemployment

Like many business and organizations, colleges and universities (especially expensive private ones that sell experience and connections as much as they do education) are desperate to ”go back to normal” in the Fall. But amid the present uncertainty of how this can proceed, lots of academic hiring has been frozen. Without a clear path to resumption of on-campus teaching in the main, we might even expect teaching staff layoffs in measure with the consequent drop in enrolment. This is a scary thought, but speaking as someone that has just tested the job market, it feels especially grim.

Let me back up to offer some comments on my experience with the academic job market this past year, before the whole global pandemic thing. I’ll begin with an anecdote. When I was thinking about doing a PhD in mathematics, the Fall before I applied I reached out to a friend of my older brother who was doing his PhD to ask for some advice on applications. His advice was pretty simple, and probably sound. He said it was best to go to the “most prestigious” school possible, “for the purpose of getting a job afterwards.” (I still have the e-mail.) I found this a bit cynical, so instead I applied to the two places that were closest to where I was living and never really looked back. On the other hand, it looks like I didn’t get a job this cycle, so, uh, touché.

This is not meant to knock my home institution. I have great confidence in my professors and classmates, and I think the education and opportunities I’ve had are top-notch. But we have inherited this system of academic “prestige” which is quintessentially American in its modes of self-perpetuation, class stratification and resource concentration. Like other sorts of inequality, it is both absurd and inescapable. We have all been trained to constantly judge and to feel judged for our pedigrees. Abolishing these biases is a necessary part of our long arc towards a more diverse and inclusive mathematical community. But this is tricky territory when you’re a job applicant, because pedigree and name association is basically what you’re peddling.

Surprise bonus challenge: trying to graduate and secure a job in “just” five years when six is standard at many institutions, especially elite ones. An extra year is more time to develop a research program, present at conferences, and plug into a network before you even start applying. Applying for jobs takes a frustrating amount of time which eats into everything else. Part of the exercise is useful – it is a chance to meditate on your accomplishments, think about what has value, and develop your communication skills. Another part of it is dull and mind-numbing. And then it is at turns brutalizing (strong language warning), making you feel puny and worthless to the club in whose service you have just spent several devoted years.

Here’s another piece of advice I have received multiple times, probably sound: “apply everywhere.” I didn’t do this either: 50-60 total (they call this a “medium amount,” nowadays), mostly research postdocs at large universities where there were people I could imagine collaborating with, in places I could conceive of living. The issue is that many of these were departmental postdocs, meaning there are often hundreds of applicants from all areas of math vying for perhaps just one position. At that point, search committees just don’t have the bandwidth to consider everyone fully, so the shorthands of prestige and the personal connections they may have to your advisor and recommenders become even more significant.

Of course, a fancy institution or a famous advisor are not a golden ticket. We all have to work extremely hard in this business to stay afloat, and part of that work is schmoozing and forging connections. So let me give some advice of my own: if you are going to self-sabotage by, say, avoiding conferences that require air travel on sheer environmental principle, then you should come up with other creative ways to connect with people outside your university. It’s easy to slide into the view of networking (*ack*) as vanity and rank careerism, but if you decide to make it about seeking human connection, it can be the most rewarding part of the job. Mathematics alone may not always sustain you. People, more likely, will.

So anyway, we have this fundamentally weird human market situation: hundreds of people that have very intensive and specialized job training, most of whom would surely excel in the jobs they want, and a bottle with a very skinny neck they are all trying to get through. How is it that so many of us are invited into graduate school where there seems to be abundant funding to support us to attend conferences and summer schools and where we are plied with coffee and sugary treats like creatures whose mental energy is a precious resource: “Look how many problems we have! Will you please help us solve them?” And yet once we have made our modest contribution, some number of us will be expelled from the main quarters of the research enterprise to make room for the next round. Or we can become data scientists.

Aperçu de l’image

I can’t claim to be annoyed or surprised by any of this. Actually, one thing was annoying. For many positions, I never received an e-mail or saw an update to the status on mathjobs even after their searches were concluded. So you might be left hoping for longer than you should, waiting indefinitely unless you go after them. Given how easy it would be to just say that the position’s been filled, and how many separate follow-up e-mails a clear status update would obviate, this move feels lazy and shortsighted on the part of hirers, and I wish they wouldn’t do it. Ok, but otherwise, fine, I guess finding a (good) job is just as hard as they said it would be. To those that have been successful, you have my congratulations.

Extension

The frozen job market, research interruptions, and the specter of recession are affecting all disciplines, so graduate students at many universities (including mine) are organizing to ask for funding extensions for their assistantships. See this op-ed for just a sample of how this is going. If the current disruptions have you concerned about your status in your program, and you aren’t hearing anything from your university, I think it is wise to mobilize with other graduate students to let your administration know! Some of the decision-making on these matters will be out of anyone’s hands, but it never hurts to communicate.

I am fortunate that my department has said it will guarantee at least one more semester of assistantship to me and others who have asked. I had been feeling ready to move on, but I can’t say for sure that I would have a job if not for coronavirus, and so this may be for the best.  If nothing else, I guess at least I get to keep writing for this blog — I’ll take it.

Disclaimer: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

Comments Guidelines: The AMS encourages your comments, and hopes you will join the discussions. We re- view comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in Jobs | Tagged | Comments Off on Lockdownward mobility

Careful what you wish for…

Posted on March 30, 2020 by Aram Bingham

So around two months ago, as the novel coronavirus was just breaking in the Western media, I wrote a post bemoaning the culture of carbon-intensive academic travel. Funny — here we are, barely a quarter of the way into the year, and every conference in sight (including those I had planned to attend) is either cancelled or postponed for the foreseeable future. Let me say for the record: this is not what I had in mind. Sorry, everyone. Still, it’s hard not to feel some awe about the fact that air pollution over China has been dramatically reduced over the past weeks, and that the canals of Venice are shimmering their crystalline Adriatic blue thanks to the lack of boat traffic. My goodness, what happens when we all just stop for a bit?

But more importantly, what are the consequences for mathematics? ; ) While some are predicting that all of the at-home time we have in store will lead to another baby boom, knowing this audience I am going to predict… A MATH BOOM! Let’s face it, there is perhaps no sector of the population more suited to social distancing than mathematicians. Avoidance of public gathering is basically our lifeblood. There’s a reason some of our most renowned and productive research stations are isolated deep within the Black Forest, or in the middle of the Canadian Rockies. Unlike Ariel, we don’t want to be where the people are, because people have a well-known habit of messing with my concentration. This lockdown, we will be getting busy at home reading the papers in that special pile on our desk that only seems to grow in an ordinary semester, finishing proofs of tricky technical lemmas, and polishing up our pre-prints. I would bet on an observable uptick in arXiv uploads resulting from this whole situation, at least for us, the tribe of the portable research lab, pencil and notebook.

Disclaimer #1: Contrary to popular stereotype, mathematics is very much a social and collaborative activity, and there is no reason to believe that isolation is a beneficial precondition for its pursuit.

On the other hand, many of us are busy gearing up to face the bugbear we’ve steadfastly avoided for years — online teaching. After participating in a few trainings about (insert commercial video-conferencing service), I have to say it’s honestly better than I thought it would be, and even the dinosaurs of the department seem to be taking to it pretty well. It seems at least the basic task of communicating course material to students will be achieved at a reasonable level, though I’m still a bit skeptical of the substitute systems for evaluation and feedback on coursework. I’m concerned that the extra distance and technological hurdles will prevent students from taking the opportunities to talk one-on-one with their instructors, and also that instructors may fall into patterns that reduce their levels of availability. Then again, there may be some students that find the modes of electronic communication less intimidating than, say, showing up to office hours. Plus, blah blah millennials blah blah digital natives blah, right?

Disclaimer #2: I know there are many people out there (workers of many stations that do not have the luxury of working remotely, and healthcare workers in particular) for whom this pandemic is no frivolous affair. I wish them health and safety, and I hope my levity does not offend them.

While educators are making the online transition for our courses, it’s a little perplexing that more conferences aren’t doing the same. There are likely others, but I only know of one conference which was planned to be in-person and is making the switch to virtual (and it’s run by hardworking graduate students, on top of it all). I think this is great for at least three reasons. First, as early-career mathematicians, the ceasing of all conference activity for a long period could be injurious to our employment prospects, if indeed there are any jobs left when all the coronavirus dust has settled. Second, this will be a good excuse to have social contact, feel more normal, and less alone. Talk of virtually reviving the graduate student colloquium in my department has come up on these grounds. Third, thinking more long-term, I think something really cool could come out of this. The conference organizers are giving speakers the option to record presentations to upload, and such presentations, if well prepared, could be a real asset to people’s research profiles.

Think of having on your webpage not just a vague one or two paragraph description of your research plus links to papers, but also a 20-minute accessible video presentation on your work. This could take forms much more creative than your typical recording of a chalk talk at a conference, and really open up the world of math research — both amongst ourselves and to outsiders. I realize this is not a radically new idea: the promise of The Internet 2.0 was basically that we were all supposed to become content creators. As is often the case, academicians lag behind culture; but the video conference is a great excuse to think hard about how to produce good expository media about one’s own research.

I imagine this not as a way to supplant conference talks or research papers, but to supplement. The culture of academic publishing is such that research papers read dry and obscure to those outside the subfield, and (as a senior researcher and journal editor recently told me) there are (some) good reasons for this. But very often this style is a tool for exclusion. I am sure I am not alone in having had the experience of a professor handing me a paper or telling me to go read one as a way to make me buzz off. They don’t expect you to come back, and often they are right.

Lucid, informal, and big-picture explaining is usually reserved for talks, but on the one side, we can’t always make it to everyone else’s talks, and on the other (as Andrés points out), it’s a lot of work to put together a well timed and executed talk, only to do it once. It would be nice to be able to capture all of the thought and effort that goes in for posterity. I mean, your own work is the math that you should care the most about presenting — there’s no reason that the online calculus and algebra cartels should have a monopoly on well-produced, expository math videos.

Disclaimer #3: I have no idea how to produce any such media at present, but if I figure out anything cool, I will be happy to share. There was one tip in Mahrud’s recent post, and probably Mohamed Omar knows a thing or two as well.

Much is being made about how this is online education’s big opportunity, but this might also be a kick in the pants to the channels of research dissemination. I expect it’ll be rough at first, but when this dry-run is over, we may have some hard questions to face about how we’ve been doing things. At my university, (at time of writing) we are currently on pause to regroup with classes resuming on-line next week, and the virtual conference is next weekend, so I can’t yet really make any judgements on the tenability of these digital alternatives. In the meantime, I would be interested to hear how it’s going for graduate students in other places. Is your institution handling this pandemic very differently? Did anyone’s semester/quarter just get postponed or cancelled entirely? Is anybody being forced to move? What about plans to take quals/graduate/find a job? Are international students’ visas or applications being affected? Is your advisor taking advantage of the situation to really ghost you now?

I’m sure it’s still too early to really understand what this will all mean, so in the interim I wish everyone peace and safety, and that we can all treat one another with compassion. I hope to see you at the next virtual conference.

 

 

Disclaimer: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

Comments Guidelines: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in Conferences, Mathematics Online, Technology & Math | Tagged , | Comments Off on Careful what you wish for…

Teaching in the Time of Coronavirus, Part I

Posted on March 26, 2020 by Richard Wong

Hi all,

2020 has been a complicated year so far, and things are only going to get more complicated as the COVID-19 pandemic. I’ve been thinking a lot about teaching recently, (as I’m the instructor for a class of undergrad learning assistants) and today I’m here to share my thoughts on how our teaching methods should change as we transition to remote teaching platforms. Later this week, I’ll also share my experiences with the Zoom platform – both as a speaker, and as an audience member.

First of all, no matter what technologies you or your university plan to use, remember that not all students will have access to the internet, or even internet with sufficient quality to participate in videoconferencing technology. It’s therefore imperative that you make your lesson as accessible as possible through as many forms of media as you can. This might include:

  • posting your lecture notes online
  • making lecture slides (and posting them online)
  • recording your lecture

As a quick note on slides, I learned recently that Google Slides offers a closed captioning service that works amazingly well (much better than YouTube’s, for sure). So even if you use Beamer to make slides (as I do), I recommend porting them over to Google Slides just for the closed captioning ability.  Here’s a quick picture of how to enable it:

How to enable closed captioning on Google slides

Also, check to see what services/platforms/support your university is offering (for example, UT Austin is integrating Zoom, and I also learned that Hangouts Meet is offering free premium services as well).

Secondly, know that it’s difficult to suddenly switch to online/remote learning (especially in the middle of the semester!), and that the quality of education will suffer. But we can try to make the best of it that we can. In particular, my recommendation is to focus on engaging your students, rather than on trying to keep up with the pace of content in a normal semester.

It can be difficult to promote student engagement during a video lecture, but one thing you could try using are survey/polling tools such as Slido, Kahoot, etc. (UT Austin has a service called Instapoll). These tools can be used to engage students and to check for understanding on a basic level. However, I would recommend that you don’t assign grades/points to these assessments as again, not all students have the internet access/equipment to participate.

Other ways to engage students include using mediums outside of lecture, such as forum/discussion posts, email, etc.  These are all additional means for students to ask questions and engage with the material at their own pace.  You should encourage your students to use whichever means they prefer, and when possible, tailor your lectures to address the topics/questions that they bring up!

Teaching in the time of the coronavirus will take more time and energy than we might be used to, but in times like these, students will remember and appreciate the compassion and effort we put in!

 

 

Disclaimer: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

Comments Guidelines: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in Advice, Math Education, Math Teaching, Uncategorized | Comments Off on Teaching in the Time of Coronavirus, Part I