Ideas and Strategies for TAing Inclusively and Equitably Online

This blog post is based on a talk that I gave at the Inequity in STEM seminar at UT Austin.  The key ideas come from this Center for Organizational Responsibility and Advancement webinar, led by Dr. Frank Harris III and Dr. J. Luke Wood at SDSU.  However, I have supplemented their ideas to show what we, as grad student TAs, can do to be inclusive and equitable.

As the end of the summer break nears and the fall semester approaches, we, as grad student teachers and TAs, need to prepare for the coming online or hybrid semester.   If you TAed during the spring semester, take the time to reflect on your own experience with the transition to online classes. In particular, think about whether the tools and techniques that you used were effective.

However, for both new and returning TAs, another important thing to think about is whether your online teaching practices are inclusive and equitable. Teaching online is a completely different experience from teaching in person, and it’s not enough to just use your in-person teaching practices on Zoom. On the other hand, it’s also important to not get excited and carried away with new technology (aka Zoom breakout rooms) – you need to carefully consider whether your students have access to the resources and hardware/software to use these technologies.

If you are a course instructor with control over your syllabus, I recommend using the framework of Universal Design for Learning in re-designing your course for online instruction. However, if you are a TA without the power to make these changes, I recommend thinking about the four following tenets in your online teaching practices:

  • Accessibility
  • Building Community
  • Intervention
  • Empathy & Race-consciousness

I will discuss each of these tenets below, and provide a non-exhaustive list of suggestions that go with these tenets.  Please feel free to comment and share any other suggestions you might have!

Disclaimer: Most of this blog post is written in race-neutral language.  This is because most of these suggestions are about inclusivity and equity, and will improve the learning experience for everyone, regardless of race, gender, socio-economic background, etc.  However, we should not ignore the fact that race and cultural identity can be a barrier to accessing resources and opportunities in education.  Therefore, you should read these suggestions with a focus on how they can help counteract and overcome systemic racial inequity.


The first and most important tenet of online teaching is accessibility – in order for your students to learn, they must have access to the course materials.

However, this does not just mean recording your Zoom lectures!  Not all students have access to fast, reliable internet.  Therefore, you should offer accessible, low-data and mobile-friendly materials (such as accessible pdfs).

Similarly, not all students will have access to webcams, microphones, or even a quiet workspace.  Furthermore, you should consider how to best use your synchronous time.  Plenty of good videos teaching calculus exist already, so you should consider using active learning techniques, instead.

  1. Anonymously survey your students about their resources and needs.  In particular, things you should ask include:
    • What technology/software do they have access to?
    • Do they have reliable internet access?
    • Do they have a quiet/safe workspace?
    • Are there accommodations they might need?
    • What are the student’s course goals?
  2. Use both synchronous and asynchronous materials and activities.
    • Make use of discussion forums like Canvas/Piazza.
    • Use active learning techniques.
    • Vary the activities you use, and split your synchronous time into smaller (5-15) minute segments.
  3. Use transcription services.
    • For example, Google Slides offers a free(!!) live captioning feature. Powerpoint also has this feature.
    • Youtube and Zoom also have transcription features for videos.
    • Find out if your university offers transcription services – UT Austin does!

Building Community:

We lose a lot of things in the virtual learning format that are normally taken for granted – for example, we lose having a shared, physical space.  It’s a lot harder to see non-verbal cues to measure student engagement/interest.  Similarly, it’s a lot easier to get distracted online.

Therefore, it’s worth examining our models for learning.  This post in particular is based on the Community of Inquiry” model for (online) learning.  This model posits that the educational experience not only requires a teaching presence and a cognitive presence (aka teacher and student), but also a social presence (aka discussion with peers). Therefore, it is important to build community in your online teaching.

  1. Build student communities that exist beyond class hours:
    • Zoom breakout rooms are not enough!
    • Encourage students to collaborate and create class notes using Google docs.
    • Use discussion forum platforms such as Canvas or Piazza.
  2. Encourage students to form study groups/connect via social media.
    • For example, Groupme is extremely popular at UT Austin.
    • Suggest the use of social contracts for accountability.
  3. Encourage participation in office hours and/or other tutoring services.
    • In particular, encourage students to attend in groups!


Along with building community, another important facet of online teaching is to reach out to students before they are at risk of dropping out or failing.  It is especially easy to stop engaging and/or attending class in the online format.  Furthermore, students from under-represented groups may struggle with seeking help, so it’s especially important to take the initiative to reach out.  Showing that you notice and care can make a world of difference.

  1. Track participation/engagement weekly through low or no-stakes check-ins:
    • Possible tools include polls, Canvas posts, etc.
    • You could ask students to share weekly highs/lows, share their pets, or other ice-breaker games.
    • Make sure you are doing this with accessibility in mind!
  2. Continue to survey students about their needs.
    • Re-evaluate your discussion section goals biweekly or monthly.

Empathy and Race-consciousness:

Finally, the last tenet to keep in mind is empathy and race-consciousness.  It’s important to humanize yourself, and connect with your students, especially in the time of the COVID-19 pandemic.

  1. Acknowledge the difficulties of the pandemic.
    • Know that it affects different people and groups in different ways.
    • Promote self-care resources.
  2. Be accommodating/flexible (within reason).
  3. Be available:
    • Have regular office hours.
    • Respond to emails/messages promptly.

You should also be aware of your own actions – don’t downplay the difficulties that they may be facing, but instead be empathetic and accommodating.

Furthermore, you should be conscious about the examples and people that you choose to talk about!  For example, calculus may have been invented by Newton and Leibniz, but ideas in calculus existed before them, and calculus has been refined and developed by people afterwards.  Furthermore, you can also include people that applied calculus to solve real-world problems.

  1. Be aware of your own actions:
    • Validate, affirm, and empower your students.
    • Avoid microaggressions.  Your behavior can have an adverse impact on others, even in the absence of malicious intent.
  2. Be race-conscious in the examples you use, and the mathematicians you mention. Some suggestions include:
    • Kerala School of Mathematics
    • Maria Agnesi
    • Katherine Johnson, Dorothy Vaughan, Mary Jackson
    • Annie Easley

See below for resources that can help you find other examples of mathematicians to highlight in class.

Further Resources:

In my talk, I highlighted and mentioned several resources available at UT Austin – in particular the Faculty Innovation Center.  I highly recommend learning about the resources and support your institution offers, and reaching out to ask questions.

Resources for inclusive and equitable online teaching practices:

Resources for examples of mathematicians to bring up in your classes:


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 com- mercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in Math Teaching, Social Justice | Leave a comment

The history is the history

Pre-script: This is absolutely not the moment for centering white perspectives in public discourse. That being said, everything I understand about the fight for equality and justice leads me to the conclusion that the responsibility to dismantle whiteness falls uniquely to white people. This post goes with this aim in mind, in solidarity with the uprising to end systemic anti-black racism.

1. Eponymy

Academia does not enjoy the fame of a position on the vanguard of social change. It is at once deflating and encouraging: the horror that must be witnessed, the massive public outcry it takes to incite our institutions to issue statements and undertake reflection that might to lead to action. On the other hand, the fact of these statements and the promise of action signal that our institutions, or the people that steer them, have tuned in, and it gives the opportunity to channel momentum to enact meaningful changes.

I have a proposal for a modest change. It may seem like something that doesn’t directly help the stakes of marginalized people in our community, but (1) it provides something everyone can do (2) it requires little effort and (3) I think it does have a direct impact on all of us. Further, there is a special role for the current generation of young mathematicians to play here, as we will soon become the primary bearers of mathematical culture. I’m talking about the monuments we have in mathematical language which appear to celebrate bigots and racists.

When a society names something for a person, whether it be a land, a town, a building, or an idea, the society is communicating its values. That person made some contribution which the society appreciates, and in return the society honors the individual with the transmortal fame of eponymy. By continuing to use the person’s name as the name of something more permanent, the culture celebrates that person and their legacy.

Over time, cultural values change, and a society may come to find that the once-sacred values a person stood for are no longer acceptable, let alone worthy of honor. Recently, this has become manifest in the efforts to remove Confederate monuments and symbols from public places and the names of Confederate generals and politicians from military bases, municipalities, and universities. The calculus here seems fairly simple. Slavery is abhorrent. We do not want to celebrate (or appear to celebrate) the history of slavery in this country. The psychic injury caused by the prominence of such monuments, not to mention the hypocrisy of their continued presence, outweighs any historic value they might tender. So they must come down. It is farcical to claim that the dislocation of inanimate sculpture amounts to an attack on heritage – heritage of the aggressor culture.

Academicians in general, and mathematicians in particular, have an obsession with eponymy. We also have what seems like a nervous compulsion to insist that as professionals, we are apolitical, and kindly stop bothering us trying to slip politics in with our morning theorem-precursor. This tenet of mathematical orthodoxy, always reminds me of Stephen Colbert, in character as Stephen Colbert, prefacing any discussion of race by making sure his interlocutor knows that he is “colorblind.” Nothing, of course, could be further from the truth. Mathematics was for many centuries the domain of the priest and the patrician – sophisticated leisure for the leisure class. Since then we have variously been court scientists to emperors, patronees of dukes and kings, and in more recent memory, mercenaries of Wall Street, surveillance capitalism, the modern warfare machine, and the retooling of this machine for local police forces to use on citizens with extreme prejudice. Proposition: Mathematicians are not political. Corollary: The pope is not catholic.

If we can dispense with that small issue and accept that everything we do is bound up in the power relations of our society, the political import of our habits and conventions can begin to come into relief. To avoid further pontification, let me just lay out some facts, and you see if you can spot the same issue I do.

2. Some American mathematicians

Benjamin Peirce (1809-1880)

Known as in:

  • Peirce’s criterion: rule for eliminating outliers from a data set
  • Peirce decomposition: decomposition of an algebra as eigenspaces of commuting idempotents
  • Benjamin Peirce Fellow: what Harvard calls its math postdocs.

Less known:

  • Defender of slavery, especially as it allowed an elite to pursue the sciences. (Josiah Lee Auspitz (1994) The Wasp Leaves the Bottle: Charles Sanders Peirce)
  • “My constant text now is I have seen slavery and I believe in it.” (Peirce, quoted in: Louis Menand (2001) The Metaphysical Club, §7.3)
  • “No man of the African Race has ever shewn [sic] himself capable of any advance in the mathematical sciences. If therefore we would insist upon it that the knowledge of God in the physical universe was the duty of all men and that this knowledge could only be acquired through mathematics, and that therefore any man of that race should be compelled to become a student of mathematical science we should labour in vain. We might as well hope to wash out his colour, as we should be attempting to prevent the order of God’s creation.” (Peirce, quoted in Menand, §7.3)

Charles Sanders Peirce (1839-1914) (son of Benjamin Peirce)

Known as in:

  • Peirce’s law : ((P → Q) → P ) → P (axiom that implies the law of the excluded middle)
  • the Peirce arrow: ↓ (symbol for NOR or “not or” in logic)
  • the Peirce triangle: like Pascal’s triangle but counts set partitions

Less known:

  • Shared his father’s views on race and slavery throughout his life. (Menand, §7.3)
  • Fond of quoting racist syllogisms to illustrate the limitations of classical logic. (Menand, §7.3)

Robert Lee Moore (1882-1974)

Known as in:

  • Moore space: a developable regular Hausdorff space
  • the Moore plane: example of a completely regular Hausdorff space that is not normal
  • the Moore method: teaching methodology in which students are only given definitions and theorems and must supply the proofs themselves
  • Robert Lee Moore Hall: home of the mathematics department at UT-Austin

Less known:

  • Strongly in favor of segregation, made many documented racist remarks. (Albert C. Lewis (2002) The Beginnings of the R. L. Moore School of Topology, p. 10)
  • Refused to teach black students or listen to black mathematicians lecture. (Mac McCann (2015) Written in Stone: History of racism lives on in UT monuments )
  • Frequently made anti-Semitic and misogynistic remarks. (Reuben Hersh & Vera John-Steiner (2011) Loving and Hating Mathematics: Challenging the Myths of Mathematical Life, p. 279)

The inclusion of Birkhoff and the individuals below is not intended to equate anti-Semitism, Nazism, or the Holocaust with systemic and anti-black racism in the US. Instead, I feel that the naming issue they present is too related to ignore. They also represent my point of entry to the more universal questions I wish to pose (see below), so I’m hoping an awareness of their biographies may function similarly for others.

George David Birkhoff (1884-1944)

Known as in:

  • Birkhoff factorization: a decomposition for matrices with Laurent polynomial coefficients
  • Birkhoff’s axioms: some postulates for Euclidean plane geometry
  • Birkhoff interpolation: a method of polynomial interpolation of point sets
  • Birkhoff’s theorem: (there are several)
  • G. D. Birkhoff prize: given jointly by the AMS and SIAM for applied mathematics

Less known:

  • Made consistent and documented anti-Semitic remarks (Reinhard Siegmund-Schultze (2001) Rockefeller and the Internationalization of Mathematics Between the Two World Wars: Documents and Studies for the Social History of Mathematics in the 20th Century,  p. 64)
  • Opposed Lefschetz’ election to AMS presidency because he was Jewish, believed he would use the position “to work positively and strongly for his own race. They [Jews] are exceedingly confident of their own power and influence in the good old USA.” (Birkhoff, quoted in: Steve Nadis and Shing-Tung Yau (2013) A History in Sum: 150 Years of Mathematics at Harvard (1825-1975), p. 83)
  • Accused by Jewish scientists and mathematicians (Einstein, Wiener) of anti-Semitic hiring practices while chair at Harvard. (1912-1944) (Nadis & Yau, p. 82)
  • Speaking on his fear of a flood of immigrant scientists in the pre-war period, Birkhoff defended his purpose of protecting jobs for American mathematicians. (Nadis & Yau, p. 81)
  • “[Birkhoff] speaks long and earnestly concerning the ‘Jewish question’ and the importation of Jewish scholars…. He is privately (and entirely confidentially) more or less sympathetic with the difficulties of Germany. He does not approve of their methods, but he is inclined to agree that the results were necessary.” (Letter of Warren Weaver (1934), quoted in Siegmund-Schultze, p. 200)

3. Some German mathematicians*

Ludwig Bieberbach (1886-1982)

Known as in:

  • Bieberbach’s inequality and Bieberbach’s conjecture (now de Branges’ Theorem) on univalent holomorphic functions
  • Fatou-Bieberbach domains, which are biholomorphically equivalent to $\mathbb{C}^n$

Less known:

  • Nazi Party and Sturmabteilung (Nazi paramilitary group) member.
  • Actively campaigned for the removal of Jewish colleagues from universities (e. g. Landau, Schur).
  • Founded and promoted a nationalist Deutsche Mathematik which sought to racialize mathematical tendencies.

Oswald Teichmüller (1913-1943)

Known as in:

  • Teichmüller spaces: moduli for complex/hyperbolic structures on a surface
  • Teichmüller character: a kind of character of $(\mathbb{Z}/q\mathbb{Z})^\times$
  • Teichmüller cocycle: a certain obstruction in Galois cohomology, named by Eilenberg-MacLane
  • Inter-Universal Teichmüller Theory: Mochizuki’s name for his notorious work in arithmetic geometry

Less known:

  • Was a “dedicated Nazi.” Joined Sturmabteilung (SA), Nazi Party (1931).
  • Initiated boycotts of Courant, Landau, while a student at Göttingen.
  • Collaborated with Bieberbach on the application of Nazi ideology to mathematical thinking.
  • Participated in cryptographic work and the invasion of Norway for the Wehrmacht (Nazi armed forces).
  • Killed in battle during German retreat from Soviet Union (Sep. 1943).

Erich Kähler (1906-2000)

Known as in:

  • Kähler manifolds: complex manifolds with closed Hermitian 2-forms, named by Weil
  • Kähler differentials: generalization of differential forms to schemes
  • K3 surfaces: smooth complete surfaces with trivial canonical bundle (The other K’s are Kodaira and Kummer; also named by Weil, who at least had sense enough not to call them, well…)

Less known:

  • Committed German nationalist, volunteered for military service 1935, served for all of WWII becoming a POW (1944-47).
  • Defended the Reich for years afterward, keeping a Nazi navy flag in his office.
  • Believed that the news of Auschwitz came from the Russians intending to defame Germany.

Ernst Witt (1911-1991)

Known as in:

  • Witt vectors: provide a model for the p-adic integers
  • Witt’s theorem: on quadratic forms, extending isometries
  • Poincaré-Birkhoff-Witt theorem: gives a monomial basis for the universal enveloping algebra of a Lie algebra
  • Hasse-Witt matrix: describes the Frobenius map on a curve over a finite field

Less known:

  • Active Nazi Party and SA member, under the influence of Teichmüller.
  • Worked for the cipher department of the Wehrmacht.
  • Though not documented as outspokenly anti-Semitic, Witt took advantage of cooperation with Nazi administration for the benefit of his career.

Other mathematicans with known and documented Nazi affiliation:

  • Wilhelm Blaschke (1885-1962): Blaschke product, Blaschke selection theorem, Blaschke conjecture
  • Helmut Hasse (1898-1979): Hasse diagram, Hasse-Witt Matrix, Hasse principle, Hasse-Weil zeta function
  • Gerhard Gentzen (1909-1945): Gentzen sequent calculus, Gentzen’s theorem

*… and more. See: Sanford L. Segal (2003) Mathematicians under the Nazis.

4. Reckoning

This issue first came to my attention early in graduate school near the beginning of a lecture by a mathematician and teacher I admired. The speaker, Jewish, and having to communicate a mathematical abstraction named for one of the men above (I think it was Hasse), pronounced the concept, defined it, stated the eponym and then inserted the unexpected phrase, “… who, by the way, was a Nazi…”  It need not be explained that white cis male identity, and especially of the WASP variety, acts to insulate the bearer against such small traumas. So imagine my wonder, realizing for the first time that it must be a very odd and troublesome thing to have in one’s livelihood, the constant necessity of naming and honoring individuals who willfully participated in, and even afterwards defended, a system that degraded, enslaved, and murdered your ancestors.1

What do we do about this? What does it mean to honor the intellectual contributions of persons we find morally reprehensible? Is everybody just OK with it? Can we separate the mathematical from the political?

On the one hand, I think we can, but only in the following limited sense. I wouldn’t suggest that we suddenly stop studying some areas of mathematics just because they were touched by white supremacists. For instance, in my outsider’s understanding, Maryam Mirzakhani has some beautiful theorems in an area known as Teichmüller dynamics. Why would we let her forerunner’s Nazism debase the value of her work? Furthermore, as scholars concerned with our history, we should keep track of who contributed which ideas and when, and try not to let our political biases color the scientific record.

But we can do all of this without having these names constantly on our breath. Frankly, the usage is at times entirely senseless and gratuitous. Take the Peirce arrow – I mean, I’m sorry, but it’s a freaking arrow. Does it really matter if he was the first one to ascribe a particular logical meaning to it? Notation choices get trophies now?

I am tempted here again to draw the parallel with monuments to slavers, Confederate soldiers and politicians, and segregationists. But Our Problem is actually a little bit trickier than that: in the other context, the statues honor individuals precisely for their contribution to upholding a racist social order, making the moral imperative to remove them much clearer. In Ours, the conferred honor appears skew to any objectionable aspects of the honoree’s character, and so renders the orthodox agnosticism towards the content of that character defensible.

But here’s the thing: language is powerful. It can cause alienation and injury, and we ignore this to our collective peril. Consider the black (1) undergraduate who sits in combinatorics class listening to their white professor go on about the “stars and bars”; (2) graduate student who goes to work and study every day in Robert Lee Moore Hall; (3) postdoc who is lent the “honorific” Benjamin Pierce Fellow (people with this title on your CV – hi). Does it seem to them like mathematics is trying to become a more diverse and inclusive place?

So again, what do we do about this? As much as we might like to tear down the monuments to bigots and fascists and erect new ones for the heroes and martyrs of the moment, I don’t think this is the right answer for mathematics. We might easily be led down the path to witch-hunt and a new noxious puritanism, or alternatively earn ourselves the absurd task of trying to distinguish between “full-blown racists” and mere collaborators.

Instead, we can accept the idea that individual, flawed human beings might not be fit for the immortal ideals of our collective imagination, and stop using the names. But not just the names of racists, misogynists and the like: stop using all the names. We can construct better, more poetic and descriptive names from the bare elements of language as replacements for the myriad dead white men. In many cases, alternatives already exist, and we merely have to insist upon a preference. For example, a “Hasse diagram” is also called an “ordering diagram” (or also a “poset graph”), and the latter actually tells you what information the diagram communicates. The deployment of a name as argot seems here a deliberate attempt to alienate outsiders, or at least gate-keep by forcing indoctrination into the cult of white male worship upon those that wish to persist.

Coming up with descriptive names for some of the lemmas and theorems will require a bit more finesse, but we’ve done it before (here’s looking at you, snake lemma), and I bet we can do it again. We can also stop naming new things for people immediately, and this is where the younger generation can play a key role – if we build no more pipelines, we will eventually stop burning oil. If you discover/invent a new mathematical object, have the courage and scholarship to find a name that actually conveys some of its meaning. Don’t name it for your adviser, your hero, or the author of some paper you read. Your idea is more perfect than they can ever be.

Our use of surnames as jargon has truly reached levels of self-parody. You may have noticed that it was impossible for me to briefly mention the work of the mathematicians above without recourse to several other proper names. And scientific eponymy is fraught for reasons beyond the political. To cite just a few:

  1. Concepts are often not named for (all of) their originators, adding confusion rather than clarity to historical record.
  2. Full proper attribution in naming to all parties involved is much less wieldy, and (again) conveys much less information than succinct descriptive terminology (compare: the Albert-Brauer-Hasse-Noether theorem and the four color theorem).
  3. Persons’ names become overburdened, compounding the confusion (as in, um, which Euler’s theorem? Which Borel?).
  4. Bias which denies the contributions of women and other systematically excluded groups (*euphemism: hidden figures*) inevitably permeates the practice.

Moving away from our problematic love affair with eponymy seems like a pitiful first step in view of the challenges our society faces. But it isn’t nothing. I find it liberating because we also don’t have to ask permission from anyone in charge for this, unlike (it seems) so many other issues of injustice. It is rather an ancient tradition of youthful rebellion for a generation to differentiate itself in language (the “youthful” part is at this point, however, for many of us, debatable).

We can’t remake the history, and we can’t expect our elders to transform the system for us all at once – it is the system in which they, after years of their own struggles and pushes for change, now occupy a comfortable status. But we can transform the language, a little bit at a time. Onward, friends – to Wikipedia!

 [1] I do not know of any mathematicians that enabled, supported or vociferously deny the Armenian genocide. ↩︎

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 com- mercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Posted in Math History, Mathematicians, Mathematics in Society, Social Justice | Tagged | 3 Comments

Math Students Hunt For Errors in False Proofs!

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 | Leave a comment

Let’s Take Responsibility For Our Math

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.

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Lockdownward mobility

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.


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.


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.


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.

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