Math for Crime Fighting


Credit: Tony Webster/Wikimedia Commons

On New Year’s Eve 2019, abc13 News posted a story about using mathematics to tease out the details of a crash that killed a father and two young boys. The driver who hit them was sentenced to 60 years in prison for their deaths.

“We were able to clearly show that (he was) driving almost double the speed limit and being intoxicated. Remove either one of those and I don’t think this crash happens,” Lt. Paul Adkins of the Texas Department of Public Safety told article author Marla Carter.

Reading about this case made me think about the different ways mathematics can be used to analyze or solve solve crimes. Here are just a few of the pieces available on the topic.

If you haven’t already, check out Anna Haensch’s “Some Math About Guns,” which she wrote in 2018 for this blog.

“Catching criminals with maths” by Hugo Castillo Sánchez, David Pérez Esparza, Rafael Prieto Curiel and Sanaz Zolghadriha.

This post for the Chalkdust Magazine blog covers the mathematics of organized crime networks and gang rivalry, bloodstain pattern analysis, how Newton’s law of cooling can be used to calculate time of death and more.

Plus Magazine articles

Plus Magazine, “an online magazine which aims to introduce readers to the beauty and the practical applications of mathematics,” has some interesting articles about math and crime:

“Crime fighting maths” by Chris Budd

This post describes how math can help law enforcement catch getaway cars, determine the origin of contaminants illegal dumped into water, and more.

There is also a section of that post that discusses solving inverse problems pertaining to crime scenes and accidents, a topic that is covered further in “Inverse problems save the day” by Chris Budd and Cathryn Mitchell.

“Police and thieves” and “Police and thieves continued” by Marianne Freiberger

These articles describe Andrea Bertozzi’s research on modeling home burglary patterns and “what we can learn” from her model.

This model “consists of two interlinked equations (partial differential equations to be precise) which describe how the attractiveness value of a location and the density of burglars at a location change over time (depending on a number of parameters),” Freiberger wrote.

“Interestingly, the equations you get have the same form as those describing reaction-diffusion processes you see in chemistry or biology: here two substances spread out (diffuse) through space and react with each other when they meet…The way to think about reaction-diffusion in a crime context is this. Both the risk of crime (given by the attractiveness value of a location) and the density of burglars diffuse through space — they are like the two chemical substances. And when risk meets burglar the two can interact, causing a crime. Using this idea, Bertozzi and her colleagues built a more sophisticated model in which targets can also move around. This means they can represent, not just houses, but also cars or people. So a wider range of crimes, not just burglaries, can be represented,” Freiberger added.

“Law and Order: MVT” by Evelyn Lamb (published on her Roots of Unity blog and posted to her website)

This post is definitely more lighthearted than the other ones I’m mentioning here.

Come for the amusing opening lines (“In the criminal justice system, velocity-based offenses are considered especially unimportant. In New York, the dedicated detectives who investigate these minor misdemeanors are members of an elite squad known as the Moving Violation Team. These are their stories.“), stay for the plot twist ending in this post where Law and Order meets the Mean Value Theorem.

Want to connect with us or share something you would like to see us cover on this blog? Reach out in the comments or on Twitter (@writesRCrowell)!

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Advocating for Mental Health at JMM

Photo by Natasha Spencer from TotalShape.

Mental health, which encompasses our emotional, physiological, and social well-being, has been a topic I’ve wanted to write about on this blog for a while. I was hesitant at first because I felt unqualified to address the many factors that holistically impact our mental health. However, I couldn’t pass the chance to encourage those attending the Joint Mathematics Meeting to go to the session “Mental Health in the Mathematics Profession” organized by Justin Curry (SUNY Albany) and Mikael Vejdemo-Johansson (CUNY College of Staten Island) tomorrow Wednesday, January 15, 2:15–3:45 pm. Curry and Vejdemo-Johansson along with Julie Corrigan wrote about their own experiences with mental health in their opinion piece, “Mental Health in the Mathematics Community“. Among the authors, we see the challenges faced by being diagnosed with bipolar disorder, depression, and post-traumatic stress disorder (PTSD). As highlighted in the article mental health is a problem that is very prominent in the academy and especially prevalent in students from underrepresented backgrounds. 

“Mental illness is a widespread problem, but it has a uniquely devastating presence in the university. In a recent international study of 2,279 masters and PhD students, 39 percent were evaluated as having moderate to severe depression, compared with 6 percent of the general population. Multiple studies have also shown that rates of depression and anxiety in graduate students who are women, people of color, or LGBTQ are higher than among those students who are not. This makes individuals in these groups, who are already underrepresented in STEM programs and consequently STEM careers, at increased risk. Failure to address mental health problems in the university makes addressing inequality in society at large harder to do.”

I encourage you to read their stories, what helped them, and what didn’t. Mental health does not come in one size fits all and supporting each other will depend greatly on the individuals. When searching for articles at the intersection of mathematics and mental health one of the most commonly discussed topics is math anxiety. In “On Math Anxiety“, Anna Haensch gives a glimpse of the research on its impact in math performance and shares links to the experiences of many students. As she mentions,

“By reading the brain functioning of math anxious and math non-anxious people while performing simple arithmetic problems, the research shows that people are better at automatic problem solving when the parts of their brains associated with math anxiety aren’t activated. So that feeling you get when someone asks you to quickly multiply two numbers and you just stare at them, tearing up, like a sad deer in headlights…that’s real. So we know it’s real. And we know that a lot lot lot of people feel it (disclaimer, sometimes I have really had bad math anxiety…sometimes math still makes me cry) but that doesn’t change the fact that everyone has to get through some amount of math education. This means we need to teach math in a way that minimizes the stimulation of that anxious brain and maximizes the release of those glorious math fueled dopamines.” – Anna Haensch

And it is real! During grad school, it became very apparent to me how my anxiety (and in particular math anxiety) got in the way of my studies. Many people think that anxiety is a mindset but for me, it manifested in very real physical symptoms too. A week before any big assignment or test, I would have consistent nightmares along which got worse as deadlines closed in. It got to the point I was incapable of sleeping before tests which meant which not only made me more prone to getting sick but by the time the test arrived my brain was already overwhelmed. In the past, math anxiety has been associated with being “bad” at math. It makes sense that you would be anxious about areas in which you don’t feel you can’t succeed in, however, a recent study at University of Chicago by Jalisha Jenifer, Kyoung Whan Choe, Christopher S. Rozek, Marc G. Berman, and Sian L. Beilock mentions that math anxiety can lead to math avoidance even if rewards are offered. The article “Fear of math can outweigh promise of higher rewards“,

 “People often say that being anxious about math is just a byproduct of being bad at it. Our research shows that isn’t true,” Beilock said. “Even when math-anxious people are capable of doing math, they avoid it, which means that educators and parents have to think about how we can lower math anxiety in our kids or we are going to miss students who are capable of success in math and science and just stay away from it.”

Opening the doors to conversations around mental health is crucial to decreasing the stigma around it. Seeing the article in the Notices of the AMS showcased to me the value of opening up and being vulnerable about your experiences. Mental health challenges don’t only affect our students but they have a big impact on faculty and staff as well. This is also worsened by the fact that training to identify these challenges and intervene is scarce. Hilal Lashuel highlights this in his article “The mental health crisis must be met with an accepting campus community” and suggests that making mental health a strategic priority, providing regular assessments, and training on intervention would be steps to addressing these challenges.

“But we forget that faculty and university staff also struggle with mental health challenges. Their mental health is usually overlooked, perhaps due to their small number compared to students. They are also less likely to speak up or admit to experiencing mental distress because they fear the stakes for their reputation and career are high.[…] What people fail to realise is that the faculty and staff charged with caring and educating their children are not trained in mental health awareness and intervention. Although universities expect their faculty – the people who have daily contact with students – to play a big role in addressing the mental health crisis, most fail to offer staff the training needed to do this job, adding to their struggles to manage their own mental health. ” – Hilal Lashuel

While I can manage my anxiety better now thanks to therapy, it still very present in my life. I am grateful to have been surrounded by understanding peers that worked hard to remove the stigma about mental health in my department and faculty that were very understanding of my challenges. Maybe someday I’ll feel comfortable sharing my own story and for now, I am thankful for all those who have brought this topic to the forefront of our community. Reaching out for help when I needed it was crucial to my well-being and finding the right resources for you can be hard. There are many resources and support groups available in the Depressed Academic Blog, if you are struggling, reach out.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)

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The Math Values Blog: A Tour

Have you checked out the Mathematical Association of America’s Math Values blog? The site includes posts about inclusivity, community, communication, teaching and learning, and more. Please join me on a blog tour highlighting some Math Values posts that I find noteworthy.

“Calculus and Virtual Reality” by Katie Haymaker

For this post, Haymaker interviewed Nick Long and Jeremy Becnel, both of Stephen F. Austin University in Texas. They discussed the Calculus and Virtual Reality (CalcVR) project.

“Often the largest hindrances to student success in multivariable calculus courses are the student’s inability to visualize the curves, surfaces, and vector fields, as well as the disconnect that this causes between the geometric interpretation and the algebraic calculation. While there are many great tools that are freely available (like CalcPlot3D) to help students understand these multivariable objects, the rendering is still a two-dimensional picture of a three-dimensional object. In order to show these objects with visual depth, we created a virtual reality app that is available for free on smartphones (Android or iPhone) and requires less than $5 in additional hardware costs for students. We have created interactive lessons and demos for multivariable calculus topics with the ability for the user to input their own expressions and explore the related figures with inexpensive integrated Bluetooth controllers,” the researchers told Haymaker.

Also within the post, Long and Becnel shared with Haymaker how the project has impacted students, advice for other researchers applying for NSF DUE grants, and more.

“Do Math and Chess Make You a Better Problem Solver? Teaching Math for Life in a Wicked World” by Keith Devlin

You have probably heard the buzz before: playing chess supposedly helps people become better problem-solvers. “Does taking a logic course at college make you a better reasoner? How about algebra? Or calculus? Or playing chess?” Devlin asked.

“Despite being a mathematician who concentrated on mathematical logic for my Ph.D. studies and many years of research thereafter, I always had my doubts. I harbored a suspicion that a course on, say, history or economics would (if suitably taught) serve better in that regard. Turns out my doubts were well founded,” he wrote. He shares a potentially controversial idea: that learning mathematics is “ideal preparation for, well, doing mathematics; but [is] not, on its own, particularly conducive to success in solving problems in other domains.”

He then explains the difference between “kind problems,” which can be solved by choosing and applying rule-based procedures, and “wicked problems,” which fall outside the bounds of that approach. He closes with a list of suggestions for providing students with what they need to grapple with wicked problems, beginning with putting together diverse teams of students so they can tackle problems together.

“Questioning Final Exams” by David Bressoud

With many folks on winter break, it seems timely to mention this post. Bressoud also asked readers to share their thoughts on exams here.

He begins by discussing Chapter 8 of The Years that Matter Most: How College Makes or Breaks Us by Paul Tough. I haven’t read the book, but it seems like one I should add to my reading list. Bressoud highlights a major issue with final exams. In his words? “The problem that I have is how much depends on that final exam.”

He explains his rationale for usually restricting final exams to no more than 10% of a student’s grade (“there are no options for allowing students to improve a grade on a final exam”), then discusses how he builds “at least three major projects into each course.” It’s important that students receive detailed feedback early on in the course of these projects, he noted. He discusses how to provide this feedback, especially if class sizes are large.

“Equal vs. Fair” by Dave Kung

Kung’s post begins with a scene from 10 years ago in which he asked students to find a partner for an activity. “As the room buzzed to life with students pairing up, most people quickly found someone,” he wrote. “But I watched as the one black student in the back corner slowly made his way past already-partnered-up students, eventually meeting up with the other lonely soul: a 40-year-old former carpenter returning for his degree,” he noted. Kung analyzes the difference between equal and fair treatment of students. He then shares how awareness of these issues, along with focusing on inclusion, can help us create a more just mathematical community.

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Holiday Math Treats

The holidays are a perfect time to unwind, reflect, and spend time with loved ones. For me, it is also a great time to browse the internet for fun activities to do. In this post, I highlight some of the mathematics inspired holidays internet treats you can explore and share with others!

Holiday Inspired Crafts and Puzzles

In her blog, “Math = Love”, Sarah Carter compiled a list of “Must Share: Math-y Christmas Ideas” that are ideal for the classroom but also great to do at home. I loved the idea of making icosahedron ornament balls as decorations for Christmas trees or your office. 

Icosahedron Ornament Balls from the “Math = Love” blog.

If you are looking for Christmas and Hanukkah puzzles take a look Almost Surely Math’s blog post “5 Holiday Math Puzzles”.  One of my favorites is the 2n-Menora puzzle.

The 2n-Menorah: In an alternate universe, instead of the flask in the temple lasting for just 8 days, it lasted for 2n days for a positive integer n. Thus, Hanukkah in that universe is celebrated for 2n days, and the menorah has 2n+1 candles (including the shammash):

A 2n-Menorah illustration from Almost Surely Math.

On the first day, 1 regular candle is lit and also the shammash. On the second, 2 regular candles and the shammash. And so on until the 2n-th day where 2n regular candles are lit alongside the shammash. How many candles do you need for the whole of Hanukkah in that universe? (Note that the shammash is just the name for the candle put in the center, which is traditionally used to light the other ones. No candle is reused.)

Best Math Moments of 2019

If you are a fan of lists like me,  you’ll enjoy reading about the best math “treats”/moments of 2019. In the Biggest Math Breakthroughs of 2019″, David Linkletter highlights that 2019 was a great year for getting closer to answering old questions and developing new techniques.

“In 2019, math seemed to have many mainstream moments—and that’s not including the viral problems that made us want to rip our hair out. This year saw a steady stream of answers (or at least partial answers) to tough questions that had puzzled mathematicians for decades, as well as new techniques that captured our attention in a big way. Here are the numbers—and the minds behind them—that mattered most this year.” – David Linkletter

Also, as described by Bill Andrews in  “The Year in Math and Computer Science”,

“For mathematicians and computer scientists, this was often a year of double takes and closer looks. Some reexamined foundational principles, while others found shockingly simple proofs, new techniques or unexpected insights in long-standing problems. Some of these advances have broad applications in physics and other scientific disciplines. Others are purely for the sake of gaining new knowledge (or just having fun), with little to no known practical use at this time. Other fun insights into the world of numbers this year include finally discovering a way to express 33 as the sum of three cubes, proving a long-standing conjecture about when you can approximate irrational numbers like pi and deepening the connections between the sums and products of a set of numbers.”  – Bill Andrews

Some of the big math moments have also been featured in previous blog posts such as  (Re)Discovering Identities, “With Category Theory, Mathematics Escapes From Equality”: Rachel Crowell’s Take, and “Karen Uhlenbeck: Congratulations and Thank You”.

Getting ready for 2020

Book lovers rejoice! If you were already preparing your 2020 “to-be-read” list, take a look at Evelyn Lamb’s “Math Reading Challenge” in which she has provided 12 prompts along with suggestions to help you find mathematics-related books for the coming year. My favorite three prompts are,

  1. A math book that helps you make something
    Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist by Ellie Baker and Susan Goldstine
    Making Mathematics with Needlework, edited by sarah-marie belcastro and Carolyn Yackel​​​​​​​
    Crocheting Adventures with Hyperbolic Planes by Daina Taimina
  2. A nonfiction math book written by a woman
    Beyond Infinity: An Expedition to the Outer Limits of Mathematics by Eugenia Cheng​​​​​​​
    Mathematics in India by Kim Plofker​​​​​​​
    Power in Numbers: The Rebel Women of Mathematics by Talithia Williams
  3. A math-related book published the year you were born
    For me, that’s 1983. Your mileage may vary.
    Discrete mathematics: A Computational Approach Using BASIC by Marvin Marcus
    Invitation to Geometry by Z. A. Melzak

For the last prompt, if you are born in 1990 like me, you might try reading “Reaching for Infinity” by Stan Gibilisco or “Journey through Genius: The Great Theorems of Mathematics by William Dunham. You can find inspiration on the Goodreads group or if you want to try getting “Free Springer Science Books” you can use the R code provided in the Learning Machines blog to update your library.

As we close out the year, I wish you all happy holidays and look forward to the many new math treats to come. Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)

Posted in Blogs, Mathematics and the Arts, people in math, Publishing in Math, Recreational Mathematics | Tagged , , , , , | 2 Comments

A Tour of the Chalkdust Magazine Blog

Tall 100 year old evergreen tree

Decorating for Christmas but running behind? The Chalkdust Magazine blog has a post about tangled strings of lights, formulas for “perfectly” decorating trees, and more. Image credit: Yuvarani N (Wikimedia CC)

Chalkdust Magazine (“for the mathematically curious”) and the associated blog are a treat. Anna wrote a post last year in which she described the magazine as “filled with as much mathematical goodness as a fresh unopened box of Hagoromo “Fulltouch” chalk.” Now I’m here to share links to a few of my favorite pieces on the blog.

Christmas lights, trees and maths

Earlier this year when we purchased our artificial holiday tree, my partner and I had a great debate about the relative merits of pre-lit versus unlit trees. I’m more old-fashioned — I would rather hang the lights myself and have the option of rearranging them in different configurations from year-to-year.

This blog post written by Hugo Castillo Sánchez in 2018 discusses connections between tangled strings of lights, knot theory and the second law of thermodynamics. As a last minute tree decorator and the self-appointed “official light string detangler” of our house, I appreciate the explanations in this post about how tangles form and why it’s often true that if one tangle is present, there are more tangles to be found.

Speaking of last minute decorations, if you’re like me (planning to decorate a tree but running behind) and want mathematical insights for decorating your tree, this post also includes “treegonometry” formulas for “perfectly” decorating a tree. Those formulas were developed in 2015 by members of the Sheffield University Maths Society.

Constructing the cover of issue 10

The post shows each step needed to create a tiling art project mimicking the style of cover artist Samira Mian, whose cover image was inspired by the work of Ali Reza Sarvdalir, a Persian geometer and architect.

The maths ‘black box’

In this 2018 post, John Pougué Biyong, who was in graduate school studying mathematical modeling and scientific computing at the time of its writing, wrote about his experiences. He describes studying Ebola virus propagation during the 2013 outbreak in West Africa and researching connections between droughts and food insecurity in Mauritania households.

His post also discusses a few disconnects between mathematics and people. “Looking back at my French middle and high-school years, I cannot remember a lecture during where a teacher put a mathematical concept into a more realistic context,” he wrote, adding. “As the years went by, the more complex the tools, the fewer the illustrations… when it should actually work the other way around. Consequently, most pupils would lose track of mathematics because it doesn’t speak to them anymore. And maybe it has never done.” He also noted, “As a maths student, I know as a fact that most people do not know what mathematics deals with and what it is for.”

He rounds out the post by describing the importance of representation and communication, ending with this inspiring line:

“At the end of the day, [mathematics] is all about passion and love so, until the sun sets, let us keep nurturing ourselves all together.”

A few other posts that caught my attention cover the mathematics of brewing, a ponytail shape equation and fractional polygons.

Happy Holidays, everyone!

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(Re)Discovering Identities

In November, I ran across a very interesting article in QuantaMagazine “Neutrinos Lead to Unexpected Discovery in Basic Math by Natalie Wolchover. She described the discovery that three physicists — Stephen Parke (Fermi National Accelerator Laboratory), Xining Zhang (University of Chicago) and Peter Denton (Brookhaven National Laboratory) had made about eigenvalues and eigenvectors while studying neutrinos.

“They’d noticed that hard-to-compute terms called “eigenvectors,” describing, in this case, the ways that neutrinos propagate through matter, were equal to combinations of terms called “eigenvalues,” which are far easier to compute. Moreover, they realized that the relationship between eigenvectors and eigenvalues — ubiquitous objects in math, physics and engineering that have been studied since the 18th century — seemed to hold more generally.” – Natalie Wolchover

Neutrinos, which are sub-particles that interact with matter weakly, have oscillations between different types that can be described by eigenvalues and eigenvectors. In particular,

“The mathematical description of how neutrinos interact with matter involves square arrays of numbers called matrices. Every matrix has a set of characteristic numbers called eigenvalues; and along with each eigenvalue goes a direction in space called an eigenvector.” – Peter Lynch, “Particle physics gives maths potentially powerful new tool”

With the help of Terrence Tao and Van Vu, the Eigenvector-eigenvalue identity (shown in Figure 1) was proven! Formulas to convert from eigenvalues to eigenvectors exist but they can be hard to compute. This “new” identity states that if you have a square Hermitian matrix (such as the matrix associated with neutrino scattering), one can relate its eigenvalues to its eigenvectors through the eigenvalues of its minors (i.e. a sub-matrix of the original matrix with some rows and columns removed).

Figure 1. From Terrence Tao’s blog post.

While the discovery was exciting on its own, they also noticed that similar versions of this identity were independently discovered by others after this article was published. In his blog post, “Eigenvectors from eigenvalues: A survey of basic identities in linear algebra”, Tao describes,

“Within a few weeks we became informed (through private communication, online discussion, and exploration of the citation tree around the references we were alerted to) of over three dozen places where the identity, or some other closely related identity, had previously appeared in the literature, in such areas as numerical linear algebra, various aspects of graph theory (graph reconstruction, chemical graph theory, and walks on graphs), inverse eigenvalue problems, random matrix theory, and neutrino physics.” – Terrence Tao

For example,  What I love about this article is how it portrays two ideas that resonate with my experience in mathematics:

(1) Mathematics is full of discovery (and rediscovery) even when studying well-known objects.

(2) Nature is indeed written in the language of mathematics and often mathematics is also written in the language of nature.

As Mike Lawler describes in his blog, “Sharing the Eigenvectors from Eigenvalues paper with my son”, this discovery is also a nice way for those learning linear algebra to play with a result that relates to what they cover in class.

“I think this new paper is an incredible lucky break for anyone teaching linear algebra now or in the future. It really isn’t that often that a new math paper has a result that is accessible to young students.” – Mike Lawler

An advantage of being a mathematician is that the more that you immerse yourself in math the more you see it all around you. At least for me, even the most personal self-reflections have a math flare to them. As a new decade approaches, I’ve seen a lot of Twitter posts reflecting on what they have accomplished and how they’ve changed over the last decade. This made me ask myself, “how has my experience of mathematics changed during the last 10 years?”

Back in 2009, I was a sophomore barely starting to learn the vastness of math. I would go to the library and promise myself I would come back one day and be able to read any math book in the stacks. I used to think mathematics as a sorta abstract and above all else. When I started to meet other mathematicians and learned of their experiences, how they played with concepts, and their own passion for the subject, mathematics became much more than ideas, it became about the people and the world around me too. Fast forward to now, I’ve realized how naive I was and what a beautiful dream that was. In the video The Map of Mathematics”,  Dominic Walliman captures the richness and evolution of math and as he shares,

“Now the thing I have loved most about learning maths is the feeling you get where something that seems confusing finally clicks in your brain and everything makes sense. Like an epiphany, kind of like seeing to the matrix” – Dominic Walliman

And it clicked,  I started thinking about matrices, and in particular, eigenvalues and eigenvectors. Historically, the prefix eigen, has been a cause of debate. As described in Math Origins: Eigenvalues and Eigenvectors by Erik K. Tou,

“By the middle of the 20th century, there were at least five different adjectives that could be used to refer to the solutions in our particular type of matrix equation: secular, characteristic, latent, eigen, and proper. In general, though, two naming conventions dominated: eigen- (from Hilbert’s German writings) and characteristic/proper (from Cauchy’s French writings and von Neumann’s translation of eigen-). In the United States, the peculiar prefix eigen– won the debate.” – Erik K. Tou

Even in many languages, the idea behind this concept remains the same: eigenvalues (and eigenvectors) tell us something about ourselves and definitely as this decade ends I will be reflecting on my own eigenvalues, maybe I’ll also rediscover new identities.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)


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On the AWM Moving Towards Action Workshop

In September, Brian Katz wrote a post on the AMS inclusion/exclusion blog about the Moving Towards Action workshop to be held on Tuesday, January 14 by the Association for Women in Mathematics (AWM) at JMM 2020. (JMM 2020 will be held in Denver, Colorado.)

“When members of the mathematics community are made to feel unwelcome in our profession, the success of mathematics as a whole is put into jeopardy,” an announcement about the workshop notes, adding “This workshop is focused on understanding and creating welcoming environments (providing actionable information and process change plans to mathematics department interested in driving cultural change at their respective institutions) so as to invite more people to enter and persist in STEM disciplines.”

The workshop, which is funded by the NSF and supported in part by the AMS, is co-organized by Maeve McCarthy, Elizabeth Donovan, Vrushali Bokil, Ami Radunskaya and Karoline Pershell. In a brief interview conducted over email, McCarthy and Bokil answered some additional questions about the event. (The following interview has been lightly edited for length and clarity.)

Rachel Crowell: Is this the first workshop of its kind?

Maeve McCarthy: Yes, this is the first workshop like this that AWM has held.

RC: What inspired the idea for the workshop?

MM: Mary Anne Holmes from the University of Nebraska–Lincoln gave a talk at Murray State about ethics in the professional societies based on her work with the American Geophysical Union. It was part of Murray State’s ADVANCE program. Two mathematicians, Maeve McCarthy and Elizabeth Donovan, heard the talk and got excited about doing a similar workshop on harassment for the mathematics community.

RC: I read on the posting about the workshop that there will be sessions related to developing an action plan for departments. Can you explain a little more about what those action plans entail and how those sessions will help participants develop those?

Vrushali Bokil: We have adopted the idea of using action plans to provide an avenue for each participant of our workshop to participate in social transformation in the mathematical sciences. The idea of using action plans is based on the work of the ADVANCE program at Oregon State University (

As part of the Moving Towards Action Workshop, all participants are requested to apply the knowledge of issues around sexual harassment in the mathematical sciences that they acquire in our workshop by proposing actions to create change within their spheres of influence. An action plan template will be provided to all participants to help them think of the actions they could take to further the work of improving the climate at their institutions or within the mathematical community.

Oregon State ADVANCE co-PI Tuba Özkan-Haller will give a talk on how action plans have been created and implemented at Oregon State University within its ADVANCE program activities and the impact that they have had at the personal, symbolic and institutional levels. Professor Özkan-Haller will also lead participants in drafting their own action plans using the provided template.

RC: What are some of the key things you hope participants will take away from the workshop?

MM: We hope to equip participants with tools for making cultural shifts in their departments and in the mathematics community.

RC: If you had to choose just three pieces of advice about how departments can be more welcoming to all, what would you pick?

MM: 1. Harassment must be stopped. 2. An atmosphere of inclusion is imperative. 3. Changes can have a bigger impact than we think.

RC: Is there anything else you would like to share about the workshop or related to the topic of improving the culture and climate in math?

MM: We’re particularly excited to have the University of New Hampshire’s PowerPlay group do a Bystander Intervention program led by Stephanie Goodwin from Wright State. It’s a powerful approach and will really get the audience thinking.

Editor’s note: Unable to attend the workshop but interested in learning more on about the topics it will cover? See the list of resources and pre-readings available here.

Posted in Current Events, Events, Issues in Higher Education, Math Communication, Math Education, people in math, women in math | Tagged , , , | Leave a comment

On the Living Proof blog

A new AMS blog — Living Proof: Stories of Resilience Along the Mathematical Journey — was recently launched. It follows the publication of the book by the same name (jointly published by the AMS and MAA). The book, which was edited by Allison K. Henrich, Emille D. Lawrence, Matthew A. Pons, and David G. Taylor, tells the stories of 41 mathematicians. It’s available as a free e-book download here.

Henrich is the editor of the new blog and Pons is the co-editor. (They are also joined by associate editors Jen Bowen, Susan Crook, Chawne Kimber and Anisah Nu’Man.) In an interview conducted over email, I asked Henrich and Pons some additional questions about the book and their plans for the new blog. (The following has been lightly edited for length and clarity.)

Rachel Crowell: For folks who might be unfamiliar with the Living Proof project, can you please share in a few sentences what you would like for them to know about it?

Allison Henrich and Matthew Pons: Living Proof is about sharing our stories—the good, the bad, and everything in between. We hope that people will find comfort in reading about others who have had experiences similar to their own. Our aim is for Living Proof to help our community to become more open and accepting by recognizing, on the one hand, struggles that are common across the community and, on the other hand, long-standing biases the community has had that have disadvantaged mathematicians with different backgrounds. We also hope the stories in the blog point the math community to places where we need to focus energy to eliminate those biases.

RC: Since Living Proof: Stories of Resilience Along the Mathematical Journey was published earlier this year, what kind of feedback have you received about it?

AH and MP: We have received incredibly positive feedback. Many folks in the community have voiced their support for us, the editors, and those authors who contributed to the book. Several people have wondered why it took so long for a project like this to come about. Students have told us in person and via email how much the stories have encouraged them or opened their eyes to situations they wouldn’t have imagined.

RC: How did you decide whose stories to include in the book?

AH and MP: We were pretty open here. We reached out to about 100 people and accepted most of the stories that were submitted to us. For most of the submissions, our editorial team worked with authors to fine-tune their writing in order to clarify what happened and what message they want to pass on to others. After submitting a partial draft to the MAA and AMS, we discovered the need for a bit more diversity in several areas. So, we sent out a second round of invitations to a smaller, targeted group of people. In general, we tried not to exclude any piece unless it really didn’t fit with the goals of the project.

RC: Can you tell me a little bit about your vision for the blog? 

AH and MP: The main goal of the blog is to continue sharing stories. The book contains 41 stories, which cannot possibly represent the variety of experiences out there. For instance, we just had our first anonymous post by someone who struggled with alcoholism and severe depression on the tenure track. The story is quite different from anything in the book, but a valuable one for people to read—especially those in our community who might be silently struggling with similar issues of addiction, substance abuse, or depression that involves suicidal ideation.

Another goal behind sharing more stories is that more themes will begin to emerge. A single story about imposter syndrome has limited impact, while 20 stories that all involve people struggling with self-doubt can help others realize just how common the phenomenon is.

We have also talked about having posts that synthesize themes in several stories. In addition, we are excited to have posts about how people are using Living Proof in their classes or with advisees.

RC: What are some of the things you hope readers will be able to take away from the blog? 

AH and MP: The takeaways are essentially the same as mentioned above for the book. We want the conversations about how to address struggles and build resilience to continue. We want to shine a light on the places where folks don’t want light shone. We want to acknowledge and appreciate the path that each of us has to walk and work to make it a little smoother for the next few generations.

RC: If you could share three pieces of advice with math students (especially those from underrepresented groups) who are overwhelmed and feeling uncertain about whether there is a place for them in math, what three things would you pick?

AH and MP:

  1. Whatever you are feeling or struggling with, it is highly likely that someone else who has had a very successful career has been there. And even though you might not be able to talk to them or read their story (although, our hope is that you can!), at the very least you can find some comfort in the fact that you aren’t the first one experiencing this particular struggle.
  2. Find mentors. Find professors, colleagues, and friends who can listen to you when you need to be heard, who can advocate for you when you need someone to speak to others on your behalf, and can give you advice when you’re not sure about how to navigate a situation. Tell them what you are struggling with. Ask them questions.
  3. Your self-worth is not defined by how well you did on a math test or what you got on the math GRE. It is not defined by what others tell you about yourself. If you are struggling with learning math, that does not mean that you are stupid or a failure. If you got an A in your math class, that does not mean that you are a genius. The more we can decouple our self-worth from our performance and accolades, the more resilient we can be to challenges and the less likely we’ll be to develop egos that will push others away from us. (For more on these ideas, see Francis Su’s story in the book and the story by Anonymous in the blog from November 2019.)

RC: Are you open to folks contacting you if they’re interested in contributing their stories for the blog? If so, what is the best way for them to get in touch? Are there any guidelines they should keep in mind about the types of stories you’re looking for, beyond the general theme of resilience along the mathematical journey?

AH and MP: We are absolutely interested in folks contacting us.  We are pretty open about what types of stories we are looking for. People are welcome to contact us to discuss ideas before they start writing if that works best, or they can write something, send it to us, and we can get back to them with general feedback before an official acceptance/editorial process begins. Folks can feel free to reach out by email to Allison Henrich (, Matthew Pons (, or any of the other members of our editorial team (Jen Bowen, Susan Crook, Chawne Kimber, and Anisah Nu’Man).

Posted in Blogs, Book/App, Issues in Higher Education, Math Communication, Math Education, people in math, women in math | Tagged , , , | 1 Comment

Category is…A Tour of Math3ma’s Blog

I am a huge fan of Tai-Danae Bradley’s blog Math3ma. Why? In her blog, Tai-Danae explains concepts related to Category Theory and many other fields of math with beautiful illustrations in an accessible way. Math3ma was launched in 2015 when Tai-Danae was in her second semester of graduate school and has been going strong ever since. Tai-Danae, was also a co-host in PBS Infinite Series”, has written a collection of introductory, expository notes on applied category theory, “What is Applied Category Theory?”, and was featured in the “Mathematically Gifted & Black” website. A while back on Twitter, I read a post asking for advice on how to start a blog as a graduate student and was inspired to revisit Math3ma and chat with Tai-Danae about her blog. In this tour, I hope to give you a glimpse of the blog’s style, content, and insights from Tai-Danae herself!

As she describes in her post “What is Category Theory Anyway?”, Category Theory provides a bridge between different areas of mathematics and its objects (see Figure 1). Informally, “each has objects in it (set theory has sets, group theory has groups, topology has topological spaces,…) that can relate to each other (sets relate via functions, groups relate via homomorphisms, topological spaces relate via continuous functions,…) in sensible ways (composition and associativity).”

Figure 1: Example of categories showcased in Math3ma’s “What is Category Theory Anyway?”

Similarly, Tai-Danae’s blog and its great content is a bridge between many wonderful areas of math and it’s readers.

VRQ: Can you tell our readers a bit about yourself and your blog?

Tai-Danae Bradley: “I’m a 6th—and final! —year PhD candidate inmathematics at the CUNY Graduate Center. My research interests lie in the (non-empty) intersection of quantum physics, category theory, and machine intelligence. I also have deep admiration for ideas within algebra, topology, and homotopy theory. Some quick facts about my blog: Math3ma is pronounced “mathema” as in mathematics. Mathema is a Greek word that means “a lesson.” The domain was owned by an Italian tech company. I didn’t want to give up the name, though. That’s why the “e” became a “3.” The blog’s official logo is an “M” surrounded by little electrons. I double-majored in math and physics as an undergrad, so the logo is a nod to my interest in both subjects.”

Figure 2: Math3ma’s Logo.

VRQ: What is the inspiration behind your blog?

Tai-Danae Bradley: “Math3ma began as a tool to help me adapt to graduate school. I knew that the transition from an undergraduate program to a PhD program would be a challenging one for me. The blog’s About page opens with this idea: The learning process often involved writing. That is, I learn best by putting my thoughts on paper. That’s because the process of writing helps me to slow down and think carefully about the mathematics. This is when the ideas have a chance to slowly simmer and marinate to eventually become “aha moments,” which are hard (impossible?) for me to get otherwise, without putting in this concentrated time and effort. I developed writing as a study skill pretty early on—in middle school, I think—so it naturally carried over into high school and college and later to graduate school. In particular, after my first semester I’d already amassed lots of mini expositions I’d written for myself. So I had all these little essays laying around and eventually decided to share them online in case other students could find them helpful, too. “Math3ma was originally created as a tool to help me transition from undergraduate to graduate level mathematics. Quite often, I’d find that the ideas of math are hidden behind a dense fog of formalities and technical jargon. Much of my transition process was (and still is!) learning how to fight through this fog in order to clearly see the ideas, concepts, and notions which lie beneath.”

Figure 3:  From “Learning How to Learn Math”, Tai-Danae shares: “In the third line, I’ve written, “…ask yourself…what is it that they all have in common?” I suspect this may be the origin of my affinity for category theory, a unifying language in mathematics.”

VRQ: Has blogging changed how you view math? If so, how?

Tai-Danae Bradley: “Blogging hasn’t changed how I view math, but my view of math affects how I blog. This goes back to the idea mentioned earlier: “Quite often, I’d find that the ideas of math are hidden behind a dense fog of formalities and technical jargon.” The articles on Math3ma are my personal attempts at fighting through the fog. I believe that much of advanced mathematics can be made accessible to a wide range of people, even though it may feel inaccessible because of the sophisticated language. So I spend a lot of time distilling ideas for myself. As I mentioned above, this takes a lot of work that often involves putting my thoughts on paper. Occasionally, I’ll polish up some of these writings for a blog post. Blogging, then, ultimately began as a tool to help me gain access to the simple ideas that lie beyond the misty fog. This is one of the main reasons I wanted to go to graduate school—to learn how to see mathematics clearly. Math3ma is a public record of this personal endeavor.”

VRQ: Any advice for other mathematicians interested in creating their own blog?

Tai-Danae Bradley: “The blog’s most frequently asked question is how I make the illustrations. It’s super easy! I draw them by hand with pen and paper, snap a photo with my phone, Airdrop to my computer, and edit (add color, etc.) with Photoshop. In the early posts, I used a pen tablet but that didn’t last long. I like the personal touch of a real pen. So, I’m not sure if this counts as advice, but if you want illustrations then old-fashioned pen and paper is an affordable place to start! Another thought I often hear is, “I really want to start a math blog, but I just don’t have time to post regularly.” I don’t have advice on time management, per se, but I will share something that I’ve personally found helpful: don’t worry about frequency! Or rather, I don’t worry about frequency. I blog only when I have time, and I think this has worked out well so far. My blog articles are really written for me, so there is no internal (or external) pressure to publish on a certain schedule. If you enjoy mathematics, and you enjoy sharing it, then I think others will be just as delighted to share in your joy, whether it’s once a week, once a month, or once a year.”

Some of her favorite blog posts include:

1.”A First Look at Quantum Probability (Part 1 & Part 2)”, where she shares some simple mathematics that’s a combination of linear algebra and elementary probability theory (see Figure 4).

Figure 4: Comparison of classical and quantum probability from “A First Look at Quantum Probability”.

2.  “The Yoneda Perspective/Embedding/Lemma (a three-part series), which is one of her favorite theorems,  and says, informally, that a mathematical object (a set, a group, a vector space,…) is completely determined by the set of all mappings into or out of that object (see Figure 5).

Figure 5. Illustration of the relationships showcased in “The Yoneda Perspective/Embedding/Lemma”. 

3. “The Tensor Product Demystified”, where she illustrates through examples and pictures, how you can make new vectors from old using the direct sum and the tensor product (see Figure 6)!

Figure 6: Illustration from Math3ma’s “The Tensor Product Demystified”.

Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)


Posted in Blogs, Category Theory, Math Communication, people in math, Theoretical Mathematics, women in math | 1 Comment

Joining the 3D Printed Revolution

Photo by Ines Alvarez-Fdez.

While browsing the math blogosphere on Twitter, I found myself diving into the wonderful 3D printing posts. Back in 2014, Evelyn Lamb wrote a post in this blog called “The Revolution Will Be 3D Printed”. Inspired by the title, I was curious to explore how 3D printing has evolved in the last few years and some of its uses in the classroom.

3D printing gives us a way to bring an abstract concept to life and interact with it in new and exciting ways. This field has been around since the 1980s and has kept growing ever since, but why did it become so popular? As mentioned by Ki Karou in “Back to the Future: 3D Printing and the Future of Math Education”,

“Although the field of 3D printing has its roots in the 1980s, it surged in popularity recently thanks to decreased costs (printers can be found in the thousand dollar range) and people’s astonishingly creative uses of the devices. News articles regularly come out with new uses for 3D printers, from the 3D printed car to the 3D printed wrench used in the space station, and even the exciting possibility of printing human organs.” – Ki Karou

In Mike Lawler’s blog post,  “Ten 3D Printing math projects to help students explore math, he shares a collection of projects that explore a wide range of mathematical ideas with some interesting applications. Some exciting examples include seeing the relationship between geometry and shadows (inspired by Henry Segerman), using tilings as cookie cutters (inspired by Laura Taalman) and revising the volume of a sphere.

As a fan of differential equations, I was excited to read the article, Riding the “Wave” of Affordable 3D Printing”, where Nate Barlow, Colin Huber, and Olivier Montmayeur share the applications of 3D printing in Partial Differential Equations (PDEs). Here they provide examples of problems in which 3D printing lends itself nicely to classroom explorations and also how it might be useful to introduce ideas in research groups.

“Currently, we are using 3D prints in lectures for demos, for workshops, and also for homework problems. When introducing our research to new group members or talking at conferences, 3D prints have become the perfect tool for showing the difference between a linearly stable, convectively unstable, and absolutely unstable response to a localized initial disturbance”.

Not only students benefit from 3D printing magic. It also can help mathematicians understand structures that otherwise might be missed. In his article, “Can’t Imagine Shapes in 4 Dimensions? Just Print Them Out”, Luke Whelan explains,

“For the past couple of decades, mathematicians have increasingly relied on digital imaging to see complex shapes. But certain characteristics and symmetries are just not obvious until you look at a physical representation. A digital rendering, even one you can rotate, is, after all, a just a series of 2-D images. When trying to study a shape in 4-D space, much less 3-D, even more is lost.” – Luke Whelan

Also, 3D printing has been used to replace or support coral reef systems in that have been affected by bleaching or by other weather events,  manufacturing medical equipment and surgical devices that are adapted to patients needs, and many more applications.

Another important benefit of 3D printing is to make math accessible to those who have visual impairments. As Chelsea Cook, a physicist,  shares in her 2013 TedTalk, using 3D models made mathematics come to life in her Multivariable Calculus class.

“When I took Multivariable Calculus a couple of years ago, I developed a partnership with Chris Williams from the dreams laboratory. We worked together to create three-dimensional models of the shapes and graphics needed for the coursework. Again, the models made the math come alive. I could perceive every detail of a model much clearer than a tactile graphic. I made the shape work for me to reach my goal.” – Chelsea Cook

If you want to get ideas on how to introduce 3D printing in your classroom, Jamie Back’s blog post, “How I Introduce 3D Printing in My Classes” provides a great guide. In her article, “The Resurgence of 3D Printers in Modern Learning Environments”, Erika Gimbel provides many free resources for educators who are interested in incorporating 3D printing to their curriculum such as,

  • MakerBots  free 3D-printing educators’ guidebook with STEM-based learning projects, an educators community forum, and more than 100 lessons on its educational website, Thingiverse.
  • The STEM:IT curriculum created by VariQuest, whose project-based lessons include coding and 3D design and provide instructors poster templates, 3D print files, and step-by-step guides.
  • Robo, which designs 3D printers for classrooms, teamed up with to offer K–12 educators printers with more than 240 built-in 3D STEM lessons.

Whether you are an enthusiast, teacher, or researcher there are many wonderful resources out there to get started. MakerGirl,  a non-profit organization that inspires girls to be active in STEM, mentions that are many apps, toys, and websites that promote STEM learning. In a recent post, they highlight some of their favorites including Morphi, Thinkercad, Toybox which provide ways to learn 3D design. Also, there are many geometry labs and makerspaces around the world that can serve as a resource. Are you ready to join the revolution?

Do you have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)

Posted in 3D printing, Applied Math, Interactive, Math Education, Recreational Mathematics, Visualizations | Leave a comment