Perkins has been creating posts for this challenge since March 16 and encouraging people to post their creations to social media. She described it as “just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy.” I haven’t participated in the challenge yet (I saw a few posts about it on Twitter, but hadn’t had a chance to look it up until recently), but a lot of these look cool and I’m hoping to try them. For instance, I’m looking forward to trying the isometric illusions (15) and the decagon and Pride flag (75).
Some of her posts for the challenge have focused on recent events. She wrote a Black Lives Matter post. “If you do any #mathartchallenge do this one,” she wrote. She also wrote in early June about future plans for the challenge:
“The Math Art Challenge has been on hiatus for about a week now. Mostly because it’s jarring to see folk happily engaging in math art while protestors are getting arrested. I couldn’t conscionably post things about the Hilbert curve, knowing it would divert time and energy that we need focused elsewhere.
I am keenly aware that a lot of white educators are doing more harm than good right now. Often because we’re moving too fast in an attempt to assuage guilty feelings that are hard to sit with. I am trying to let myself sit with and consider those feelings while also making sure that I am taking thoughtful, productive action and planning to be in this for the long haul. Because we need to be here beyond this week. Especially white folk. Especially white educators.”
She went on to write about her thoughts on how she can contribute to dismantling white supremacy, both inside and outside of the challenge. Among those things, she plans to spend the summer “updating, revising and adding to the Mathematicians Project.”
Perkins has written several posts about the project for the MTMS blog. In the first post (from 2016), she describes how the project came to be:
“I was giving a lecture on Pythagoras. Most of the class was giggling, having just learned that this mathematical giant was afraid of beans…One of my students, who rarely participated in class, raised his hand to ask a question.
‘Yes?’ I said, eagerly looking forward to engaging this hard-to-reach student.
‘Ms. Perkins,’ he said, ‘Why do we always talk about white dudes?'”
She wrote about how she could have sidestepped or dismissed the student’s question, but instead decided to probe further:
“Knowing this particular student identified strongly with his Mexican heritage, I asked, ‘Would it matter to you if I showed you a Mexican mathematician?’
He paused, got a weird look on his face, and responded with one of the most depressing questions I’ve ever heard: ‘Do you think there are any?’
I assured him that there were, but when he asked who they were, and I came up with nothing, his suspicions were confirmed…The fact that I didn’t know even the name of one Mexican mathematician, but I did know that Pythagoras was afraid of beans, spoke volumes about which mathematicians I valued.”
The project was born when Perkins researched Diego Rodriguez before talking to her students about him and his contributions to math. “My student was so excited that he stood up at the end and yelled, ‘Take that, white dudes!’ He had found a role model, and for the rest of the year frequently talked about Rodriguez as a point of pride,” she wrote.
Just Equationsis a California based project dedicated to advancing math-related policies that give students the quantitative tools they need to advance in college and beyond. The project hopes to achieve this through research and analysis, strategic communications, convening of relevant stakeholders in education, and providing expert advice. It was founded by Pamela Burdman whose recent research has focused on re-thinking the role of mathematics in educational equity. As described in their website,
“Just Equations is a project of Community Partners. We partner with research institutes, equity advocates, educators, and other experts in advancing educational equity across the high school-to-college pipeline. Our foundation supporters sustain our work and deepen our roots.
A growing body of evidence points to the need and potential for redesigned math policies that reduce, rather than reinforce, inequities in K-16 education. At stake is not just math learning, but the broader architecture of opportunity that is shaped by math requirements. When educational requirements are arbitrary, outdated, or unfounded, they create barriers rather than gateways to students’ success.”
I was very excited to see that the project has a blogin which they share insights into the intersections between mathematics, equity, and current events. In this post, I will give a glimpse of some of their more recent posts.
In this post, they discuss how college math requirements grew in the 1970s along with enrollment. While demographic changes allowed more people of color in higher education it also aligned with the introduction general requirement which in turn led to the increase of remedial courses. In particular, as math departments grew, these courses contributed to racial stratification and prevented students from completing their college degrees.
“At their height about 10 years ago, remedial courses represented more than half of math enrollments at community colleges nationally. These seemingly benign attempts to help students learn math in fact served to prevent millions from progressing toward a college degree: Students placed at the lowest levels of remedial math had less than a 10 percent chance of completing a credit-bearing math course, and even lower chances of earning a college degree. Even students with weak preparation were more likely to complete a required math course if they were actually permitted to enroll in one (rather than a gauntlet of remedial offerings) and given support.”
Opposite to their intention, these courses did not help students succeed in college but rather became gatekeepers. The question becomes, how do we dismantle the built-in inequities?
“Rather than penalizing students for not passing a test of math content they hypothetically might use one day, more colleges are focused on supporting students to learn math that is actually relevant to their aspirations.
To actually reach the goal of educational justice and “re-purpose mathematics as a tool for liberation”, as Just Equations called for in our recent statement, far more must be done, beginning well before students arrive in college. To that end, we are encouraged by the voices of mathematics organizations around the country that are elevating the importance of dismantling the traditional architecture of math opportunity.”
As the pandemic continues, this is an extremely relevant post in how we should reconsider what is really essential in assessing students. In the math blogosphere, many conversations have centered on how to prevent students from cheating, facilitate student engagement, and alternatives to testing through “correctness” only. This post emphasizes that both traditional forms of assessment such as standardized testing “often fail to address the uneven playing field that results from racial and socioeconomic disparities” and contributes to a culture of ranking students. It encourages promoting a growth-oriented approach to learning and provides some suggestions on how to achieve that.
“Under distance learning, an emphasis on “answer getting” makes even less sense, since students can easily turn to calculators or online materials. Therefore, despite the challenging conditions for learning, the pandemic may prove to be fertile ground for cultivating more growth-oriented approaches to assessing mathematics that equity-minded math instructors in schools and colleges are already embracing. These include:
Instead of trying to prevent students from “cheating,” design your test (or homework) knowing that students have access to a range of tools and resources.
Provide opportunities for students to improve their grade through feedback and revision, just as they might revise a paper in a composition class.
De-emphasize tests, while emphasizing the multiple ways students have to demonstrate learning during class — including discussions and projects.”
These suggestions, while particularly useful under distance learning, should be considered in our in-person classrooms as a way to refocus student learning around growth. Because in the end, isn’t mathematics more than a snapshot in time?
Traditionally, diversifying math pathways often looks like providing students with a wide variety of choices in how they pursue their math education and to make it more relevant to their goals. In this post, they discuss how new pathways must be rigorous so that they don’t end up diverting students from STEM fields and promoting patterns of inequity. It is not enough to present or create additional options without ensuring there is equitable access to those pathways. In particular, if not all students are equipped or supported to make optimal-decisions about their aspirations, they might be steered from a pathway based on their previous experiences. While some universities are no longer required to offer remedial courses, many still do and can lead to students enrolling in them even if it’s not the optimal choice for them.
“This ignores the way institutional structures (in this case, course offerings) may combine with self-perceptions to steer students’ choices. Those who have had discouraging experiences in mathematics and were led to believe they are not “math people” may tend to shy away from algebra-intensive math courses, even if they could have succeeded in them. If not carefully implemented, new mathematics pathways could become another such institutional structure, perpetuating patterns of tracking students, especially students of color, out of STEM fields.”
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).
Mark Chubb writes the “Thinking Mathematically” education blog. He has taught grades 5-8 and serves as an instructional coach for the DSB of Niagara in Ontario, Canada. He’s also an Additional Qualifications instructor. Here are a few highlights from the blog.
At the beginning of the post, Chubb links Tracy Zager’s 2016 post on the same topic for her “Becoming the Math Teacher You Wish You’d Had” blog. Zager wrote about her daughters’ first days of second and fourth grade math classes, which both began with math tests.
She discusses how teachers can balance the need to assess the knowledge of new students so they can plan their instruction accordingly against the need to “set a tone and climate for mathematics…[by] build[ing] community and trust and relationships and an atmosphere conducive to collaboration and risk taking and inquiry and learning.”
In Chubb’s post, he notes three common points teachers have discussed with him about distance learning this school year that will impact classes next fall:
Learning over the past few months has not been ideal for many students;
Learning about our students’ thinking has been difficult, at best, for us, making it difficult to sequence learning, consolidate big ideas, and use various students’ thinking to drive conversations; and
There will be a huge discrepancy between how much / what students have learned over the past few months
“What first moves we make when school returns matters more this year than ever. This leads me to wonder, will our decisions be driven by thoughts of how to fill gaps or how to build a community of learners?” he wrote.
He then discusses issues that can occur with the “gaps driven” approach and suggests other ways of “thinking about how to start all new learning with experiences that will help bridge current understandings with what your students will be learning, [which] will need to be a focus.”
Should all students learn the same things? Should they learn different things based on their abilities and readiness? This post explores these questions and more.
“Instead of seeing the issue as simply whether or not we want a classroom of students to be doing the same things or if we should be providing some students with different things, we should also consider what is actually being learned by the students,” Chubb writes. He presents a matrix with four options about student learning: everyone is doing the same tasks and learning the same things, everyone is doing the same tasks but learning different things, everyone is doing different tasks and learning different things and everyone is doing different tasks but learning the same things.
After going into further detail on each of those points, he discusses broader ideas about what it means to take an equity stance in mathematics (“we both believe that every student can achieve, and understand that every student might need different things from us”), and how we can aim for equity by expanding “who is considered a math student” and “what is accepted as mathematics.”
In this post, Chubbs describes using exit cards to determine how individual students are learning and thinking. He discusses four purposes of exit cards and offers sample exit card prompts that could be used to fulfill those purposes.
For instance, “Write a question you’d like to ask or something you’d like to know more about” is a prompt designed for meta-cognitive reflection/connection. “Create 2 addition questions, one that is easy to solve mentally and one that is harder. Use a number line to explain how to answer both. What makes one of the questions harder?” is a question targeted towards concepts. “How many ways can you solve 68 + 18? Explain each way. Which was the most efficient for you?” is a prompt targeted towards procedures and “Phillip explained that 100cm2 is the same as 1m2. Explain why he is correct/incorrect” is a prompt focusing on clarifying misconceptions.
Have comments to share? Want us to consider writing a future post about your blog (or a blog you like)? You can reach us in the comments or on Twitter (@writesRCrowell).
This question is on the front of my mind and is followed by how is anti-racism in mathematics practiced? The differences in how members of underrepresented groups, especially those who identify as Black and African American, are treated in the mathematical community, and our society as a whole is glaring. Protests condemning the murders by the hand of the police of George Floyd, Tony McDade, Ahmaud Arbery, and Breonna Taylor has led mathematicians to ask professional organizations and institutions to take a stand. In particular, through concrete action and by building better support structures to address the many ways systemic racism plays a role in our community.
First and foremost, one must acknowledge that mathematics is part of a societal system that is inherently racist. In this post, I want to share some of the resources that have helped me reflect on how to grow as a better ally, to understand how organizations and institutions promote racism, and what actions could/should we be taking to dismantle racism as a community. There are several resources out there that I encourage you to share and engage with, these are just a few.
“Nonetheless, one thing is clear: if mathematics is political (and also racial and gendered), then we must be on the side of justice, whatever that may look like. In other words, if mathematics can be antiracist, then it ought to be.[…] I don’t pretend to have the answers to the questions I am asking. This small sampling suggests a handful of possibilities for mathematics as, say, an intersectional, anti-racist, and class-consciously feminist enterprise. In any case, if we can agree that mathematics can operate as whiteness, then we have a moral duty to ask how mathematics might be otherwise. There is much work left to do. With the strength of our combined mathematical creativity, what might we come up with if we dared to imagine?”
What does anti-racist mathematics look like? And, how is anti-racist mathematics practiced? It is our responsibility to make sure that these questions do not become a passing trend but the foundation in which we build our community. In The Aperiodical, Samuel Hansen sharesResources for Anti-Racism and Social Justice in the Mathematical Sciences,a definition of anti-racist from Ibram X Kendi, author of How to be Anti-Racist and This is what anti-racist America would look like. How do we get there?.
“There is no such thing as a “not-racist” policy, idea, or person. Just an old-fashioned racist in a newfound denial. All policies, ideas, and people are either being racist or antiracist. Racist policies yield racial inequity; antiracist policies yield racial equity. Racist ideas suggest racial hierarchy, antiracist ideas suggest racial equality. A racist is supporting racist policy or expressing a racist idea. An antiracist is supporting antiracist policy or expressing an antiracist idea. A racist or antiracist is not who we are, but what we are doing at the moment.” – This is what an antiracist America would look like. How do we get there?by Ibram X Kendi.
In their post, they lists many of the resources that have been shared in social media including the statements of support to the Black Lives Matters movement by organizations, readings, list of anti-racist mutual aid projects you can donate to, organizations and projects focused primarily on the mathematical sciences you can become a member of, or otherwise support and sponsor, and actions you can take, scaffolded anti-racist resources, among others. For example, you can support the National Association of Mathematicians (NAM), as mentioned in the statement of support of the Black Lives Matter movement, their organization has made a priority promoting the excellence and mathematical development of all underrepresented minorities.
“NAM was founded in 1969, one year after the assassination of Dr. Martin Luther King, Jr. sparked widespread protests throughout the nation, similar to the ones we are seeing today. Indeed, NAM’s founding was a direct result of the marginalization of black people within the professional mathematics community, which then and now serves as a microcosm of the society in which we live. Over 50 years since NAM’s founding, despite the lessons of the civil rights movement, we still see systemic racial inequities in education, economic prosperity, criminal justice, and public health. Today, it should be clear to us all that the consequence of ignoring these racial inequities is dire.” – NAM’s Statement on the Death of George Floyd
On June 10th, there was a call join the Strike for Black Lives. In the post, #ShutDownMathin the inclusion/exclusion blog makes the great point that in these we must avoid ally theater and focus on the actions that will tackle systemic racism in mathematics.
We can hold conferences, panels, read, and discuss as we acknowledge this conversation is long overdue. Our community is in dire need of action at all levels. For example, a group of mathematicians has urged the community (and professional organizations) to stop using predictive-policing algorithms and other models. As discussed in the Nature article, Mathematicians urge colleagues to boycott police work in wake of killings, this is due to the widely documented disparities on “how the US law-enforcement agencies treat people of different races and ethnicities”. Predictive policing, a tool aimed at stopping crime before it occurs, is only one of many ways mathematics can promote racism through algorithmic oppression. As mentioned by one of the coauthors of the letter, Dr. Jayadev Athreya,
“In recent years, mathematicians, statisticians, and computer scientists have been developing algorithms that crunch large amounts of data and claim to help police reduce crime — for instance, by suggesting where crime is most likely to occur and focusing more resources in those areas. Software-based on such algorithms is in use in police departments across the United States, although how many is unclear. Its effectiveness is contested by many.
But “given the structural racism and brutality in US policing, we do not believe that mathematicians should be collaborating with police departments in this manner”, the mathematicians write in the letter. “It is simply too easy to create a ‘scientific’ veneer for racism.”
While exploring resources on Twitter, I discovered an initiative aimed at department chairs to brainstorm and share ideas on how departments can become anti-racist places for the community. You can participate and look at the resources provided at Academics for Black Survival and Wellness (June 19 – June 25) which was organized by a group of Black counseling psychologists and their colleagues who practice Black allyship. Also, you can sign-up to join Math Chairs for Racial Justiceby June 23, and find a brief description below.
“Over the next two months, we will be gathering in small groups to read Ibram X. Kendi’s How to Be an Anti-Racist. Weekly discussions (starting as soon as possible) will give you space to brainstorm how you might work to make your department an anti-racist place – a community that is not just open to all people, but one that actively supports and empowers students, faculty, and staff from groups historically underserved by the mathematics community. All discussions will be facilitated by mathematicians with experience tackling issues of racial justice in mathematics.”
In the field of math education, which has a long history with tackling and understanding racism in the classroom, a recent article by principal Pirette McKamey. In What Anti-racist Teachers Do Differently, McKamey emphasizes that,
“Anti-racist teachers take black students seriously. They create a curriculum with black students in mind, and they carefully read students’ work to understand what they are expressing.[…] To fight against systemic racism means to buck norms. Educators at every level must be willing to be uncomfortable in their struggle for black students, recognizing students’ power and feeding it by honoring their many contributions to our schools. Teachers need to insist on using their own power to consistently reveal and examine their practice, and seek input from black stakeholders; they must invite black parents to the table, listen to their concerns and ideas, and act on them.”
In a lot of ways, this thinking should be adopted beyond K-12 and into higher educations institutions as well. A lot of the resources I shared start or end with an acknowledgment that we must learn, we must do better, we must grow. This is a process that has been happening in subsets of our community but it must become part of the bigger narrative of who the mathematics community is and strives to be. I wanted to end this post with a quote from the book ‘So You Want to Talk about Race’ by Ijeoma Oluo. Join the conversations, follow and listen to diverse voices of Black mathematicians, join the fight to make mathematics an anti-racist place for all, and when you do remember: it is the system of racism that we must fight.
“Ask yourself: Am I trying to be right, or am I trying to do better? Conversations about racism should never be about winning. This battle is too important to be so simplified. You are in this to share, and to learn. You are in this to do better and be better. You are not trying to score points, and victory will rarely look like your opponent conceding defeat and vowing to never argue with you again. Because your opponent isn’t a person, it’s the system of racism that often shows up in the words and actions of other people.”
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).
Figure 1: Obtained from Rage the Blackboard postin honor of Maryam Mirzakhani.
This wonderful illustration was made by mathematical physicist and illustrator Dr. Constanza Rojas-Molina (who sometimes also goes under the pseudonym E. A. Casanova for her illustrations).
Originally from Chile, she is a Lecturer at the CY Cergy Paris University in France. She is also the author of the blog The RAGE of the Blackboard, where she interviews female scientists and reflects on life in academia.
Rage the Blackboard is divided into four sections: the main part of the blog, Blackboard Whisperers, The Questionnaire, and Art&Science. Each section features different styles of interviews or graphic summaries. For example, one questionnaire features is that of Francisca Onaolapo Oladipo (see Figure 2) a Computer Scientist in Nigeria, and a participant of the 2017 edition of the Heidelberg Laureate Forum.“She developed educational software to help girls that couldn’t attend school (or weren’t allowed to) in some parts of her country”.
One of the first things that caught my attention on the blog was the name itself. As she describes on her blog,
“The blog’s title makes reference to an angry blackboard, but also to the RAGE Theorem, named after the mathematical physicists D. Ruelle, W. Amrein, V. Georgescu, and V. Enss. Imagine an electron moving in some material, like a metal surface or block.
Mathematically, one can describe how the movement of the particle evolves in time and space, using a wave function to represent the probability that the particle is somewhere at a given time (the quantum analog of its position in space), and using a linear operator (called Hamiltonian) to represent its energy, where the effect of the environment on the particle is encoded.
The RAGE theorem relates, roughly, the time evolution of the wave function with the spectrum of the operator. More specifically, with the spectral measure, an object that encodes the nature of the spectrum. This theorem is a beautiful example of how something more concrete and “physical”, like the dynamics of the particle (will the particle stay or will it go?), is associated to a more abstract notion as the spectral measure of a linear operator acting on a Hilbert space.”
If this explanation of the RAGE Theorem has piqued your interest you can also find a fantastic illustration of it (see Figure 3). What I love about this illustration, it’s the way it decomposes the different aspects/components of the theorem: authors and years, the statement with a small summary of its components, along with some its motivation.
Figure 3: The Rage Theorem illustrated by Dr. Rojas-Molina.
I was so curious about the inspiration behind it, that I reached out to Dr. Rojas-Molina to get to know more about what motivated her to illustrate and start her blog.
VRQ: Can you tell our readers a bit about yourself and your blog?
CRM: I’m a mathematician. I’m originally from Chile, and I have moved a lot. I did my graduate studies in France, and after visiting for a while in Slovenia, I did my postdocs in Germany. I was a lecturer in Germany and now, in France. I work on random Schrödinger operators and Anderson localization, a topic in the field of mathematical-physics that combines analysis, probability, and physics.
I’m also an illustrator, whenever I’m not dealing with operators or writing grant proposals. I combine all of my interests in a blog called The RAGE of the Blackboard (RAGE as in the RAGE theorem), where I interview established female mathematicians and write about academia and maths. In this blog, I write and illustrate the articles myself, and I think of it also as a playground for experimenting in science communication. Lately, I’ve been using other social media platforms for my work, like Twitter and Instagram, but I still work on articles for my blog. Even if it looks like I’m not very active there, I have a pile of material waiting for my next holidays to get ready for the blog!
VRQ: What is the most interesting thing you’ve learned through blogging?
CRM: I’ve learned a lot from my interviewees. All of them taught me something. One thing stands out: their definition of success is very different from the usual one. It had more to do with having a balanced life and a satisfactory experience with research and teaching, than with awards and competition. They were compassionate, they thought of their students when thinking of teaching and their collaborators when thinking of research. It’s a very human take on professional success, and it’s what I aspire to. I would like to add one more thing. What I’ve learned overall:
Doing the blog has taught me a lot about processes, starting from an idea until reaching a finished outcome. I intended it as a playground to experiment and it’s exactly what I did. I learned about how to run an interview, recording, transcribing the audio, communicating with the interviewees, selecting the text, trying to make a coherent story.
I learned to promote the articles in social media, to go out, and show my work and the work of my interviewees. To illustrate the articles I read a lot about my interviewees and their research! I browsed their research articles and sometimes even their Ph.D. theses. I learned a lot about how women mathematicians run their professional and personal lives, how they find balance, and how academia works.
Not everything worked out well, and I also learned from my mistakes. I learned to be less naive, to know somethings might go wrong, and to organize myself accordingly.I also learned a lot about myself. What is important to me and what is not, who inspires me, and what is the academic I want to be.
VRQ: I am fascinated by your art, what motivated you to become an illustrator?
CRM: I’ve always been interested in drawing. Since I can remember, I was always drawing everything around me. We all start like that, but some people stop. I just kept doing it, and I always loved to hear and to read stories, so comics were my favorite medium. At some point during my Ph.D., I started drawing what I would see around me: academia, maths, and the lack of women in science.
The Ph.D. can be frustrating at times, so drawing helped me process and cope with it. Those are still the main topics in my illustrations because it’s what I’m exposed to in my daily life, and what interests me the most. I’m not interested in simply mirroring my experience on paper, but to make a reflection, and criticism when necessary. Because there’s beauty, but it’s a very imperfect beauty. Scientists are human, after all!
VRQ: Do you have advice for other mathematicians interested in creating their blog/illustrations?
CRM: I would suggest finding your community. Even if it’s you and one friend. Being a scientist and science communicator is still rare in the academic environment, so it can be a very lonely experience. Besides that, there isn’t something in place to help with the transition out of academia in case you want to do science communication full time. But actually, there are communities of science communicators out there and there are many events for people with similar interests, so go out and meet them! It’s great to have people to give you objective feedback and constructive criticism. And lastly, remember: “finished, not perfect” (a quote by illustrator Jake Parker).
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).
Álvaro Lozano-Robledo at the Grand Canyon. His new blog is A Field Guide to Mathematics. Credit: Marisa Gioioso.
A Field Guide to Mathematics is a blog by Álvaro Lozano-Robledo, an associate professor of mathematics at the University of Connecticut. He launched the blog this February. It focuses on “stories about mathematics, students, professors, mathematicians, abstract nonsense, research, papers, publishing, and academia,” according to its description.
In an interview conducted over email, Lozano-Robledo answered questions about the blog. (The following interview has been lightly edited for length and clarity.)
Rachel Crowell: What inspired you to start a math blog?
From time to time, people have liked my stories that they have found on my website (mostly “The Importance of Being Bounded,” since it is in English) and have asked if I would write more. I always said “Yes, some day when I have more time.” Well, now in confinement and distancing mode, I have found more time to write.
Recently, I decided to share the stories I am writing in a blog that may be more accessible to people and enter the “blogsphere” to connect with other people writing blogs and writing about math.
RC: On your blog, you describe it as “stories about mathematics, students, professors, mathematicians, abstract nonsense, research, papers, publishing, and academia.” That is a pretty broad set of topics. Is there anything more you would like to share about the types of pieces readers can expect to find on your blog in coming months?
ALR: Sure! My goal is to write about mathematics from a mathematician’s point of view, but not about technical topics. Rather, I’m hoping to write about what it means to be a mathematician, in a way that both mathematicians and non-mathematicians can enjoy and relate to. I’m not sure if I’m achieving my goal, but that’s the focus! For now, I’m just writing the stories that I feel like writing at the moment and those that I am ready to share now.
However, I do have a more global view of the collection of stories that I’d like to put together. They span the entire life of a mathematician, from undergrad, grad school, postdocs, tenure-track to a tenured/permanent position, and include topics about learning math, doing research, discovery, failure, publishing, etc.
In addition to the main theme of the blog, I’d like to include “interludes” of fiction that are written for the sake of writing and entertainment.
RC: What do you envision as the target audience for your blog?
ALR: Continuing with the narrative of the previous answer, I have two audiences in mind.
I’d like to reach non-mathematicians that are curious about what a mathematician does, and how a mathematician works on proving theorems.
I’d also like to reach mathematicians, particularly “mathematicians in training,” who may want to read stories from the point of view of a more senior mathematician. I’m hoping they will relate to these stories or learn useful information about, say, what it’s like to be tenured or what it’s like to be a working mathematician and a parent in a household where both parents work and split childcare evenly. I hope the ‘realism’ in the writing helps people understand that we all struggle sometimes, that we have all gone through tough times and happy times during our careers and that almost all of us fight impostor syndrome.
RC: Other than your own blog, what are some of your favorite math blogs and why?
ALR: I am actually quite interested in the Blog on Math Blogs, because I keep finding out about blogs I didn’t know about or reminding myself of blogs I have not checked out lately. The blog by Matt Baker is excellent. Lately, I’ve been obsessed with Not Even Wrong, particularly the post on the abc conjecture with what I consider the most important comment section in the history of blogs and comment sections! The back and forth between Taylor Dupuy and Peter Scholze is especially gripping.
I also follow the AMS inclusion/exclusion blog, because I learn so much and I feel that I need to keep reading what they write in that blog to be a better member of the community. It is just very important stuff and they are doing a great job covering these very difficult topics. On a related topic, “Alice’s Adventures in Number Land” is an incredible set of stories that are so eye-opening that anyone who is in the business of math should be reading very carefully. After every entry, I am like, “wow.”
I love Jordan Ellenberg’s “Quomodocumque” blog, because I love his style of writing, his ideas and the way he thinks about things.
Now that I have a blog, I am discovering other blogs that I like. For example, I found Anthony Bonato’s recent entry on the pandemic so inspiring that I changed plans for my latest entry and spent a huge amount of time recreating my last 60 days of social isolation in one of my entries in my blog (the Logbook entry).
RC: Out of the posts you have written so far, which one is your favorite and why?
ALR: That’s like asking who is your favorite child! Ha ha. At the risk of hurting the feelings of my other entries, I have to go with the post about Quijote. “El Quijote” is my favorite book of all times, and the only non-math book that I have read more than once. In fact I’ve read it many times. And I had so much fun writing that entry, because I read a bunch of chapters from the Quijote once again, first in Spanish, and then in English, so that I could learn from a translation how the more archaic Spanish had been translated into English. Anyway, I do not expect most people to love that piece, but if anything, I hope it drives some mathematicians to read El Quijote, because it is so much fun, and so incredibly clever, that it is just amazing.
Quijote entry excluded, I think my other favorite piece was the “Love Letter to Birders,” which the reader may surmise is more of a love letter to my brother than anything else. The piece explores the connection of doing research in very specialized fields. I think it’s something that many scientists can relate to: when our passion is misunderstood by a large amount of the population, even our friends.
RC: Are there any suggestions or resources you would like to share with people who are considering starting their own blogs or who have just started one?
ALR: I would love to see more writing by mathematicians! Go ahead and write! It doesn’t need to be a technical piece. I’d love to read more about personal experiences. I’d love to see our field being more humanized.
Want to share feedback or ideas for future blog posts? Reach us in the comments or on Twitter (@writesRCrowell)!
Dr. Bastian Rieck is a senior assistant in the Machine Learning & Computational Biology Lab of Prof. Dr. Karsten Borgwardt at ETH Zürich. He is interested in understanding complex data sets and topology-based machine learning methods in biomedical contexts. Especially, those related to developing tools for personalized medicine.
In his blog, which has been active since 2006, he shares his musings on interesting topics related to programming, his research interests and projects, and many “how-to” posts. What I like about this blog is that it has a very nice balance between sharing the experience of being an academic, providing advice for other researches, and diving into topics related to machine learning and programming. In this tour, I’ll give you a glimpse of some of his most recent posts.
In this post, Rieck shares the story of his experience as an undergrad taking an advanced mathematics course. He describes what I feel many of us have experienced at some point in our careers in which you wonder how your knowledge stacks up with that of your peers. In this class, he found himself in awe of his peers which seemed to understand all the concepts quickly even when they were introduced. This led to feeling increasingly out of place. He then recalls how his professor, by being honest about his limited knowledge on a subject, really changed his perspective. In a footnote, he even highlights how this particular interaction became a changing point in his career!
“There is a power in being as honest and outspoken as Prof. Kreck was. Here is this proficient and prolific member of THEM, and he could have just made up something on the spot to make me feel dumb. Instead, he chose the intellectually honest option, and made it clear that this is the normal state of affairs in mathematics (or any sufficiently complicated topic). I relish the fact that such a small action could have such a profound impact on one person, and I am grateful that I dared pose my question.
In the years since, in my own dealings with researchers, I never once feigned knowledge when I was not feeling sufficiently confident about it. I think it is important to be honest about what you know and what you do not know. Ignorance is not a moral blemish—pretending to be smarter than you are is (just as choosing to remain in a state of ignorance is).
So the moral of this story is: do not be afraid of not knowing or not understanding something.”
Similarly, I appreciated his honesty in describing this experience. It made me reflect on similar instances in my career and how, by being vulnerable when we don’t understand something, we can humanize ourselves to our students and peers.
This post caught my attention with the beautiful illustration of ‘The Land of Middle Math” (see Figure 1) by Prof. Dr. Franka Miriam Brückler. In this post, he argues that machine learning as an ever growing-field would benefit from having a structure of communicating among its different branches. Especially, since this can be a difficult task even though the branches share commonalities. He discusses some solutions including creating something similar to the Langlands Programme, which aims to study the connections between number theory and geometry. I love his analogy where he describes the program as the ‘Rosetta Stone’ for mathematics.
“The individual branches of mathematics are represented as different columns on the stone. Each statement and each theorem have their counterpart in another domain. The beauty of this is that, if I have a certain problem that I cannot solve in one domain, I just translate it to another one! André Weil discussed this analogy in a letter to his sister, and his work is a fascinating example of using parts of the mathematical Rosetta Stone to prove theorems.”
Figure 1: The land of Middle Math drawn by Prof. Dr. Franka Miriam Brückler. Obtained from blog post.
He argues, that the main benefit of a program like this would be to make as many connections among results in different fields to avoid in a sense over specializing in the tools that as researchers are created.
“The classical way of writing a machine learning paper is to present a novel solution to a specific problem. We want to say ‘Look, we are able to do things now that we could not do before!’, such as the aforementioned learning on sets. This is highly relevant, but we must not forget that we should also look at how our novel approach is connected to the field. Does it maybe permit generalising statements? Does it shed some light on a problem that was poorly understood before? If we never explore the links, we risk making ourselves into toolmakers with too many bits and pieces. Looking for the general instead of the specific is the key to avoid this—and this is why machine learning needs its own version of the Langlands programme. It does not have to be so ambitious or far-reaching, but it should be a motivation for us to investigate outside our respective niche.”
In this post, Rieck highlights how similar the choices designers make in creating an installation script for a program, researchers who develop packages also make are. In particular, the dangers of providing misleading parameters or defaults to users.
“It dawned on me at some point that we, i.e. researchers that develop a software package in addition to their research, are doing precisely the same thing. We create a software tool for solving a certain problem. It might be an itch that we want to scratch, or it might be software that is related to our research—in the end, we all write some code in some language to produce some kind of value. How often do we think about the dangers of the API that we are exposing, though?”
I found this post super helpful in talking to my students in my machine learning class about important considerations when training a model. Many machine learning models are implemented in the Python library scikit-learn and come with a set of defaults that when misunderstood or misused could lead you to draw incorrect conclusions. For example, he discusses that by default when training a Logistic Regression model, one may choose to alter how the algorithm changes the model to improve its performance on a new data point by using a technique called regularization. However, applying this technique to the data should be the user’s choice and could affect the reproducibility of results.
“In the worst case, it might trick users into believing that they did not employ regularisation when in fact they did: when comparing to other methods in a publication, it is common practice to report the parameters that one selected for a classifier. A somewhat hidden assumption on the model can be very problematic for the reproducibility of a paper.”
He ends by discussing the benefits of having parameter defaults (and that by no means they should be removed!) and provides tips on how to address setting default parameters for complex algorithms.
Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Resources to share? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)
Dynamic Ecology is a group blog by Jeremy Fox, an ecologist and evolutionary biologist at the University of Calgary, Brian McGill, a macroecologist at the University of Maine. and Meghan Duffy, an aquatic and disease ecologist at the University of Michigan. Invited guest posts are also occasionally published on the blog.
“We post ideas, opinions, commentary, advice, and humor that we think might be of interest to our fellow academic ecologists and ecology students,” the bloggers note. While the blog itself isn’t math-themed, there are many posts on the blog discussing math topics or ones that are relevant to folks working in a variety of STEM fields. Here are just a few interesting posts from the blog.
Duffy wrote this post in 2017, but it’s still informative. Three groups of people — 271 introductory biology students (at the beginning of the semester), 349 readers of the blog, and, more specifically, 225 readers of the blog who identified as ecologists — were polled about how much math they thought geneticists, ecologists and evolutionary biologists use in their work. Duffy shared the following results:
“75% of incoming Intro Bio students think geneticists use a “moderate” or “substantial” amount of math. But only 33% think ecologists do.”
“64.7% of Dynamic Ecology poll respondents think geneticists use a “moderate” or “substantial” amount of math. 78.5% think ecologists do.”
“80% of DE poll respondents who identified as ecologists said they use a “moderate” or “substantial” amount of math.”
“In other words: there is a really big difference between the amount of math that students just starting Intro Bio think ecology will involve vs. how much ecologists say it involves,” Duffy wrote.
“I’ve been thinking about how I will talk about this with students. I think that, at the start of the population ecology lecture, I will tell them that there’s something that often surprises students: ecology involves math. I will note that most people haven’t been exposed to ecology before taking the course – it was certainly true for me that I never thought about ecology before getting to college. I think that, as a first year college student, I didn’t really know what ecology was, but probably had a vague sense that it was what you see in the nature videos on PBS. It definitely did not occur to me that it involved math…My hope with this is not to scare [the students], but to better prepare them for what is coming,” Duffy added.
I also enjoyed reading the comments on this blog post. For instance, commenter Art Weis wrote:
When I teach Ecology, the first words out of my mouth are always the ecology in at it core a quantitative science. Each and every aspect of ecology can be boiled down to the question “under what conditions does the net reproductive rate of a population exceed 1.0, and what are the conditions where it doesn’t. Similarly, in the evolution course, the most basic question is when does the net replication rate of a locus exceed 1.0.? In any particular case the answer can be due to deterministic or stochastic processes, but, the key question is greater than or less than 1.0.
This post also left me wondering what can be done from the math instruction side of things to inform more students about connections between ecology and math.
This 2014 post by McGill builds on Fox’s “What should ecologists learn LESS of?” post (in which asked readers to “name the one thing you think it’s most important for ecologists to learn more of, and the one thing you think ecologists should learn less of, in order to free up time for them to learn more of whatever it is you think they should learn more of”).
“More math skills was a common answer of what should be prioritized,” wrote McGill, who notes in his post that his bachelor’s degree is in math. McGill then shares his thoughts on bridging the gap between the math he thinks ecologists should know and the offerings that are available to them through many university math departments:
I often get asked by earnest graduate students what math courses they should take if they want to add to their math skills. My usual answer is nothing – the way math departments teach math is very inefficient for ecologists, you should teach yourself. But its not a great answer.
He explains that a student would usually have to take “7 courses over and above 1st year calculus to get to all the material” he thinks “a well-trained mathematical ecologist needs!” His phrasing comes across as a bit strong to me in certain sections (such as this sentence: “This is an extraordinary waste of time since over half of what is taught in those classes is pretty much useless in ecology even if you’re pursuing deep into theory”). However, I think his overall message — that the current math offerings aren’t meeting the needs of ecology students — merits consideration and brainstorming about how to enact changes that will benefit these students.
McGill closes out the piece by listing the topics he thinks well-trained mathematical ecologists need to know and discussing different options for delivering instruction on those topics. One alternative that I don’t see listed but that I wish universities would consider? Interdisciplinary courses co-taught by ecologists and mathematicians. I understand that could come with significant logistical challenges, but I think that if departments could make it work, it would be a great option for students. The post also drew (as of this writing) 44 comments from readers, which also enhance the discussion.
Now, as much as ever, many of us are seeing the lines between our work and home lives blur. In this post, guest writer Greg Crowther, a biology instructor at Everett Community College in Washington, wrote about his decision to pursue therapy:
“Again and again, I devote unusually large amounts of time to certain work-related tasks, leaving less time for sleep, exercise, family, friends, and so on. You name it, I’m neglecting it (at least intermittently). If this lament sounds like a humblebrag, well, I don’t mean it as such. I don’t like the health-neglecting, people-neglecting version of myself, and I’m about to get professional help.”
In the comments section, readers share many helpful experiences and insights about workaholism and pursuing their passions while also tending to their mental and physical health.
This week I dived into the math blogosphere and found this cool blog Matt Baker’s Math Blogby Dr. Matt Baker, a professor, and Associate Dean at Georgia Tech School of Mathematics. This blog was featured back in 2013 in Evelyn’s post “How Quadratic Reciprocity Is Like Dealing Cards“. There she talks about a blog post in which Baker uses a deck of cards to describe quadratic reciprocity, a theorem in modular arithmetic that gives condition for when it’s possible to solve a quadratic equation modulo prime numbers. What caught my attention about this blog is that it has been active since 2013 and covers a wide breadth of topics including but not limited to “number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics”. As described in the about me page by Baker himself,
“Many of my recent papers are kind of long, and I’m hoping to post overviews of what’s in them and why a person might hypothetically care. I also want to post some new perspectives on older papers of mine, for example streamlined proofs or links to related work. The blog won’t be just about my own work, though: I also want to highlight recent preprints that I find exciting and share my thoughts on them. In addition, I hope to revive some old chestnuts from the past which I think deserve to be better known. I also want to share some thoughts about teaching in the 21st century with the hope of starting interesting and/or valuable dialogues. Finally, I hope to share some of the simple joys I find in math problems with beautiful solutions or things that are just plain fun. So hopefully there will be something for everyone in this blog — well, not everyone but you know what I mean.”
In this post, I will give a glimpse of some of his most recent posts.
In this post, Baker talks about the many different systems to mentally calculate the day of the week on any given date. He reflects on a discussion he had with John Conway, about the pros and cons of these systems. Here he covers two systems, the Gauss-Zeller algorithm (i.e. Day of the Week = Month code + Day + Year Code + Century Adjustment (modulo 7)). and Conway’s Doomsday Method (i.e. Day of the week for Doomsday = Year Code + Century Adjustment (modulo 7)). Both these methods rely on encoding the year and century adjustment of the date. As he mentions, calculating the year code is one of the most intensive aspects of these methods and provides alternatives to speed up the calculations such as the Lewis Carroll’s method, Mike Walters’s “Easy Doomsday” method, The “Odd + 11” method which he describes in detail. What I enjoyed the most about this post were the many examples to practice mental calculations and the detailed explanations of each method.
As a tribute to colleague and friend Robin Thomas who passed away last March from Amyotrophic lateral sclerosis (ALS). In this post, Baker shares some personal remarks about his friendship with Thomas and two of his most famous theorems. This post gives a glimpse of Robin Thomas beyond his math which I deeply appreciated and it’s a great way to remember him by. The first theorem he tackles is Thomas, Robertson and Seymour’s 1993 proof of the Hadwiger’s conjecture, which is a generalization of the four-color theorem, for graphs without a $K_6$-minor. The second theorem which he highlights is their classification of the forbidden minors for linklessly embeddable graphs. He states, “a graph is called intrinsically linked if every embedding in $\Bbb R^3$ contains a pair of linked cycles, and linklessly embeddable otherwise.” In this description of the theorem, he explains that the Petersen family of graphs that are intrinsically linked provides a link (pun intended) between a connection the minors of a graph and a graph being linklessly embeddable. Mainly, he states that “the theorem of Robertson-Seymour-Thomas asserts, conversely, that a graph with no minor belonging to the Petersen family is linklessly embeddable.”
“Of course, I’ve barely scratched the surface here, both in terms of Robin Thomas’s contributions to mathematics (he published over 115 papers from 1984 to 2019) and on the subjects of graph colorings and graph embeddings. But I hope this little panoply helps highlight some of the marvelous contributions of Robin Thomas (and John Conway) to the subject.”
I enjoyed reading this 2018 blog post with a neat biological application. Baker recounts chatting with a cancer researcher, Iswar Hariharan, and learning about an interesting problem he had been thinking about for a while. Centrifuges, a laboratory device that separates liquids by density by spinning test tubes, must be balanced to avoid being damaged. In this context, balanced means that “the center of mass of the collection of test tubes coincides with the center of mass of the centrifuge itself”. He poses the following question,
“If you spend a lot of time balancing centrifuges and have a mathematically curious mind, the following question might naturally arise: For which pairs (n,k) with 1 ≤ k ≤ n can you find a way to balance k identical test tubes in an n-hole centrifuge?”
Throughout the post, he provides the details on some special cases of configurations of test tubes, discusses Iswar’s conjecture which states that “you can balance k identical test tubes, 1 ≤ k ≤ n, in an n-hole centrifuge if and only if both k and n-k can be expressed as a sum of prime divisors of n”. Curiously, by translating the question into a problem about linear relations between roots of unity he found it was proven in 2010 by Gary Sivek.
He is also a mathemagician and author of ‘The Buena Vista Shuffle Club’, a book dedicated to magic tricks. I took a look at the introduction and found a great description of his magic.
“My magic tends to appeal more to the mind than to the eyes. It’s primarily card magic, frequently with some kind of mathematical principle happening in the background. But I try not to limit myself by viewing these general characteristics as constraints; on the contrary, I’m constantly testing boundaries4 to see if I can challenge myself with something unfamiliar. If you’re willing to come along for the journey, I hope you’ll enjoy the diversity of effects and methods which you’ll find in these pages.”
As he mentions in the article ‘The Magic of Math’: “There’s a lot of math in card magic,” he said. “Just like with a recipe, you might be able to follow the recipe and execute it, but you may not know enough about how it works to vary it. With card magic, I know enough to be able to combine principles in new ways and jazz around with existing effects.” Many times mathematics has seemed truly magical to me. Through his blog or his magic, Baker takes us through a pretty neat journey of mathematical discovery.
Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Resources to share? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)
Marilyn Burns’s math blog has been around since 2015. Her posts cover a wide range of topics, including math games (some of which appeal to kids and adults alike), teacher resources, math and children’s literature, and more. Please join me on a tour of just a few interesting posts on her blog that might be fun or useful to engage with, especially while staying at home during these unprecedented times.
Burns explains how to play the “Factors and Multiples” game. It’s supposedly for kids ages 7 to 16, but I found it fun enough to play several rounds myself. It’s designed for two players, but I found the online, interactive version enjoyable even when I was playing against myself. Players are given bubbles with positive integers between 1 and 100 written on them. The first player chooses an even number less than 50 from the group to start the string. The next player must then choose a number that is either a factor or a multiple of the first number to continue the string. Play continues in this manner until one of the players is unable to find a number that can be used to continue the string. That player loses.
The game is hosted on the University of Cambridge’s NRICH website. The NRICH project “aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice,” according to its website, which contains other games, projects and resources (for teachers and students) sorted by age/academic level.
I often think about how the quality of math questions impacts students — for better or worse. In this post, Burns reviews three components of good math questions presented in the book “Good Questions for Math Teaching: Why Ask Them and What to Ask, K–6″ by Peter Sullivan and Pat Lilburn.
“These features sing to me. Good questions require more than remembering a fact or reproducing a skill. It’s possible for students to learn by answering the question. There may be several acceptable answers,” Burns wrote.
She then shares some of the sample questions from the book, along with the rationale behind them and tips for tweaking them to meet different students’ needs. She also points out that there is a companion book focused on questions for middle school students.
In this post, Burns presents a simple game (along with variations) that can engage people of all ages. “Race for 20” seems like a great game to play with family, especially since many of us are now spending extended time periods at home. There are only three rules in the basic version.
Choose who will start and then take turns.
Starting at 0, when it’s someone’s turn, they can add 1 or 2 positive numbers to the string of numbers. For instance, if the first players says “1,” the second player would say either “1, 2” or “1, 2, 3.”
Whichever player gets to the number 20 wins.
Burns presents options for making the game more accessible to kids who are still learning how to count to 20, ways to make the game more concrete and more. At the end of her post, she shares connections between the game and game theory.
“Race for 20 fits into the category of the game of Nim. For more information, there’s lots online. Here’s a definition of Game Theory that I’ve cobbled together from a slew of online choices: Game theory is the study of how and why people make decisions. It is the branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant’s choice of action depends critically on the actions of other participants. Game theory has been applied to contexts in economics, business, and biology,” she wrote.
In an Educational Leadershippiece on the “Math Solutions” website, Burns explains what math menus are and how they can be used in classrooms. “A math menu is a list of math options posted for all to see. The options can include problems, investigations, games, and other activities that promote students’ understanding, support their reasoning, or provide practice with the content and skills they’ve been learning,” Burns wrote in the Educational Leadership piece. The approach also seems like it would be helpful to parents who are currently educating their kids at home due to the pandemic and are looking for additional ways to supplement their instruction.
She explains that these menus can be used to respond to three big questions from teachers: “What can I do with students who finish their math assignments more quickly?” “How can I free up time to work with students who need extra help?” and “How can I differentiate experiences to support struggling learners while also meeting the needs of students who need additional challenges?”
In her blog post, Burns describes how she responded to some questions about the approach posed by Jill Downing, a Title 1 Educator with the Helena Public Schools in Montana. Burns also shares some of the written responses students have shared about their experiences with math menus.
Want to share an idea with us? Reach out in the comments below or on Twitter (@writesRCrowell)!
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