Azimuth: A Tour

John Carlos Baez blogs at Azimuth, the official blog of the Azimuth Project, which “is a group effort to study the mathematical sciences for ‘saving the planet.'” Anna and Evelyn mentioned Azimuth in previous posts on this blog (such as “Planet Math” and “Solidarity with Scientists”). Please join me on a tour of a few of his newer posts.

“Open Systems: A Double Categorical Perspective (Part 1)”

This post is about the Ph.D. thesis “Open Systems: A Double Categorical Perspective” by Kenny Courser, who is one of Baez’s students. Baez notes that Courser “has been the driving force behind a lot of work on open systems and networks at U.C. Riverside.”

The post includes nice, easy-to-follow pictures. Baez wrote:

His thesis unifies a number of papers:

• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 995–1027.

• John Baez and Kenny Courser, Coarse-graining open Markov processes, Theory and Applications of Categories 33 (2018), 1223–1268. (Blog article here.)

• John Baez and Kenny Courser, Structured cospans. (Blog article here.)

• John Baez How to turn a Petri net into a category where the morphisms say what the Petri net can do., Kenny Courser and Christina Vasilakopoulou, Structured versus decorated cospans.

The last, still being written, introduces the new improved decorated cospans and proves their equivalence to structured cospans under some conditions. For now you’ll have to read Kenny’s thesis to see how this works!

Linear Logic and Petri Nets”

This is Baez’s summary of “Linear Logic Flavoured Composition of Petri Nets” by Elena Di Lavore and Xiaoyan Li, who wrote the piece as a guest post for The n-Category Café.

“This first post of the Applied Category Theory Adjoint School 2020 presents the approach of Carolyn Brown and Doug Gurr in the paper A Categorical Linear Framework for Petri Nets, which is based on Valeria de Paiva’s dialectica categories. The interesting aspect of this approach is the fact that it combines linear logic and category theory to model different ways of composing Petri nets,” Di Lavore and Li wrote.

Baez’s post uses an example to show what Petri nets are. He then shows three ways to form categories using Petri nets. He wrote that he and Jade Master have focused on the first two, while the post by Di Lavore and Li describes the third approach.

Getting to the Bottom of Noether’s Theorem”

In this June 29 post, Baez wrote “Most of us have been staying holed up at home lately. I spent the last month holed up writing a paper that expands on my talk at a conference honoring the centennial of Noether’s 1918 paper on symmetries and conservation laws. This made my confinement a lot more bearable. It was good getting back to this sort of mathematical physics after a long time spent on applied category theory. It turns out I really missed it.”

He wrote that his paper focuses on just one of the two theorems from Noether’s 1918 paper: Noether’s theorem. Furthermore, he noted that his paper “studies the theorem algebraically, without mentioning Lagrangians.”

“In talking about Noether’s theorem I keep using an interlocking trio of important concepts used to describe physical systems: ‘states’, ‘observables’ and `generators,'” Baez wrote. After explaining what these concepts are, including differences between observables and generators, he wrote:

When we can identify observables with generators, we can state Noether’s theorem as the following equivalence:

The generator a generates transformations that leave the
observable b fixed.


The generator b generates transformations that leave the observable a fixed.

In this beautifully symmetrical statement, we switch from thinking of a as the generator and b as the observable in the first part to thinking of b as the generator and a as the observable in the second part. Of course, this statement is true only under some conditions, and the goal of my paper is to better understand these conditions. But the most fundamental condition, I claim, is the ability to identify observables with generators.

In the rest of the post, he explains more about what that means and how it relates to his paper.

Finally, Baez has shared a diary containing many of his tweets and Google+ posts about math, physics, his travels and more. It’s more than 2,000 pages long and includes content from 2003 to July 2020.

Want to reach out with ideas or feedback? Leave a comment below or reach out on Twitter (@writesRCrowell).

Posted in Category Theory, mathematical physics, planet math | Tagged , , , , , , , , , , , | Leave a comment

Francis Su’s Blogs and Rough Drafts for Math

I was recently looking around on Francis Su’s blogs (the Mathematical Yawp and his new one that’s hosted on his website). Though his blogs have just a few posts each, each of those posts packs power.

For instance, while he wrote the post “Race, Space, and the Conflict Inside Us” (originally published in MAA Focus) before the 2016 election, his words are just as relevant as our nation grapples with racism and police brutality and prepares for another election:

Talking about race is hard. Our nation is wrestling with some open wounds about race. These sores have been around a while, but they have been brought to light recently by technology, politics, and an increasingly diverse population. And regardless of the outcome of the U.S. presidential election, we will all need to work at healing these sores, not just in our personal lives, but in our classrooms and in our profession.

In a different and more recent post on his new blog, he wrote about “7 Exam Questions for a Pandemic (or any other time).” He wrote that post in April while he was considering what questions to ask on his final exams. Su wrote:

I speak often about how mathematical teaching often overemphasizes teaching specific facts or procedures, while underemphasizing all that goes into building mathematical explorers who have the habits of mind and confidence to solve problems they’ve never seen before…In other words, we often overemphasize building skills rather than building virtues…

Even after writing a whole book about the way the proper practice of math can build virtue, and even after aspiring to teach math in this way, it dawned on me that these virtues have not appeared much in my student learning goals or the way I assess student learning.

Su goes on to suggest sample exam questions assessing persistence, curiosity, imagination, disposition toward beauty, creativity, strategization and thinking for oneself. This post resonated with me, because I still remember an open-ended final assignment from one of my linear algebra classes in which I was asked to describe connections between what we had learned in linear algebra and my life outside of school. That question encouraged me to build different virtues, including thinking for myself. I remember what I wrote about and how much I enjoyed the assignment.

Su’s “Teach math like you’d teach writing” is the first place that I’ve encountered Amanda Jansen’s Rough Draft Math, which sounds like an excellent book.  He writes that Jansen is “pushing back against the (unfortunately common) way of teaching math at the K-12 level that primarily expects students to memorize or compute things, and makes no effort to connect to the ways that students are beginning to make sense of the ideas…Thinking is everything in mathematics. Thinking is where joy is to be found, when you truly grasp an idea and understand it.”

He adds:

I really appreciate Jansen’s framing of encouraging what she calls rough draft thinking. As she defines it:

“Rough draft thinking happens when students share their unfinished, in-progress ideas, and remain open to revising those ideas.”

He then offers suggestions for encouraging “rough draft thinking,” such as offering partial credit on assignments when students suggest strategies (even if they don’t solve the problems) and sharing your own rough draft thinking.

In a post for the Stenhouse Blog (associated with Stenhouse Publishers), Jansen and co-authors Megan Wickstrom and Derek Williams write about rough drafts in online math class. They wrote:

Whenever we learn anything new, bringing a spirit of rough drafting to the process can open us up to be more free to try on new ideas. We can ask students to share their first drafts, then we can discuss the ideas as a class to provide new insights on the drafts. During these discussions, it’s helpful to maintain a stance of curiosity and seeking to understand rather than judgment while we help each other’s ideas evolve. Students can then revise their initial thinking and document how their thinking shifted…

Rough draft thinking is just as useful online as it is in face-to-face mathematics learning environments. According to Megan, “In my experience, posting online can feel pretty final and students feel pressure not to share ideas because they don’t think they are worthy of posting. Although students still begin the course with some hesitations, within a couple weeks of the course, students are posting ideas, ponderings, confusions, and extensions. It is very exciting to see!”

Posted in Current Events, Issues in Higher Education, K-12 Mathematics, Math Communication, Math Education, people in math | Tagged , , , | Leave a comment

National Association of Mathematicians posts on the Math Values blog

The National Association of Mathematicians (NAM) has six contributors on the MAA’s Math Values blog. They are Jacqueline Brannon-Giles, Jamylle Laurice Carter, Leona A. Harris, Haydee Lindo, Anisah Nu’man and Omayra Ortega.

The NAM is “a non-profit professional organization in the mathematical sciences with membership open to all persons interested in the mission and purpose of NAM which are: promoting excellence in the mathematical sciences and promoting the mathematical development of all underrepresented minorities.”

Here are a few important recent posts by and about the NAM on the Math Values blog:

“Lift Ev’ry Voice: Supporting DVC Umoja Students in Math “

This post is by Jamylle Carter, professor of mathematics at Diablo Valley College. She conducted focus groups with some of her students who “identified partly, if not solely, as African-American” and were part of the DVC Umoja Learning Community.  Her goal was to understand what factors contributed to their “academic transformations.” In this post, she focuses on four qualities instructors had that helped these students: compassion, connection, comfort and challenge. Carter shares some quotes from her students (whose names were changed for privacy) about how these qualities impacted them.

Black Lives Matter: A Message from the MAA Committee on Minority Participation in Mathematics

This post is by Carrie Diaz Eaton, chair of the MAA Committee on Minority Participation in Mathematics.  She wrote:

To the Black mathematics community:
You are an important part of mathematics. We see your anger at police brutality, police murder, and active racism all against Black bodies and lives. We see that this extends beyond George Floyd and Breonna Taylor. We see COVID-19 is taking your lives disproportionately. We see the absolute dearth of Black mathematicians in our community. We are actively failing you at every turn as a society and as a mathematics community. We kneel together with you. #BlackLivesMatter

“Mathematics instruction and research do not happen in a vacuum,” she reminds the broader math community, adding “We cannot be effective mathematics teachers if we think that students all enter the classroom with the same sense of value and safety. We cannot be effective colleagues if we think that all of our colleagues enter academia with the same sense of value and safety. We need to actively work to become anti-racist as individuals and collectively in our workplaces. In doing so, we must hold ourselves and our academic institutions accountable for the continued oppression of Black students, staff, and faculty.”

Eaton then describes the efforts of her committee, along with five actions we can take to make a difference.

“Black and Excellent in Math”

Haydee Lindo, assistant professor of mathematics at Williams College, wrote this piece. It opens with some stark statistics:

Only 4% of Bachelor’s degrees in Mathematics (1007 of 24,293) were awarded to Black and African American students in 2016…Of the 1,769 tenured mathematicians at the math departments of the 50 United States universities that produce the most math Ph.D.s. approximately 13 are black mathematicians.

“It is difficult to speak honestly about the fact that living, working and studying in Predominantly White Institutions (PWIs) or primarily white spaces are often a fraught experience for Black students and professionals. This is compounded by the fact that our fellow students, colleagues, and mentors sometimes do not see, or fail to acknowledge, racial discrimination when it occurs. Such discrepancies in awareness and perception are an issue inside and outside of academia,” Lindo added.

She discusses microaggressions, overt aggression toward students of color, and stereotype management. She wrote that “black mathematicians and engineers remain successful by progressing, ‘from being preoccupied with attempts to prove stereotypes wrong to adopting more self-defined reasons to achieve.’ The truth is that our happiness and continued achievement may rely on the realization that our excellence and accomplishments may not, and more importantly do not, need to be validated by anyone but ourselves. This, of course, is much easier said than done.”

Sisters, How Y’all Feel? Brothers, Y’all Alright?

This post is by Anisah N. Nu’Man, assistant professor of mathematics at Spelman College. She wrote it in May for Mental Health Awareness Month, but the topic remains important as ever now. The post was inspired by a question that one of Nu’Man’s students posed before one of her linear algebra classes. It discusses a variety of issues that impact mathematicians of color at all career stages.

With regards to her undergraduate students, she wrote:

Notably, the conversation with my students about mental health started in one of my upper-level mathematics courses at Spelman College, a historically Black college for women. As math majors, these students are, in some sense, entering our profession. It feels important to appreciate how their experiences, as young female Black mathematicians, will inform the ways they experience this profession. During the conversation, I recognized that this unique classroom setting allows for discussions on the intersection of mathematics, gender, and race within academia – from the undergraduate experience to that of a tenured professor – and the impact this intersectionality can have on one’s professional and personal life.

She added “If one is considering the mental health of Black women in mathematics, one must be aware that identity markers, such as ‘Black’ and ‘women,’ add to the conversation of mental health…I know from personal experience that having these layered identities, of being a Black woman mathematician, can add stress in what can already be a stressful profession.”

She describes some of the challenges Black mathematicians face and steps for addressing mental health issues in the mathematics community, especially within communities of color.


Posted in BlackLivesMatter, Current Events, Issues in Higher Education, K-12 Mathematics, Math Communication, Math Education, people in math, women in math | Tagged , , , , , , , , | Leave a comment

Fractal Kitty Blog: A Tour


Figure 1. Fractal Kitty Logo by Sophia Wood.

Fractal Kitty: Making Sense of the Abstract, is a blog created by Sophia Wood and edited by her daughter, where she shares an assortment of fantastic math content. What caught my attention was the great number of math illustrations (both in gif and comic form) and activities hosted on the website.

She is also the author of Marie’s Atlas, a middle-grade mathematical fantasy trilogy where the main character goes on adventures and solves math and science problems. The motivation for creating her website lies within sharing the educational practices have worked for her with parents and students, as she describes,

“This website is meant to provide parents and teachers with resources, ideas for exploring, and ways to have fun with mathematics. I have found through the years that I get a lot of parents and students needing various forms of help in math. I started this website to share what has worked in my practice. Growth mindset is a core belief for me. I truly believe that mathematics is a field of study that anyone can grow in. It takes time, but that time can be fun, full of passion, and integrated with projects and applications.”

An interesting feature for me was the list of 52-week hands-on mathematics activities compiled all through last year. Also, if you are a cat lover, you will enjoy the many comics featuring cats and math (see Figure 2).

Figure 2. ‘Good Gordian! – Knotty Kitty’ by Sophia Wood.

The activities showcased in her blog touch many areas of math. In particular, I was amazed at the posts where she creates games inspired by mathematical ideas. For example, in a recent post, she discusses that inspired by the recent QuantaMagazine article, Math of the Penguinsshe and her daughter created a board game, Hex-a-Huddle Board Game.

Figure 3. Hex-a-Huddle Board Game by Sophia Wood. (Top) Penguin game piece. (Bottom) Board game grid.

As described in Math of the Penguins, penguins tend to  “arrange themselves as if they were each standing on their own hexagon in a grid” to keep themselves (and the huddle) warm. In this game,  players have 13 penguins, a wind tile, and a hexagonal board (see Figure 3). In each turn, Wood captures the efficiency of the geometric ways penguins huddle for warmth by exposing the penguins to a wind chill, assessing the losses of ‘hearts’ on the board, allowing penguins to waddle to a new position, and recover (i.e. gain hearts).  After two full rotations of the wind tile (i.e. full game), or one rotation (i.e. short game) the players with most hearts wins! I was so intrigued by the ideas behind the blog, that I reached out to Wood to know more about her work.  

VRQ: Can you tell our readers a bit about yourself and your blog? What inspired the name, Fractal Kitty?

Sophia Wood: “I am a mother of 3 and work as a math specialist for Silvies River Charter School in Oregon. I have a BS in Math and have tutored for about 2 decades. I have also worked as a systems engineer for 11+ years specializing in analysis and algorithms. I started my blog a little over a year ago to start sharing activities, ideas, and horribly dry comics. My plans currently are to grow my generative art activities for Algebra students, post math-oriented world-building lessons for STEAM, and continue with the overall flow of comics, curiosities, and GIFs. I have recently been restructuring it for new content as well. My big accomplishment this month was to finish the 52 weeks of hands-on math activities that I set out to do a year ago when the blog began. My 16yr old daughter is my editor, and we often play with math and art as a family. Fractal Kitty came from the doodle you see (Figure 1). I am a spinner, knitter, and fiber artist and always feel like I am either tangled in math or yarn. I often doodle to think, and as I was planning this blog, Fractal Kitty was born. ”

VRQ: What is the most interesting thing you’ve learned through blogging?

Sophia Wood: “I have learned that blogging pushes me to continue to play and innovate. I often start on one curiosity to find myself down a rabbit hole with the Cheshire Grin. These rabbit holes are what often guide me on life long adventures in learning. When I discover something new, I often go on the quest for who discovered it first – I have this picture in my mind of people rediscovering patterns throughout human history. Another lesson I have learned: I dropped posting for a while when my mom moved in for chemo in January through May. I cherish the time that we had together, and I would say to any blogger that ebbs and flows are part of life, so allow them to be part of your blogging as well.”

VRQ: What motivated you to become an illustrator?

Sophia Wood: “I fractured my skull in high school and went deaf in my left ear. With the deafness came an ambulance siren of ringing. It is always there and I will never hear silence again. To cope, I started painting, writing poetry, and playing music more.  Hitting my head is one of those tests in life that I have 20/20 hindsight gratitude for. I think that without the hearing loss and tinnitus, I may not be who I am today. All those midnight art adventures have evolved into more.”

VRQ: Do you have any advice for others interested in creating their own blog/illustrations?

Sophia Wood: “There is never a better time than now. You never know who you will touch, inspire, or change (including yourself). I love how  a growth mindset has changed perspectives on math, music, art, and so much more. If you put time into sharing your knowledge and creativity with the world, it will grow. I know it can make you feel vulnerable, but I often tell learners that with math we get comfortable by being uncomfortable”.

VRQ: After finishing the 52-week hands-on mathematics challenge, are there any new directions/projects you are thinking for the blog? 

Sophia Wood: “So there are a thousand directions I’d like to go, but only so much time. I am hoping to post materials from classes I have facilitated, plan to facilitate, or am facilitating –  (The 52 weeks is from a long time of facilitating hands-on math). I hope to have the following posts in the next year:

  • Scripting Algebra (10 posts) – Algebra through the lens of generative art in p5.js.  It’s not practical, but it’s beautiful. I think that beauty is what will attract more people to math.
  • Worldbuilding (10 posts) – Build a world through maps, cultures, governments, and technology while integrating STEM. Learners’ worldbuilding can lead to a deeper understanding of math, storytelling, games, character building, and art.
  • GIF design (5 posts) –  How to make GIFs to demonstrate concepts, learning, and ideas.
  • Math through Fiber (10 posts) –  Spinning, knitting, crochet, wool appliqué, quilting, and more will dive deep into combining the beauty of math and fiber.

In addition, I plan to continue doodling, investigating curiosities and sharing my love of math.  I am always open to suggestions or needs. I know a lot of kids (and adults) need meaningful activities right now. I hope to contribute.  I would say that with blogging as with life, you go with the flow. I have no idea where my curiosities will lead me, but I hope to share them as I discover wonder in this world.”


Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).

Posted in Applied Math, Blogs, Math Communication, Math Education, Mathematics and the Arts, people in math, Publishing in Math, women in math | Tagged , , , , , , , , , , | Leave a comment

Robert Talbert’s Blog: A Tour

The Fall semester is upon us! While searching for blogs that focused on teaching (and learning), I was happy to find Dr. Robert Talbert’s blog where he shares his ideas on how to keep up with the ever-changing world of higher education. His blog has been around in various forms since 2005 and covers topics at the intersection of teaching, learning, technology, and faculty work. As he describes in his blog,

“I am a Professor in, and the Chair of the Mathematics Department at Grand Valley State University in Allendale, Michigan USA. I teach a couple of classes per year, keep active with research, and (mostly) manage a large department of 30+ faculty members and hundreds of math majors in a rapidly-changing world for higher education. This blog is where I put my often half-baked but always whole-hearted ideas out on display about how I am making sense of the wickedly complex issues about teaching, learning, technology, and faculty work (and sometimes broadsides on higher education as a whole).”

In this tour, I will summarize the lessons learned from some of my favorite August posts, especially those regarding the flipped learning environment and how to use this in an online setting. While not the focus of this post, I also found his building a Calculus series very insightful and filled with great ideas to implement. You can read more in Building Calculus: The toolbox.

What does flipped learning have to do with online learning?

Flipped learning is a pedagogical methodology that aims to “flip” and the learning environment by having students learn concepts at home and use the classroom space to practice what they’ve learned. As said simply in this handout, it means students do ” school work at home and homework at school”. It also explains the acronym FLIP that stands for its four pillars Flexible Environment, Learning Culture, Intentional Content, and Professional Educator.

The idea of a flipped classroom became more prevalent in my mind as I attempted to teach online in the Spring and found that the time spent to both lecture and practice in a dynamic way felt insufficient. In this post, Talbert argues for the benefits of the flipped method, in particular, that flipped learning

  1. Optimizes face-to-face and synchronous time.
  2. Not only is predicated on student responsibility and self-regulation, it gives practice and training in these areas.
  3. These environments are structured yet flexible, which makes them well suited for our current situation.
  4. Provides a balance between structure and flexibility.

In the current times, I think these are great arguments in favor of using this type of pedagogy as many of us transition into online learning. He concludes with a powerful statement that while we may use different learning models, these are all trying to achieve the same thing; create a flexible yet structured active learning environment for students.

“As more faculty rediscover what flipped learning has to offer in these times, it makes me think that all of these models — flipped, hybrid, online, blended, hyflex, etc. —  are really just different expressions of the same overall pedagogical idea: A pedagogy that optimizes for active learning at the most crucial moments, prioritizes and codifies student self-regulation, and balances structure with flexibility. That’s a powerful combination that all students deserve.”

Flipped learning and self-teaching

What I enjoyed about this post, is that after reading, the previous one I was immediately how do we incorporate self-teaching? Is this a realistic goal for our students? How can we facilitate this process in intentional and meaningful ways? Talbert takes us back to his 2014 series and dives into some of these questions.  He describes a ‘common’ problems with flipped classrooms, one that I’ve encountered myself,

1) students can have a hard time adjusting to non-lecture classes and feel they are teaching themselves (so, what’s the point?), and

2) the fact that this can lead to negative instructor feedback from the students. I’ve also thought that these problems are solved by students getting on board with active learning, however, as Talbert points out,

“The flipped classroom does not automatically provide those sorts of outstanding learning experiences. What it provides is space and time for instructors to design learning activities and then carry them out, by relocating the transfer of information to outside the classroom. But then the instructor has the responsibility of using that space and time effectively. And sometimes that doesn’t work. In particular, if there’s no real value in the class time, then the students are not mistaken when they say they are teaching themselves the subject, and they are not wrong to resent it.”

Students may have good reasons to be skeptical of the use of the class time and, if they share this with you, it is worth looking into what is going on in your class and adjust. He mentions that if we are handing students just a ‘rule book’ to follow as they play the “class” game,  we need to reassess and work with the students to shape their learning experience.

 If the answer is that we’re handing students the rulebook and telling them to learn how to play the game this way, then students have a legitimate beef. In this case, it’s time to give class time a makeover, of sorts, so that students are actively involved with you while working with each other (or by themselves, or some combination) on crucial learning experiences.

Reading this blog gave me tons of ideas to incorporate into my courses. Some of the other post I enjoyed reading include Mastery grading and academic honesty, Research report: What are the biggest barriers to online learning?andModels for the FallAlso, another one of his post was also feature by Evelyn Lamb in this blog back in 2015, which you can check out here.

Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).

Posted in Blogs, Current Events, Issues in Higher Education, Math Education, people in math | Leave a comment

The Math ∩ Programming Blog

I’m a new reader of Jeremy Kun’s Math ∩ Programming blog. However, it didn’t take much scrolling before I read a post mentioning a tool I’ve wanted to find for quite a while and hadn’t even realized it.

In “Contextual Symbols in Math” post, Kun shared a link to Detexify. This tool guesses the names of symbols based on drawings of them. I’ll admit that when I first visited the site, I was pretty skeptical about whether the tool would work for me. I wondered how well the tool would perform if the drawing input was messy, because I find it tedious to try to draw well on a computer screen. So, I tried it out by quickly drawing a few symbols that I already knew the names of. I was impressed to discover that it was able to correctly guess the identities of the symbols based on my chicken-scratch-style drawings. Next, I made my drawings a little bit worse and was surprised that the tool still offered correct guesses.

This is exactly the kind of tool I find useful when I’m reading papers in areas of math and science that I’m less familiar with. Sure, I can often figure out what an unfamiliar symbol means by looking around online or asking researchers in the field, but this tool could cut down a lot of time. The tool does spit out multiple options for the same drawing, but at least that narrows down the list (and based on the symbols I tried, it seems pretty simple to identify which one is the match).

In addition to the main content on the blog, Kun’s site has an extensive list of primers on math and computer science topics, including ones from abstract algebra, computing theory, topology, coding theory and more. I think this section would be useful to students or anyone who would like to brush up on certain topics.

Many of his posts are also tagged as “general” and cover a wide range of topics. I enjoyed reading “Math Versus Dirty Data,” in which Kun provides a snapshot of his experience working as an engineer at Google. He wrote that the hard part of his job isn’t working on large optimizing problems:

The real hard part is getting data. Really, it’s that you get promised data that never materializes, and then you architect your system for features that rot before they ripen.

There’s a classic image of a human acting as if they’re throwing a ball for a dog, and the dog sprints off, only soon to realize the ball was never thrown. The ball is the promise of freshly maintained data, and recently I’ve been the dog.

When you don’t have good data, or you have data that’s bad in a known way, you can always try to design your model to accommodate for the deficiencies. As long as it’s clearly defined, it’s not beyond our reach. The math is fun and challenging, and I don’t want to shy away from it.

He goes on to describe in detail how bad or inadequate data affect his work in myriad ways and what he has been doing to counteract some of those effects. I think this post is particularly relatable now, as we have been seeing on a large scale the impacts that inadequate data have had on the world in the context of the pandemic.

Kun ends the post by stating “We let dirty data interfere with our design and architecture, now we’re paying back all that technical debt, and as a consequence there’s no time for our human flourishing. I should open a math cafe.” Which left me dreaming of visiting a math cafe. Are those a thing? (The first several results Google gave me with those search terms related to online math learning games for kids that let them run their own imaginary coffee shops.)

I was intrigued by “Bezier Curves and Picasso,” although it’s definitely on the longer side of posts. After enjoying this post and deciding to mention it here, I realized that Brie Finegold also wrote a post on this blog about that post back in 2013. I guess it goes to show that posts about the intersection of math, art and dogs have long-lasting appeal.

Want to share feedback or ideas for blogs we could cover in the future? Reach out in the comments or on Twitter (@writesRCrowell)!

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Tanya Khovanova’s Math Blog: A Tour

Dr. Tanya Khovanova is a mathematician whose research interests lie in recreational mathematics, combinatorics, probability, geometry, number theory. Currently, she is a Lecturer and PRIMES Head Mentor at the Massachusetts Institute of Technology (MIT). 

In To Count the Natural Numbers, Emily Jia (former 2016 AWM Essay Contest winner and a recent graduate in Math and Computer Science at Harvard) writes a fantastic essay where she interviews Khovanova. I was very appreciative to read about her personal story, career path in mathematics,  and the motivation behind creating her blog. In particular, the excerpt below resonated with me deeply, 

“Having struggled with writers block, Tanya started a blog that changed her life. She began to take English lessons, and stopped being afraid of writing papers. When she wrote about mathematical topics on her blog, she could write 3-4 posts and have enough material for a paper. Finally, she realized, “I wasn’t successful before as a mathematician because I was always doing what people told me to do.” Gelfand gave her the problem for her first publication, and afterwards she followed her then-husbands’ interests. She had picked a job in industry that she didn’t enjoy but, finally, this blog was a chance to turn this around. For the first time, she learned to follow her heart. And her heart led her to recreational mathematics: a mix of combinatorics, geometry, probability theory, and number theory that resembles puzzles instead of abstract math” – From To Count the Natural Numbers

Her blog features a great number of neat puzzles. Some of which have been highlighted in some of the previous posts on this blog (e.g. On the mathematical wedding controversy, How Math Can Help You Avoid Talking about Politics at the Holidays, and Hot Hands and Tuesday’s Children).

In this tour, I hope to give you a glimpse of the blog’s content and review two of my favorite posts. What I love about many of her posts is that they highlight joint projects with her students from MIT’s PRIMES STEP (Solve–Theorize–Explore–Prove), a program aimed at middle schoolers who like solving challenging problems. Khovanova’s blog posts are a great segway to the articles that dive deeper into the projects. 

The Blended Game

In this post, Khovanova discusses a game that her students from the PRIMES STEP program invented where they mix the rules of two games: Penney’s game and an original game by the same group called The Non-Flippancy game.  As described in the post, Penney’s game has two players, Alice and Bob, that individually select separate strings comprised of coin flip outcomes (i.e. H for heads and T for tails) of a fixed length n. They toss a fair coin repeatedly until one player’s selected string appears in the sequence of tosses and they are declared the winner. 

In contrast, the non-flippancy game does not require a coin, instead, players alternately select a flip outcome deterministically according to a “flip” rule. Again, whoever’s string appears first in the sequence of choices wins. The blended game is a combination of the previous two games where now when Alice’s and Bob’s wanted outcomes coincide, that is the outcome they receive, similar to the No-Flippancy Game. If not, they flip a coin.

“For example, suppose Alice selects HHT, and Bob selects THH. Then Alice wants H and Bob wants T, so they flip a coin. If the flip is T, then they both want Hs, and Bob wins. If the first flip is H, they want different things again. I leave it to the reader to see that Bob wins with probability 3/4. For this particular choice of strings, the odds are the same as in Penney’s game, but they are not always the same.”

She concludes that this game has the interesting property of non-transitive cycle of choices of length 6. You can read more about it in the arXiv papers  The No-Flippancy Game and From Unequal Chance to a Coin Game Dance: Variants of Penney’s Game. Students Co-authors: Isha Agarwal Matvey Borodin Aidan Duncan Kaylee Ji Shane Lee Boyan Litchev Anshul Rastogi Garima Rastogi Andrew Zhao.

Set Tic-Tac-Toe

This post brought many great memories from my time as a graduate student. The game SET was popular during our math-related outreach activities and was a favorite among my peers. In the SET game, for each of four categories of features (i.e. color, number, shape, and shading), a player must spot three cards that display said feature as all the same (or all different) to make a set. 

Figure 1. Example of the SET game cards. These three cards are considered a set since all their features are different.


In this post, Khovanova illustrates what is called a magic SET square , which is “a 3 by 3 table of SET cards such that each row, column, diagonal, and anti-diagonal is a set”.  This square is a fantastic combination of magic squares (i.e. an arrangement of numbers in a square in such a way that the sum of each row, column, and diagonal is one constant number) and the SET game. 

As she explains in the post, her students invented a version of tic-tac-toe,  played on the 9 cards that form the magic SET square. It was super exciting that in this version of tic-tac-toe ties are impossible, and the first player can always win. What amazed me was the idea of combining three different games in one for a completely new experience. You can read more about it in the arXiv paper The Classification of Magic SET Squares to see an overview of the game, and its properties. Student Co-authors: Eric Chen, William Du, Tanmay Gupta, Alicia Li, Srikar Mallajosyula, Rohith Raghavan, Arkajyoti Sinha, Maya Smith, Matthew Qian, Samuel Wang.

Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).



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Farewell, Roots of Unity

Last month, Evelyn Lamb shared her last “Roots of Unity” blog post. Photo courtesy of Evelyn Lamb.

Last month, Evelyn Lamb (former co-editor of this blog) shared her final post for her Roots of Unity blog, which was part of the Scientific American blog network. I’m sad to see such a fantastic math blog come to an end, but it had a good run! And, I’m eager to see what exciting new math pieces Evelyn will write in the future.

In an interview conducted over email, Evelyn reflected on her time writing the blog, shared some advice for others who want to get started with or get better at writing about math, and more. Continue reading

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Aleph Zero Categorical Blog: A Tour

The Aleph Zero Categorical: There can only be one blog is written by Canadian mathematician Dr. Jason Polak. The blog started back in 2011, when Polak began his Ph.D. as a way to “showcase abstraction and its beauty in the realm of pure mathematics, especially in algebra.”

Its tagline was inspired by the show “Highlander” and relates to the blog’s title since “there can be only one countable aleph zero categorical model up to isomorphism.” His research interests began in ring theory, module theory, p-adic groups, automorphic representations, logic, and combinatorics and recently have shifted towards ecology and conservation. It was hard to pick which post to write about! I love that the blog features a content tab that lets you see all the posts in alphabetical order and those whose topics involve multiple posts.

For this tour, I will summarize some of the most recent posts and hopefully give you a glimpse of the blog’s style and content. What has been most amazing to me is the combination between older and more recent research interests. I definitely got pulled in by a post about ecology and ended up reading more about spectral sequences. You
can also see some fantastic bird pictures on Bad Birding, a joint blog he created with Emily Polak.

Some misuses of science

In this post, Polak discusses some of the ways people can misuse science (i.e. using science to harm society, individuals, or the environment) and our responsibility to recognize and address them. He provides different examples like to support science out of complacency, science to support ideologies and beliefs, and science that harms living organisms.

“Anyone who is trained in science has the knowledge and ability to recognize many of these issues. Such knowledge entails a responsibility to make decisions based on our understanding. These decisions, both personal and professional, do not have to be predicated on what is right, since we may not know what that is. But they do have to be made with the desire to understand and discover the truth even if that truth is uncomfortable, and connect with others in order to share our limited understanding in the hopes of creating a better environment for all living organisms on this planet.”

Measuring biodiversity, Part 1: Difficulties

As a math ecologist, I was really eager to read this post which discusses how we measure biodiversity. Is it the number of species that count? If so, how do we decide how to count them?  Should we consider other information such as order, family, and genus level classifications? For example, using the number of species in two lakes does not take into account relative differences among the species.  He shares two great pictures of an Australian White Ibis (Threskiornis molucca) and Straw-necked Ibis (Threskiornis spinicollis) observed by the author in Darwin’s George Brown Botanical Garden. Both appear very similar even though they are different species from the same family and genera. An approach, for example, to compare the diversity in two lakes, is making a table that lists the last sighting of birds species into genii, families, and orders. While there are many ways to summarize this table, one of them called the Shannon entropy, measures how much information is stored in the probability distribution for each lake. As he explains,

“The higher the entropy, the more evenly the vector is distributed. The maximum entropy is obtained when the distribution is uniform in which case the Shannon entropy of that vector is log(n). The idea is that the more evenly distributed the probability distribution is, the more diverse the area with respect to the subset of organisms you are studying.”

Polak concludes that a way to improve biodiversity measures one needs better data which he hopes to analyze in more detail in futures posts.

Wild Spectral Sequences Series

In these multi-post expositions of proofs that use spectral sequences, Polak illustrates that these sequences are ‘safe’ and in fact, can be used in a variety of examples. The post assumes the reader is familiar with spectral sequences so I dived to find a ‘big picture’ idea of what these are.  In the notes, Spectral Sequences: Friend or Foe?“,  Ravi Vakil describes spectral sequences as “a powerful book-keeping tool for proving things involving complicated commutative diagrams.” Through six posts/episodes, Ep.1 Snake, Ep.2 Five, Isomorphism!, Ep.3 Cohomological Dimension, Ep.4 Schanuel’s Lemma, Ep.5 Lyndon-Hochschild-Serre, and Ep.6 The 3×3 Lemma he gives examples of spectral sequences being used in ‘toy examples’. While the concept of spectral sequences was new to me, I appreciated seeing the main ideas behind how these sequences allow us to “clean-up” messy proofs. As he concludes in Ep.4,

“Notice that once we get used to spectral sequences, they can help remove a lot of the clutter that comes with ridiculous proofs that contain sentences of the form ‘let $x\in X$, then  $f(x)\in Y$ is in the kernel of…’, which are exceptionally hard to read.”

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 (@VRiveraQPhD).


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An Arbitrarily Close Tour

Annie Perkins, a math teacher for Minneapolis Public Schools, writes the arbitrarily close blog. Here are just a few of the interesting/exciting/compelling components of her blog.

#MathArtChallenge posts

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.

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