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).

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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.

Posted in BlackLivesMatter, Current Events, Issues in Higher Education, K-12 Mathematics, Mathematics and the Arts, people in math, Recreational Mathematics | Tagged , , , | Comments Off on An Arbitrarily Close Tour

Just Equations Blog: A Tour

Just Equations is 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 blog in 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.

“Deconstructing the Mathematics Industrial Complex”

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.”

Learning Math Virtually: What’s Essential in Assessment?

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?

Promises and Pitfalls of Diverse Math Pathways: Examining Equity in Students’ Course Choices

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).

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“Thinking Mathematically”: A Tour

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.

How Not to Start Math Class in the Fall – 2020″

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.”

“Taking an Equity Stance in Math Class”

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.”

“Exit Cards – What do yours look like?”

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).

Posted in Blogs, Current Events, Interactive, Issues in Higher Education, K-12 Mathematics, Math Education | Tagged , , , , , , | Comments Off on “Thinking Mathematically”: A Tour

What does anti-racism in mathematics look like?

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.

Back in January, Dr. Tian An Wong asked can mathematics be anti-racist? in the AMS inclusion/exclusion blog, he concludes,

“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 shares Resources 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, #ShutDownMath in 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. 

“So yes, go to Black Lives Matter protests, donate to bail funds for protestors, use hashtags to express your outrage at police brutality, but be prepared to commit for the long haul. Donate to NAM (or better yet, get your department to become a departmental member!), donate to Mathematically Gifted and Black, donate to Data 4 Black Lives. Get your department to read anti-racism books. Design your classroom around rehumanizing principles that center your Black students. Change your hiring practices. Think about how you may be complicit in gate-keeping by accepting the status quo. And given that the current national focus is on the police state and how it’s implicated in the murder of Black people, demand that your colleagues stop contributing to the development of algorithms of oppression. Demand that we stop rewarding work that supports policing, inequality, and surveillance. Be intentional and mindful about mentoring graduate students. Read this letter in its entirety. And then do something.”

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 Justice by 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).

Posted in BlackLivesMatter, Blogs, Current Events, Math Education, people in math | Comments Off on What does anti-racism in mathematics look like?

Rage of the Blackboard: A Tour

I’ve always found great beauty in the way illustrations can convey a lot of information in a succinct, elegant, and beautiful way. I am a big fan of art especially when it intersects with math.

While on Twitter, I ran into a wonderful illustration of Field Medalist Maryam Mirzakhani on May 12 (see Figure 1). Her birthday was chosen as a day to celebrate women in mathematics (you can read more in: A Holiday For Celebrating Women In Mathematics,, along with this excellent summary of the Gender Gap in Science Project).

Figure 1: Obtained from Rage the Blackboard post in 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”.

Figure 3: The Questionnaire of… Francisca Onaolapo Oladipo  from Rage The Blackboard.

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.

Image describes the statement of the Rage Theorem.

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).

Posted in Applied Math, Blogs, Current Events, Math Communication, Mathematics and the Arts, people in math, Physics, Recreational Mathematics, women in math | Tagged , , , , , , | Comments Off on Rage of the Blackboard: A Tour