## Musing Mathematically: A Tour

Musing Mathematically is a blog written by Nat Banting, a mathematics teacher, and mathematics education lecturer at the Department of Curriculum Studies at the University of Saskatchewan, Canada.  His blog, which began back in 2011, is centered around the ideas behind teaching and learning.

As he describes on his website, “his academic interests include the ecological and biological roots of cognition (particularly pertaining to the instigation and observation of student problem-posing), the decision-making of teachers and students with(in) high-density mathematics classrooms, and student (and teacher) impressions of probability.” In this tour, I will give a glimpse of some of my favorite recent posts on the blog.

Thoughts on Thinking Classrooms

In this post, Banting discusses his thoughts on the ideas presented in the book  Building Thinking Classrooms by Peter Liljedahl. In this context, a thinking classroom, as defined by Liljedahl himself, means,

A classroom “that’s not only conducive to thinking but also occasions thinking, a space inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion.” – Peter Liljedahl, Building a Thinking Classroom in Math

It has 14 elements, that if your curious, you can see in this wonderful sketchnote by Laura Wheeler. Banting states in his post, ‘I don’t have a thinking classroom’. I was taken aback a bit by this, since I for one, believe many of the mathematics teachers I know strive to create thinking classrooms. He does, however, mention that the statement comes with three qualifications which I encourage you to read. The one that caught my attention the most was the second one in which he points out how the idea of “Thinking Classroom” get’s simplified to  “any class with students standing at whiteboards in random groups.”

The structure above serves as an example of two of the elements of a “Thinking Classroom”:  (VNPS) and (VRG). I found this very insightful since with a lot of active learning scenarios. I know, I’ve counted on these structures creating a conducive environment for learning that is lost in traditional lectures. As Banting  puts it elegantly,

“However, those structures do not teach; they amplify potential teaching moments. We, as teachers, still need to harness them, and Peter develops many ways to do so in the sections of the book. And so when it is proposed that I run a “Thinking Classroom,” I am always careful to interrogate what the proposer has in mind, because I think we have the responsibility to ensure that the term “Thinking Classroom” doesn’t strictly refer to the structure(s) of VNPS and VRG and leave the teaching behind.”

Probability Quizzes with No Questions

In this post, Banting begins sharing his encounter with the following question:

If you chose an answer to this question at random, what will be the chance it would be correct?

A) 25%

B) 50%

C) 60%

D ) 25%

What I appreciated about the post is the idea that, while questions like this are can lead to a lot of mathematical arguments,  it is not the answer itself the end goal. As he mentions,

“The point of the exercise is not to complete the exercise, it’s to dwell a while in the complexities it offers. By constructing the argument, you interact with notions of odds, randomness, probability, and the like.

To engage help students to step out of the usual problem-solving approach (i.e. recall a similar situation and apply it to a new context),  Banting proposes shifting the focus of answering a question to puzzling with it by not offering any questions at all (see Figure 1).

Figure 1. Sample questions from quiz from ‘Probability Quizzes with No Questions’ by Nat Banting.

For example, in a ten-question quiz, seven out of those 10 questions will have the correct answer be C, you may ask students to circle the correct response and ask them to reflect about what are the chances that, in fact, that would happen? What I love most about this idea, is that it throws the student into grappling with uncertainty in a very familiar scenario. You can use mathematics as a way to demystify this feeling of uncertainty. As Banting mentions,

“Mathematics is suggested as a way to dissect the feeling of uncertainty, and this act of justification becomes the focal point. I mean, of course it does. What is left to argue about the solutions of questions 1-7? They were simply the vehicle to encounter an experience, and, in this way, those seven “questions” were never the questions to begin with.”

Since I am a big fan of board games and art, I also enjoyed looking at Project: QuaranTiles and hope to try them out soon. In addition, to Musing Mathematically, he is also the curator of FractionTalks.com, a really neat website that fosters creative ways to visualize fractions. I loved exploring this website, especially, since oftentimes one of the most memorable early mathematics experiences for students is learning fractions! The ideas behind this website also lead to great implementations of these for the classroom such as Marie Brigham’s Fraction Talk War for fourth-graders and the creation of the Fraction Wars Cards by Carla Dawson

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).

## Mean Green Math Blog: A Tour

The Mean Green Math Blog: Explaining the whys of mathematics is a blog by Dr. John Quintanilla, a professor of mathematics at the University of North Texas (UNT).  It has been around since 2013, and its name,  ‘Mean Green’, is an ode to one of the symbols of UNT.  This blog is for future mathematics teachers, alumni, colleagues,  friends and family, along with teachers who mentor other teaches.   As he describes on the blog, the purpose of the blog is to dive into the why behind the math.

“This blog does not aim to answer common student questions like “How to factor this polynomial?” or “How do I solve for $x$ in this equation?” (There are plenty of excellent websites out there, some listed on my page, that give good step-by-step instructions of such problems.) Instead, this blog aims to address the whys of mathematics, providing readers with deeper content knowledge of mathematics that probably goes well beyond the expectations of most textbooks. As well as an audience of current and future secondary teachers, I also hope that this blog might be of some help to parents who might need a refresher when helping their children with their math homework. I also hope that this blog will be interesting to students who are interested in learning more about their subject.”

In this post, I will share some of the posts that caught my attention, in particular, those aimed at engaging students.

Engaging Students Series

As part of a capstone course for secondary mathematics teachers, he asked his students to come up with ideas on how to engage their students with mathematics topics. What appealed to me the most about this assignment was the structure provided to the students.  Instead of  lesson plans, students had to come up with three different ways to catch their students’ interests. As you’ll see in the examples, the type of engagement activities varies  for each topic.  With the permission of the students, we get to see their work and draw inspiration from their ideas! Below are some of my favorites,

Engaging students: Deriving the Pythagorean theorem

Former student, Haley Higginbotham, shares how as a teacher she would create an activity to involve her students. She presents a visual proof of the Pythagorean theorem using a hands-on activity. What I found super interesting her answer to the question: how has this appeared in high culture?

“The Pythagoras tree is a fractal constructed using squares that are arranged to form right triangles. Fractals are very popular for use in art since the repetitive pattern is very aesthetically pleasing and fairly easy to replicate, especially using technology.” (see figure below).

Pythagorean tree created by Guillaume Jacquenot. Picture obtained from Wikimedia Commons.

She concludes by discussing how to incorporate technology in the activity and shares  how she would use an activity that allows students to drag the different sides to see that the Pythagorean relationship holds no matter how the sides of the triangle change.

Engaging students: Solving linear systems of equations with matrices

The next idea comes from former student Andrew Sansom. In this case, he explores an interesting word problem that students can do to practice solving linear systems with matrices.  He discusses and walks the reader through the solution to the following problem,

Map of Denton showing the set-up for the system of equations by Andrew Samson . Obtained from

“The Square in Downtown Denton is a popular place to visit and hang out. A new business owner needs to decide which road he should put an advertisement so that the most people will see it as they drive by. He does not have enough resources to traffic every block and street, but he knows that he can use algebra to solve for the ones he missed. In the above map, he put a blue box that contains the number of people that walked on each street during one hour. Use a system of linear equations to determine how much traffic is on every street/block on this map.”

Based on the diagram above, you can build an equation for each intersection that has the sum of people walking in and out as equal, rewrite the system in standard form, represented as an augmented matrix, reduce the matrix using Echelon form, and voila! You find that the best place to advertise is in Hickory Street between Elm and Locust Street. He also provides his thought on the are the contributions of various cultures to this topic and shares some of the history of solving systems of linear equations.  Below is an excerpt,

“Simultaneous linear equations were featured in Ancient China in a text called Jiuzhang Suanshu or Nine Chapters of the Mathematical Art to solve problems involving weights and quantities of grains. The method prescribed involves listing the coefficients of terms in an array is exceptionally similar to Gaussian Elimination.

Later, in early modern Europe, the methods of elimination were known, but not taught in textbooks until Newton published such an English text in 1720, though he did not use matrices in that text. Gauss provided an even more systematic approach to solving simultaneous linear equations involving least squares by 1794, which was used in 1801 to find Ceres when it was sighted and then lost.”

Predicate Logic and Popular Culture Series

Similar, to the goal of the last series of posts, the Predicate Logic and Popular Culture series has a great number of examples (with different sources and complexity) to make predicate and propositional logic more appealing to students.  As part of his Discrete Mathematics class, he presented students either with a logical statement (which they had to translate to actual English) or gave them a famous quote to translate into predicate logic.  This was so fun that I ended scrolling for a while just to find my favorites. Below are some that caught my eye,

• Predicate Logic and Popular Culture (Part 189): Mana

I was captivated by the idea of using song lyrics to practice! Especially, since in this example is a song from a Mexican band, Mana, which I listened to growing up.”Let  W(t) be the proposition “At time t, you want me as I am,” and let R(t) be the proposition “At time t, you reject me for what I was.” Translate the logical statement:

$$\forall t <0, (\neg W(t) \wedge R(t)).$$

This matches a line from the Spanish-language song “Tengo Muchas Alas / I Have Many Wings.”

• If you are a fan of Star Wars you might remember this quote from Yoda from “Star Wars Episode I: The Phantom Menace.”

“Let $L(x,y)$ be the proposition “$x$ leads to $y$.” Translate the logical statement:

$$L(fear, anger) \wedge L(anger, hate) \wedge L(hate \wedge suffering)$$. Can you guess which line the statement above refers to? Check out the post for a video clip with the answer.

• Predicate Logic and Popular Culture (Part 182): MoanaIn the same spirit, you might recognize the following line from the movie Moana.

“Let $P$ be the set of all people, let $L(x)$ be the proposition “$x$ is on this island,” and let $K(x)$ be the proposition “I know $x$.” Translate the logical statement:

$$\forall x \in P(L(x) \Rightarrow K(x))$$ Can you guess which line the statement above refers to? Check out the post for a video clip with the answer.

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 (.

## #BlackWomenRockMath: An Interview

If you’re looking for an exciting new blog to check out, look no further. Kaneka Turner, Deborah Peart, and Dionne Aminata recently launched #BlackWomenRockMath. In an interview conducted over email, we discussed why they started the blog, what they have planned for it and more. (The following interview has been lightly edited.)

Rachel Crowell: Why did you decide to start a blog now?

Kaneka Turner, Deborah Peart, and Dionne Aminata: We have been working together for about 2 and a half years and have had the opportunity to present together at conferences on varied topics around mathematics. We recognized that we have a lot in common and that our beliefs around equity in mathematics are aligned. During the pandemic, many things have been brought to light, and it is clear that Black communities are impacted at an alarming rate. Collectively, we have a deep and profound understanding of and experience with racism and decided that now was the time to speak up. Individually we have always had a lot to say, but by approaching it together we found the boldness to say so much more. We hold one another accountable and desire to lift up other Black women so they can find their voices. This is the platform we are hoping to create.

RC: What are some of your main hopes and goals for the blog?

KT, DP and DA: Our main goal is to create a community that recognizes the brilliance of Black women in math education and sets the stage for securing a legacy for Black girls to be inspired and walk in confidence as doers of mathematics. We hope to disrupt systems and break patterns, so Black women have the opportunity to lead and Black girls have the opportunity to shine. We hope everyone will begin to see the power we hold and the brilliance we share and open doors for so many who have gone unseen or unheard for far too long.

RC: Who are you hoping will read the blog?

KT, DP and DA: We are hoping that educators, school leaders, and parents are reading the blog, but we also hope that people with the power to support our mission to make a difference will also take notice. Educators and people in the field are the obvious choice, but we also want the community at large to recognize that our voices need to be heard because we (Black women) have valuable contributions to make in the field of math education.

In the first post on the blog, you wrote “A little over 2 years ago our paths crossed when we joined Illustrative Mathematics as lead writers for the K-5 Math Curriculum. This was a rare space and opportunity. We immediately recognized the weightiness of our roles and the need to support each other. The reality is, Black women are not typically asked to use their expertise in mathematics to co-design a national math curriculum. We were not brought to IM to address the diversity, equity, and inclusion components of the curriculum. We were hired for what we know about mathematics, and this was unprecedented.” How did you feel to be asked to use your expertise in math to co-design a national math curriculum?

KT: I felt honored to be selected because I was familiar with IM’s work and believed it was important to align with a company that had a reputation of producing quality materials. I didn’t know they had a plan to focus on supporting marginalized communities through their materials, but I was glad to know that I would be connected with this positive presence in math education.

DA: I believed that I needed to take this opportunity, especially if they were planning to serve students in marginalized communities. My work had always been with students of color or students living in poverty, so I thought it was vitally important to offer my perspective to support the development of materials for these children. I was hopeful that IM’s vision aligned with mine, a vision of a future of mathematics with a shift in curriculum towards inclusion.

DP: When I was offered the position, I was in disbelief. It seemed too good to be true. I had always thought of curriculum writers as something other than me; it never occurred to me that I could be one. It was exciting to be a part of something that would reach thousands of students. It was rewarding work from the beginning because I felt inspired by the people with whom I would have the chance to collaborate. It was a dream come true because I had wondered how I could be a part of a larger mission to impact math education for more students. Changing lives 1 class at a time was great, but I longed for such an opportunity as this.

RC: What are some of your favorite math or math education blogs?

KT, DP and DA: We don’t have a long list of math education blogs that we subscribe to, but we agreed that Krisin Gray’s blog “Math Minds” and Theresa Wills’ “Where THERE’S A WILLS, there’s a way” are both favorites. During the time we have been writing, we have read several books as a team around content and pedagogy. We have also started to listen more and more to podcasts, which is something we plan to launch for BWRM in 2021.

RC: Is there anything else you would like to share?

KT, DP and DA: Something that is important to all of us is that we want people to know that this is a heart project. Because of what we’ve experienced growing up and working in education, we have decided to do something to make a change. Our motivation and inspiration comes from the vision of a future where little Black girls know they rock math and boldly say it with pride. We overcame our math trauma and became something wonderful, so we hope to ease the path for those coming after us. We believe that Black women rock math because Black girls rock math! Now it’s time for the world to know.

Want to give feedback or suggest ideas for future blog posts? Reach out in the comments or on Twitter (@writesRCrowell).

## The TODOS Blog: A Tour

We are almost midway through Hispanic Heritage Month (September 15 – October 15)!  This month marks a national holiday in the United States that began as a way to promote the history, contributions, and culture of Hispanic-Americans. The month wouldn’t be complete without recognition, reflection, and celebration of the contributions of Hispanic and Latinx individuals in the mathematical sciences.

In last year’s post, A Tribute to Hispanic Heritage, I talked about several initiatives, articles, and blog posts that shared some of the history and  the challenges still ahead for members of the Latinx/Hispanic community. This year marks the fifth anniversary of an initiative close to my heart,  Lathisms (Latinxs and Hispanics in the Mathematical Sciences, www.lathisms.org).

“Since 2016, Lathisms has featured 122 diverse mathematicians, highlighting one per day during US Hispanic Heritage Month, which is celebrated September 15–October 15. The website, which has been visited more than 250,000 times since its inception, also features some of the honorees in podcast interviews by Evelyn Lamb, and each honoree is featured in freely downloadable posters.”

I encourage you to read the profiles of each honoree unraveled daily on the website along with this AMS notices article five of this year’s honorees.

This year, I wanted to highlight another fantastic organization TODOS: Mathematics for ALL. As described in their website, TODOS mission is to advocate “is to advocate for equity and high-quality mathematics education for all students— in particular, Latina/o students”.

“TODOS: Mathematics for ALL is a professional organization that advocates for equity and excellence in mathematics education for ALL students – in particular, Latina/o students. Founded in 2003 and with over 800 members from across the country, TODOS Members know that Equity and Excellence in Mathematics Matter! We promote Social Justice in Mathematics education and provide high-quality resources to help reach our Mission and Goals.”

TODOS was established in the years 2000-2003 as a result of the Equity and Diversity Advisory Committee (EDAC)  sessions at the National Council of Teachers of Mathematics (NCTM) annual meetings. The themes of these sessions revolved around organized issues of teachers of Hispanic/Latino students.

The website of the organization is full of resources for mathematics educators, parents and families (in both English and Spanish), webinars, a podcast, a blog, and many more. In this tour, I give you a glimpse of some of TODOS blog posts. I’ve learned so much by reading these posts and found their resources insightful.

Latinidad in the US, Latinx, Latina/o, or Hispanic?: Geographies of Oppression, Race, Gender, and Language.

In this blog post by Carlos López Leiva, Silvia Llamas-Flores, and Kyndall Brown, the authors describe the importance of naming an identity, in particular the naming of the Latina/o community in the United States. They state very clearly at the beginning of the post that rather than propose solutions, they hope to open the conversation around naming identities.

Naming an identity is something many of us may feel strongly about, especially as it relates to our relationship with the United States. As stated in the post, naming my identity has been a way to name the community I belong to. However, as an identity that captures a group with a lot of diversity, we may never have a consensus on a single word that captures all of our experiences. However, the post highlights that naming those identities depends on the historical and current context in meaningful ways and that as educators we must acknowledge the identity of our students beyond mathematics as a way to make our interactions more meaningful.

“As educators, we must challenge places of marginality (Aguirre et al., 2013). We must also learn about, acknowledge, and nourish the intersectional identities of the students with who we work.

When we self-identify, we often make use of language to name those identities according to a context. In the case of Hispanic, Latina/o, Latin@, Latinae, and/or Latinx people in the U.S., the changes in these names or words have been linked to linguistic and political perspectives.”

These terms  (Hispanic, Latina/o, Latin@, Latinae, and/or Latinx people) have changed as a way of challenging  paradigms on many different axes of identity such as gender, race, land, residency status, language, among others. The authors conclude with a call to unity towards inclusivity and ask the readers to share with them which term is more relevant to identify this diverse group and what should that decision be based on.

Ethnomathematics: Mathematics de TODOS

In this post, Carlos LópezLeiva, Kyndall Brown, and Silvia Llamas-Flores, dive into the world of Ethnomathematics, as they define (and describe) it as,

“Ethnomathematics is a term introduced by Ubiratàn D’Ambrosio (1991) from Brazil to describe the techniques used to explain, understand, and cope with reality in order to survive across diverse communities. Ethno relates to the members of distinct groups identified by cultural traditions, codes, symbols, myths, and specific ways of reasoning and inferring (D’Ambrosio, 1985). So, ethnomathematics refers to the way that members of various cultural groups mathematize their own reality because it examines how both mathematical ideas and practices are processed and used in daily activities (D’Ambrosio and Rosa, 2017, p. 288).”

This definition blew my mind, especially as I think about how we engage with math and the world around us. The goal of this post is to showcase the relevance and practice of ethnomathematics in and outside the classroom. In terms of the first goal, the authors mention that ethnomathematics can help us engage with a culturally-responsive approach to teaching mathematics. They share great resources on mathematics community-based approaches in and outside the U.S. and works at the intersection of ethnomathematics  and culturally responsive mathematics education.  Some including the Navajo Nation Math Circles, Current and Future Perspectives of Ethnomathematics as a Program, Mathematics of the Americas, and Google site, lesson plans on ethnomathematics. I encourage you to take a look and explore all these amazing resources!

Finally, as the elections draw near, in their post, The Mathematics of Voting and its Consequences: Ideas for Mathematics Lessons, they present resources on the historic link between democracy and voting, current issues that relate to inequity in voting, and mathematics lessons on voting. For example, using mathematical modeling to explore the link between voting and the Common Core Standards of Mathematics.

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).

## 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!

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.

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.

$\Updownarrow$

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.

## 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.”

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!”

## 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.

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.”

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.

## 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, she 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).

## 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 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 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).

## 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)!