iRashida: A Tour

is a blog written by Rashida Hakim, a junior computer science student at Caltech, where she shares “problems and ponderings in physics and math, with a sprinkle of computer science”.  In particular, she is interested in the algorithms used to implement machine learning and artificial intelligence. I was excited to find her blog and enjoyed reading many of her posts. In this tour, I summarize my favorite posts and talk to Rashida herself, to get to know more about the motivation behind her blog.

Putting #1 First – Deriving Benford’s Law

In this post, she discussed Benford’s law is named for physicist which gives a distribution for the first digits observed in datasets. In particular,  it states that the probability that some digit d will be the first digit is $log_{10}(1+ \frac{1}{d})$ (see Figure 1).

Figure 1. Plot of Benford’s Law vs the first significant digit of a set of physical constants. Public Domain.

She takes the reader through the intuition of the law using the circular slide rule, its generalization to other digits, and provides an example of how it has been used in the real-world as a way to detect fraud.  She ends the post by reflecting that one of the most interesting facts is that this law doesn’t apply to just numbers. For example, the natural numbers do not seem to follow this distribution. However, it says something about the real-word.

“The circular slide rule explanation exposes that exponential sequences and other multiplicative sequences should follow Benford’s Law. But does that apply to data sets of physical constants, or heights of mountains, or the masses of celestial bodies? All of these examples, and many more, have been observationally shown to follow Benford’s Law. Maybe the real discovery that Benford and Newcomb made isn’t just about digits, but about how multiplicative and exponential growth is inherent to so many unexpected corners of our universe.”

Mathematicians’ Many Hats

Riddle: “There is a line of 100 mathematicians, each positioned so that he can see all the hats in front of him but none of the hats behind him (nor his own hat). Each of the mathematicians is randomly assigned a colored hat, which is either red or blue. Note that there do not have to be exactly 50 of each color (since each mathematician’s hat color is independent of the others). Now the mathematicians are asked to guess the color of their own hat, starting from the back of the line and everyone can hear the guesses. Assuming that they are allowed to confer beforehand to discuss a strategy, what is the maximum number of mathematicians who can guess correctly?

In this post, Rashida discusses how to solve this and variations of the riddle with logic and a bit of set theory. Turn out that in this more simple case all but one mathematician (the first) can guess their hat color correctly by using the formula below (see Figure 2).

Figure 2. Obtained from the post Mathematicians’ Many Hats.

She describes two variations of this riddle and their solutions. First,  replacing 100 with a countably infinite number of mathematicians, and secondly, what happens when no information is being carried forward when the mathematicians say their own hat color.

RH: I’m currently a junior at Caltech studying computer science, but I started my blog as a high schooler. I started my blog as a way to remember cool math, physics, and computer science concepts that I learned through school projects or on my own. I like to do fairly long-form posts on a single subject that approach the subject from multiple angles. A concept inspires me to write a blog post when at first it seems simple (or is easy to introduce) but under the surface, there is mathematical complexity that I can really dive into. Over time, my blog posts have evolved to include more simulations and code (rather than pure math/physics) to match my evolving interests.

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

RH: The most interesting thing I’ve learned through blogging is that you have to understand a concept really well (much better than you initially think!) to explain it to other people. It always surprises me how many unanswered questions pop up in my brain in the process of writing a blog post. And still, I sometimes get questions from my readers that I haven’t even thought about!

VRQ: What are some of your favorite blog posts you’ve written?

RH: One of my favorite posts that I have written was about Benford’s Law – which explains why some datasets have first digit distributions that aren’t uniform. I like that post because it is about a fairly unintuitive concept that shows up in a lot of places. Just recently, I got a lot of hits to that post because Benford’s Law was being (incorrectly) applied to find discrepancies in the 2020 election data. I hope after reading the post, people have the mathematical knowledge to debunk that particular bit of ‘fake news’ for themselves. Another of my favorite posts is about the optics of why rainbows show up. Most people know the basics of rainbow formation – water droplets in the air refract sunlight. But when you closely study the physics of just a single drop of water, you can learn so many new and interesting facts about this natural phenomenon. I like the post a lot because it taught me that it is worth it to go deeper, even on subjects I think I already understand.

VRQ: What are some of your favorite math blogs, if any?

RH: I love Ben Orlin’s ‘Math with Bad Drawings’ for insight on both math and math education. Also, it is often hilarious (if, like me, you’re the kind of person who laughs at geometry puns). I also enjoy David Richeson’s ‘Division by Zero’ because his posts are very interactive and make difficult geometry concepts accessible.

VRQ: Do you have advice for other students interested in creating their own blog?

RH: My advice is to focus on writing posts that excite you and that make you feel like you’re learning while writing them. That way, even if you’re not getting much engagement from readers, your blog is still a valuable learning experience for you.

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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Mathematical Gemstones: A Tour

is a blog created by Dr. Maria Gillespie (Colorado State University) whose research interest lies in combinatorics, with applications to Algebraic Geometry and Representation Theory. One of the aspects I like most about the blog is the fact that the posts are organized by level of difficulty using gemstones (i.e. Amber, Pearl, Opal, Saphire, and Diamond) as indicators.  It was really interesting to see how the writing and explanations varied according to the levels and their intended audience.

I was curious to know more about the inspiration behind the ‘Mathematical Gemstones’ blog so, in this tour, I hope to give you a glimpse of two of my favorite posts and insights from Dr. Maria Gillepsie herself!

(Sapphire)

In this post, Gillespie discusses a solution to the following voting problem:

Suppose two candidates, A and B, are running for local office. There are 100 voters in the town, 50 of whom plan to vote for candidate A and 50 of whom plan to vote for candidate B. The 100 voters line up in a random order at the voting booth and cast their ballots one at a time, and the votes are counted real-time as they come in with the tally displayed for all to see. What is the probability that B is never ahead of A in the tally?

She discusses first how to enumerate all ballots as sequences and then finds the solution using the crystal functor (F1)  to count the ballot words by counting the chains of the F1 graph. I found the diagrams in the post very insightful, especially for those new to the topic but familiar with combinatorics.

Another fun post was the idea of constructing Pythagorean triples (i.e. a triple of positive integers (a,b,c) with a²+b²=c²) on a sphere! By parametrizing all triples via geometric means, one can show that (r²−s²,2rs,r²+s²) is a Pythagorean triple on the unit circle for any integers r and s.   However, questions about finding Pythagorean triple on the unit sphere remain open, and she shares,

“There is some hope, however. In this paper by Hartshorne and Van Luijk, the authors show that there are infinitely many Pythagorean triples in the hyperbolic plane, using the Poincare Disk model and some nice hyperbolic trig formulas combined with some Eulerian number theory tricks. So Pythagorean triples are not the sole property of flat Euclidean space.”

Maria Gillespie:  I’m an assistant professor at Colorado State University, currently working towards tenure. I started my blog when I was a graduate student at UC Berkeley.  At the time, there was just so much new mathematics that I was trying to learn, and I know I absorb things best when I explain things to others.  So it started out as a way for me to learn new concepts in graduate school, by typing them out to an online audience. At the same time, I’m also passionate about bringing the joy of mathematics to everyone around me – sometimes I learn or remember a fascinating nugget of truth in mathematics, and I just want to yell it to the universe and share that joy with as many people as I can.  So I decided to put together a blog that could accomplish both of these goals at once.The way I accomplished this was to make each post a self-contained “mathematical gemstone” – a shining example of mathematical beauty and truth.

Some gemstones could be “harder” than others, perhaps assuming a higher level of mathematical background for the intended audience.  In order to help the reader determine which posts would be appropriate for them to read, I sort posts into five levels of “gemstone hardness” according to actual measures of how hard real gemstones are, starting from Amber (one of the softest gemstones) and going all the way up to Diamond (a very hard stone).  Here is a description of the levels, from my About This Blog page:

• Amber: This category contains posts that anyone with very basic elementary or middle school mathematics background can appreciate.
• Pearl: For Pearl posts, some high school courses or early college courses may be helpful in understanding the content.
• Opal: These posts are aimed at undergraduates with some basic first-course background knowledge in algebra, analysis, discrete math, or topology.
• Sapphire: These gemstones would be appreciated by mathematics graduate students or professors, or those with at least an undergraduate degree in mathematics.
• Diamond: The hardest type of gemstone. These posts are highly specialized, containing content that only mathematicians who have studied the topic in depth will have the background to understand.

I later added a “Miscellaneous” category that allows me to share thoughts or discoveries that are not strictly mathematical, but which are related to mathematics as a discipline.  This category includes things like LaTeX tips, book recommendations, thoughts on social issues in mathematics, and most recently, doing math in a pandemic.  To summarize, my blog serves as an organized platform for me to share ideas and thoughts that could be enjoyable or helpful to other mathematicians and scientists, to students, and to the general population.

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

Maria Gillespie: People can get REALLY upset when you try to suggest new ways of doing or explaining elementary mathematics.  I have a post on the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), which is often used as a tool in middle and high schools to teach the order of operations.  In the post, I explain why I feel the acronym can be confusing to students, and the incorrect answers you can get by interpreting it too literally. That post really must have hit a nerve among some readers for some reason, because I have never been at the receiving end of so much hatred and vitriol in the comments!  It just goes to show that even mathematics can become controversial if it’s out there on the internet.

VRQ:  What are some of your favorite blog posts you’ve written?

Maria Gillespie: I’d have to say the most fun one I’ve ever written was Can you prove it … combinatorially?in which I prove Binet’s formula for the Fibonacci numbers directly using a combinatorial proof, without any induction or generating functions (the usual methods).  It was just so satisfying and fun to find a combinatorial proof of a formula that involves the square root of 5, as intricate as such a proof may be.My favorite among the “soft topics” is the recent series of four posts I wrote on doing math in the pandemic, starting with (it links to the others there).  Writing all of that out really made it hit home how very much we’ve all learned from being forced into a virtual setting this past year.

While the pandemic has been a horrible natural disaster, the silver lining I see is that there’s a lot of opportunity for some of these tools to still be used after the pandemic is under control, and I’m excited to see where things go in the future.Finally, the posts that have turned out to be the most useful are the ones I wrote early in graduate school on the basics of my research area, in symmetric function theory and Schubert calculus.  I’ve looked back at them countless times to remember formulas, and other grad students in my area have appreciated them as a resource as well.

VRQ:  What are some of your favorite math blogs, if any?

Maria Gillespie: I have found so many of Qiaochu Yuan’s posts on his blog Annoying Precision so useful.  They’re well-written, precise, and often have exactly the proof of some fact that I was trying to look up at the time.  I also like since I think he has really good ideas about open access and the future of mathematics publishing.

VRQ: Do you have advice for other mathematicians interested in creating their own blog?

Maria Gillespie: My main piece of advice is to just go for it!  Write about what you’re passionate about, or something you’re learning at the moment, and post it without too much worry about who will see it and whether it’s perfect.  Blogs are an excellent outlet for mostly-correct mathematics (or mathematics-related topics) that doesn’t have to be peer-reviewed but can add a lot of value to the world.  And more likely than not, your thoughts will be valuable to someone. As for the technical side, WordPress is your friend and it will take care of you.

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)

“Combinatorics and more”: A Tour

Gil Kalai writes the “Combinatorics and more” blog. I find many of his posts on the blog to be detailed and nicely structured. Here are just a few of the recent ones I enjoyed.

I always think it’s interesting to explore which big research questions attract a lot of interest. For those who aren’t familiar with polymath projects, this post describes what they are and gives updates on potential polymath projects that Tim Gowers suggested on his blog in 2009. Kalai also suggests some possible future directions for polymath projects and asks “meta questions,” such as “What is the ideal platform for a polymath project?” and, my personal favorite, “Are polymath projects inviting in terms of diversity of participants?”

The “To cheer you up in difficult times” posts

So far, Kalai has created 19 “To cheer you up in difficult times” posts, including this one about a proof of the Erdős-Faber-Lovász conjecture uploaded to arXiv by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus and
To cheer you up in difficult times 13: Triangulating real projective spaces with subexponentially many vertices” about another new paper posted to arXiv by Karim Adiprasito, Sergey Avvakumov, and Roman Karasev.

This post is replete with interesting bits of information about recent papers, videos from some of Kalai’s lectures and even “a small taste of quantum poetry for the skeptics.”

Besides all of the interesting posts by Kalai, there are also a bunch of guest posts worth checking out. Here are just a few:

“Dan Romik on the Riemann zeta function”

“Recently when I was thinking about the Riemann zeta function, I had the double thrill of discovering some new results about it, and then later finding out that my new ideas were closely related to some very classical ideas due to two icons of twentieth-century mathematics, George Pólya and Pál Turán,” Romik wrote in the beginning of the piece.

In this post, Bárány explains limit shapes and the limit shape theorem, limit shapes for polygons in convex bodies and more.

Want to share ideas or feedback? Reach out in the comments or on Twitter (@writesRCrowell).

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Math Walks: A Tour

Math Walks is a blog created by secondary math teacher Traci Jackson. It started on March 27th to encourage math discussion on neighborhood walks during the quarantine.  I was so excited to find this blog that brings such a playful atmosphere to learning math.    Inspired by a mom and her kids doing a PE class on a walk, she was inspired to find a way to bring math to daily walks!

“If I could leave a little bit of math on daily walks, I could not only give parents a way to incorporate some math, but maybe I could try and change math into curiosity, wonder and problem solving, even just a little.” – Traci Jackson, in  the About section

As a person who loves going on walks, especially, since this year has been spent a lot in online meetings, I found her blog full of playful invitations to do math.  You may suspect that like my last post, this intersects in wonderful ways with my hobbies and a love for math. And, who doesn’t love playing with chalk?

It was so well received, that translations in French and German are now available. She shares, how excited she is about people creating their own math walks.

I was thrilled to see other people duplicating drawings or creating their own. It was suddenly not just my neighborhood! Although watching my neighbors take pictures and problem solve as family was equally interesting, since almost every time I walked down my street I could see problem-solving in action. I got several texts and a few emails asking “Is that you leaving math everywhere?” I guess I have a reputation.” – Traci Jackson, in  the About section

You can see many walks share by others by following the hashtag #mathwalks on Twitter. If you enjoy puzzles, beware, this blog will have you puzzling the day away.  Below I share some of my favorites puzzles:

“Graphing Stories” based on the website with the same name, which is a collaboration between Dan Mayer and BuzzFeed.

“Climbing Stairs” based on Curriculum Burst 28: Stair Climbing by Dr. James Tanton.

“Crack the Code” based on the by Rajesh Kumar.

In her blog, she provides helpful how-to’s to begin your own math walk.  If you are not great at free-hand drawings you can bring cutouts of the shapes you will use, use string as a compass, and a nice tip is to bring an index card with the problem drawn out.  The puzzles come from a wide variety of sources which you can read more about by clicking on the pictures.

Her blog was also featured in How Sidewalk Math Cultivates a Playful, Curious Attitude Towards Math, in which she remarks,

“The perception of math is a set of sterile problems but in reality, it describes all the patterns of our world. … [Sidewalk math] opens the conversation to what math is. It engages people who wouldn’t do math ordinarily,” Jackson said. She believes that the visual element plays a role in that engagement. Stanford professor Jo Boaler has advocated for the learning benefits of visual math for years, but Jackson said it remains an under-explored dimension of math instruction. Though we often think of math as numbers and letters, Jackson sees it as a way of viewing the world, and using images can unlock new connections.” – From How Sidewalk Math Cultivates a Playful, Curious Attitude Towards Math

I couldn’t agree more! What a wonderful way to share math with your community. So, this Spring, take your math for a stroll, and if you bring your chalk with you, share it with your neighborhood.

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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On the “Reflect, Revise, Repeat” Blog

Bonnie Basu, a secondary mathematics teacher in California, writes the “Reflect, Revise, Repeat” blog. She started the blog in June 2020. On Twitter, Basu describes herself as “trying to teach teenagers to think mathematically for a quarter of a century.”

There are currently nine posts on the blog. Here are a few of the ones I recommend checking out:

This is the first post that Basu wrote. She describes why she decided to start the blog now after thinking for a decade about writing one.

She wrote that instead of focusing on the number of years she’s been teaching, she typically reflects on her growth. “Am I making the same mistakes as I did during my first years? Or even last year? Why did I make those mistakes? What did I learn from those mistakes? Who can I turn to for mentorship to help me grow as a teacher? And I try to do this continuously,” she noted. She used this idea to come up with the name of her blog.

“My Plan” posts

Last fall, Basu wrote some “My Plan” posts, including “My Plan: Week 1,” “My Plan: Week 2,” and “My Plan: Week 3 and 4” about how she handled distance learning for the first month of the semester. There are points scattered throughout that I thought were great.

For instance, in her Week 1 post, she wrote:

“The only way of communicating with families is email or individual phone calls. That made it extremely challenging and frustrating that our automated phone system wasn’t set up so we can at least call en mass.  (By the way, it’s still not set up and it has been 5 working days.) I have been calling home, but it’s a slow go.  I can get through about 5 each day before I have to stop. Most take a lot of time because I am fluent in only one language – the one I am currently writing in. And the district doesn’t have a department solely dedicated to helping teachers translating which is very strange considering how many of our families are more comfortable speaking a variety of other languages.”

Another thing that resonated with me was her discussion of how disconnected students have felt from their teachers and each other, along with her ideas for building community in a virtual setting.

In her Week 2 post, something that stuck with me was her discussion of her desire to help students become more independent and the challenges that come with cultivating that independence with distance learning. She wrote:

“Over 80% of the student population is highly dependent. And mostly because of all the hand holding that goes on from when they are little.  Just because kids are from a vulnerable and disadvantaged area, does not mean they cannot.  But that is the mindset that so many educators have.  If kids are given the opportunity and have scaffolding in place, success will happen.

I help students become more independent when in the classroom, but I am really struggling with it virtually.”

By her Week 3 and 4 post, Basu started using the phrase “distance surviving” instead of distance learning. She wrote “I feel like I am riding a rollercoaster in the dark.  (I don’t like rollercoasters nor do I like the dark.)” She discussed a situation where many kids didn’t complete an assignment and how she handled that.

“Find the Good”

This is the first post Basu wrote in 2021 and I think it’s a good one to discuss in closing out my post. She opens with this relatable paragraph:

“I have had many challenges in my teaching career, but nothing like I experienced over the last 9 months (as every single teacher). Every day was like starting all over. I had no idea what I was doing and was just trying to survive each day while keeping my students’ education and social-emotional well being at the forefront.”

She details the importance of teachers knowing their audience and making slow changes with students to give them time to acclimate. Reflecting on the previous semester, she wrote that she had “wanted to weave in assignments that would support a healthy mental state and allow them to forget the world, even if it was for a few minutes.” Based on the feedback she received from her students, it sounds as if she met that goal.

She also wrote about switching gears from giving weekly assignments she hoped would teach her students time management to assigning activities that were more fun, such as ones in Desmos.

Have an idea or suggestion you would like to share? Reach out in the comments or on Twitter (@writesRCrowell).

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Playful Invitations: A Tour

Playful Invitations: Inspiring Ways to Teach Early Mathematics, is a blog written by Dorie Ranheim. Its goal is “to inspire parents, caregivers, and educators of preschool children to intentionally teach math using natural materials.” By using “loose parts”, backyards, playgrounds, and parks become great places for teaching and learning math. As described in the blog’s about page,

“My playful invitiations to learn math eventually extended beyond our home to our backyard and nearby park. During our time outdoors I realized I could showcase the beauty of real, natural materials and how inspiring, meaningful, and relatively easy they are to acquire. Overall, I hope to share ways adults can intentionally teach preschool math using these beautiful natural materials.”  – Dorie Ranheim

In this tour, I will summarize some of its most recent posts. Many of them can be used as guided activities, and Ranheim provides a helpful guide on   One of the aspects I appreciate about the activities is that they are all centered around play. Many of the posts consist of three parts: Prepare, Invite, and Play, and some include reflections and extensions to the activities.  As she remarks,

“The blog posts are simply suggestions. There are MANY ways to develop these math skills. My hope is that reading the blog will inspire you to find opportunities in your daily life to teach math to preschool children.If it is playful, meaningful, and reasonably challenging then chances are the learning will stick!”

Numbers

In this post, Ranheim shares some of the ways that during last Spring, she and her children became more entune with nature and spent some time thinking of long-term projects.

We watched the bare branches bud and blossom, now we celebrate trees bursting with bright green leaves. […] Here are a few ways my children have explored math during this time at home:

• Practice number identification and formation using loose parts.

• Write numbers on river stones using water and brushes.

• Pattern using various colored rocks. Sometimes the simplest activities and materials seem to hold their attention the longest!

I found it related a lot to what I’ve done since the quarantine. In a very similar what, I discovered patterns in one of my hikes.

My own exploration of patterns at the beach.

Measurement

Invite: Today I thought we could trace our bodies so we can see how big we are. (After tracing one or more bodies) I wonder whose body is the tallest/longest?

This invitation is inspired by the following quote,

To compare objects, children begin by using nonstandard units (“My table is more than four hands long”) and then move to using standard units (“The table is almost three feet long”). Comparing fairly is an important concept for young children. – Juanita Copley, 2010

This made a lot of sense to me! Once you learn the standard units of measurement it’s easy to forget all the other intuitive ways we make sense of measurements. If anyone has ever tried to learn a family recipe, you’ve probably encountered many non-standard ways of measuring yourself. In this activity, each child lays on the ground and draw a chalk outline of their body, afterwards they use natural loose objects of similar size (e.g. leaves, rocks, etc.) to lay them side by side and compare the lengths. Some fun extensions include introducing rulers, or filling the outline of the body, and talking about the area.

Puzzle and Spatial Position

Invite: “I’m trying to put this leaf back together! Will you help me find the perfect match to make my leaf complete?”

As a big fan of tangrams as a kid, I love that in this invitation, you introduce the idea of fractions at a basic level by transforming leaves into puzzles.

It is a great way for children to play with the idea of a “whole”. You can start by cutting leaves in half and trying to find the match, or you can extend the activity by adding more different types of leaves or cutting them in four pieces instead.  Ranheim advises that before using the leaves for learning math, they should explore their properties.

“It would also be beneficial if the child has explored the property of leaves before being asked to use them for math learning. Observe a small pile of leaves and the attributes before taking them apart.”

A personal project that has brought me great joy during the pandemic, has been to gather seeds and start my balcony garden.  I could see all the fascinating (and often subtle) ways math seeped into my gardening.

The joy of learning and caring for my small garden.

What drew my attention to this blog, were my conversations with my best friend and early-childhood educator, about the ways math should be tied into how we experience nature around us. I would have loved these activities as a kid!

Also, it reminded me of one of my favorite books Braiding Sweetgrass: Indigenous Wisdom, Scientific, Knowledge, and the Teaching of Plants” by can’t recommend it enough).

“The land is the real teacher. All we need as students is mindfulness.” – Robin Wall Kimmerer, Braiding Sweetgrass

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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“Physics Buzz”: A Tour

While the “Physics Buzz” blog from the American Physical Society isn’t a math blog, there is some overlap. Here are some interesting recent posts on the site.

“Holiday Instability”

This post explores questions such as whether a Christmas tree, a Hanukkah menorah or a Festivus pole is more likely to topple over and which of the items would be the best choice for easy juggling.

“The Forces in Spilled Coffee Awaken”

This post discusses the physics of coffee stains. “Just in case you’re wondering if it is a worthwhile use of processing power, time, and money to study a stain, consider this. Coffee isn’t the only substance that’s made of tiny particles suspended in a liquid. Blood, paint, ink…understanding the way these kinds of liquids behave could have huge implications in areas from medicine to high-tech manufacturing. For example, in the future we may be able to create tiny structures with unique properties by carefully dropping a liquid filled with nanoparticles onto a surface and evaporating the liquid. In order to do this, though, scientists need to be able to accurately predict a mixture’s behavior. This requires an understanding of the forces involved,” Kendra Redmond wrote in the post.

“Star Light, Star Bright: Measuring All the Starlight (Ever!)”

This post describes how “astronomers have found a way to ‘see’ all of the starlight produced in the history of the 13.7-billion-year-old universe.”

Article author Kendra Redmond wrote:

“If you want to know how much starlight is in the universe, you might try something like measuring all of the starlight you can see, and then estimating how much is out there that you can’t see. Scientists have performed refined versions of this type of analysis, but the estimates require lots of assumptions that may or may not match reality.

The Fermi-LAT Collaboration explored this question using an entirely new approach that doesn’t rely on the same types of assumptions. Instead of measuring starlight directly, they looked at the influence of starlight on high-energy gamma rays detected by the Large Area Telescope (LAT), an instrument on the space-based Fermi Gamma Ray Telescope.”

The team estimates that over the history of the universe, stars have produced about 4*1084 photons!

“Chaos, Fractals, and Complexity: Big Ideas in the “Science of ‘Roughness'”

This post presents “a few short stories behind some of the biggest ideas in chaos and complexity theory,” including Lorenz and The Butterfly Effect, Mandelbrot’s Fractals, Complexity Theory & The El Farol Bar problem and a section about “symmetry versus roughness.”

Want to share ideas or comments? Reach out below or on Twitter (@writesRCrowell).

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A Tour of “Nepantla Teachers Community” Blog

The Nepantla Teachers Community Blog is a group blog that aims “to provide an honest and encouraging space to navigate sociopolitical situations that occur in mathematics education for the purpose of working towards justice in traditionally marginalized communities. By using the word political, we mean any situation that involves power dynamics,” according to its authors. There are six instructors on the blog’s leadership team — Esther Song (high school math specialist with the Chicago Public Schools), Chanel Keyvan (Assistant Principal at Oswego Community SD and former mathematics teacher at Oswego Community SD), Jennifer Dao (mathematics teacher at Evanston-Skokie SD), Jerica Jurado-Paz (mathematics teacher at Chicago Public Schools), Erin Berg (mathematics teacher at Lyons SD) and Crystal Penn (mathematics teacher at Fulton SD in Atlanta).

Here are a few interesting recent posts on the blog.

“Small Wins: Math & Identity”

This post, which is part of the “Small Wins” series on the blog, was written by an anonymous writer Michelle, a math teacher who describes her experience with learning from her students “how to break the rules.” Her California school district has a policy that for remote Zoom learning, students must only use selfies, Bitmojis or nothing as their profile picture. But when she required that one student chance his profile picture because it didn’t meet those requirements, he said “I don’t see why I need to change my picture. I’m just trying to learn.” After he told her that his profile picture was of his favorite rapper who had died — and changed his picture back to it after she let him into the Zoom meeting — she allowed him to keep it as his picture.

She wrote:

What does my Zoom picture policing have to do with social justice and mathematics education? Everything. Especially in a Zoom environment, where most students’ cameras are off, it is even more difficult for students to express who they are as human beings. The limited avenues for self-expression are their Zoom picture and name, which are both mediated through Zoom as a platform. When Alberto changed his Zoom picture back to the picture of his favorite rapper, Alberto had demonstrated resistance in the mathematics classroom. How can students view themselves as mathematicians if they cannot bring who they are into the classroom? Who are students as mathematicians if they cannot resist and question what it means to be a student engulfed in a larger school system during a pandemic? As we discussed in our [Nepantla Teachers Community] over the summer, students are not simply stripped of their identities when they step into the mathematics classroom, even though many wish mathematics to be an apolitical space.

I asked myself, “Why am I following this distance learning policy so closely? Which students might this policy disproportionately harm? What actual consequences are there if students don’t follow this rule?” There are nuances and complexities within all of these questions. For instance, I am a first year teacher without tenure. There have been instances of inappropriate/offensive Zoom pictures. However, in the end, I decided to let Alberto keep his Zoom picture.

The “Student Voices in Remote Learning” series is also worth checking out. The most recent post in that series is from May.

“Universal Language Part I” and “Universal Language Part II”

Like other Part I and Part II posts on the blog, part I shares “a math teacher author’s real dilemma that they have recently experienced” and part II provides “an analysis of the powers at play and the author’s response (or lack of response) to the situation.”

In part I, Melissa Adams-Corral wrote:

In the summer before my third-year teaching, our district made a decision that I thought would be a game-changer: mandating dual language education district-wide. Previously, most schools in our district operated under bilingual education models that were focused on quickly moving children to all English instruction, with many schools refusing to offer clases bilingues at all. Moving to dual language meant that the district was taking an explicit stance advocating for students to continue to develop their English and Spanish side-by-side. I remember feeling very excited and hopeful about this shift…finalmente, I thought, policy would reflect the goal I had going into teaching—pride in bilinguismo, and meaningful, relevant language and content area instruction for mis estudiantes. It was a dream come true…that is, until I saw the model that all teachers were told to follow ‘with fidelity.’

This model required that certain content areas be taught in one language only and that teachers practice and enforce strict separation of languages in the classroom. My bilingüismo doesn’t work that way—it flows effortlessly, trying to stop it is like putting a wall in the middle of a river. I grew up in a bilingual home in Miami, where my language never needed to be split in two. During the summer, I would spend weeks with my primos en Honduras, singing along to Boyz II Men and Shakira, watching movies and telenovelas. Back at home in my city, bilinguismo and latinidad was lo normal. I became a bilingual teacher in large part because my language y mi cultura are a large source of my joy, pride and hope. I imagined bilingual teaching as being the work of supporting children as they grew from similar raices.”

When she followed the policy “with fidelity,” which required that her mathematics classes be taught only in English, Adams-Corral noticed a disturbing pattern. “In every math discussion, students who were comfortable speaking in English dominated. And mis estudiantes who preferred to read and write in español? They were silent. I could call on them and ask them questions, but they would shake their heads no, refusing to speak up. I would remind them that they could share their thinking in any language, already moving away from ‘total fidelity.’ But they would sit there and wait.”

Part II tackles the response, starting with Levels of Oppression, a reflection tool created by Mariame Kaba. I’ll leave it to you to check out Part II, because it’s meant to be read after reading and pondering part I.

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A Year in the Math Blogosphere

For me, the end of the year always is a time for reflection. If you haven’t yet, I encourage you to read Rachel’s round-ups of AMS blog post Part I and Part II.

In the AMS December Notices, Dr. Katherine Thompson wrote her opinion on the role of blogs in our mathematical community in  The Place of Blogs in the Modern Math World As Thompson mentions, while there are many considerations to still be figured out (e.g. structure, defining their success,  lack of peer review) these are an extremely versatile tool that is here to stay.  I was excited to see Blog on Math Blogs has 631 subscribers and an average number of shares over the last 10 posts of 110.2. I am grateful to all of our readers for their support.

Inspired by this piece and by Dr. Jennifer Quinn’s blog (see my last post), I wanted to take inventory of the lessons learned this year from Blog on Math blogs.

This blog has been a wonderful way to keep learning about fascinating mathematics and the people behind it. The blogs I have toured stood out to me because of their sheer dedication to share and be seen. As you may have realized by now, there is a whole world in the math blogosphere that I have sought to discover. One of the greatest pleasures has been learning from the breadth of content and styles that you get when you allow people the space to be their most creative selves.

This year we toured twenty blogs, wrote about different topics and themes ranging from mental health, crime-fighting, traffic modeling, Black History Month, among others, and interviewed seven bloggers.

Authors have shared their passion for mathematics and beyond, advocated for change, and given us a glimpse of their interests and passions. It begs the question, what will be the role of math blogs in the future? In my opinion, math blogs have opened the door to content that might be inaccessible otherwise (for both mathematicians and non-mathematicians alike). I am grateful for the opportunity to interview authors who joyfully share their blogging journey. If you ask me what the role of blogging and what is its importance is in our community, I think some of our interviewees have given fantastic answers to that question. Below you will find some of my favorite excerpts from the interviews.

“The name of the blog is “Logic ForAll”, now this is what I want, all people using logic formalized or not in the daily activities. But the name is also a pun, because in Brazilian Portuguese we have a dance and a style of music called “forro’ “. I only realized very late that the music (which is great and very danceable) comes from a mispronunciation of the English expression “for all”. So I wanted my blog to be like the music, fun and enjoyable and for all. Also, if possible full of little puzzles and games that it didn’t matter if you didn’t get them. It’s not about competition, it’s about fun!” – Valeria de Paiva in LogicForAll: A Tour

“I’d like to reach non-mathematicians that are curious about what a mathematician does, and how a mathematician works on proving theorems.

I’d also like to reach mathematicians, particularly “mathematicians in training,” who may want to read stories from the point of view of a more senior mathematician. I’m hoping they will relate to these stories or learn useful information about, say, what it’s like to be tenured or what it’s like to be a working mathematician and a parent in a household where both parents work and split childcare evenly. I hope the ‘realism’ in the writing helps people understand that we all struggle sometimes, that we have all gone through tough times and happy times during our careers and that almost all of us fight impostor syndrome.”  –  Álvaro Lozado-Robledo in Field Guide to Mathematics

“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.” Contanza Rojas-Molina in Rage of the Blackboard: A Tour.

“So my biggest piece of advice is to make it enjoyable and sustainable for yourself. There are no guarantees that even very good writing will end up getting widely read, but if you enjoy it and find that it helps you learn new things or understand your own ideas better through the act of writing them down, it’ll be worth it.”  – Evelyn Lamb in Farewell, Roots of Unity

“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.” – Sophia Wood in Fractal Kitty: A Tour

“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.” – Kaneka Turner, Deborah Peart, and Dionne Aminata in  #BlackWomenRockMath: An Interview

“I have been told that what I post has resonated with folks—not just mathematicians, not just teachers, but many people experiencing this wild and crazy pandemic year.  If they find any comfort, then I consider it a success.” – Dr. Jenny Quinn in Math in the time of Corona: A Tour

I look forward to continuing to discover the treasures in the math blogosphere next year. Until then, thank you for showcasing your math, for reading our posts, and joining Rachel and me as we tour the math blogosphere. Stay safe and happy holidays!

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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A Roundup of Posts on Other AMS Blogs (Part 2)

As I mentioned in my Part 1 post, I’ve been seeing a lot of posts on other AMS blogs that have piqued my interest and really got me thinking about a variety of different subjects. As we approach the end of this interesting and oh-so-challenging year, I offer you a roundup of some thought-provoking posts on other AMS blogs.

Earlier this year, the decision was made to broaden the scope of the BookEnds blog by AMS Consulting Editor, Eriko Hironaka. If you haven’t already read about that decision and would like to, this post by Nicola Poser discusses it.

In “Interacting With Ordinary Differential Equations,” a guest post by Stephen Kennedy (Carleton College), AMS/MAA Press Acquisitions, he writes about “changing content and delivery” methods for ordinary differential equations in the context of the online interactive textbook Interacting with Ordinary Differential Equations by Sandy and Max Saperstone.

On the e-Mentoring Network blog:

“The Mentorship of Our People” by Jennyfer Galvez-Reyes

Galvez-Reyes writes about her concerns with navigating the graduate school application process and the organizations she’s found — such as Cientifico Latino and Women+ of Color Project —that provide mentorship and support. She closes with these words:

“While there is no doubt that the application process is daunting, it can also be a chance to find your people. People who will cheer you on, pick you up when you’re down, and remind you of your worth when imposter syndrome threatens to take over. It’s important to not only have mentors ourselves but also to pass on the knowledge to those coming after us. Like Toni Morrison so perfectly put it, ‘When you get these jobs that you have been so brilliantly trained for, just remember that your real job is that if you are free, you need to free somebody else. If you have some power, then your job is to empower somebody else. This is not just a grab-bag candy game.’ Reach back and help those trailing you. Pass on information you wish you had, resources you needed, job listings you know about. Mentorship and community are integral parts of succeeding in spaces that weren’t designed for people like us. Despite the lack of consideration for us and our experiences, we have an ever growing community willing to help each other into these spaces.”

Saul wrote about his experience working with children under federal custody with the Office of Refugee Resettlement (ORR), which is now on hold because of COVID-19.

“The children love it. Their eyes light up. They intrigue each other. Language and social barriers tumble. And their minds are active. The work is similar to leaving food and water in the desert for thirsty immigrants. We are not offering them a complete diet or significant sustenance. But we are keeping their minds alive until their situation stabilizes,” he wrote.

Have an idea that you would like for us to cover? Want to share what you’re most excited about for JMM 2021? Reach out in the comments or on Twitter (@writesRCrowell).

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