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Conway’s Circle Theorem: a proof, this time with words

Back in the olden days, Colin entered a proof without words in the Big Mathoff. It was mentioned, in passing, in a New York Times obituary of John Horton Conway.

Resources for Anti-Racism and Social Justice in the Mathematical Sciences

Yellow back ground with a closed black fist on the left with two rows of text next to it reading I am a mathematician / Black Lives Matter
(via WeRepSTEM)

(Author Positionality: I want to start this post by stating I am writing this from my position and lived experience as a white, male-passing queer, non-binary person who has lived their whole life in the United States of America. I am employed full-time as the mathematics & statistics librarian at a large endowment public doctoral granting university in the USA. I do not have to go up for tenure. I am a highly privileged person. I have not been perfect and I have been a part of the problem. I believe anti-racism is the way to no longer be a part of the problem. I believe Black Lives Matter.)

Note: This post will focus on the mathematical sciences within the USA as that is where my lived experience and knowledge lies. Also, this is an evolving post and resources will be added/changed over time.

Systemic racism, empowered to a great extent by white supremacy, is a part of the society of the USA. The results of this systemic racism can be seen everyday and has most recently made itself crystal clear through the killings of George Floyd, Tony McDade, and Breonna Taylor (among many others), by police and Ahmaud Arbery by a former officer, as well as the differential impact race has on COVID-19 infection rates and outcomes. The mathematical sciences (under whose umbrella I include both Mathematics & Statistics) are a part of this society and not immune to this systemic racism as can be seen day in day out in many ways, including the lack of representation of Black people in the mathematical sciences. A lack of representation that is ongoing and historic with the first Ph.D granted in mathematics in the USA was in 1862 while it was not until 1924 Elbert Frank Cox became the first Black person granted a Ph.D. in mathematics in the USA and intersectionally that Euphemia Lofton Haynes had to wait 19 more years to become the first Black woman granted a Ph.D. in mathematics in the USA. Statistical and Mathematical organizations from around the country have all made statements which are worth reading to understand where the US mathematics and statistics community is collectively at this moment, but I would like to call attention in particular to the one by the National Association of Mathematicians

NAM was founded in 1969, one year after the assassination of Dr. Martin Luther King, Jr. sparked widespread protests throughout the nation, similar to the ones we are seeing today. Indeed, NAM’s founding was a direct result of the marginalization of Black people within the professional mathematics community, which then and now serves as a microcosm of the society in which we live. Over 50 years since NAM’s founding, despite the lessons of the civil rights movement, we still see systemic racial inequities in education, economic prosperity, criminal justice and public health. Today, it should be clear to us all that the consequence of ignoring these racial inequities is dire.

NAM’s Statement on the Death of George Floyd

(you can join NAM here).

In my own journey toward removing things like false neutrality and color-blind ideology and incorporating anti-racism and social justice into my life and work I have put together a set of lists and resources which may be helpful for those who see a need to take anti-racist steps against this systemic racism. These resources are especially helpful to those with white privilege like I have, especially if you have never interrogated your privilege.

First let us define Anti-Racist:

There is no such thing as a “not-racist” policy, idea or person. Just an old-fashioned racist in a newfound denial. All policies, ideas and people are either being racist or antiracist. Racist policies yield racial inequity; antiracist policies yield racial equity. Racist ideas suggest racial hierarchy, antiracist ideas suggest racial equality. A racist is supporting racist policy or expressing a racist idea. An antiracist is supporting antiracist policy or expressing an antiracist idea. A racist or antiracist is not who we are, but what we are doing in the moment.

This is what an antiracist America would look like. How do we get there? by Ibram X Kendi

This set of Scaffolded Anti-Racist Resources provide activities, books, articles, videos, podcasts, and next steps for everyone, even if they have no prior knowledge of social justice and anti-racist concepts. ShutDownSTEM has put together a group of readings and resources for a wide range of people. They have self-care and healing resources for Black academics and STEM professionals, readings for people who are only now starting to think about race, and information for those who want to examine anti-Black racism in academia. For those looking for a more structured approach, Autumn Gupta and Bryanna Wallace have put together a 30 day Racial Justice challenge (at 10,25, or 45 minutes a day). If you are just starting your journey and looking for definitions of some foundational social justice concepts I was on a team which recently published a column featuring 15 such definitions with examples of each from the sciences. Jasmine Rice was also put together a list of 10 actions white academia can take to support their Black colleagues and students.

There are a number of readings on these topics that are related to the mathematical sciences as well. I will call out some I have personally found helpful in conceptualizing and integrating anti-racist and social justice ideas into my conceptualization of the mathematical sciences: The rehumanizing mathematics work of Rochelle Gutiérrez, the Living Proof: Stories of Resilience Along the Mathematical Journey collection, Inventing the Mathematician: Gender, Race, and Our Cultural Understanding of Mathematics by Sara N. Hottinger, Mathematics for Human Flourishing by Francis Su, Lou Edward Matthew’s on rejecting gentrification within the mathematical sciences, Tukufu Zuberi’s Thicker than Blood: How Racial Statistics Lie, AMS’s Inclusion/Exclusion Blog, the many other readings available at University of Michigan’s Math Learning Community on Inclusive Teaching site, the items on NCTM’s Math and Social Justice book and article list, and Kari Kokka’s list of Social Justice Mathematics and Science Curricular Resources for K-12 Teachers are all good, if incomplete, starting points. For more information about the history of Black mathematicians Mathematicians of the African Diaspora is a great resources, and to better understand the lived experience of Black mathematicians check out the profiles in Mathematically Gifted & Black and for the lived experiences of Latinx mathematicians the profiles in Lathisms. If you are looking for more readings that encompass all of the sciences, the pan-STEM Decolonising Science Reading list by Chanda Prescod-Weinstein provides a wide selection.

Finally, I have put together a list of anti-racist mutual aid projects you can donate to,

organizations and projects focused primarily on the mathematical sciences you can become a member of, or otherwise support and sponsor,

actions you can take,

and others projects to support which cover the rest of STEM.

There are a lot of learning and actions happening and a lot more which need to happen. I know there is still so much left for me to do and so much left for me to learn. You can reach out to me if you want to discuss these topics more. If you are starting out and struggling with the concepts or you are looking for more ways to learn more about anti-racism and social justice or you are wondering what a next step could be I can not guarantee I will have an answer but I am happy to talk with you. I hope everyone has been able to find and access the support they need, and if there is a way I can provide needed support let me know and I will do what I can.

Black Lives Matter.

IWD 2020: Books about Maths by Women

For International Women’s Day, mathematician Lucy Rycroft-Smith has read a selection of maths books by women authors, and recommended some favourites.

There’s a strange irony about being a woman in mathematics. You spend a huge amount of time and energy answering questions about being a woman in mathematics instead of, you know, using that time and energy to do or write about actual maths. We women are somehow both the problem and the solution. 

But behold: 2020 is here, and better and braver women than I have solved this conundrum.  Here are a whole host of excellent books about maths by women that you should definitely read, collected for you by another woman in maths. 

Geogebra to Cake in Five Steps

Flower design using Reflect Object in Circle tool in Geogebra
Tessellated squares design using Reflect Object in Circle tool in Geogebra

In the Aperiodical’s Big Internet Math-Off 2019, Becky Warren posted an entry about Geogebra’s ‘reflect object in circle’ tool (it’s the second article in the post). I enjoyed playing with the tool and, after making a few colourful designs, it occurred to me that one of them would make a great cake for the MathsJam bake-off. It would only work if the curves were accurate; sadly this would be beyond my drawing abilities, and definitely beyond my piping abilities. But with some help from 3D printing I thought I might be able to manage it.

Here are the steps I used to transfer the design to a cake.

That which we call an identity

I’m grateful to Jemma Sherwood and Rob Low for reading an early draft of this and for their comments thereon. All opinions are, of course, my own.

This post is inspired by something that I see crop up now and again in discussions with other Maths teachers. It usually manifests itself as a rallying cry to use ≡ in place of = in identities and reserve = for equations. My standard response is to mutter something about identities being equations and leave it at that. But in the latest round, Jemma Sherwood challenged me, in the nicest possible way, to explain a bit further. This is that explanation.

Although I’m going to state my case here, I’m well aware that there are different opinions. In matters of opinion, such as this, agreement and disagreement is less important than that all sides think. So if what I write seems to you wrong, that’s fine so long as it makes you think about why you think that it is wrong.

Small Sets of Arc-Sided Tiles

Tim has previously written guest posts here about tiling by tricurves, and is now looking at ways of tiling with other shapes.

In an earlier post elsewhere I covered some basic arc-sided shapes that tile by themselves. Lately I’ve been playing with groups of curved tiling shapes, asking a question common for me: how to get the most play value as an open-ended puzzle? This means getting the most interesting possibilities from the simplest set. “Interesting” includes variety, complexity, challenge and aesthetic appeal. “Simplest” covers not only size of set and the shapes, but also the least total information needed to describe or construct the shapes.