For many people, in particular for students from underrepresented backgrounds and identities in mathematics, it is difficult to express your thoughts and experiences on applications (why this is requires a longer reflection and current research exists… so look into it!). In fact, many people I reached out to were hesitant to contribute to this post because many felt that they were “just lucky to get them.” I believe, THIS IS NOT TRUE. While there are human components to getting chosen to receive any award, we must acknowledge that we have worked hard and deserve every chance to apply and win these prestigious awards.

Dr. Jiuya Wang (Phillip Griffiths Research Assistant Professor and Foerster-Bernstein Postdoctral Fellow at Duke University) who won an Association for Women in Mathematics Dissertation Prize in 2018 for her dissertation in Number theory, wrote to me and mentioned that she let her advisor nominate her for that award. This may seem unusual for some, but sometimes it is in our best interest to self-nominate or ask others to nominate us for the prestigious awards that exists. In winning this award, not only did Dr. Wang get national recognition for her amazing work, she writes that “she [got] the chance to say thank you to the people [she] values in public.” As a recipient of a 2017 NSF Graduate Research Fellowship (NSF GRF), I can also add that we should not be embarrassed, afraid, etc., to ask for nominations or help. When I was applying for the NSF GRF, I recognized that I did not have the strongest writing background, so I sought the help in mathematical writing from my two master’s thesis advisors and my now PhD advisor and for the personal statement I sought the help from the fellowships advisor at my master’s institution (Advice: look to see if your institution has a Fellowships Office or Office of Nationally Competitive Awards, or something similar and seek out support from them).

While I hope that some of the advice shared in this post will help current applicants submitting applications to prestigious awards, I also hope that some of the readers are also people that will consider applying in the future (advice here applies also to finishing undergraduate students). With that said, TIME AND PLANNING ARE IMPORTANT! Dr. Marissa Loving (NSF Postdoctoral Fellow at Georgia Tech) mentioned to me that she began preparing for fellowship applications “(both mentally and by purposefully building [her] broader impacts/research portfolio) years in advance.” For her, the NSF Postdoc application was already on her mind from Day 1 of graduate school. While this may not be the case for some of you, I hope that you do consider applying to the NSF Postdoc and realize that time is essential for planning. One influential piece for her NSF Postdoc timeline was some job advice listed on Dr. Chelsea Walton’s website: https://faculty.math.illinois.edu/~notlaw/JobAdvice.pdf

Below are the experiences and advice of a diverse group of students and professionals who have received prestigious awards during their time as graduate students or shortly before/after. As my friend, Dr. Loving also mentioned to me, it is also helpful to share that she along with some of the other contributors on this blog post are women and people of color who have been very successful in applying for these awards. My hope is that as you read their replies to certain comments, you can get ideas on how to better your applications, manage the application timeline, and realize that there is a support system when applying for these awards. Additionally, I hope that readers recognize that the applicants are more than “just mathematicians”: they are people who come from all backgrounds, perspectives, experiences, and mathematical/personal interests, which you see below.

Thank you to all the contributors, many of whom I consider great friends!

Contributors: Jessica De Silva, Katie Taylor, Amzi Jeffs, Theo McKenzie, Elaina Aceves, Rob Davis, Nohemi Sepulveda, Darleen Perez-Lavin, Chase Ashby, Liam Solus, Anastasia Chavez, Jiuya Wang, Marissa Loving

Awards Represented: National Science Foundation Graduate Research Fellowship, Ford Foundation Predoctoral Fellowship, Fulbright Award, Department of Defense’s SMART Fellowship, NASA Pathways, National Science Foundation Postdoctoral Fellowship

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** Name:** Jessica De Silva

** Current Position/Institution:** Assistant Professor of Mathematics at California State University, Stanislaus

** Award Received and Year:** NSF Graduate Research Fellowship awarded in 2013

__Advice for future applicants:__* *Do your best to give the application reader a good sense of who you are, what you value as a person, and how it leads to you pursuing a graduate degree in Mathematics. I suggest trying to connect the reader to you in this way at the beginning of the personal statement, that way they feel invested in you while reading the rest of your application. For the personal statement, this also allows you to have a less modest (and therefore prouder) tone when discussing your accomplishments and their broader impacts. At the end of the personal statement, be sure to include your plans and goals as a graduate student and in your professional career.

__Application timeline/schedule/goals: __

**First two weeks of September:**

- Identify who will write your letters of recommendation. Meet with them (in person) to ask if they will write one for you. Let them know that this is different from a graduate school recommendation and give them print-out information about the fellowship.
- Write a first draft of the personal statement.
- Determine the premise/title of your research proposal.

**Last two weeks of September:**

- Have at least three different faculty members (letter writers in particular) read your personal statement.
- Write a draft of your research proposal.

**Until the deadline:**

- Have peers read your personal statement to fix any last minute typos.
- Have at least three different faculty members (letter writers) read your research proposal.
- Revise and submit!

__Benefits from receiving this award__*: *The application experience in and of itself was extremely beneficial. Since the due date was early in the Fall of my senior year as an undergraduate, I already had letter writers and a personal statement ready to go for graduate school applications. The research proposal gave me a glimpse of a researcher’s perspective in identifying and motivating the questions they aim to answer. I also made sure to include a sentence in my personal statement stating that I applied for this fellowship. I believe this showed that not only do I set goals for myself, but I am able to commit to executing the appropriate steps to achieve that goal.

** Non-mathematical activities:** Spending time with my hilarious family is my favorite way to spend my non-mathematical time. When I’m not at my parents’ house barbecuing tri-tip and ribs, you will probably find me at the gym lifting weights and putting that protein to use!

__Other accomplishments:__

Co-PI on an NSF grant to hold the 2019 Pacific Math Alliance Conference (Fall 2019)

Women and Mathematics Ambassador for the Institute for Advanced Study (Fall 2017)

PureMath@SACNAS Mini-Collaboration Grant (Fall 2016)

__Additional comments:__* *Although my advice is primarily for the personal statement, don’t forget that the research proposal needs to be equally as strong. In the comments I received from the reviewers, they noted that my research proposal was not as strong as others. I strongly encourage you to seek advice from research advisors and mentors on how to best prepare this portion of the application.

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** Name:** Katie Taylor

** Current Position/Institution:** 1st Year PhD Student in Mathematics

** Graduate Institution & Research Area**: The University of Alabama, Undergraduate Research in Mathematics Education

** Award Received and Year:** National Science Foundation Graduate Research Fellowship, 2019; I did my proposal in STEM Education.

__Advice for future applicants: __

- Utilize prior applicant examples to gain ideas of how to format your application (you can find these online or possibly reach out to students who have one at your school).
- Pay attention to the formatting requirements as they are REQUIREMENTS, and there is no need for something like font or text to cause issues in your application.
- Ask your possible recommenders if they would be willing to support you at the start of your application process…mention to them that you will send along a draft of your application/completed application closer to time.
- Having done previous research will be extremely helpful in the application process as this fellowship application asks you to do a research proposal.
- Aim to find someone, even if not in your department, who has experience with the application and would be willing to read over your applications. If you have done research, your PI is also a wonderful person to have look over your application as your application will likely reference that research often.

__Application timeline/schedule/goals: __

I did not find out about the program until mid-September. That was enough time to set up a timeline, receive feedback from mentors, and really create an application that told my story. I wanted to make sure that my passion was clear as well as explain well what activities/past research I had done that confirmed this passion. I had over 5 full drafts where one or two of them were complete redos.

** Benefits from receiving this award (more than financial benefit):** This award allows you to focus on your coursework and research without an additional workload. During the decision process, it can also act as a reason for Universities to want you in their program.

** Non-mathematical activities:** I love to play piano, pet kittens, and spend quality time with my friends (game nights are frequent).

** Additional comments:** Remember you can only apply as a graduate student if you have completed no more than 12 months of graduate study at the time of application (i.e. you are a first or second year student), and you can only apply ONCE as a graduate student. Thus, if you do not feel your application is strong enough in your first year, there is no benefit to sending it in the first year as you will not be able to try again.

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** Name:** Amzi Jeffs

** Current Position/Institution:** 4th year graduate student at University of Washington Seattle, working with Isabella Novik as my advisor.

** Research area:** Convex and discrete geometry.

** Advice for future applicants:** I received the NSF GRFP after my second year of grad school. I had also applied unsuccessfully two years earlier, during my senior year of undergrad at Harvey Mudd College. I think there were two main improvements in my second application. First, I had a

Lastly: Take your application to your local campus writing center!!! I did this with mine and ended up with a far more effective and “punchy” structure, as well as improved wording and grammar.

__Application timeline/schedule/goals:__** **It gets said a lot, but you should plan to complete a rough draft (however shoddy) of you *entire *application at least two weeks before the submission deadline. This will give your letter writers something to reference and build from, which is hugely helpful. It will also give you the chance to make lots of edits. An application which has been worked over for style and flow will go a long ways.

__Benefits from receiving this award (more than financial benefit):__** **The most significant benefit of the GRFP—besides being slightly more able to afford rent in Seattle—is that it freed me from teaching duties. This gave me a huge amount of free time, which I’ve used to get involved with all sorts of exciting activities on campus and beyond (see below). With time to pursue my interests further, I was more excited and motivated to return to research, and also had the ability to travel to various conferences/talks when the opportunity arose. Another opportunity I wouldn’t have been able to take if I had teaching duties: I was offered the chance to teach a “math in society” class at Cornish College of the Arts this Fall. It’ll be my first time teaching my own course, and I’m excited to teach in the context of an art school. Class started; wish me luck!

** Non-math activities:** I’m an active organizer with UAW4121, the union of grad students and postdocs at UW. Through the union I’ve had the chance to fight for demands that benefit the whole campus: better mental health care, justice for trans students and workers, robust protections from harassment and discrimination, and better wages to name a few. The union gives us a lot more power than our graduate senate or other forms of activism: our contract is a legally binding document that we can hold the university accountable to, and with 5000 members we have a lot of collective power to enforce it!

I’m also active in the Seattle branch of Socialist Alternative, and have been involved with campaigns for renters’ rights, indigenous struggles, anti-ICE work, and more through them. The synergy between my work with UAW and my work with SA has been a huge benefit to both. vMore recreationally, I enjoy a hell of a lot of cooking, some parkour around town, and rock climbing in the Cascades.

** Other accomplishments:** During the last year I’ve managed to produce a whole lot of papers, including my first published single-author paper, which you can check out here.

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__Name:__** **Theo McKenzie

__Current Position/Institution:__** **PhD Student at UC Berkeley

__Graduate Institution & Research Area:__** **Probability theory and combinatorics

** Award Received and Year:** Ford Foundation Fellowship and NSF Graduate Research Fellowship

__Advice for future applicants:__** **Get a large set of people to look at your application. Advisors should look at it as they’re familiar with your research and known what it takes to receive a grant. You can swap fellowship applications with other STEM students to see what ideas they have. Have someone in writing/humanities look at your application to complement on your writing style.

__Application timeline/schedule/goals:__** **Over the summer I talked to my advisor about what would make sense to submit as a research project. I also made sure to have a final draft ready at least a month before the deadline so that you can get feedback. This was crucial for me.

** Benefits from receiving this award (more than financial benefit):** This has given me a lot of flexibility. I’ve had the ability to go to conferences and present my work without worrying about missing teaching duties. Also I’ve built a network of other scholars through meeting people who have also gotten the fellowship.

** Non-mathematical activities:** I’m the lead instructor for a math class at a California State Prison. I also play squash.

** Other accomplishments:** I’m proud to have won a mentorship award last year.

** Additional comments:** I would start on your application as early as you can! Things can always get crazy during the semester. Good luck!

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** Name:** Elaina Aceves

** Current Position/Institution:** Math PhD student (fourth year)

** Graduate Institution & Research Area:** University of Iowa, Topology

** Award Received and Year:** Ford Foundation Predoctoral Fellowship received in 2018-2019 academic year

** Advice for future applicants:** The most important advice I can give is to have non-mathematicians read your application and make sure they can understand the research centered essays. I think that as mathematicians we get too comfortable thinking only other people in our research area or other mathematicians will read our work. However, in a fellowship application like this, it is essential that someone from any area of study can read and take something away from your essays about your work.

** Application timeline/schedule/goals:** I began working on my application in September 2018 when I started writing all of the essays needed for the application. My adviser was very supportive and took the time to read through a draft of one of my essays every week so I was continuously improving my essays throughout the semester. Once my essays were all mathematically correct and the prescribed page limit, I took my essays to the Graduate Success Office at the University of Iowa. This office has personnel who are available to read scholarship and fellowship applications, provide feedback, and meet with students to discuss their comments and suggestions. This step was crucial because after the revisions, I was sure that my essays were readable by a general audience and still maintained their mathematically correctness especially in the Previous Research and Proposed Plan of Research essays. In regard to the letters of recommendation, I selected professors who could speak about the various experiences I had described in my essays. One of my letter writers was my adviser who could speak about my research. Another writer was a previous teaching supervisor who could discuss my teaching ability and positive reviews from students. Since I had talked about outreach to the community through Sonia Kovalevsky Day, I had one of my letter writers be the organizer of SK Day who could testify to my participation and leadership in the activities of the event.

** Benefits from receiving this award (more than financial benefit):** I am now a part of a large organization of very supportive and encouraging students and professionals. As a mathematician, I have a great opportunity to talk about mathematics to scholars from other fields of study and represent a field that doesn’t receive enough attention or appreciation.

** Non-mathematical activities:** Binge watching Psych episodes, reading fantasy books, Listening to 80’s music

__Other accomplishments:__

Alfred P. Sloan Foundation’s Minority Ph.D. (MPHD) fellowship recipient

Graduate College Iowa Recruitment fellowship recipient

Graduate Assistance in Areas of National Need (GAANN) fellowship recipient

** Additional comments:** I was the only mathematician to receive the Ford Foundation Predoctoral fellowship during my year of admittance. Please everyone, apply to this fellowship program if you qualify! Mathematics needs more representation in these national programs.

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** Name:** Robert Davis

** Current Position/Institution:** Assistant Professor, at Colgate University

** Graduate Institution & Research Area:** University of Kentucky; Combinatorics, discrete geometry, applied algebraic geometry.

** Award Received and Year:** U.S. Student Fulbright Grant. I spent the 2013-2014 working with Alex Engström’s research group at Aalto University in Espoo, Finland.

** Advice for future applicants:** The Fulbright website has statistics on applications and awards funded in recent years. You can use this data to see (very) roughly how “competitive” it is to receive a grant from a particular country. That said, don’t read too much into these statistics, as each application cycle contains a ton of unpredictable elements that are out of your control. Do keep in mind that programs in different countries may have different expectations, even though they are all Fulbright programs, so make sure that your goals mesh well with the particular program to which you apply. Make sure to reach out to your institution’s office for external grants, as they will almost certainly need to submit some information as well. They should also be able to help you out with any statements you write, which you should give yourself a lot of time to do. The people evaluating your application will very likely have little mathematical background, so you have to convey the importance of your project without getting bogged down in technical jargon. I found that very difficult to do, and I had to start my statements from scratch several times until it was in good shape. If you are awarded a grant, make sure you look into the process for obtaining a visa or residence permit promptly, because it can take a while.

** Application timeline/schedule/goals:** I brought the idea of studying abroad to my adviser, Ben Braun, in the early summer of my second year of grad school — around June 2012. I originally just intended it to be for a summer or semester, but the most straightforward way to spend some extended time abroad seemed to be through a Fulbright grant, which was a year long. Since my goal was to finish grad school in five years, I was about to enter my third year, and my last year would be filled up with teaching and a job search, I had to apply the following October, about four months later. That seemed to be an appropriate length of time, since I had to rewrite my statements multiple times. Looking back, I wish I had given myself another month or two, but four months worked.

** Benefits from receiving this award (more than financial benefit): **One of my goals during the grant period, which I was able to do, was to give talks and attend conferences at other universities that I wouldn’t normally have had the chance to visit. They gave me great opportunities to meet lots of people who work in my area and nearby areas. I learned a lot about how academia in Europe works, how to find out about more job opportunities, and a whole lot of new math. Even just the process of applying to the Fulbright was very helpful. I’ve applied for a variety of different awards and grants since then and had a much easier time writing project descriptions and summaries because of it.

** Non-mathematical activities:** The Fulbright group in Finland organized a number of trips during the year, which gave us a chance to see much more of the country than we otherwise might. Also, I lived in a building occupied mostly by students of the University of Helsinki, and I was fortunate to get swept up with a great social group. We were from all over the place: the U.S., Mexico, Ukraine, Italy, France, Spain, Germany, and lots more were in the group. We often took trips together, ate meals together, and generally got to be good friends, which really helped out with being away from home for so long. And of course, I became very familiar with saunas: going to them is practically a required activity in Finland.

** Other accomplishments:** In terms of external math awards: AMS-Simons Travel Award; NSF Collaborative Grant (co-PI with Tianran Chen at Auburn University in Montgomery), under the

** Additional comments:** If you’re reading this and have any questions about applying for a Fulbright grant that I didn’t answer, feel free to send me an email directly! I had such a great time that year and would be happy to help others how I can.

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** Name:** Nohemi M. Sepulveda

** Current Position/Institution:** Williams College Alumni

** Graduate Institution & Research Area:** N/A at the moment

** Award Received and year**: Fulbright Teaching Assistantship in Spain, 2018

** Advice for future applicants:** Go for it and apply! Sometimes these prestigious awards can seem a bit intimidating, and perhaps impossible to get, but at the end of the day everyone has something strong to offer the program, and you may be just exactly what they’re looking for. I didn’t learn about this fellowship until late in the summer before I applied, so even if you feel like the deadline might be slowly approaching, still try and get that application in! Don’t let fear keep you from applying. There’s so much potential that each and every single one of us has to offer. Simply taking that first step and deciding to apply is very important and you deserve credit for it!

** Application timeline/schedule/goals**: As I mentioned above, I actually didn’t learn or know about the Fulbright fellowship until late in the summer before applying—sometime in August to be exact. After I finished a teaching internship that I was doing in India that summer, I returned home for about 3 weeks, and this is when I was able to work on the application and on the essays. I returned back to college early September, and the campus application deadline was in mid-September. After submitting my application to my school, I had to do a little mini interview with the fellowship office where I was asked questions like “Why do you want to do this fellowship?” and what qualifications I had. They also ended up reviewing my essays and they gave me feedback on them. This was very useful in that I was then able to make some last-minute changes that later strengthened my application. After this, you’re basically in charge of submitting the application online yourself by the deadline that Fulbright gives you, which is in early October.

** Benefits from receiving this award (more than financial benefit):** The greatest benefit in receiving this award was honestly having the opportunity to live in another country for a year, and also meeting some very wonderful people in the process. I was a teacher in a private catholic school in La Rioja, Spain and I really enjoyed my time there. I taught some really cute 5

** Non-mathematical activities:** Some non-mathematical activites that I included in my application were being a college senior advisor for a program called Matriculate, volunteering at my local immigrant center, interning at the Williamstown Historical Museum, working at the Mexican Red Cross, being a teacher in India as well as in Mexico, and doing human rights research abroad in Nepal, Jordan, and Chile.

** Other accomplishments:** I received some awards from my college such as: The Class of 1951 Scholarship, The International Public Service Fellowship, and The Ware Scholarship.

** Additional comments: **Apply, apply, apply!

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** Name:** Darleen Perez-Lavin

** Current Position/Institution:** Graduate Student at University of Kentucky

** Graduate Institution & Research Area:** University of Kentucky, Number theory

** Award Received and Year:** SMART Fellowship funded by the DoD, awarded 2017

** Advice for future applicants:** Pick your top 3 labs you would like to be placed at. Do some research on each lab and try to understand what they do. Answer questions like: What type of research do they do? Who do they actively service? During these searches, you should be asking yourself: How do I fit in? Are these things something I want to work on? The application has various writing statements. Each statement should encompass your strengths. I would encourage portraying your strengths and how they would benefit the lab in their current research goals. Talk about why you want to be there as well as how you are qualified to be there.

** Application timeline/schedule/goals:** The application is due early – mid January. I highly recommend you start as soon as possible since this application has a lot of parts. To help with length and time, I set weekly goals. Even if it was a section that took 5-10 minutes, it was better than doing everything at once. It felt less taxing this way. I also had various people read over my statement drafts. Pick letter writers that will display your strengths in different areas. For example, have someone outside your department write a letter that will show a strength outside of your academic achievements.

** Benefits from receiving this award (more than financial benefit):** With this fellowship, you get a lab sponsorship. I am required to do internships during the summer while in graduate school and work at the lab for the amount of years funded. I find this extremely beneficial because I have a job when I graduate that will provide me with work experience on research areas outside my thesis work. During each summer, I get a taste of what my life will be like when I graduate. I spent this past summer at the naval lab in Charleston, SC and it was a great experience. I have a better understanding on how the lab functions and what I may be working on when I get there. You also get the benefit of getting a security clearance. This process takes over a year sometimes and heavily delays you being able to be hired at the NSA, for example, or other government agencies. If you decided to leave after your years of service, having clearance will be a bonus if you would like to move to a different government agency.

This fellowship allows you to continue teaching if you would like but does not require you to teach. This allows you to have more time to focus on your research. It helped reduced my stress level since I had less demanded requirements to do each week. I chose not to teach but I have met other fellows that teach one class each semester so they can get an academic job once they finish their time at the lab. The lab does allow you to teach while you work there but you still required to do your 40 hours a week.

** Non-mathematical activities:** (As part of the SMART Fellowship) You are encouraged to do community outreach event and attend workshops / conferences. Since I work for a naval lab, there are different rotations to apply for at the office of naval research.

(Personal) Dancing to salsa, jazz and hip hop. I enjoy yoga, hiking and paddle boarding.

** Other accomplishments:** Awarded the NSF – MSGI internship program with the Department of Energy. To be honest, I’m most proud of getting the SMART Fellowship. With the lack of support from the department since I struggled to get my prelims done, it gave me purpose, hope and sense of belonging to be award a competitive fellowship. It made me feel like I deserved to be here. I just have a different purpose.

Personal accomplishments: Doing yoga on a paddle board! I’m still working on it but it’s a rush to be able to achieve balance on the water.

__Additional comments: __

Since you will be working for the government, they only expect you to work 40 hours a week. If you work over 40, you can save those hours for time off. For example, some people work 4 – 10 hour days to have a long weekend every weekend. There are limitation and rules to follow but you get the make your work week within reason. This forces work – life balance, which I find to be important. This also allows you to do something outside of work, maybe continue your thesis work or collaborations, maybe even learn a new sport or hobby.

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** Name:** Chase Ashby

** Current Position/Institution:** Ph.D. Student at University of Kentucky and Civil Servant working in the Computational Aerosciences Branch of the NASA Advanced Supercomputing Division at NASA Ames Research Center.

** Graduate Institution & Research Area:** I’m currently a Ph.D. student at the University of Kentucky researching in the field Computational Fluid Dynamics with applications in Aerospace Engineering. My research is focused on developing approximation methods for adjoint PDE solutions on structured curvilinear meshes.

__Award Received and Year:__** **Accepted into NASA Pathways Intern Employment Program in 2019.

__Advice for future applicants:__** **Network. Network. Network. NASA employees are a passionate group of individuals who enjoy both sharing their experiences and assisting motivated students. Be willing to explore and put in the hard work. NASA truly seeks people who have a passion for research and community. Email current employees, even if you’ve never met them before. Someone will likely respond, and you may just end up with a spontaneous interview.

Last year I was a struggling graduate student, however, simply being willing to ask for and take opportunities allowed me to find a solid foundation within a year’s time. Thus, my main advice is to be daring even in the presence of overwhelming self-doubt. Knock on those unopen doors and don’t be afraid of the work waiting on the other side.

__Application timeline/schedule/goals:__** **Applications appear on USAJOBS.gov as funding becomes available throughout the year. I first scheduled a phone call with someone to simply discuss what they worked on and my interests (this was obtained from a distant family connection). I simply couldn’t hide my excitement for the work being done at NASA Ames and they decided to forward my resume to their supervisor. Within a few minutes, they emailed me to schedule a phone call, which ended up being a surprise interview for an internship and the only interview throughout the whole process. After the interview, which took place in late December 2018, they decided to offer me an internship position for Summer 2019 in early February. They remained in close contact to ensure I found housing arrangements and eventually suggested that I apply for the NASA Pathways Program. Once the link went live on USAJOBS.gov, I filled out a medium length application that took around an hour to complete, including intermittent coffee breaks. On my way to the internship, I was notified of the offer!

__Benefits from receiving this award (more than financial benefit): __

- Hired on as a Civil Servant (federal employee), which comes with Federal Employee Health Benefits, retirement plan, life insurance.
- Competitive Salary. For my position as a student trainee in engineering: $75,000.
- Most Pathways students transition to fulltime employees after graduating.
- Students can actually be converted to other federal positions after graduating, e.g. FBI, CIA, NSA, DOD, DOE.

__Non-mathematical activities:__** **Acoustic guitar, hiking/camping, reading.

__Other accomplishments:__

- First-generation college student
- Graduate Scholar in Mathematics Fellowship
- Bob Gaines Research Fellowship

__Additional comments:__** **Readers may feel free to connect with me via LinkedIn or my university email. I’m very willing to and happy to discuss my experiences in further detail.

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** Name:** Liam Solus

__Current Position/Institution:__** **Assistant Professor of Mathematics, KTH Royal Institute of Technology, Sweden

** Graduate Institution & Research Area: **University of Kentucky, Combinatorics and Statistics

** Award Received and Year:** 2016 US NSF Postdoctoral Fellowship

__Advice for future applicants:__

0. Ask previous winners of the fellowship you are applying for if you can see their application as an example. While the content of your project is fundamental and novel to you, writing a grant/fellowship application that presents your proposed work in a clear, concise, and well-supported way is formulaic. A well-structured proposal will make it easier for the reviewers to follow your points, and it can even make it easier for them to argue on behalf of your proposal when discussing with the rest of the committee. Successful proposals will often have such a structure, and there is no reason for you to re-invent the wheel. Ask people who you know have been successful if you can see their successful proposal as an example when writing your own. If you don’t know such people, ask your mentors or other mathematicians you know if they know someone who you could ask. Just like in math, grant writing should start with an informative example.

- Get detailed feedback from mentors and get it from more than one of them. Get as much feedback as you can, and if you can get it from people who have written successful grant or fellowship applications then even better! Writing fellowship/grant proposals, like mathematics, is not something are born able to do. Just like mathematics, we need to learn from people who have been doing it longer than ourselves. Using the examples of successful proposals, you have available as guidelines, write the best proposal you can, and then send it off to at least two mentors for feedback. Different readers will see different things, so having more than one take a look at it will be helpful.

- Don’t fear critique. Remember the people who you picked to read your proposal are your mentors. They are people that support you and your ideas, and they want to see you succeed. Any comments they have or changes they suggest, no matter how seemingly drastic, are aimed at helping you present your ideas in the best possible way. More comments are not a sign of your failure, but a sign that your readers really care and want you to succeed!

- Don’t be afraid to start over. Sometimes the feedback you get is going to suggest a lot of changes. So many in fact, that you may realize that you are essentially rewriting the whole proposal. Be willing to do this. Think about it like a math problem: you were trying one approach to the problem, and then your mentors came by and gave you good evidence why that approach may not work. So, try attacking the problem from a new angle based on their expert advice.

- Start early. A well-thought-out proposal takes time, and the time invested in the proposal often shows. Having a first draft months in advance of the deadline gives you ample time to let your mentors read it, and for you to make major changes. Even before that, just musing about the problems you will propose and the general structure of the proposal months before you start writing it can really help you present clear, and well-thought-out ideas.

- Always apply and apply to everything. There is no reason not to apply to a fellowship that you think you might want. In the worst-case scenario, you don’t get the fellowship, but you did learn a ton about writing a proposal. This is still better than if you had not applied at all. The experience of writing a rejected proposal still gives you insight into how to better prepare the next one. Which is why you should apply to anything that seems of interest. Just like with mathematics, the more experiences you have with writing proposals, the more success you will have on future ones.

** Application timeline/schedule/goals:** My personal application timeline starts about 6 months before the due date. At this point I’m thinking about what I want to include in the proposal, and perhaps the general structure of how I will include these things. This is also the time when I’m asking people for examples of the successful applications. During this time, I write a lot of outlines, and small pieces of what I think I’d might include, by hand. Three months before its due I try and have my first draft prepared, and then I send it out to mentors/colleagues/etc. for their opinions.

** Benefits from receiving this award (more than financial benefit):** The NSF postdoctoral fellowship really changed my future. It gave me ample time, time that I would otherwise not have, to really engage with my research. During this time, I believe I produced some of my best work to-date. I was also able to broaden my mathematical interests significantly, investing time in learning things about new and exciting fields related to the problems in my proposal. Beyond this, the NSF postdoctoral fellowship took me to Stockholm, which has since become my home.

** Non-mathematical activities:** Surfing, snowboarding, rock climbing, exercising, skateboarding, and playing violin

______________________________________________________________________________________

** Name**: Anastasia Chavez

** Current Position/Institution:** NSF Postdoctoral Researcher and Krener Assistant Professor at UC Davis

** Graduate Institution & Research Area:** University of California, Berkeley; Combinatorics

** Award Received and year:** Ph.D. in mathematics in 2017; NSF Postdoctoral Fellowship 2018

__Advice for future applicants: __

Some of the best advice I received was to make the research statement include enough evidence showing you fully grasp the difficulty level of your research goals and the necessary plan to achieve them. Personally, this meant doing a little extra work to gather examples, work out small cases, employ computer algorithms, and do whatever seemed appropriate for the tasks outlined to ensure I could show I knew what these projects really required. Also, it showed I could be realistic about my goals and what was achievable in the time period the award offered. That means, if a problem seems hard, it is ok to acknowledge that then followed by how you will address it. Be clear, concise, and honest.

Another great piece of advice I received was to find and include collaborators who were skilled in the machinery needed for the research program outlined. This, again, shows the feasibility of success for the project and that experts are interested and want to work on your problems. Plus, it means you can begin building a network of collaborators and expand your math community.

If possible, it is great to have a diverse group of folks willing to read and edit your statements. These can be people who are experts in your field and those who are tangential. Not too many of course, but enough to have a diverse opinion if your statements are well written, your research program is outlined clearly, if your program shows agency, and if the purpose of the research program is clear. Moreover, you will generate many iterations of your statements, which is great! Make every word used count!

Ask fellow applicants (perhaps fellow grad students, former grad students, new faculty) if they are willing to share their application packet with you. There is a lot of technical formatting that you need to adhere to, and documents types that are not so common. Getting examples from others, especially successful applicants, will be a great guide to developing your own. Plus, it’s great to have a work buddy to keep you on track. So, finding a comrade during the application season you can share tasks of editing, reading each other’s work, and setting deadlines to complete drafts, will make it a much more enjoyable process.

Last, imposter syndrome is something that can creep up, if it hasn’t prior to the application process. If you find yourself feeling lacking in any way around your abilities, competency, and value during this time, here’s a small exercise I use often (actually, I used it just the other day preparing for a talk!). I first recognize the nervous, anxious energy that wants to eat up my time worrying about my competency and confidence. Then, I ask myself, “Do you want to spend the next chunk of time worrying? Or would you like to take the action that will prevent the outcome you are so worried about?” Often, that builds enough strength and resolve to diminish the worry so I can complete my tasks at hand. The best part is, I have evidence that I am capable, and I can build that confidence back.

I hope this of use to you, and best of luck in your application process and award acceptance!!!

** Application timeline/schedule/goals:** I am currently finishing my 2nd of a 3 year NSF postdoc and plan to be applying for academic and possibly non-academic jobs in Fall 2021. My goals are a little vague, but I believe the next step will include elements of mathematics, education, and technology.

** Benefits from receiving this award (more than financial benefit):** The greatest benefit has been the opportunity to explore new mathematics and branch out of my graduate research while building a great mentorship relationship with my research mentor. It has also allowed me the time to dive deeply into projects on my own schedule, without juggling classes, teaching, etc. so common in graduate school.

** Non-mathematical activities:** I enjoy camping with my family, playing softball, learning the guitar, walking our two dogs Big Boy and Hope, and dancing.

** Other accomplishments:** I have an amazing, supportive partner and together we have been blessed to be parents to two incredibly inspiring and creative children. My daughters are probably my greatest gift and I’m profoundly grateful for all they teach me.

** Additional comments:** I’m a firm believer in the following advice from a mentor on how to find the right work-life balance: “You first have a life, then you find out how to fit your work in, not the other way around.” If that resonates with you, enjoy!

______________________________________________________________________________________

Now, go and APPLY! You got this!

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

As Sarah introduced, my name is Caleb McWhorter and I am the new editor-in-chief for the AMS Graduate Student Blog. I am a Ph.D. student at Syracuse University studying Algebraic Number Theory and Arithmetic Geometry. I am excited to be working with the many wonderful writers that have already volunteered their time and energy to bring you new and exciting articles. While we will strive to produce a wide-ranging collection of articles for you over the next year, we will be focusing on a few themes:

- Diversity in Mathematics, Mathematicians, and Mathematical Life: Though our lives tend to shrink as graduate students, we come from a broad variety of genders, ethnicities, ages, orientations, backgrounds, countries, universities, etc. We all also live varied (mathematical) lives. We will work to highlight the diversity of mathematics graduate students, their activities/accomplishments, and the lives they lead.
- Teaching and Graduate Resources: Graduate students have the delicate task of balancing their teaching, coursework, and research. But there are many gems out there to help mathematics graduate students along the way! We will work to highlight the resources out there for teaching, studying graduate Mathematics, preparing for qualifying exams, etc. We will also work to create original content to help graduate students complete their studies and their teaching to the best of their abilities!
- Mathematical Distractions/Tidbits: To say the least, life as a graduate student can be overwhelming. We will try to help with the stress by bringing you fun and interesting short articles, rather than always delivering you ‘heavy reads’. So look out for fun short articles including crosswords, comic strips, and quick math reads that can also be shared with interested undergraduates!

But of course, *the AMS Graduate Blog is for you*! We want to hear your ideas and hear what types of articles you would like to see over the next year. Feel free to contact me, cgmcwhor@syr.edu, or any of the other writers to suggest ideas. However the best way of seeing content that you would like is to write for the Blog yourself, see the recent advertisement calling for writers! If are interested in writing for the AMS Graduate Blog, send in an application! We would love to hear from you!

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

When I participated in the pre-orientation as an incoming first year student, I believed that the purpose was to learn some math. (A kind of pre-wash before the real brainwashing took place.) I was good with that goal. As a bonus we even got the tried and true messages about impostor syndrome and growth mindset so I was rather content. But — as I moved into my first and second year, I realized that perhaps I had missed the point. I began to think that the purpose of the pre-orientation was to learn how to effectively work in groups and continually push yourself to become an independent learner. So, when I structured my lessons, I kept this in mind and moved forward ready to impart whatever minimal wisdom I could muster.

I began the week with five key messages that I wished my classmates and I had at the start. We would revisit these messages at the end of the week and hopefully keep them in the back of our minds throughout.

**No matter what your background you deserve to understand completely. Ask questions as soon as you are confused.****If you are feeling like you are frustrated and struggling you are doing it right; these are signs of growth. Alternatively, if you are feeling comfortable, consider pushing yourself.****Everybody’s voice is important. Listen to what your classmates have to say.****Supporting each other academically and emotionally is an important aspect of first year. The older graduate students are here to help. Reach out to us.****The only person you have to prove yourself to is yourself. Avoid comparing your progress to your peers.**

At the end of the week, during a group discussion, the students expressed that “the messages, [especially the first two] empowered [them] to speak up, ask questions, and feel OK with being confused.”

The week was structured to begin with heavy group work and slowly release more of the responsibility to the individual.

**Day One (“We’re kind of a package deal”): **

Students began reading the paper. The instructions: “Read through this section with your group. Make sure that you are all staying together and understand what you are reading. A suggestion would be to read to a particular point and check–in with each other. In addition, be sure to keep track of all of your questions and your group mate’s questions.” It was really up to the students to decide how to proceed and I watched as they quickly took up all of the usual roles: the “I don’t need to write anything down, I got it” student, the “I don’t know what I am reading, but I’ll pretend just by staring at my paper” student, the “I’ll work on examples in the margins” student, and so on. I watched as they worked through the paper, keeping track of the questions going unanswered and the blank head nods that said, “I don’t quite know what you’re talking about, but, sure, I am good enough to move on.” All of these feelings were painfully familiar.

Throughout the time, we paused and answered questions as a group. I slowed us down and questioned how much we really understood of the paper. It was a rough first day, but played out exactly as I wished. This is what the beginning of first year felt like to me.

**Day Two (“I have some questions.”): **

Now that we felt the harsh realities of trying to do something at the pace of the fastest person in the group, it was time to forcefully grab a hold of our desire to understand and stand up for a more reasonable course of action.

On day two, students started by reading independently for fifteen minutes. The instructions: “Write every question that you have in the margins of the paper. Try not to spend too much time thinking about it. Just jot it down and read ahead. We will come back to it.” After a short break, in groups the students read the same section together, picking out all of their questions and answering them. Since they already thought about the questions, students were more willing to express the parts that were confusing and the questions that lingered. We progressed to feeling more comfortable with expressing confusion and answering questions without judgment. Day two was reminiscent of second semester for myself.

**Day Three (“I think for myself, thank you very much.”):**

Now for me the energy was buzzing. I am sure that for them the energy is that of pure exhaustion at this point in the week (they had been reading two papers six hours a day four three days now.) Nonetheless, I have enough energy for all of us.

Today, we spend even more time working independently. At first the instructions are the same as day two. “Read through the section making note of all of the questions that you have without thinking too hard about what the answer may be. Just get a feel for what you are reading.” But, today, the instructions come with a part two, after the 15 minute independent read: “Now, read the same section again. This time, really take the time to try and answer all of the questions that you wrote for yourself in the margins.” Finally, after a small break, the students worked in groups reading through the paper and answering any remaining questions. The conversations on this day were different. Rather than being focused on confusion, the conversations were filled with curiosity and explanation. They talked about how they had a question and were able to answer it themselves. They explained how. They asked questions beyond the reading. They moved along more quickly and with a deeper understanding. In my own mathematical journey, one could say this resembled third semester.

** **

**Day Four (“I can do this.”): **

The final day was for concluding what we learned so we spent the first half tying up loose ends and reviewing what we learned. The second half we spent reflecting on the week. I pointed out that each day was structured so that they had more and more independence in their learning. I emphasized that each student is coming in with a different background and a different comfort level with working independently. **The critical part being that everyone, no matter where the start can be successful.** It is about pushing yourself to be more independent and striving for a deeper understanding. Then, I asked some key reflection questions related to our independence goals:

- During the individual reading time was it difficult to focus on math? Were you distracted by how much was unfamiliar?
- What percentage of your questions went unanswered each day?
- How did the amount of independent time contribute to the overall group discussion?
- Which day felt the most comfortable and why?

We had a very meaningful discussion in which the students reflected on their participation in their own learning. Taking away the importance of helping each other out, both to understand the material and to gain independence. They reflected on their understanding saying that as the week progressed to more independence they could feel their understanding growing deeper. Within their words, they emphasized the importance of working through their questions before meeting as a group.

I highly enjoyed watching them take learning into their own hands and certainly enjoy watching their continued growth throughout the program.

]]>First-time learners of calculus often struggle with the notion of an infinitesimal, and considering $\frac{dy}{dx}$ literally as a fraction can lead students astray in Calculus III and differential equations, when implicit differentiation and separable equations rely on the chain rule in ways that strongly contradict any consideration of $\frac{dy}{dx}$ as a literal fraction. However, literality can be restored by considering infinitesimals algebraically as nilpotents, which is exactly the claim that one is free to ignore all but a finite number of terms of a Taylor series for a smooth analytic function. Algebraic geometry offers methods of ‘zooming in on a point’ to consider local phenomena in ways that reveal an algebraic structure to infinitesimals which can console newcomers to calculus (“just look at the linear part!”) and restore the desire to take infinitesimals literally. Here, I offer one framework for viewing infinitesimals as dual numbers which is not new, though the connection I make to complex algebraic geometry shows that what one learns in advanced graduate coursework is in keeping with a traditional undergraduate curriculum.

For $f$ a smooth, analytic function defined on the real line, the linear part of its Taylor Series approximation is given by the differential. Scheme-theoretically, one can view infinitesimal neighborhoods of a point of $\mathbb{R}$ as isomorphic to the dual numbers, or an algebraic generalization thereof. For $\mathbb{D}:= \mathbb{R}[\epsilon] = \mathbb{R}[x]/(x^2)$, one representation of the dual numbers comes from \[ \mathbb{D} \cong \{a+b\epsilon\text{ }|\text{ }a,b\in \mathbb{R} \text{ and }\epsilon^2=0 \} \cong \left\{ \left[\begin{matrix} a & b \\ 0 & a \end{matrix}\right] \text{ }|\text{ }a,b\in \mathbb{R} \right\}. \]

For $Z\subset X$ an irreducible subscheme, the $\mu$-th infinitesimal neighborhood of $Z$ is denoted and constructed by \[ Z_\mu := \{ Z, ^{\mathcal{O}_X}/_{\mathcal{I}^{\mu + 1}} \}. \]

The same construction can be carried out for $\{a\} \subset \mathbb{R}$ considered as an irreducible subvariety, even though $\mathbb{R}$ is not algebraically closed, where $\{a\}$ is given by the ideal sheaf \[ \mathcal{I} = \begin{cases} (x-a) & \text{ over any open set containing }a\\ (0) & \text{ otherwise. }\end{cases}\] This implies that the first infinitesimal neighborhood of the point $Z=\{a\}$ is \begin{align*} Z_1 := \{ Z, ^{\mathcal{O}_\mathbb{R}}/_{\mathcal{I}_Z^2} \},\end{align*}

and for $U$ any open set of the real line containing the point $a$, $\mathcal{I}$ assigns the ideal $(x-a)\subseteq \mathcal{R}[x]$ to $U$ so that over $U$,

\begin{align*}

Z_1:= \{ \{a\}, ^{\mathbb{R}[x]}/_{(x-a)^2} \} \end{align*}

and substituting $a=0$ gives

\begin{align*}

Z_1 :&= \{ \{0\}, ^{\mathbb{R}[x]}/_{(x-0)^2} \} \\

&= \{ \{ 0 \}, ^{\mathbb{R}[x]}/_{(x^2)} \}. \end{align*}

Now, the ring associated to the the point $0$ is $\mathbb{R}[x]/(x^2) \cong \{a+b\epsilon \text{ }|\text{ }a, b\in \mathbb{R}\text{ and } \epsilon^2=0 \} \cong \mathbb{D}$, which is the desired isomorphism mentioned above. To illustrate the connection with infinitesimals, **Figure 1** shows the graph of $y=x^2$ in the real 2-dimensional plane, and **Figure 2** shows the graph of $y=x^2$ “in the limit” at the point $(2,4)$. This concept can at times be hard for undergraduate students and new learners of calculus to grasp, though the fact that an algebraic structure is present in infinitesimals can be reassuring, once one understands that we have simply varied the square of the imaginary unit from -1 to solve $x^2+1=0$ over the real numbers, to simultaneously solving $x^2=0$ and $x\neq 0$.

The derivative of a function of a dual variable can be defined to agree with the real case: What is an Infinitesimal?. This has applications to, and can be interpreted with Lie algebras and tangent bundles. Matrix representations exist for higher order infinitesimal neighborhoods of a point of the real line, which encode higher order derivatives of a function matrically. Graduate students should have a good understanding of infinitesimals both in terms of Taylor Series approximations to smooth analytic functions, and of nilpotent elements in the ring of coordinate functions of a variety and their relation to each other in order to impart these concepts clearly to undergraduate students.

**Further Reading**

1. Fulton, William, and Joseph Harris. *Representation Theory: a First Course*. Springer, 2004.

2. Griffiths, Phillip, and Joseph Harris. *Principles of Algebraic Geometry*. Wiley, 1978.

3. Humphreys, James E. Linear Algebraic Groups. Springer, 2004.

4. Mora-Camino, Felix, and Carlos Alberto Nunes Cosenza. *Fuzzy Dual Numbers: Theory and Applications*. Springer, 2018.

5. Tu, Loring W. *An Introduction to Manifolds*. Springer, 2011.

Staff writers cover any topic that would be of interest to mathematics graduate students; however, we are especially looking for writers interested in writing about one or more of the following areas:

- Diversity in Mathematicians: Mathematicians come from a large variety of countries, genders/races, ages, socioeconomic backgrounds, etc. We are especially interested in bloggers that are willing to write about the barriers/challenges facing members of these communities and advice for overcoming these issues, e.g. gender/race bias, child care, accessibility issues. Further, bloggers can discuss how others outside of these communities can help.
- Diversity in Mathematics: Mathematics graduate school isn’t just about math. Graduate students and professors engage in a large variety of activities. Bloggers could post about mathematicians engaged in spreading the Directed Reading Program (DRP), researching partisan gerrymandering, volunteering in disadvantaged communities, etc.
- TeX & Beamer: Learning TeX is an arduous, lifelong journey. Bloggers could help make this easier by positing helpful tips/tricks and advice for the beginner and advanced user alike. Bloggers could also share templates or explain how to use resources like Overleaf/ShareLaTeX/GitHub effectively (including for teaching).
- Graduate Resources: The amount of material students need to learn is vast. Bloggers could help by discussing useful internet resources, such as qualifying exam repositories or pointing out things like Keith Conrad’s treasure trove of expository article.
- First Year, Middle Years, & Last Year Experience: We are especially interested in students in their first, third, or last year to write about their experiences throughout the year. This can be navigating exams, building a CV, finding an advisor, traveling to conferences, applying to jobs, etc.
- Mathematical Amusements: One thing all graduate students know is that graduate life is stressful. Help lighten the mood by posting a monthly meme, writing a fun article on a lesser known topic, linking to a good math YouTube video, writing or sharing a math poem (see ‘A Poem for Lonely Prime Numbers’), etc.
- Teaching Resources: Teaching tends to be either a breeze or a struggle. Bloggers can share projects, problems, worksheets, or general advice on the teaching aspect of graduate life. Bloggers could also write articles on topics relevant to undergraduates that a graduate student may share with students in the classroom. Shared items could even be existing resources such as WolframAlpha, Symbolab, etc.
- Interviews: Calling all extroverts (or introverts). We are especially interested in those willing to conduct interviews (via email, Skype, etc) with graduate students, professors, or other people/groups (for example Math YouTube pages such as Numberphile) of interest to the mathematical community.
- Vlogs: Mathematical stories, experiences, and advice do not have to be shared just via long articles! We are interested in ‘writers’ who are interested instead in short (3-5 minute) vlogs to convey a variety of mathematical stories, whether these be your experience at your program, you at a conference, a book review, or a short exposition on a mathematical topic. The possibilities are endless!

Writing for the AMS Grad Student Blog puts your thoughts and experiences in front of a wide audience including AMS members, math department faculty, and AMS social media followers. Your insights will be visible and helpful to your fellow math grad students and mentors around the country and elsewhere, as well as to undergraduates who may be considering grad school. Your posts can demonstrate your writing and leadership skills and create communities beyond your school, especially if your readers share blog posts and offer feedback in comments!

The position requires excellent communication skills, a commitment to posting at least once a month, and monitoring and responding to comments. The posting process itself is done in WordPress, a free and open source content management system for blogs. Familiarity with WordPress is a plus. AMS blogs are hosted on blogs.ams.org, and AMS staff liaisons help promote awareness of the blog and the blog posts on ams.org, AMS social media and via other outlets.

Applicants for this position are requested to provide a sample of their writing (from a blog or for a similar audience), CV, and their reason for interest in being a staff writer, along with a vision statement for the blog (such as examples of topics for blog posts). The AMS requests applications by September 6, 2019 to membership@ams.org.

]]>**Start by briefly describing your 1dividedby0 project**.

Well the basic idea is that it’s a website devoted to how to *actually* divide by zero. Not just in the limit, not getting around it by using calculus, but legitimately actually do it.

**What is better about your approach rather than merely taking the limit?**

It’s not so much an instance of “better”. It’s more that division by zero is something that I remember learning is supposedly impossible and that never sat right with me. The general belief amongst teachers and the general public is that you can sort-of-kind-of do it if you use limits, or if you get around it in some way, but that actually dividing by zero is impossible.

Ever since learned about imaginary numbers being invented to allow another supposedly “impossible” operation (taking the square root of a negative number), I couldn’t help but think, why can’t you come up with some other kind of way to allow division by zero? Some other number that we just don’t have yet?

When you think of division by zero in terms of limits, you get two possible answers of positive and negative infinity. But if you could somehow find a way to join the two ends of the number line — as if it wrapped around somehow — and make it so that positive infinity and negative infinity are connected at some new number you didn’t have before, then that new number could be a legitimate answer to division by zero.

As it turns out, that’s actually not that farfetched a concept, and it’s made possible using a branch of math called projective geometry, It’s actually a pretty well-known thing among mathematicians, but you never see it mentioned in any textbook below the undergraduate level.

So the purpose of the 1dividedby0 website is to lay down the foundations of how it works, in as simple and visual a way as possible, so that any curious high schooler (or maybe even younger in some cases!) could understand it. The site also shows how understanding division by zero helps make the rest of high school algebra make sense, but it also tries to explain why mathematicians would be reluctant to divide by zero in the first place.

**How did you come up with the idea to create this website?**

Every summer at Georgia’s Governor’s Honors Program (GHP), I teach a course called “To Infinity and Beyond”, where we look at topics like cardinal and ordinal numbers, p-adic numbers, superreal and hyperreal numbers, surreal numbers, infinite series, set theory, measure theory, logical paradoxes, inversive geometry, and even a little bit of algebraic geometry… it’s *so* much fun and the students love it.

One of the lessons I’ve always taught as part of that course was in fact division by zero, and lots of students said that was their favorite part of the course. That’s what got me thinking, what if I made a website where I could share this with a larger audience?

Michael Hartl’s website www.tauday.com was a *huge* part of that inspiration as well. In his Tau Manifesto, Hartl takes a so-far-universally-accepted idea of $\pi$ being the fundamental circle constant, and challenges it little by little, leading the reader through the thought proces, until the conclusion is inevitable. That was the sort of feel I tried to convey with my own website about division by zero.

**Say more about how you used these ideas with high school students.**

Well, it started when I was working at a tutoring center, and I found out about the whole “unsigned infinity” thing myself. I started showing some of my students who were struggling with things like trigonometry and rational functions how to think of some of the things they had trouble with using unsigned infinity, and that “light bulb moment” happened with them.

One moment that I remember that really spurred me to try this with a student was when one of my students had trouble with a specific problem: finding the cotangent of 90 degrees.

They tried evaluating it as $\frac{1}{\tan(90^{\circ})} = \frac{1}{\text{undefined}}$, so they thought it was undefined. I explained the line that I’d always been told, that in that one special case you couldn’t say that $\cot\theta = \frac{1}{\tan\theta}$, and that instead you had to say

\[\cot(90^{\circ}) = \frac{\cos(90^{\circ})}{ \sin(90^{\circ})} = \frac{0}{1} = 0.\]

The reaction I got was “Why does math have to be full of all these stupid rules and exceptions? Why can’t things just work and make sense?”

And at that moment, I decided “you know what, let’s look at this another way.” And I showed them how to think of $\tan (90^\circ)$ as unsigned infinity, so that \[\frac{1}{\tan(90^\circ)} = \frac{1}{\infty} = 0,\] and suddenly it all made much more logical sense. They never missed that question again.

So from then on, when I taught students in Precalculus and Calculus, I figured, why hold that knowledge back from them, when instead I could have them see that it does make the rest of their mathematics make sense?

I also was excited to be teaching something supposedly “controversial”! But a huge part of that was to emphasize to them the idea that math is something you can play with and ponder about, and break the rules and see what cool stuff happens when you do.

**You’ve mentioned that these ideas are fairly well known by mathematicians, even though K-12 teachers are not generally aware of them? Are they written about in university level textbooks?**

Well, besides obviously showing up in books about projective geometry, in real analysis and topology books, you’ll often see it described as the “one-point compactification of the real line”, and the complex analogue involving the Riemann sphere is a pretty central thing to find in a complex analysis book. At the graduate level, once you start getting into algebraic geometry (which I’m really interested in!), it’s pretty common to include points or lines at infinity.

**Going back to mathematics educators not being aware of these things: A colleague of mine recently said that her son came home one day from school asking how to add infinity plus infinity and she told him infinity was not a number but a concept. I know for a lot of teachers, that is what they tell their students when they ask about these sort of things. What would you say to a student who is trying to add infinity plus infinity?**

If a young student asked me how to add infinity plus infinity, I would most likely ask them what they meant by that. How do they understand infinity? What do they think infinity plus one should be? What about infinity minus one? Infinity times three? What do they see in their head when they think of all this? Let them hash things out, and give validation to the mathematical system that they’re building in their head, and also give them some things to think about that could lead them down new mathematical rabbit holes.

I vehemently disagree with the notion that “infinity isn’t a number, it’s a concept”. Two is a number and also a concept. If you mean “infinity isn’t a badly-named so-called ‘real’ number” (thanks Vi Hart for that phrase: https://www.youtube.com/watch?v=23I5GS4JiDg ), then that’s fine. Neither is the square root of negative one, but that doesn’t keep it from being its own kind of number.

Another great example there is with the $0.999… = 1$ debate. Some kids will accept the various proofs out there (like multiplying both sides by 10 and subtracting), but others have this idea in their head that $0.999…$ is just the tiniest bit away from 1, infinitely close to it, but not quite there. Instead of “What part of ‘that’s how it is’ don’t you understand?”, take the opportunity for the student to explore that idea of things being infinitely close together. Maybe they’ll end up coming up with something interesting and not unlike the hyperreal numbers.

**This reminds me of Robert Ely’s article (https://www.jstor.org/stable/20720128) about nonstandard student conceptions, where he discusses a student who believed strongly in the idea of infinitely small numbers, and had an internally consistent logic for how those numbers worked.**

He probably calls it that to link it with nonstandard analysis by Abraham Robinson, who came up with the whole hyperreal numbers thing!

**What kind of lessons can curriculum designers take away from your project? How might we be able to design curricula that encourage breaking the rules? **

Well, for one, I’d love to see “you can’t do such-and-such” replaced with “you can’t do such-and-such YET”. Let students know that there is more to math, that there’s always another way of thinking about things, and maybe even hint at what it might be like. Or even phrase it as a “What if you could? What might that be like?”

Infinity is too rich and beautiful a concept to deny our students the fun of playing with it. It already captures their attention, why shut that curiosity down?

**If you have questions for Bill Shillito, feel free to post them in the comments. You can also follow him on Twitter at: ****https://twitter.com/solidangles**

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**Questioning:** There’s an old saying that math instructors should “always answer a question with a question.” I had numerous opportunities to watch the ways in which instructors would respond to student questions. Direct answers tended to make the encounter about the teacher and taught the student that mathematics was about rules and procedures, while asking the student “what do you think?” kept the encounter student-centered and lead to more conceptual understandings. This semester, I am going to watch carefully what happens when I give different types of responses to students in order to fine tune the way in which I offer responses.

**Alignment:** Another thing that I had an opportunity to observe was the ways in which instructors structured their classes, particularly in the ways in which they aligned their lectures, in-class activities, homework, and examinations. One key take-away from this: giving problems that are completely different from what was done in-class and on the homework in order to assess student thinking on novel problems can backfire if you are primarily grading based on correctness or adherence to the way you as an instructor think the problem should be solved. It seems like there are two directions here; one is to give problems on tests that look a lot like what was done in the homework and the review, and two is to grade in ways that take into account students’ creative reasoning and attempts towards a solution. I generally go for the former, but am interested in experimenting more with the latter. In particular, I am thinking of using what Annette Leitze refers to as an analytic rubric, where each stage of the problem-solving process is graded separately.

**Metacognitive Modeling:** One key aspect of graduate classes in mathematics was watching instructors solving problems during lecture where they explicitly elucidate the decisions they were making, as in, *why did they do this and not that, what happens when you reach a dead end*, etc. A key observation here is that although it is important to model realistic problem solving in this way, doing too much of it can sometimes lead to a lack of student confidence in the instructor. It is okay for the instructor to be stuck once in a while to show students what that is like, but at the same time it is important to spend adequate time preparing for class and to occasionally work harder problems out ahead of time so that students are confident that you know the material that you are teaching.

It was invigorating to my practice to have the opportunity to take mathematics content courses at the graduate level, especially since as a mathematics education doctoral student, I ordinarily only have the opportunity to take courses on pedagogy, history, theory, and curriculum rather than actual mathematics content courses. I am excited to begin this next semester with new ideas and thoughts about how to approach my own teaching.

]]>In this post I will talk about the basic components of a class activity, and illustrate them with an extended example. (*I am borrowing these from the context of English teaching! As an ESL instructor I used to look for ways to improve my teaching skills, which lead me to Task-Based Learning (TBL). The category below is from “Designing Tasks for the Communicative Classroom” by David Nunan. It is fascinating to be able to apply it to math education!*)

There are 5 ingredients that go into a purposefully crafted class activity;

**1. Goal**. We already discussed this above. Of course, you must have a desired result in your mind for the activity. It can be anything from getting the students to know each other and begin to communicate math among themselves, preparing them for future work, to ambitious aspirations such as leading them to discover the fundamental theorem of calculus! Whatever your goals are, they must be there to begin with.

**2. Input. **The raw material/data that class will be provided. In a math class we can expect copies of exercises and instructions, but also ropes and scissors and card boxes, say in a topology class. As an instructor, you must prepare the equipment beforehand.

**3. Setting. **Just like a movie set. Is everyone at the board or should they be sitting in round tables of three? Do groups talk to each other? Is the teacher at the board?

**4. Roles. **Talking about movies, we have to assign roles. Will each student in a group be assigned certain jobs, such as the calculator expert, the presenter, etc? What is the teacher’s role? Will they be the central figure, or try to minimize their influence? Deciding on these roles reduces the risk for confusion and increases efficiency.

**5. Activities.** With the set prepared, let the action begin. Make sure the instructions are clear and students know exactly what to do. Do not start with vague tasks that basically say “OK, go on and prove that every graph has a maximal sub-tree!” Break the job into smaller tasks that increase in difficulty. This will give the students a sense of success each time they fulfill one of the steps.

As the promised example, here is a copy of one of the worksheets I have designed for my ordinary differential equations class this summer:

**Goal**: The goal of this worksheet is to review the concept of the derivative of a function from calculus. *By the end of this worksheet you will be able to:*

*define derivative both algebraically and with graphs;
*

**Input**: Copies of these sheets for each student. A calculator per group.

**Setting**: Isolated groups of three.

**Roles**: Once a strategy is determined, one student uses the calculator, one records the numbers, and the third syncs them and corrects mismatches. The teacher circulates to observe progress and provide hints, but not answers.

**Activities**: Follow the instructions in the order written, and complete the tasks.

I may omit some of the above from the actual copy that the students receive. I usually make an instructor and a student version. The activities begin with:

*Section 1: Flashback to Calculus 1; the Derivative*

*Recall that*…

Then later comes the first task:

*Task 1. Let $ f(x)=1+x-2x^{\frac{2}{3}}\ .$ We have $ f(1)=0$. Which do you expect to be true? Circle your choice:*

*a. $f(1.1)>0$*

*b. $f(1.1)<0.$*

And the activities go on…

If the obscurity of writing up class activities was one factor keeping you from experimenting with the amazing world of non-lecture based math education, I hope this will be the little nudge and nod you were waiting for. I really urge you to try it at least for one topic in a semester. Good luck! And don’t forget to share your experience here.

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Secondly, I have learned that professors who have only been in research-style departments have no idea how to explain what is going on in other types of departments. Either that, or they don’t like telling the truth. And so I threw my thinking cap on and decided I would venture into the world and learn about these varying departments with my own eyes! Here is what I (as objectively as possible) saw and learned.

But wait … before we begin, let me explain how I view different departments. On one hand, we have departments that are heavily invested in teaching, but have little to no research component. On the other, we have departments which are all about the research, and where (while there may be a teaching component) your ability to teach is of secondary concern. All departments fall somewhere in between these two extremes. Now, let’s dive into some examples!

**A small, private college **** **

I first visited the University of the Cumberlands. The math department consists of four faculty members and the university is largely focused on teaching. The professor I shadowed explained that when he arrived, the college told him, “Teaching is the number one part of your job. If you would like to do research, that is nice, and we would happily encourage that as long as it does not get in the way of your responsibilities as a teacher.” I believe the most telling part of all of this visit was the schedule and which day they wanted me to visit.

Here, the professor asked me to come on a day that consisted of teaching. I met him at 8am on the way to his first class. The class sizes were relatively small (under ten students) and the first class was student-led. The students presented proofs and the professor helped them correct or beef up their arguments. This was one of the first places students experience writing proofs at this university. The second class of the day was more teacher-focused, but still with intermittent class participation. After class, we went back to the professor’s office and he explained that he spends the rest of the day planning for things coming up on the horizon, and then heads home.

Since teaching is the main focus at this school, there is little to no research aspect. There is one professor (out of four) that attends the occasional conference, but beyond that they primarily focus on their students. Also, professors tend to teach the same classes every year with minor adjustments based on students’ needs. In general, the department is very flexible and listens to their students closely. If someone wishes to take a class that is not offered, the department tries their best to accommodate those wishes.

One thing that I learned over the course of these visits was that a 4:4 schedule is not conducive to research. I forget exactly where I heard this, but I learned that even if new faculty come in wanting to do research, a 4:4 teaching load makes this extremely difficult. Not to say they wouldn’t feel fulfilled – teaching is a highly rewarding profession, and if you find yourself in a department filled with respect and common goals, work can be quite delightful.

**As close to center as possible?**

Next, I was interested in finding a school that emphasized both teaching and research equally. After asking multiple people for input, it seemed to me that they best option would be Macalester College. So, off to St. Paul, Minnesota I went. In my pre-correspondence with the professor I shadowed, she expressed that it would be great if I could come over the course of two days. One more teaching-focused day, and one more research-focused. I ended participating in a multitude of activities – I observed several classes, watched the students practice their capstone presentations, sat in on a Skype research meeting, and observed an independent study in the professor’s office. To be honest, I didn’t know I was on a time change until the end of day two, when I needed to make it to the airport on time. So, you could say they kept me pretty busy! I will try to highlight the parts that I found most illuminating.

I must put a disclaimer here before continuing (as many people gave me the same when I said that I visited). Macalester is a top-tier private school, which means that to get a job there right out of graduate school is fairly uncommon. It is more natural to have gone through a post-doc before hand; more on that later.

All the courses I sat in on at Macalester had one thing in common – student engagement. In every class, there was a chunk of time where the students were working in groups and asking each other questions. The professors walked around and spoke with each group in order to clear up confusion and gain feedback on where the class should go next. Since the class sizes averaged about twenty students, this was a manageable task. Because the environment was very much about the students, when the professor was presenting new material, there was little hesitation by students to ask questions and be active learners.

The professor that I shadowed also had a strong research agenda. She Skyped with collaborators every week and was planning on speaking at several conferences in the near future. One thing that I learned was how important it is to respect your research time. Especially when you want to do right by your students, it is easy to let your research time be overtaken by teaching duties. As in mathematics, while teaching you always feel like you could be doing more. But unlike mathematics, the deadline to accomplish teaching goals has an almost immediate turnaround. As soon as a class is over, you might already be preparing for the next one, with a deadline of less than two days away! So, put research time in your schedule, tell others you are busy, shut your door, and enjoy your time with math.

Sitting in on the research meeting was quite eye-opening. The professor I shadowed was a junior faculty, and all the junior faculty participating in her collaboration decided to meet for an extra hour before their actual meeting. During this first hour, the conversation was not unlike the meetings that I have with my fellow graduate students. Their energy was high and curious. Each of the participants was shouting out questions and pointing to places that didn’t make sense. The group worked together in a quick and cohesive way to try and provide insight on the confusion, often reaching a point where the whole group stopped to ponder. After a quick bit of silence they vowed to look at things again before next week and moved on to the next question. As the first hour came to a close, the conversation was relieved of the high energy and they talked about upcoming conferences until the senior faculty joined.

In comparison, the second hour was much calmer. Each person took on their respective roles, and one by one they moved through the to-do list. During this meeting, they were making revisions to a paper that they submitted. They moved as one elegant research team all the way through the hour. At the end, they created new goals for the next week and finished by talking about upcoming conferences. It was nice to realize that what we do as graduate students is not so different to what professors do with collaborators.

Another important cultural aspect of Macalester was its community. The department fosters a supportive environment and believes in creating life-long learners, both within their students and their faculty. While walking around the hallways, the professor I shadowed would stop to talk to students and other professors from many departments. For example, when we went to lunch as a department, they provided thoughtful advice to a first-year professor. And, when professors would talk in the hallway, they shared tricks of the trade. Everyone respected each other and the experiences that they shared.

**Postdoc – on your way to a research institute **

We are now at the other end of the scale – research institutes. Since going this route typically involves getting a postdoc and since I already attend a research institute, I felt like it might be of more interest to shadow a postdoc. To do so, I shimmied my way over to Michigan State University. The postdoc I shadowed asked me to come on a day in which he had no teaching duties. We started the day in his office, where he explained the research project he is working on. This took about an hour and a half and he moved from explanation to working on the problem rather seamlessly. Then he paused, looked at the time and said, “I guess I should answer emails.” While answering emails another postdoc stopped by to say good morning. We spoke and I learned that doing a postdoc is not solidifying your life into a research path. Some postdocs begin to realize that they really like teaching and working more closely with students and the department is supportive of these ventures. They both expressed how they like the freedom to do math all day and have relatively desirable departmental responsibilities.

The plan for the day included research time, meeting with his advisor, attending departmental tea, and going to a seminar talk. As it turned out we really only went to tea and did research since the other activities were canceled. His advisor asked if he should come in, but the postdoc I shadowed explained that it wasn’t really necessary to meet. The research is more independent and really it is just to check in. It frequently happens that they cancel their meeting, especially when the weather is not the best. So, we ended up doing research all morning, going to lunch, research until tea, and then research after tea. During that time an undergraduate stopped by and had a conversation with the postdoc about a question he was thinking about. Later, the postdoc I shadowed explained that saw his role as being a mentor to the graduate and undergraduate students. We ended the day around six. He explained to me that as a postdoc it is easy to allow yourself to feel that you should be putting in many extra hours and working really hard because you are trying to build a research agenda, but it is really important to find a work-life balance. And while his postdoc experience began with long nights, he would not recommend it and really tries to respect his home life.

I am truly grateful for all of people that allowed me to shadow them for the day and to the departments for being so welcoming. I learned more than I could have expected both about the departments and also about what questions are important to me. I would say the message that was consistent about all of these visits was the importance in finding a department that is right for you and your style. Once you get there, identify your primary goal and responsibility and put your efforts towards achieving it. But, more importantly, identify your secondary goals and responsibilities and work hard to keep them in balance with your primary one. This will not only increase your ability to be a strong part of your community, but also create deep and lasting respect for your own time.

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