## What I wish I knew in the first two years

At the end of last summer, I had the pleasure of leading a pre-orientation for incoming graduate students. We had four three-hour meetings over the course week and our only main objective was to read through “Six ways to sum a series” by Dan Kalman.

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.

## What is an Infinitesimal?

A guest post from Reginald Anderson at Kansas State University.

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.

## The AMS invites applications for staff writers of the Graduate Student Blog

The AMS invites applications for staff writers of the Graduate Student Blog, a blog by and for graduate students in mathematics, beginning in the fall 2019 semester. Staff writers share their experiences and challenges as graduate students, solicit posts, and encourage all mathematicians to join the community of followers and to post comments.

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!

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.

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## How to Divide by Zero: An Interview with Bill Shillito

For this post, I interviewed a colleague about a new project he is working on: a website where he encourages his readers to consider the possibility of dividing by zero.  Bill Shillito has a Master’s degree in Secondary Mathematics Education and currently works as both an academic tutor and an independent curriculum designer. In this interview, we discussed his “How to Divide by Zero” website, which can be found at:  https://www.1dividedby0.com/ Continue reading

## Teaching Lessons from a Summer of Taking Mathematics Courses

Although I am a mathematics education graduate student and am not required to take mathematics courses as a part of my PhD, I had the opportunity to take several mathematics graduate courses this summer as part of a Mathematics Graduate Certificate for Teachers program, which aims to bring mathematics graduate courses to current classroom teachers in order to certify them to teach college-level mathematics.   I wanted to summarize some of the lessons I learned about teaching from these classes that I can bring into my own practice as a mathematics teacher.  These fall into three categories: questioning, alignment, and metacognitive modeling. Continue reading

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