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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Last summer, I had the amazing opportunity to participate in the Mathematical Association of America (MAA) MathFest. Each summer at MathFest one of the largest communities of mathematicians, students, and enthusiasts comes together to talk about the latest advances in mathematical research and education. One of my most memorable experiences was participating in the session * Great Talks for a General Audience: Coached Presentations by Graduate Students. *Here graduate students get coached by more experienced communicators on how to make their research accessible to a general audience, give a 20 minute presentation, and receive individual feedback by an undergraduate student and faculty member. Here I want to highlight some of the insight I gained through this experience.

**Ask yourself, who is your audience?** Are they kids, young adults, non-mathematicians, scientist, non-scientist, a combination? When addressing an audience you may not know its exact composition. I’ve always thought the best thing to do what to teach my audience my work. Make it accessible by treating it as a student-teacher interaction. However, in the limited time available may be impossible to achieve! Instead, **advertise your work. ** Talk about what drew you to your problem, what are the big ideas, why do you find it interesting?

**Less is more.** For anyone who has met me, they know when I am excited about an idea I tend to ramble. I used to think my presentations needed to stand on their own. Every detail I thought was important should be precisely stated in writing. Truth be told, audiences have a limited attention span and a presentation is a tool to highlight ideas. Your use of slides or board space is what brings your talk to life! Make it your story.

**Avoid jargon. **When you spend a lot of time talking to specialist in your field, it is hard to imagine changing how you talk about your work. However, using layman language is necessary to make your ideas engaging outside of your field. You can use a picture to build ideas. Be generous with using more simple examples that motivate your result. I am sure at many stages of your academic moment you’ve experience that ‘A-ha!’ moment. Give that gift you your audience.

**If you’ve found yourself eager to learn more about communicating mathematics, I encourage you to apply to MathFest (deadline April 30) this summer! Below are additional resources:**

- AMS Communicating Mathematics in the Media: A Guide
- MAA Audience Awareness
- MAA Mathematical Communication
- Steven Strogatz, “Writing about Math for the Perplexed and the Traumatized”
- Follow other math communicators on social media, some examples on Twitter include: Tai-Danae Bradley (@mathma), Dr. Eugenia Cheng (@DrEugeniaCheng), Steven Strogatz (@stevenstrogatz) and many more!