**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!

In this chapter, Rands, McDonald, and Clapp take up Casid’s ideas and apply them to classrooms. They argue that classrooms are arranged so that “students will enter the classroom on time, take their seats, remain stationary throughout the class so the knowledge imagined to reside in the instructor can follow the sight lines from teacher to student in a timely fashion into the minds of the students, and finally leave class” (p. 153). In my math classroom, the chairs default to facing the smartboard, with an assumption that there is one teacher who stands at the front of the classroom, lecturing, and that only that one person writes on the board. In fact, the smartboard sometimes gets confused if two people are holding a pen at a time, so it is challenging for two people to be working problems on the board at the same time. I have to intentionally ask students to move into groups; although they will sit near their group, they do not sit in groups by default.

Likewise, class occurs within a fixed range of time; if students arrive early for class, they have to remain in the hall; if they stay late in class, another class of students will enter. Students from two separate classes are not supposed to interact; even if students enter for the next class, they do not ask the previous class what was covered. The digital smartboard automatically erases at the end of class, meaning that the possibility that the next class will get to see and discuss what was on the previous class’s whiteboard is foreclosed. There is a clearly designated time for mathematics and aside from homework, the rest of the hours of the day are not for mathematics. Once class is dismissed, although students sometimes come up to ask me questions (rarely about mathematics, usually more about their grades or their worries about the exams), they do not continue to discuss mathematics with each other; the time for discussing mathematics is over, delimited into this narrow, confined space.

Rands et al. proposes that instructors should challenge norms in the classroom around time, proposing that classes could meet at unusual times, such as 6am or 11pm, on different times each day or for different lengths of time each class. Classes could also meet in different spaces; they propose that teachers “go on a field trip, conduct class at a coffee shop or a park, or hold class outside” (p.164). I have asked why we do not have evening or early morning classes, and the response is that they are not as popular anymore now that online classes are available and that classes are only to be scheduled at times that fit student preferences.

I am thinking, too, about ways in which I could change the way time is organized within in my classroom. One thing I have been doing is to forego the traditional break in favor of a more open classroom, where students can get up and attend to personal matters as they arise, particularly when students are working in groups. Another strategy I am trying is being available by email on evenings and weekends when students are actually working on homework, rather than only checking email during office hours.

If we want to transform our teaching, thinking about how we (together with our students) use time and space both inside and outside of the classroom is essential. In future posts, I will discuss additional articles that I have consulted about time and space in the context of education.

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$$ \forall x, \ \ \forall y \ \ \ d(f(x),f(y)) \leq L d(x,y) \ .$$

The definition makes sense as long as a distance is defined on the spaces. This makes Lipschitz maps highly versatile. Almost every space you deal with daily is either a priori a metric space (a set plus a distance function) or can be made one by endowing it with a distance. (Check for instance the word metric on a group that builds metric spaces out of groups, opening the doors to the beautiful topic of Geometric Group Theory.)

We will, nonetheless, limit ourselves to studying (some of) the properties of Lipschitz maps between Euclidean spaces. Properties?! Look at the definition again, what else could we expect of a Lipschitz map? What else could that condition possibly impose on $f$?

One of the most surprising facts about Lipschitz maps is the following:

**Radamacher’s Theorem:** If $f: \mathbb{R}^n \longrightarrow \mathbb{R}^m$ is Lipschitz, then it is differentiable a.e. (i.e is differentiable everywhere on $\mathbb{R}^n$ except maybe on a set of Lebesgue measure zero.)

Not surprised? Let’s look at the simplest case where $f: \mathbb{R}^1 \longrightarrow \mathbb{R}^1$ and all we know about $f$ is this:

$$ \forall x, \ \ \forall y \ \ \ |f(y) – f(x)| \leq L |y-x| \ .$$

Let’s divide to get an equivalent statement

$$ \forall x, \ \ \forall y\ (\neq x) \ \ \ -L \leq \frac{f(y) – f(x)}{y-x} \leq L \ ,$$

which means that all difference quotients are bounded. Radamacher’s theorem asserts that this boundedness (alone) implies the existence, for a.e. $x$, of the limit

$$ \lim _{y \rightarrow x} \frac{f(y) – f(x)}{y-x} \ \ \cdot $$

Not at all obvious!

For a proof of Radamacher’s theorem see Geometric Analysis [1] notes of Piotr Hajlasz.

Another nice property of Lipschitz maps is their extendibility:

**Theorem (McShane): **Assume $A$ is a subset of a metric space $X$ — so, it is a metric space itself — and $f:A \longrightarrow Y$ is Lipschitz. Then there exists a map $F: X \longrightarrow Y$ such that $F(a)=f(a)$ for all $a \in A$, $F$ is Lipschitz with the Lipschitz constant (the $L$ in the definition) the same as that of $f$.

This property shows that the differentiability result above holds if $\mathbb{R}^n$ is replaced with an open $\Omega \subseteq \mathbb{R}^n$.

By Radamacher’s theorem, a Lipschitz map $f: \mathbb{R}^n \supseteq \Omega \longrightarrow \mathbb{R}^n$ has derivative a.e on $\Omega$. Thus, its Jacobian is defined a.e. which leads us to another theorem.

**Theorem (Change of Variables) : **Assume $ \Omega \subseteq \mathbb{R}^n$ is open, $\phi : \Omega \longrightarrow \mathbb{R}^n$ is Lipschitz, and 1-t0-1. Then for any measurable set $E \subseteq \Omega$ and any measurable function $u: \mathbb{R}^n \longrightarrow \mathbb{R}$

$$ \int_E (u \circ \phi)(x) \ |J_{\phi}(x)| \ dx= \int_{\phi (E)} u(y) \ dy \ , $$

provided that $u\geq 0$ or one of the integrands is integrable. Measurability of the integrands is part of the assertion of the theorem as is the fact that the integrability of one of the integrands implies that of the other.

There are many versions of this theorem and many different proofs in different generalities. A nice and complete proof (of a more general case) can be found in [2].

In the realm of integration, the **area formula** and **co-area formula** hold for Lipschitz maps. (Theorem 3.23 in [1])

Note how much easier it is to show that a map is Lipschitz than is to verify for instance its differentiability and (local) integrability of the derivative. Some form of integrability/boundedness is required of the derivative of a function to satisfy the change of variables formula. This observation tells us that Lipscitz maps are in fact “nicer” than differentiable maps. Their derivatives are smoother and tamer, in some sense. The following theorem is a testimony to this notion.

**Theorem (Federer): **Let $f: \mathbb{R}^n \supset \Omega \longrightarrow \mathbb{R}$ be Lipschitz. For any $\epsilon > 0$ there exists a $g \in C^1(\mathbb{R}^n)$ such that

$$ \mathcal{L}^n (\{ x \in \Omega: f(x) \neq g(x) \}) \ < \ \epsilon \ .$$

Observe that $C^1$ maps are locally Lipschitz by the Mean Value Theorem because on compact sets the derivative is bounded. So, the closest cousin to Lipschitz functions are $C^1$ functions. In fact, many proofs about Lipschitz maps can be reduced to verifications for $C^1$ cases.

Moving to the next property on our list, **the Luzin N property:** A Lipschitz map from $\mathbb{R}^n$ to $\mathbb{R}^n$ that takes measure zero sets to measure zero sets. This interestingly implies that any measurable set is mapped to a measurable set. Of course, for a differentiable map (not $C^1$) this does not have to hold — another reason why Lipschitz maps are more than being a.e. differentiable.

Since the definition of a Lipschitz map is through the distance function, of course Hausdorff measures must come into the picture somewhere:

**Theorem: **If $f:X \longrightarrow Y$ is Lipschitz between metric spaces with Lipschitz bound $L$, then for any $E \subseteq X$, and any $ s $,

$$ \mathcal{H}^s (f(E)) \leq L^s \mathcal{H}^s (E) \ ,$$

which in particular gives the Luzin N property above.

Finally, I finish with a fact that my advisor (Piotr Hajlasz) and I verified recently:

**Theorem: **Let $f:\mathbb{R}^m \longrightarrow \mathbb{R}^n$ be Lipschitz. If the derivative of $f$ at a (single) point has rank $k$, then $Df$ will have rank at least $k$ on a set of positive $m$-measure.

Again, note that the property is an elementary fact if the map is $C^1$.

The V-shaped $y=|x|$ is not that broken after all!

If you know of some other facts about Lipschitz maps that you find cool, please write in the comments!

References:

[1] Geometric Analysis (notes) Piotr Hajlasz, University of Pittsburgh.

[2] P. Hajlasz, Change of variables formula under minimal assumptions. *Colloq. Math. *64 (1993), 93–101.

Fix a finite dimensional field extension $K/\mathbb{Q}$. It turns out that there is a canonical *ring* associated to $K$, which we’ll denote $O_K$, called the *ring of integers *of . Specifically, $O_K$ is defined to be the set of all elements of $K$ which are solutions to monic, integer polynomials. (As a sanity check, one can check the ring of algebraic integers of $\mathbb{Q}$ is $\mathbb{Z}$, which provides motivation for the term for the term “ring of integers”.) For example, the ring of integers of $\mathbb{Q}[\sqrt{-5}]$ is $\mathbb{Z}[\sqrt{-5}]$, but on the other hand, the ring of integers of $\mathbb{Q}[\sqrt{5}]$ is $\mathbb{Z}[\frac{1 + \sqrt{5}}{2}]$.

The next logical step is to ask what properties that rings of algebraic integers have. One might hope that the ring of algebraic integers is a unique factorization domain (UFD). However, in $\mathbb{Z}[\sqrt{-5}],$ we have that $2*3 = 6 = (1 + \sqrt{-5})(1 – \sqrt{-5})$, and it’s not too hard to show that the above equation gives two distinct factorizations of 6. However, one might notice that when passing to *ideals* in $\mathbb{Z}[\sqrt{-5}]$, then $(6)$ factors as the product of prime ideals

$(6) = (2, 1 + \sqrt{-5})(2, 1 – \sqrt{-5})(3, 1 + \sqrt{-5})(3, 1 – \sqrt{-5})$

and moreover, this factorization is unique. One can then go onto show that if $O_K$ is the ring of algebraic integers for some finite dimensional field extension $K/\mathbb{Q}$, then for any nonzero ideal $I \subset O_K, I$ can be factored *uniquely *as the product of prime ideals. This leads to the notion of the Dedekind domain, which generalizes this property.

Moreover, one can argue that one can make a *group* of these elements by including an extended notion of ideals known as *fractional ideals. *These are $O_K$-submodules $J \subset K$ for which there is an $r \in O_K$ with $rJ \subset O_K$. This is a group with a product operation similar to that of the rational numbers, so that $\frac{I_1}{J_1}*\frac{I_2}{J_2} := \frac{I_1I_2}{J_1J_2}$.

From this notion, one can define the *(ideal)* *class group* of the ring of algebraic integers $O_K$, defined to be the quotient of the above group by the group generated by all nonzero principal ideals. The class group tells us many facts about the associated field and its algebraic integers – it’s a good exercise to check that the ring of algebraic integers is a principal ideal domain if and only if its associated class group is trivial.

One of the first cool facts about this is that the class group is always a finite group! This also develops the subject of class field theory, the study of Galois extensions of $\mathbb{Q}$ whose Galois groups are abelian over $\mathbb{Q}$. This can be used to prove the Kronecker-Weber theorem, which says that for any abelian extension $K/\mathbb{Q}$ (i.e. any Galois extension $K/\mathbb{Q}$ for which $Gal(K/\mathbb{Q})$ is abelian), there is a cyclotomic field containing $K$. In short – the class group of a number field is a rich object worth studying!

]]>However, through our training as mathematicians, are we asking ourselves: How are we contributing to this diversity? How do we create environments that embrace the identities of those who “do” mathematics? Are we making mathematics accessible and inclusive? These are questions that I ask myself, but would be interested in making them part of a larger narrative.

When I started thinking about these topics I found myself overwhelmed with the knowledge that this affects the lives of many on a daily basis. This is part of their personal story and in many ways an unavoidable part of their journey to become mathematicians. It affects my students who look into who does mathematics and may not see someone who they can relate to. It affects my peers and professors who may be the first “blank” or the only or the few “blank” in their classrooms, at a conference, or departments.

Then the question became, what do I do? I searched high and low for a magical answer and found that … it’s complicated. But, I think there are certainly small things we can do to open up the conversation as individuals and as a community. Here are a few things I’ve found useful, and I hope that this list helps others who wish to start discussing some of these questions:

**Create spaces for conversation.**- It may look like small coffee chats with peers, one or two conversations with faculty, or participating in conferences that facilitate these conversations. For example, the Society for Advancement of Chicanos/Hispanics and Native Americans in Science (SACNAS) is a great one! Others that come to mind are Latinxs in the Mathematical Sciences, Field of Dreams Conference, and Blackwell-Tapia Conference.
- As part of our AWM student chapter, we have created a Teaching and Diversity Seminar where we bring speakers to tackle some of these questions. We look in neighboring departments and we look for passionate individuals within our fields. One of our speakers this semester, Dr. Rochelle Gutierrez has a piece on this blog I encourage you to read. Other great speakers we’ve engaged with as part of our TA (teaching assistant) training include Aditya Adiredja and Esther Enright.

**Learn, keep learning, and challenge your assumptions.**Join a reading group, attend talks about these topics, or follow blogs by diverse mathematicians. Some of my favorites are Francis Su’s article, “Mathematics for Human Flourishing” and Piper Harron’s blog, The Liberated Mathematician. We may never have all the answers but we can become more aware that our roles as mathematicians extend beyond our discipline. Sometimes this looks like being mindful of our biases or empathizing with the experiences that are not similar to our own. It could mean being a voice and it could mean passing the mike to let other voices be heard.**Be honest, listen, and take care.**Embrace and share your stories as part of what makes you a mathematician. The challenges and triumphs. This is difficult if you find yourself in an environment that doesn’t embrace the identities you bring to your mathematics. Having open and honest conversations require listening to others with no judgment and accepting that their experiences have a place in our community. Some will be uplifting and others not so much. But it is important to create a space to share both. These conversations may be challenging so always seek to take care of yourself as well.

Mathematics is a beautiful field that blossoms with our own unique perspectives and experiences. Let’s work towards opening those conversations, let’s challenge our assumptions and foster the growth of a more diverse mathematical community … together.

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The idea that we can inculcate students into the practices of a discipline like mathematics relies heavily on theories that were developed in the early 1990s. In their 1991 book Situated Learning: Legitimate Peripheral Participation, Jean Lave and Eitenne Wenger propose an idea called legitimate peripheral participation. They suggest that learning is the process by which one comes to be a part of a community of practice; learning is a process of coming to negotiate the social meanings, moving one not towards central participation (since there is no definitive center) but rather to full participation. Legitimate peripheral participation, Lave and Wenger propose, is a theory of how people learn rather than a theory of how to teach. Schools teach many things other than just subject matter, and students participate in many different communities of practice, not just that of the mathematics classroom.

There is something qualitatively different about the mathematics in the mathematics classroom and the mathematics that professional mathematicians do. A professional mathematician works primarily to generate new knowledge rather than to merely learn past knowledge. Although students taught with discovery learning methods have occasionally discovered new theorems or invented new mathematical ideas, that is not the primary purpose of mathematics education, particularly at the K-16 level. A professional mathematician does not work in groups of three to four for fixed periods of time like the way a student might work in a Complex Instruction- or groupwork-based classroom; they have to learn how to structure their own time and to attend professional colloquia and conferences in order to present their ideas. Professional mathematicians also use very different tools than students; they would be more likely to be seen reading a professional journal than a textbook, and use more advanced specialized software such as GNU Octave or Sage instead of (say) a handheld graphing calculator.

The idea of legitimate peripheral participation, therefore, is tricky when you try to apply it to the K-16 classroom. We are not necessarily enculturating students into becoming mathematicians; our students come to us with many different goals. The classroom as a community of practice is a very different space in which full participation means to take part in an active role in classroom discourse, to make and defend ideas, and to begin to develop one’s own ideas while also mastering the fundamental concepts of mathematics. The norms we establish and the experiences that we give students in the classroom help to make them full members of the mathematics student community of practice. By instituting the Standards for Mathematical Practice in our classroom, we provide the foundation for students to begin as legitimate peripheral participants in their future studies and careers.

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