Mathematical Inclusion Principle: Emmy Noether

This is a post in an ongoing series which features people, events, and organizations that demonstrate and/or support the importance of inclusivity and diversity in the mathematical sciences. I will share these posts with the students in courses I teach, and I encourage you readers to do the same. The series title may be considered a "play on words" with the method from combinatorics known as the "inclusion-exclusion principle".

Emmy Noether born March 23, 1882; Erlangen, Germany died April 14, 1935; Bryn Mawr, PA, USA

Amalie Emmy Noether was born in 1882 in a city in the state of Bavaria in Germany. Her father, Max Noether, was a professor at the university in Erlangen, although his daughter's achievements in mathematics would far surpass his, in time. Throughout high school, she studied English and French, and took certification examinations to become a language teacher. Although her exams were deemed sehr gut ("very good"), she actually decided to continue her studies at the university instead.

It was not at all common for females to study at university at this time in Germany. In fact, the university's administration had said that this would "overthrow all academic order" and, during Emmy's time, she was one of only two women out of almost 1000 students. [1] In fact, she was only allowed to audit courses, sitting in on lectures (if the professor even allowed this) without officially enrolling. Still, she gathered knowledge and passed the graduation exam in 1903 [2].

Thereafter, Emmy Noether was devoted to mathematics. She attended lectures by some of the most well-known mathematicians at the time (all men, of course) at the University of Göttingen in 1903, and then returned to Erlangen to work on a thesis, which she published in 1907. Fortunately, the rules for women were being changed, and she was allowed to complete this work and receive her degree, which was summa cum laude, mind you.

For the next 25 years, she worked and taught in universities in Germany and Russia. At first, she stayed at Erlangen, occasionally substituting for her father when he was ill. In 1915, she returned to Göttingen to teach, at the behest of world-famous mathematician David Hilbert, who intervened on Noether's behalf to ensure the sexist university policies could not prevent her from doing so. One faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" Hilbert's retort shows that his interest was in the mathematical prowess of the candidate and nothing else: "I do not see that the sex of the candidate is an argument against her admission as privatdozent. After all, we are a university, not a bath house." [3] It was at Göttingen that she produced her most famous work about symmetry and conservation laws. (You can read more about that work here in Science News [4].) And, although it sounds wonderful that Noether was provided this opportunity to teach, was able to apply for habilitation (something like tenure), and received a citation from the Prussian Minister for Science, Art, and Public Education, it is important to note her professorship position came with no salary! 

Emmy Noether visited Moscow State University from 1929-1930 to teach and work with mathematicians there on their research in abstract algebra. Many of her colleagues, all of whom were well aware of her talent and passion for mathematics, were frustrated at the university systems and cultures that prevented her from receiving proper recognition and compensation for her work. While the wider world was rather discriminatory, at least we can say that her corner of the mathematical world recognized her as an outstanding individual, regardless of gender. Indeed, at the 1932 International Congress of Mathematicians, in Zurich, Switzerland, Noether delivered a plenary address. Furthermore, that same year, she shared the Ackermann–Teubner Memorial Award with Emil Artin, in recognition for their "advancement of mathematical knowledge". [5]

Unfortunately, the rise of Nazism and widespread anti-Semitism in Germany impacted the university system and led to the removal of Noether, a Jew, from her teaching position. The very same Prussian Ministry for Sciences, Art, and Public Education, from whom she had received a glowing citation earlier in her life, sent her a notice: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen." [6] Luckily, scientific colleagues in the rest of the world were working to help scholars like Noether, and she found a position at Bryn Mawr, an all-female liberal arts college in southeastern Pennsylvania. While in the United States, she also visited and lectured at Princeton University and their Institute for Advanced Study.

Emmy Noether passed away somewhat suddenly in April 1935, at the age of 53. Doctors had found a pelvic tumor and an ovarian cyst during surgery. While on bed rest, she suffered a high fever and died. Doctors were not entirely sure what had happened, citing a possible infection as the cause of the fever and untimely death. Support from the mathematical and scientific community was swift: Noether's colleagues and other eminent scholars wrote their condolences and tributes, including Albert Einstein and Herman Weyl. Below is a quote from Einstein's letter published in the New York Times:

In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. [7]

Why do I write this now? Well, I think Nature put it best with their blog post last week [8]:

Celebrate the mathematics of Emmy Noether: An algebra pioneer who faced discrimination deserves wider recognition on the centenary of her namesake theorem.

From the article, because they summarize it all better than I could:

The results that Noether published 100 years ago were, for her, a rare foray into physics, in which she was not particularly interested. Albert Einstein had just developed his general theory of relativity, and was struggling to understand how energy fitted into his equations. [German mathematicians David] Hilbert and [Felix] Klein were working on it, too, and asked Noether for help. 

That she did help is an understatement. Noether’s expertise in symmetry led her to discover that the symmetries of a physical system are inextricably linked to physical quantities that are conserved, such as energy. These ideas became known as Noether’s theorem (E. Noether Nachr. d. Ges. d. Wiss. zu Göttingen, Math.-phys. Kl. 1918, 235–257; 1918).

 As well as answering a conundrum in general relativity, this theorem became a guiding principle for the discovery of new physical laws. For example, researchers soon realized that the conservation of net electric charge — which can neither be created nor destroyed — is intimately related to the rotational symmetry of a plane around a point. The impact was profound: those who created the standard model of particle physics, and the researchers who attempt to extend it, think in terms of Noether’s symmetries.

All of this is to say that mentioning Emmy Noether now is timely. Her work 100 years ago was outstanding and groundbreaking, and scientists today still find it influential. Moreover, the struggles and discrimination she overcame throughout her life may serve as a lesson for our times: that someone can overcome those obstacles, but wouldn't our world be better off if they didn't have to?

Mathematical Inclusion Principle: Maryam Mirzakhani

This is a post in an ongoing series which features people, events, and organizations that demonstrate and/or support the importance of inclusivity and diversity in the mathematical sciences. I will share these posts with the students in courses I teach, and I encourage you readers to do the same. The series title may be considered a "play on words" with the method from combinatorics known as the "inclusion-exclusion principle".

Maryam Mirzakhani born May 12, 1977; Tehran, Iran died July 14, 2017; Stanford, California

The story of Maryam Mirzakhani's life is at turns inspiring and heartbreaking. She was born in the capital of Iran and achieved academic success at an early age, becoming the first female Iranian student to earn a gold medal at the International Mathematical Olympiad and the first Iranian student to obtain a perfect score at that competition [1]. However, her sixth grade teacher "discouraged her interest in mathematics, noting that she was not particularly talented, not at the top of the class" [2]. She earned a bachelor's degree in mathematics from Sharif University of Technology in Tehran and subsequently emigrated to the United States, obtaining a PhD from Harvard University.

Everyone who writes about Mirzakhani notes her talent, passion, and commitment to both her mathematical work and the communication of that work. For example, an article in the New York Times Magazine's series "The Lives They Lived" [3] and an article in Quanta Magazine [4] both describe and portray her penchant for visualization, drawing, and "doodling", perhaps an informal act that the lay public may not associate with high-level, abstract mathematical research. Her three-year-old daughter would remark, "Oh, Mommy is painting again!", and Mirzakhani herself remarked that "the process of drawing something helps you somehow to stay connected".

The Fields Medal is sometimes referred to lazily in the media as the "Nobel Prize of Mathematics" but this is inaccurate for a few reasons: it is only given out every four years, it has an (arbitrary) age cap, and it hardly represents the diversity of mathematical research and the people who do that work. (See Michael Barany's recent comment in Nature [5] for further information.) In any event, Maryam Mirzakhani was the first (and still only) female recipient of this award, so bestowed in 2014, seventy-eight years after the award's inauguration. The most recent class received their medals in August 2018, and Caucher Birkar earned Iran that nation's second such prize. However, the historical list of recipients is heavily dominated by North American and European nations, and white males, overall [6]. This is hardly the image of mathematics that we in the community wish to convey, but it is a fact. 

Unbeknownst to many, Maryam Mirzakhani was battling breast cancer at the time this award was announced. Her closest friends and colleagues were aware of her struggles and even helped to keep the press from making this illness the main story at the Fields Medal ceremony [7]. She passed away in July 2017 at the age of 40, at the time a professor of mathematics at Stanford University [8]. It's hard to overstate the outpouring of support from the mathematical community upon her death [9]. She will never be forgotten. 

Why do I write this now? Why is this the first post in a series about inclusivity? In today's political climate, it should be obvious, but I will state it anyway. Maryam Mirzakhani is an exemplary American. She found a talent and pursued it, against all odds. Mathematics was her passion and she traveled across the globe to continue her studies and share her work with others. However, an executive order passed by President Donald Trump, and recently upheld (in part) by the Supreme Court of the United States, would have made it difficult for someone like Maryam Mirzakhani to emigrate to America, simply because of her country of origin [10]. Never mind her intellect, her passion, her humanity, her obvious abilities to contribute to the vibrancy of America's scientific community. Never mind all that. The current administration would not bother to consider these attributes merely because she comes from a "Muslim-majority country" [11].

Now, you may say: "How do you know the administration would deny her visa? The ban doesn't apply to all immigrants from those countries." To this, I say: "You're missing the point." The executive order was obviously designed based on those countries being Muslim-majority and nothing else, so why would there have been any reason to allow someone like Mirzakhani to enter the country? At what point will we admit that it is utterly ridiculous that someone can say, "You can't prove I'm racist", and feel like they mean it, despite the fact that they are so obviously racist?

In any event, my point is that Maryam Mirzakhani is a treasure of humanity, that she is now lost, that the current culture of our country (and, indeed, the world) did not appreciate her when she was alive, and that the current politics of our country would likely never have allowed her entry in the first place. I don't want to live in that world, so I will tell everyone I know about Maryam Mirzakhani and her contribution to the history of humanity. She was a stellar human being. She succeeded by fighting obstacles. She was a mathematician. She will be missed.

College: what should it be?

This post is intended for both college students (especially first-year students) and college instructors as we begin another academic year. 

This past summer, I spent a lot of time reflecting on my experiences as a student and teacher. (I've been one, the other, or both for literally my entire adult life.) I spend a little time on this kind of reflection every summer: pondering the previous school year, appreciating what went well, thinking about how to make my courses better, etc. But I spent even more time doing so this past summer, perhaps because I've now been teaching at college full time for 5 years and I needed a mental break for a couple months, and perhaps because I now have an infant daughter (7 months) and I'm feeling especially philosophical about life and learning.

As part of that reflection, I read two books about education, both of which I highly recommend: Educated: A Memoir, by Tara Westover (historian and writer); and College: What It Was, Is, and Should Be, by Andrew Delbanco (Professor of American Studies at Columbia University). I was struck by Dr. Westover's stories in Educated, particularly by how unfathomably different her life has been from mine, apart from being academics by trade. And I was moved by Dr. Delbanco's passionate writing about the history, present, and future of colleges in America and what they can and ought to do for us as a society. 

The following quote is from the introduction to Dr. Delbanco's book:

At its core, a college should be a place where young people find help for navigating the territory between adolescence and adulthood. It should provide guidance, but not coercion, for students trying to cross that treacherous terrain on their way toward self-knowledge. It should help them develop certain qualities of mind and heart requisite for reflective citizenship. Here is my own attempt at reducing these qualities to a list, in no particular order of priority, since they are inseparable from one another.

1. A skeptical discontent with the present, informed by a sense of the past.
2. The ability to make connections among seemingly disparate phenomena.
3. Appreciation of the natural world, enhanced by knowledge of science and the arts.
4. A willingness to imagine experiences from perspectives other than one's own.
5. A sense of ethical responsibility.

These habits of thought and feeling are hard to attain and harder to sustain. They cannot be derived from exclusive study of the humanities, the natural sciences, or the social sciences, and they cannot be fully developed solely by academic study, no matter how well "distributed" or "rounded". It is absurd to imagine them as commodities to be purchased and delivered to student consumers. Ultimately they make themselves known not in grades or examinations but in the way we live our lives.

 Delbanco, A. (2012) College: What It Was, Is, and Should Be. Princeton University Press. p. 3-4

I had this list of qualities in mind throughout the rest of my reading of the book and while I prepared my courses for the fall semester that officially starts tomorrow. These lofty ideals of character and citizenry appeal to me. But I wonder how much I, as a teacher, am able to engage with students on these ideals, let alone encourage them to cultivate habits and behaviors associated with those ideals. Moreover, I wonder whether students are thinking of a similar list, or something else entirely. Take note that nothing in the quote above is remotely related to "employability" or "marketable skills". And yet, this is what I imagine many students are thinking of as they enter college. They know that a college degree is somehow required to "get a good job" and "succeed in life". But what do those goals even mean? And if we teachers don't have those ideas on our list of goals, why should we be teaching college anyway? What's the point?

So, it feels important to share this list publicly as a way to say that the list above truly is important to me, both as a teacher and as a citizen of this world. Those qualities are vastly more important to me than job skills or a grade point average, and I would hope that most other teachers would express a similar opinion. Yes, ultimately, I want my students to "succeed in life", under whatever definition they have individually for "success". But, I truly believe that pursuing the ideal qualities in that list will help lead to success and so much more.

In conclusion: students, that list of qualities is important to me and I will try to keep it in mind all year. I doubt that every class meeting, every assignment, or every conversation will be directly related to that list. It won't feel like it to you, nor to me. But I want that list to percolate through everything we do. I want you to consult the list every once in a while and reflect about your time at college and your life experiences, your behaviors, your personality, your culture. I will do the same. Together, I hope that we can push each other to be better people and to cultivate the ideals and habits represented in that list. In other words, I hope we can collectively pursue education in the broadest sense of the term.

And fellow teachers, I hope you will also keep that list in mind as you prepare and teach your courses, as you mentor students, as you give advice and grades and letters of recommendation, as you serve on college committees and do research and ... everything else that we do. Take time now and then to reflect on why you teach. Think about not only what you hope your students will learn from your courses, but also what you hope to learn from teaching those courses and working with your students.

Best wishes to all for a happy, productive, meaningful, and educational semester!

Just For Fun: What's on my office door?

When I'm not actively in class, preparing for class, doing research, or writing up results, I find myself trawling through Twitter and reddit and stackexchange reading about math and education. (I know, that's a lot of  math, but it's my life!) Recently, I've been reading several blog posts about the #MTBoS hashtag. In so doing, I learned a few things:

• What it actually stands for! "Math Twitter BlogoSphere"
• That lots of people want to change it. #iteachmath was proposed by Dan Meyer and supported by many. FWIW, I support this change. (Personally, I always thought it had something to do with Boston, but maybe that's just because, as a native Bostonian (ok, Cantabrigian, technically), I was primed to think so.)
• That I don't blog nearly often enough, or hardly at all, these days!

So, this post is partly an attempt to get back into a "blogging state of mind". No more thinking to myself, "You know, I should write about XYZ that happened today," only to never actually do it.

This post is also partly inspired by Sara Vanderwerf's blog post "What is math? What do mathematicians do?" wherein she describes classroom activities to conduct during Week 1 to encourage students to self-identify as mathematicians, among other things. One of the activities is centered around answering the questions in the post's title:

Mathematics is the study of patterns.

Mathematicians notice patterns.

Mathematicians describe patterns.

Mathematicians generalize patterns.

In that post, she links to a poster created by Greta Bergman based on this activity (and described in a blog post). I liked the poster so much that I decided to print it out and post it on my office door. And that led me to this idea: I should write about the cartoons and images I've posted on my office door and explain why I found them interesting enough to serve as the face of my office.

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1. "Monday Punday" cartoon.
   
   
   
   
   This is a fantastic site with a weekly cartoon that embodies a particular pun. Your goal is to figure out the pun, and you can type your guess to confirm whether you're correct. I found this particular cartoon a few years ago and found it clever, tricky, and fun. I won't ruin the solving experience for you (since that's all the fun, really), so here's a link to that specific day so you can check your guesses: http://mondaypunday.com/133
   I've walked down the hall and caught passersby staring at this image trying to figure it out. Students, faculty, and staff alike have been halted in their tracks by this fun puzzle!
   

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2. (a+b)² is NOT a²+b².
   
   
   
   
   This is, of course, based on a common student error with binomial expansions. The apparent underlying assumption is that the Distributive Property is somehow a statement about any operation and parentheses, not specifically about multiplication over addition. (Indeed, this phenomenon is so widespread that I asked about it on matheducators.stackexchange to try to better understand where it comes from and what we can do to effectively nip it in the bud.) I assure you that, several times, I have been working on a problem with a student during office hours when they asked me to check their work and I responded by wordlessly pointing to this image on the door until they realized what they did.
   

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3. A proof without words that 1+2+3+...+n=(n+1 choose 2).
   
   
   
   
   I found this image in a MathOverflow thread. In fact, as you can see, it's an animated gif, not just a static image. The animation cycles over several pairs of dots from the bottom row to demonstrate that there's a bijection between the set of such pairs and the set of single dots in the triangular array above the bottom row. I hand-wrote some details on this printout because I don't assume anyone would piece together what this is a proof of. In short, this can be a "Proof without Words" but it certainly should not be a "Proof without Symbols". I posted this on my door because it's one of my favorite proofs and I like to point to it during discussions with students to demonstrate that there are often several different ways of proving one statement, each of which may yield different insights.
   

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4. Calvin & Hobbes comics.
   
   
   October 10, 1986; source: gocomics.com
   
   
   source: gocomics.com
   
   
   
   This is a comic strip I've loved since childhood. Wikipedia tells me the strip was syndicated from 1985 to 1995, and I turned 11 in 1995, so I surely read some of these as they were published. I also distinctly recall reading "reruns" in the Sunday funnies and I had a couple of anthologies that I reread several times over. (I still flip through them now and again.) What I don't recall is how often Calvin's math class and homework are mentioned! I forget how I found these particular strips (I may have just googled "calvin hobbes math" and picked my favorites) but I think they're especially emblematic of Bill Watterson's brand of humor.
   

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5. How to Study Math: "Don't just read it -- fight it!"
   
   
   
   
   This is a quote from Paul Halmos as illustrated by the webcomic Abstruse Goose. Halmos is one of my favorite mathematical authors: his prose is outstandingly clear and engaging. I highly recommend his books and essays, especially: his texts Naive Set Theory and Measure Theory; his autobiography, I Want to Be a Mathematician; and his essays "How to Write Mathematics", "How to Talk Mathematics", and "What is Teaching?". Those essays were especially inspiring to and influential on me during graduate school when I realized I wanted to be a college math teacher. Anyhow, the full quote (below) serves as inspiration for any student of mathematics, whether they're struggling through a basic algebra course or butting heads with the unknown during creative research. But I find it especially useful for students learning to read and write proofs for the first time.
   "Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
   

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6. Mathematics is the Study of Patterns. 
   
   
   
   
   This is the poster that I mentioned earlier, the genesis for this very post. I just taped this up today! This is an idea I've shared in a lot of my courses, but I've never stated it so elegantly, nor put it into poster form. I look forward to pointing to this poster a lot during conversations with students.
   

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7. Biostatistics.
   This is a flyer created by our Math Department to promote an interdisciplinary major we offer that combines Biology and Mathematics. Students take advanced statistics courses and intro- to intermediate-level biology and complete a capstone project that synthesizes their knowledge. We are trying to graduate more and more students in this major and we think that a big problem is lack of awareness. It's common for students to enter college wanting to be a Biology major, and we think that many would actually fit into this program well, if they only knew it were an option! So, we are trying to spread the word.
   

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8. Best EC Learn Course Site.
   These are two certificates I received from our college's Academic Technology group for creating the "best course management site" of the year, two years in a row. I'm proud of these accomplishments! I do put a lot of time and effort into making sure my course sites aren't just repositories for grades and documents. I want students to engage with the site at home to complete assignments, effectively access a variety of resources, and really use it as a supplementary tool for their learning. (Two years ago, I talked the AT group into integrating Piazza with our management system, and for that they gave me the "Piazza Delivery Man" award. I appreciated the pun even more than the honorable mention!) I posted these certificates to brag a little bit, I suppose, but also to signal to students that I do care about their learning. I'm not creating these sites to win awards; I'm doing so to help them learn.

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So, that's what's on my door. What's on yours and why?

"You ask me a question": Part 3 of N. Becoming a math teacher.

Introduction: See this previous blog post for more information about this practice of allowing students to ask me a question on their quizzes.

My Thoughts: I love doing this "you ask ME a question" thing on quizzes because it lets me see what my students are wondering about. I try to encourage them to narrow their focus on material related to our course, and I sometimes give minimal to zero credit if a question is too far afield or not well-formed. (Sometimes, I tell the student to come talk to me in person and then give them credit retroactively.) But, I don't exercise this policy too stringently because I like to see what they'll ask. Sometimes, it's about me! I think this is because they can see my enthusiasm for math both in and out of the classroom. In the context below (a math course for almost entirely History, English, and Communications majors), this may in fact (unfortunately) be the first time they've encountered a teacher who really enjoys the course and wants the students to enjoy it, too. So, I let them ask things about me and my life. I know that there are some Education majors in the class, too, so I hope that I can sometimes inspire future teachers, even if they won't be teaching mathematics.

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Context: "Math of Everyday Life" (quantitative reasoning course for non-STEM majors).

Question: I'm curious as to what led you to become a math professor. Was that always a goal for you?

My Answer: As a kid, I was “good at math” but, looking back, I realized I was just quick with numbers and liked following rules and formulas. Over a long period of time, I have come to realize that this is not at all what math is all about, it’s just a very small part of it.

I studied math and physics in college because I liked understanding how the world works and liked the idea that we can prove things (using logic in math, or using experiments in physics). I realized that I didn’t really like doing the lab work required for physics and that I was only interested in it because of the mathematical theory involved. So, I went off to grad school to get a PhD in math, with a focus on applying mathematical methods to physics. (Specifically, I studied partial differential equations, the Finite Element Method, and the Navier-Stokes equation. Those are things you can google.)

That didn’t go all that well. I took a lot of advanced courses in math and did alright, but I started feeling jaded. I didn’t feel an impetus to do research. I realized that I liked learning as much as I could because I was fascinated by the knowledge and wanted to let everybody else in on the wonderful secret that math is beautiful. I wasn’t really interested in taking that gained knowledge and applying it to my own research problems or writing papers or anything like that. I just wanted to talk about math with other people and help them learn.

So, I got really into teaching. At my school (Carnegie Mellon), grad students are expected to serve as Teaching Assistants, working under a professor (who would run the big lecture classes) to teach small sections of students on days they don’t have a lecture. (These are called "recitations".) I really loved this and found myself spending way more time prepping for teaching and working with students than I spent on my own homework and research. After a disastrous oral exam with my research advisors that I totally bombed, I thought about dropping out of grad school. I had a few dark months where I zombied my way through life, unsure of what to do and whether it had all been a waste of time.

Amazingly, at that time I learned about a different degree that my school offered that I had never really heard of: Doctor of Arts (D.A.) It’s like a PhD: it’s a doctorate, and I had to take the same advanced coursework. But, the dissertation is not original research and scholarly articles for a journal. Instead, it’s expository, it’s based on sharing existing knowledge in a new and creative way (instead of forging new knowledge, like for a PhD dissertation).

I ended up writing a textbook for a popular math course at my school that everyone has to take called “Concepts of Mathematics”. The math department let me, as a grad student, teach one of the 100+ student lecture classes, which is very rare. And I was able to “test run” my book along the way to get feedback from students and make it better. They still use that book, 4 years after I’ve graduated. And, I use the book to teach MATH 2109 Discrete Methods here at Emmanuel!

"You ask me a question": Part 2 of N. Mathematical historians.

Introduction: See this previous blog post for more information about this practice of allowing students to ask me a question on their quizzes.

My Thoughts: I particularly liked this question because it shows the student is interested in pursuing mathematics after college but perhaps not necessarily as a researcher or teacher. It's a trite but fair question: "Jeez, what am I gonna do with a math degree?" I worry that this question is so common from students (and the general populace) because we math teachers don't do enough to show students the variety of careers and passions that one can pursue, having studied mathematics. Certainly, teaching and academia are available, but there are so many others! Especially in a field like this, where one can feel overwhelmed easily by the complexity and difficulty of its content, it's extremely important to remind students that they truly can find their own place in the world of mathematics and that we're here to help them do so.

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Context: Real Analysis course (for advanced math majors).

Question: How much of a market is there for math historians?

My Answer: Depends on what you mean by "market". They certainly exist! There are many fascinating books published every year about the history of math, its ideas, its development, and its people. Several of these writers are academic mathematicians who have turned to the history of math as their subject material, while others are historians who are enthused by mathematics.
I strongly recommend any books by:

• William Dunham (especially Journey Through Genius)
• Keith Devlin (especially The Man of Numbers about Fibonacci)
• Charles Seife (especially Proofiness)

There are also some great biographies of particular mathematicians. I recommend:

• A Beautiful Mind (about John Nash)
• The Man Who Loved Only Numbers (about Paul Erdos)
• Logicomix (graphic novel about Bertrand Russell, Georg Cantor, Gottlob Frege, Ludwig Wittgenstein, and many others)

I also happen to belong to a SIGMAA (Special Interest Group of the Mathematical Association of America) devoted to the History of Mathematics.

So, there are plenty of math historians out there, and there will always be plenty of things for them to investigate, study, and write about. If you're interested in the sociocultural aspects of mathematics, its development over time, and the people who have shaped it ... by all means make a career out of it. Many people have!

"You ask me a question": Part 1 of N. Abstract thought and occupational mathematics.

Last year, I started conducting regular quizzes for my courses online, through our course management system. The regularity, structure, and significance vary from course to course, but I now do this in all my courses. The ease of this (at least compared to quizzes on paper) has allowed me to cultivate another practice: I now (frequently) add an extra question at the end of quizzes that says something like this:

You ask me a question. This can be about recent course material or something that is related to this course.

This has been enlightening and entertaining well beyond what I expected and I wholeheartedly recommend it!

The online submission of quizzes allows students to type their thoughts thoroughly, and I believe the fact that they're taking the quiz in a setting outside of class encourages them to ask more honest questions. From my perspective, this also allows me to respond to questions individually, directly, and thoroughly.

I end up sharing a select subset of questions and my answers in a document online (but I anonymize the questions to prevent potential embarrassment). This can show the students that they have a question similar to lots of others and should not be afraid to ask anything. This process also helps me "check the pulse" of the course. (For instance, I might observe a common theme among several questions and use this to guide my planning.) And, honestly, I think it's great fun! Overall, I believe that students subsequently feel more engaged with both me and the course, and it surely forces me to reflect on my teaching and consider improvements.

I'd like to use this blog to share some questions I've been asked and the answers I've provided. Since I don't know how often I will do this, I declare that this is the first of some arbitrary number, N, of entries.

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Context: Real Analysis course (for advanced math majors).

Question: In what way does a course like Real Analysis, or any that aims at developing deep abstract thought in terms of mathematics, apply to the occupational field of mathematics? 

My Answer: Interesting question! In some sense, I don't have to say much here, because your question boils down to, "How does studying mathematics prepare one for a career in mathematics?" In that case, all I have to say is, "Well ... what ELSE should we do instead? Not study math?!"

But I think your question belies an attitude that deep, abstract thought is somehow not useful, or not applicable. And I have to flat out disagree with that sentiment. Personally speaking, I feel that I have become a better and better mathematician as I've aged, and this has come from both a combination of applying my knowledge to solve problems and studying deeper and deeper into the theory of math. Without one or the other, I would be doomed as a mathematician, and I feel like this is a universal phenomenon. Even the most tried-and-true "applied mathematician" has to study its theoretical aspects for many years, and this isn't just to be difficult or to "weed out" poor students or anything like that. It's because it's genuinely useful and important. It further develops one's understanding of what one is doing and puts things in perspective.

In a broader, sense, as well, a course like this is teaching you several important general skills:

1. An appreciation for the history and philosophy of mathematics and its people. In our course, we have mentioned several famous mathematicians that have contributed to the development of humanity's collective knowledge. We are drinking from the fountain of that knowledge, and I think it behooves us to marvel at its beauty. You may not find any one particular theorem we study all that fascinating, but taken as a collection, its a remarkable body of knowledge that human beings have put together.
2. Problem-solving. This idea is trotted out in just about every mathematics course, but I think that's for a good reason. Think about the homework exercises and quiz problems you've done in this course. Haven't they helped you become a better logical thinker? You've had to grapple with ideas you've never even heard of before, putting them into logical order and making sense of them, and then writing down your ideas about them to explain them to others. Surely, this has helped you become a better thinker and expositor, right?
3. General content knowledge to be used in other areas. You may be surprised by how widespread many of the key ideas and results from this course are. Sequences? All over math! Functions and continuity? You bet! Derivatives? Ummm, calculus amirite? Just because we're looking at these ideas from an abstract viewpoint doesn't render them meaningless. If anything, it gives them even more meaning and, I'd argue, a profound beauty.

All of these skills are important for occupational mathematics.