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!