Long Live The Blank Slate

It’s the first day of school. I always loved this feeling as a kid. You’ve got your shiny new notebooks, freshly sharpened pencils, and your first day outfit all ironed and ready to go. Nothing can really compete will that feeling of having a totally blank slate, a reset, a fresh start. This will be the year you go home and copy all your notes after class, this will be the year you ask thoughtful questions. This will be the year you finish your homework ahead of schedule. This will be the year that you dazzle the academe with your brilliance.

Unsurprisingly, as a professor I still experience many of those feelings. A new batch of classes, a fresh crop of students, I’m bristly with ideas about alternative assessment strategies and non-traditional classroom models. On the first days we feel each other out, as we jointly embark on this magical journey into the unknown.

Of course in our modern day the unknown quality of the journey is slightly compromised. The omnipresence of social media and anonymous online forums steal some of the mystery. And then there’s the greatest blight of them all: RateMyProfessor.com.

This is a site that collects brief narrative reviews of thousands of professors by their students. Some reviews are helpful “definitely buy the textbook, it helps a lot,” to lewd and ridiculous “her class was ok but mostly I just stared at her butt.” I’ve often wondered about some of the biases that appear in these reviews, for example, do butts come up more often in reviews of female faculty? Luckily for me Ben Schmidt swooped in this year with his blog Gendered Language in Teaching Reviews. Schmidt scraped the data from 14 million RateMyProfessor reviews to study the occurrences of particular words across genders and disciplines.

As reported on NPR earlier this year, men are far more likely to be “brilliant,” especially when they are philosophers, while women are more likely to be rated as “friendly.” In mathematics, we see some really egregious (although unsurprising) gender splits with the words “genius” and “funny.”

It looks like physics, chemistry, and math professors are most likely to be "too smart"...whatever that means.

It looks like physics, chemistry, and math professors are most likely to be “too smart,” unless they’re women.

But don’t worry, not everyone thinks we’re too smart, math professors also have the highest incidence of the word “stupid,” with it showing up 160 times per million words of text regardless of gender.

And if you’re curious, the word butt doesn’t really seem to follow a distinct gender pattern, but mathematicians seem to rank quite low as compared to the other lab sciences.

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Promoting Diversity and Respect in the Classroom

For a lot of us, the new school year is just around the corner. We’re getting ready for new classes and a new group of students. We have plenty of learning goals for our students and subject-specific material to think about, but we also need the classroom to be a place where all our students are welcome and are treated fairly.

A mosaic of pictures from people using the #ILookLikeAnEngineer hashtag on Twitter. Image: Emma Pierson. Click to go to high-resolution, zoom-able version.

A mosaic of pictures from people using the #ILookLikeAnEngineer hashtag on Twitter to combat sexist and racist assumptions in engineering. Image: Emma Pierson. Click to go to high-resolution, zoom-able version.

David Kung, math professor at St. Mary’s College of Maryland and director of Project NExT, gave an inspiring keynote address at the Legacy of RL Moore Conference in July. It’s a must-watch. I’ll wait.

In the talk, Kung calls on us to be honest about the current state of affairs, which is not good when it comes to representation of women and some minority groups in most STEM fields, and reminds us that math classes are sometimes gatekeepers for the rest of STEM. Our actions in the classroom can affect whether people become physicists, doctors, or engineers.

Kung’s talk is just one of the things I’ve seen recently about the intersection of math, teaching, and bias. Adriana Salerno of Bates College, in one of her last posts for PhD+Epsilon, writes about her desire for the classroom to be “structured in a way that empowers students or that makes them capable of resisting oppression and changing power structures.” Like her, I am tempted to think math is neutral and pure, not sullied by society’s prejudices, and I appreciate reading her ruminations on how to shake that idea off.

Both Kung and Salerno are critical of the way organizations that promote IBL use R. L. Moore in their branding. I was until recently unaware of his appalling treatment of black students. Raymond Johnson, the first African American to earn a degree from my grad school alma mater, Rice, writes about his undergraduate experience with Moore at UT:

Moore, his method and his work are highly thought of in the mathematical world. When he died, there was a laudatory article in the Math Monthly, a publication of the Mathematical Association of America. There is also a major MAA project on the legacy of R. L. Moore. The image of R. L. Moore in my eyes, however, is that of a mathematician who went to a topology lecture given by a student of R. H. Bing. Bing was a student of Moore. The speaker was what we refer to as Moore’s mathematical grandson. When Moore discovered that the student was black, he walked out of the lecture.

One person’s shortcomings in one area do not erase good things they do in another, but the pain and bitterness in Johnson’s writing made me think about the fact that attaching this person’s name to a pedagogy can be one of those little things that makes people feel unwelcome. Moore was not racist in some abstract way; he clearly and deliberately made it harder for some individuals to succeed than others, and those individuals have faces, names, and stories. They remember their treatment by this person others hold in such esteem.

Darryl Yong, mathematician at Harvey Mudd and the school’s associate dean for diversity, has re-launched his blog with a post about radical inclusivity. He writes, “My message to all educators: not attending to diversity and inclusion concerns in the classroom is the same as allowing your classroom to continue propagating the discrimination and bias that exists in our society. We have to actively combat discrimination and bias in our work as educators.”

None of us want to feel like we’re racist, sexist or anything else-ist, but we all have implicit biases. It hurts to be told or to admit to ourselves that we have these biases, but it hurts more to have your education and progress impeded as a result of these biases. These biases often don’t manifest themselves in overt racist or sexist actions, but small differences in how we treat different groups of students can accumulate into a force that pushes some students forward in math and some students out. And being a member of an underrepresented group does not mean you can’t be biased against that group. Last year, a story about sexism in hiring made a big splash. “John” got hired more often and with a higher salary than “Jennifer” did, even though they had the same resumés. One of the takeaways of the study was that both men and women on hiring committees were biased against Jennifer. That is a bit demoralizing, but it has a silver lining. No one needs to feel like they’re being singled out when we call on people to examine their biases. We can all fall prey to them, and we should all think about how we can do better.

If you’re reading this, I’m sure you would never tell a woman she shouldn’t be majoring in math because she’s a woman. (If you would, hi, nice to meet you, now stop saying that crap.) But you probably have some implicit biases that cause you to treat women a little differently from men and black people a little differently from white people. I don’t know how to stop doing that, but I honestly believe that being aware of our cognitive biases instead of soothing ourselves with the comfortable reassurance that we aren’t racist or sexist is a huge first step. Whether I am grading, having a class discussion, or writing a recommendation letter, I try to ask myself fairly frequently whether I would act or feel the same way if the student were a different gender or race. I hope that the small step of asking the question keeps me from unthinkingly slipping into biased behavior.

It’s all well and good for me to think about my actions in the classroom, but I also want my students to think about diversity and respect. Yong’s post about being welcoming on day 1 has gotten me thinking about how I should address the issue of diversity in my syllabus and on the first day of class. The fact is, in some ways this discussion is purely academic for me. I teach in Utah, and the demographics of my classroom reflect the (very white) demographics of the state. I can’t make my school more racially diverse, but in addition to striving to treat the few underrepresented minority students I do have equally, I can explicitly encourage my students to think about diversity and respect in my classroom and other aspects of their lives.

How are you promoting diversity and respect in your classroom?

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Math Fought The Law, And The Law Won

Photo courtesy of stock monkeys.com

Photo courtesy of stock monkeys.com

Math is full of laws: group laws in abstract algebra, the law of sines in trigonometry, and De Morgan’s law in set theory, to name a few. And occasionally, the law is full of math. That was the certainly the case in recent patent dispute at the London Court of Appeals, as covered by The Independent.

Here’s the TLDR: two drug companies were arguing over a patent. Company A has a patent for a solution containing between 1 and 25 percent of a certain compound. Now company B has manufactured a very similar solution, containing .95 percent of the compound in question. But everybody knows that .95<1 so company B is obviously in the clear, right?

Wrong. The judge eventually decided that any number larger than .5 is actually the same as 1, since we can round .5 up to 1, and apparently this judge has no love for non-integers.

My immediate reaction as a mathematician is that this could all have been avoided if Company A had just used interval and set builder notation. A quick recap in case it's been awhile since you've seen interval notation. There are two types of intervals, closed and open. The closed ones have square brackets, like [1,25], and the open ones have round brackets, like (1,25). The first contains all numbers between 1 and 25 including 1 and 25, and the second contains all numbers between 1 and 25 excluding 1 and 25.

The whole point of interval notation (in my mind) is that it takes away any and all possibility for ambiguity. If I say that my solution contains m percent of some compound, where m is in the interval [1,25], I truly mean that the smallest possible value for m is 1 and the largest value is 25. For example, the number 0.9999…9. (that’s just some long string of nines), which by any convention of rounding would round to 1, is still, itself, smaller than 1 and therefore not part of the interval [1,25]. Because of course, significant digits aside, you can round and truncate wherever you please. So to say that anything larger than .5 is really the same as 1 is a bit arbitrary, why not say anything larger than .49 or even .499, you get the idea.

So I guess the upshot is this: when making large business deals, use the most rigorous language possible to describe numbers, because you can’t count on some guy in a powdered wig to do it for you.


Correction: I initially said that .9999… repeating nines forever was less than 1, but as several apt commenters pointed out, if it really goes on forever forever, that’s just 1 — a true but unsettling controversial fact the internet loves to argue about! So let’s say it’s .999…9 for some really long but finite amount of nines, then we’re ok.

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Dimensions of Flavor

We talk a lot about visualizing mathematics, and we can even listen to it sometimes. But it can be hard to get the other senses involved, especially taste. Last year, I was delighted with Andrea Hawksley’s tasty and attractive Fibonacci Lemonade, which makes the Fibonacci numbers and golden ratio tastable. Her post about Fibonacci lemonade starts like this: “How would one make mathematical cuisine? Not just food that looks mathematical (like math cookies), but something that you truly have to eat and taste in order to experience its mathematical nature.”

A collection of points in beerspace is called a "flight." Image: Quinn, Dombrowski, via Flickr.

A collection of points in beerspace is called a “flight.” Image: Quinn, Dombrowski, via Flickr.

I recently ran across a similar idea from Nathan Yau at Flowing Data. “Data plus beer. Multivariate beer.” (By the way, if you don’t already follow Flowing Data, you probably want to rectify that immediately.) Fibonacci lemonade has two variables: lemon juice and sugar. Beer has a few more degrees of freedom in the types and amounts of grains, hops, and malt.

Many of Yau’s data visualizations involve maps and demographics, so it’s not a surprise that for his first foray into mathematical libations, he chose to make beer recipes based on statistics such as the ethnic makeup, population density, and education levels of different counties. In the end, he brewed batches that represented Aroostook, Maine; Arlington, Virginia; Bronx, New York; and Marin, California. He writes:

Here’s what I eventually settled on.
1. Population density translates to total amount of hops. The more people in a county, the hoppier the beer tastes.
2. Race percentages translate to the type of hops used. For example, a higher rate of white people means a higher percentage of the total hops (determined by population density) that are Cascade hops.
3. Percentage of people with at least a bachelor’s degree translates to amount of Carapils grain, which contributes to head retention.
4. Percentage of people with healthcare coverage translates to amount of rye, which adds a distinct spicy flavor.
5. Median household income translates to amount of Crystal malt, which adds body and some color.

Did it work? Yau didn’t run a randomized control trial, but he says the beers definitely tasted different, and he had some tasting notes notes relating to the population density, healthcare coverage, and median income of the counties the beers represent.

I am coming to this idea from the point of view of a mathematician rather than a data journalist, so something I love about the idea of multivariate beer, Hanna Kang-Brown’s census spices, and other data gastronomification, as Tom Levine calls it, is that it is a natural way to explore the idea of dimension without going the Flatland route. (Flatland is great, don’t get me wrong, but it’s good to have extra tools at our fingertips.) It seems that most practitioners are interested in the way such concoctions can help people understand real-world data, but I like the potential for use in the strictly mathematical realm. Who knows? Flavors that represent shapes or polytopes? Could you taste the prime factorization of a number?

Yau says he is out of the multivariate brewing game, but if anyone is interested in doing an experiment in mathematical flavor, I’m a willing and able taste tester.

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In Praise of Teamwork

Left to Right: John Jones, Rachel Davis, and Christelle Vincent work on the LMFDB.

Left to Right: John Jones, Rachel Davis, and Christelle Vincent work on the LMFDB.

Part of what makes math blogging so interesting is that it helps to build connections between the people creating math and those consuming math. The evolution in math blogging and blossoming of math on twitter has done a great deal to dispel the crazy myth of math as a solitary pursuit, or worse yet, of mathematicians as weirdo loners. Mathematicians, just like other scientists (or humans for that matter), like to work together.

This sort of working-togetherness and community mathematics can come in many shapes: collaborative research, math blogging, and open source software initiatives, to name a few. I was first inspired to think about this by a wonderful portrait of Terrence Tao in The New York Times this week, calling attention to some of Tao’s exceptional work in collaborative math and mathemematical outreach.

But then those feelings were further amplified when this week found me at the LMFDB workshop in Corvallis, Oregon where I am sitting directly at the heart-center of an incredibly cool community math project. So, being here as I am on the front lines, I wanted to share a bit about the process. I’ve written about the LMFDB before, but to recap, it’s an online database of L-functions and “friends.” The database is open source, edited mostly through github and the kind and selfless hearts of so many contributors.

So here we are, 33 mathematicians, 33 laptops, 7 days and an unlimited desire to classify and sort things. On the first day, David Farmer, one of the LMFDB founding fathers asked that we begin by sorting ourselves according to what we felt we could contribute. He then recommended that we start by “pair programming.” Yes, this is when you sit next to someone and write code together on one laptop. Farmer said, “you might think this would cut productivity in half, but on the contrary, it doubles it since fewer errors are made.” So you see: teamwork.

This is how we spend our days, small groups clustered around tables pair programming whatever pieces of the LMFDB has sparked our interest. Some people are adding new sections to the database, perhaps a whole new wing dedicated to modular forms of half integral weight. Other people are working on the exposition of the database, writing concise descriptions of the objects for non-experts nested in knowls on the page. Some people are skimming the database for typos and html errors to make the whole thing more good-looking — seriously, nobody wants to get the L-functions from an ugly website, right?

After spending the entire day making changes and building new things and all the while programming in pairs, we have an end-of-day report. This is when the collaboration really kicks off. Each small group gives a brief recap of what they’ve done, and it is submitted to the jury of 33 for approval. Everything that goes into this database has passed before the community and been the subject of some intense and thoughtful scrutiny, from small changes (like, maybe these query boxes should be left aligned?) to huge ones (like, do you think we should change our entire labeling scheme?) gets a full-blown conversation. So you see: community.

I think the LMFDB project is an interesting example of extreme community collaboration in mathematics, but it is certainly not unique; this sort of community exists around lots of open source software initiatives. And of course this type of intense collaboration can also exist around the good old fashioned doing of mathematics.

Ok, I know what you’re thinking, “why are you telling me all this on a blog about math blogs?” Because, dear reader, I think this is the point of it all. Blogging abut math, and blogging about math blogs, or even blogging about blogging about math blogs (like I’m doing right this second), is all about brining the community together. So while writing about the LMFDB conference is not directly writing about a math blog, it feels like a friend of math blogging. And I think it’s important to remember why we blog about math.

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PCMI Blog Roundup

Earlier this month, I had the opportunity to give a cross-program talk at PCMI, the Park City Mathematics Institute. I talked about how doing math online can help us reach others in the math community, building bridges between teachers, researchers, and recreational math enthusiasts and reach those who think math isn’t for them. I talked about both social media and blogs as places where online math happens and some suggestions for how to talk about math so other people will listen.

PCMI'ers march in the Park City July 4th (Dimension) Parade. Image: Wendy Menard.

PCMI’ers march in the Park City July 4th (Dimension) Parade. Image: Wendy Menard.

PCMI is really several programs in one. There is a research program with a graduate summer school, a secondary school teachers program, a program for faculty at undergraduate teaching-focused institutions, and an undergraduate summer school. (There might be more, but I think those are the groups I’ve encountered. It is large. It contains multitudes.) When I attended in the past, the instinct was for people to associate with others in their group, but some gentle nudges towards cross-program socialization led to some interesting and fruitful conversations. I think it’s easy to have tunnel vision as a participant in any of the programs, so those nudges really help people make connections and think about the broader math community they belong to. Incidentally, these are two of the things I find most gratifying about doing math online. Programs like PCMI are rare and short; the Internet, for better or worse, is always there. I’d never cross paths with math teachers who live thousands of miles away if I didn’t do it online.

With all that online math talk, I thought it would be nice to share some of the blogs written by this year’s PCMI participants. It turns out the teacher program is one step ahead of me: they keep a list of participants’ blogs here. So I’ll just include a few blogs by the people I met either in person or online during my brief trip up into the mountains.

I was happy to see that my AMS blogging pal Adriana Salerno, who writes PhD Plus Epsilon, was at PCMI this year. She was even kind enough to include my talk in her week one roundup post. While I didn’t get to meet him, I also heard about Dagan Karp, AMS blogger, Harvey Mudd math professor, and leader of the Undergraduate Faculty Program at PCMI.

I got to meet one of my online math pals, Ashli, known to me on Twitter as @mythagon and the author of Learning to Fold. I had lunch with her, Wendy Menard, who writes Her Mathness, and Dylan Kane of Five Twelve Thirteen. Both Menard and Kane blogged about the PCMI teacher program this year. Kane posted about a short talk he gave on the #MTBoS (math twitter blog-o-sphere), a cool online conglomeration of people blogging and tweeting about math and teaching. Later, I met Anne Paoletti Bayna on Twitter, where she shared her math Tumblr, paomaths. Menard and Bayna both blogged/tumbl’d about making conic sections with piles of salt. I’ve never done that before, but there’s a PCMI page about it (pdf). I’ll have to try it.

I really wish I had been able to stay and talk more with other math bloggers at PCMI (and of course see Henry Segerman’s talk on 3D printed geometry the day after mine). If I missed any PCMI blogs, please leave a note in the comments or find me on Twitter. I’d love to connect online, even if we didn’t get to meet face-to-face.

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Talking ‘Art of Mathematics’ With Its Creators

Left to Right: Volker Ecke, Anna Haensch and Christine von Renesse.

Left to Right: Volker Ecke, Anna Haensch and Christine von Renesse.

This week I was at the Inquiry Based Learning (IBL) Workshop in San Luis Obispo and I had the rare and wonderful occasion to sit down for dinner with a great team of bloggers and get to know them a bit better. That team was Volker Ecke and Christine von Renesse, two of the masterminds behind Discovering the Art of Mathematics (DAoM), an enormous blog, video, and teacher resource network for inquiry based teaching.

Since not everyone just came form an IBL workshop, I’ll give a quick’n’ dirty rundown of what it entails. The general idea is that an IBL classroom is genuinely student-centered, that is to say, the students do almost all of the speaking during class time. Moreover, the students determine the pace and content of each lecture, by uncovering course material through inquiry. This means that eventually, students have the sensation (and ideally the reality!) that they have discovered the entire curriculum on their own. That’s my take-away, but for a better description, check out this from DAoM.

Often, educators reach a point in their evolution where they realize that the traditional lecture format is not getting them the results they desire. For Ecke this realization came early in his career, “When I realized that lecture wasn’t going to work, I thought, well, how else can I structure my class? And IBL was just that!” For von Renesse, who grew up in Germany, teaching in this style came naturally. “I was dying to teach the I was used to in Germany, so I knew I wanted to teach IBL from my background,” she says.

So Ecke and von Renesse, both professors at Westfield State University, teamed up with fellow faculty members Julian Fleron and Phil Hotchkiss to revamp their existing courses at WSU, and in the process created Discovering the Art of Mathematics.

One of the greatest challenges of IBL, say Ecke and von Renesse, is asking the right questions. “How do you make someone discover what they need to discover,” von Renesse says, “it has a lot to do with personality stuff.” The right questions need to be asked of the right student, Ecke says, “A good question is asking `How do I support a student in their struggle.'” The team offers a traveling workshop in which they coach instructors in good question asking. But if you’re curious to see what it might look like, you can watch Ecke guide students in doing the impossible — solving the Rubiks cube.

Of course it’s also important that students ask plenty of questions. Von Renesse addresses this in the blog post “Curiosity — A Culture of Asking Questions.” She discusses the gentle balance that naturally exists between relaxation, curiosity, and anxiety, and how to maintain that in the classroom. The DAoM video library has plenty of great examples of this, but watching von Renesse wrangle directional derivatives with a group of Calc III students is a beautiful example.

It was such a treat to sit down with these bloggers and learn more about their project and their passion for teaching and learning. I’ve only mentioned a small fraction of what you can find on DAoM, there are also e-books, assessment tools, content ideas, and the list goes on. If you’re planning to dip your toe into IBL this fall, I recommend you stop by for some inspiration!

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Specifications Grading Redux

An alternative grading system. Image: Sage Ross, via Flickr, h/t Robert Talbert.

An alternative grading system. Image: Sage Ross, via Flickr, h/t Robert Talbert.

Last December, I wrote about specifications grading, an idea I first saw on Robert Talbert’s blog Casting out Nines (Co9s is ending, so you can find new posts at rtalbert.org) and wanted to try out in my class. Talbert has blogged about his experiences using the system a few times; you can read some of his specs grading posts here and the rest here. You can also listen to a Teaching in Higher Ed podcast of Nilson discussing the system. Now that the semester’s over, I wanted to let you know how it went for me as well.

First, a word about the book, Specifications Grading by Linda Nilson. I decided to read it rather than just base my grading scheme on Talbert’s and others’ blog posts about it. I hoped that it would give me some nuts and bolts advice for how to make it work in my class. It did, but I almost put the book down after reading this on page six:

What the faculty reap for their endless hours of grading are more grading protests and conflicts with students than ever before. Of course, the reasons behind this student behavior lie largely in the values and beliefs of the Millennial generation and their parents, such as their consumer attitude toward higher education, their distaste for academic learning and the life of the mind, their alienation from standard teaching methods, and their sense of entitlement to high grades in light of high tuition costs.

I am a Millennial, albeit on the old side, and this broad generalization about my generation rubbed me the wrong way. Aside from the fact that it offended me personally, I had serious qualms about taking classroom management advice from someone with such a negative, adversarial view of students and other people my age, especially someone who would say we have a “distaste for the life of the mind.” I do see some amount of entitlement and a consumer attitude, but I have had almost no students who were not interested in learning something. I’m also not at all convinced that Millennials have these traits to a larger extent than members of any other generations. I mention all this here because I hope that Nilson and other educators with this viewpoint will reconsider the way they think of their students (and some of their colleagues). 

That said, I do think the book has good advice overall, and I like the idea that employing specs grading will make assessment a less adversarial process. I used specifications grading for my math history class this past semester, and it was a mixed success.

Some background on my class: this course gives students a writing credit, so the majority of assessment was based on written work. I had 43 students registered on the first day of class, a slight increase over last semester, and my attrition rate was significantly lower; by the end of the course, I had about 40% more papers to grade than I did last semester. Some of the problems I had implementing my grading scheme and with the semester in general were related to the unexpectedly large size of the class. A writing class with 40 students in it is, at least for me, nearly untenable.

Another complication not related to specifications grading was that I had students turn in most of their work on the course website powered by Canvas and used the grading tool there for the first time. I had two problems: Canvas inexplicably lacks a way to save feedback while you’re writing it but before you want to give it to your student, and it logs you out without saving after a certain amount of time. A few times, I was grading during office hours and had a student visit. By the time I was done with the student, the feedback I was working on was gone. The other problem was that some students could not access the feedback on Canvas. This seemed to be a browser issue on their end, not something I could control, but it threw a serious wrench into getting feedback to them in a timely fashion, especially because they sometimes didn’t realize there was supposed to be more feedback.

I used a blended grading scheme, which Nilson calls a synthetic option in the book. The two major projects (a group project and a research paper) were graded using a fairly traditional writing rubric that produced numerical grades. The average of those two grades determined the final grade with additional requirements for each letter grade. These included a certain number of math homework assignments and blog posts that were graded complete or incomplete, as well as turning in paper and project proposals and a paper draft on time.

I allowed resubmissions of one of the homework assignments because a number of students had more trouble than expected on it, but the rest of the homework assignments were one-shot deals. Students could revise blog posts until they were passing. I tried to have a rule that the next revision had to be finished within a week of my giving feedback on the last revision, but I had trouble enforcing that deadline (see below).

I used Nilson’s suggestion of tokens/free passes to turn in assignments late in the hopes that it would cut down on students begging for extensions. I gave each student a total of three for the semester, and there was no bonus for unused tokens. For most assignments, a pass was a one-class extension, and for blog posts, it was a one-week extension. In general, I think the passes helped, but they didn’t eliminate extension discussions altogether. I don’t know if I just didn’t give students enough free passes or if they should have been able to earn more or if I just need to be less of a pushover.

Because I required assignments to be turned in by a certain date for students to get a certain grade in the course, some assignments became high-stakes. If a blog post wasn’t on time, a student couldn’t get an A; if the rough draft of the final paper wasn’t on time, they couldn’t get a B. I am not used to classes like this where a student can’t do more work to make up for a bad week later, and for that reason I sometimes had trouble enforcing deadlines.

I had trouble giving homework assignments an incomplete grade when they weren’t quite up to the level I wanted but reflected a significant amount of effort. Perhaps next time I could borrow Talbert’s idea of Mastery-Progressing-Novice and allow progressing students to resubmit the work. In my case, I would probably use different labels. Something like Passing-Effort-No effort would better reflect the fact that I want students who try the homework in good faith to get a chance to learn from their mistakes.

In reflecting on the deadline and homework complete/incomplete problems I had, I wonder if part of what made them difficult for me is that in traditionally graded courses, you don’t immediately see the consequences of your actions in your final grade. In reality, people who didn’t turn in the first blog post in last semester’s class were unlikely to get an A, but they didn’t immediately know this. Likewise, people who consistently turned in late assignments or homework that was not high quality would get lower final grades, but the effect didn’t happen immediately.

One of the benefits of specs grading that Nilson mentions is that students can to some extent choose what grade they want. Most of my students were shooting for an A, but some only wanted a B or C and chose the assignments they did accordingly. In the future, I like the idea of having students signal to me what grade they would like in the class so I won’t worry about the students who don’t turn in some early assignments.

On the other hand, sometimes a student did all the complete/incomplete work to get a grade that was higher than the grade their paper and project earned; they did all the assignments required to get an A, but their major assignments were B material. Some of these students felt like the rules had been changed on them, and I wish they hadn’t felt that way. One unfortunate aspect of the schedule of the course was that the traditionally graded work came towards the end of the semester, so students were sometimes unsure of where they stood.

I had students turn in rough drafts of their papers about two weeks before the papers were due. I gave a lot of written feedback on the papers, but I did not return them with a rubric that showed what grade they would have gotten if they had been turned in as final versions. I am sure this would have helped my students, and I will try to do that if I teach a course like this again. I also think Talbert is right on the money when he says that giving students more examples of passing and non-passing work would be helpful.

Another overarching problem was that I don’t think students really bought into the specs grading idea. If I use it again, I might spend some time in class or in assigned reading explicitly talking about how the system works and the advantages it has over traditional points grading. I had a syllabus quiz for my students at the beginning of the semester to make sure they understood how the grading scheme worked, but I think I could have made assessment feel more like a partnership if I had let them know exactly why I thought specs grading would be better for both them and me.

I was curious about how specs grading would affect the final grades in my class. I think they were a bit higher than last semester. Most of this I attribute to students (and myself) having a clearer idea of what work needed to be turned in at what time in order to get the grade they wanted, so there were fewer students who thought they could scramble at the end to make things up. For example, they couldn’t get an A or B without turning in a rough draft of the final paper on time. I required a rough draft last semester as well, but I deducted points from the assignment if they didn’t have one. This semester, I got rough drafts from more students, and I think that improved their final papers substantially. I was somewhat torn by the rough draft requirement; it hurt two students who wrote good papers but hadn’t turned in a draft, but I think it helped more students than it hurt.

If you’ve used specs grading in a math class, I’d be curious to know how it went for you.

Posted in Issues in Higher Education, Math Education | Tagged , , , | 8 Comments

An Ugly Song That Definitely Won’t Get Stuck In Your Head. I Promise.

If you’ve ever had the hook from Call Me Maybe stuck in your head for three days, you know the power of a well-patterend musical hook. It’s these patterns in music that speak to our human brains and often help us differentiate good music from bad, and beautiful from ugly. Patterns, of course, suggest that math is at play, and this is no surprise. The connection between math and music has been well-studied from the geometric ideas of Pythagoras, the symmetries in music, and even music as topology.

What I want to talk about is how to use math to make the ugliest music possible. If patterns register as “beautiful” in our minds, then a completely pattern free tune would be the least beautiful. But to actually compose a totally pattern free piece of music is not an easy feat. Remember, pattern free is distinctly different from random (the latter being relatively easy). An interesting solution to this problem came to us by way of 1950’s work of engineers John P. Costas and Solomon W. Golomb. Costas and Golomb weren’t trying to write ugly music, rather, they were trying to solve a problem in sonar signaling. Turns out the same principles that make an ugly sound also make a great sonar ping. The solution was a combination of the Golomb Ruler, or its multi-dimensional cousin, the Costas array. These tools give a method for generating points that are not random (obviously), but are completely pattern free.

Using these tools, the mathematician Scott Rickard engineered the ugliest song every written. The basic idea is quite clever. Beginning with an 88 by 88 grid, starting in the far left column, he moves along the columns filling in boxes in rows corresponding to powers of 3. So in each column n he marks the 3n-th block. And when the numbers get too big, he reduces them modulo 88. This gives a totally pattern free sequence of points, and hey, also a handy representation of notes on the 88 key piano! He debuts this hideous (yet mathematically clever) opus in TEDx Talk: The Worlds Ugliest Music. You probably won’t be humming the riffs all day, but it’s cool to hear.

The AMS has a great link-roudup of other blogs and videos about math and music, and since it’s still the month of June, find something beautiful to enjoy and have a happy Fête de la Musique!

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Getting Warmer…

I’m currently teaching a summer school for high school students. Our main focus is number theory and its applications to cryptology, but I like to start each morning with some kind of warm-up math puzzle or game. I know plenty of fun math stuff, but I’ve never worked with high schoolers before, so I took to the blogs to find some good activities for that age group. The Internet has almost as many puzzles and games as it has cat pictures, so the sites I’m highlighting here are just the tip of the iceberg.

An element of {cats}∩{math}. Thanks, Internet! Image: Jimmie, via Flickr.

For the first day, I wanted an activity that would get the kids working together a bit and introducing themselves to each other. A bit of searching, I came upon Dan Meyer’spersonality coordinates” activity. Meyer write the must-read math teacher blog dy/dan (which, by the way, I can’t decide how to pronounce, but I suppose that’s the point). His activity had students in a group label themselves on a coordinate axis by how much of two different traits they had. I didn’t use that activity but one I found in the comments: break people up into groups of size 2n and have them come up with yes-or-no questions so that each person in the group has a different set of answers. I only did it with groups of four students, and I had students mix up a couple of times to meet new people and come up with different traits. The next time I use this activity, I will probably ask them to get into groups of eight after playing once or twice in groups of four.

Sam Shah’s blog Continuous Everywhere but Differentiable Nowhere has some nice problems and puzzles, including the sack problem that nerdsniped me a while ago. Math Munch, “a weekly digest of the mathematical internet,” has also been featured on this blog before. It doesn’t just focus on puzzles and games. There’s a strong art component as well, and the curators usually include some web-based interactive activities. Periodically they run interviews with mathematicians, teachers, and artists. I especially enjoyed the Q&A with Carolyn Yackel, who just sounded so enthusiastic about abstract algebra that I wanted to go find some symmetry groups.

A new-to-me blog that’s been a good puzzle source is Math=Love by high school math teacher Sarah Hagan. Don’t tell my students, but I think I’ll be using the 1-4-5 square puzzle challenge next week, and I might talk about happy numbers at some point. Very helpfully, Hagan often includes logistical information about how she made the puzzles or games work in the classroom and ideas to make them go more smoothly in the future. She also shares links to other sites with math games and puzzles Now that I’ve been reading the blog for a few months, I’m a bit embarrassed that I didn’t start reading it earlier. Hagan is very well known in the math teacher blogging world. Aside from the puzzles and games, she shares a lot of helpful tips about running the classroom and reflections on her teaching practices.

Futility Closet isn’t strictly a math blog, but it has tons of fun puzzles. The jeweler’s observation caught my eye recently. Why must every convex polyhedron have at least two faces with the same number of sides? It’s a simple question with a short, clever answer, but I think students will have fun trying to figure it out.

I’ve found some activities in other places as well. Last Thursday, we made a level one Menger sponge using leftover supplies from MegaMenger in October. The students had heard about fractals from a guest speaker earlier in the week, so we talked a little more about how something could have a non-integer dimension and figured out the fractal dimensions of the Cantor set and the Menger sponge. The seven penny game from The Proof and the Pudding by Jim Henle was fun (my review of the book is here), and I’ll be using Matt Parker’s Things to Make and Do in the Fourth Dimension later in the program (see my review here). I might even try to make a domino circuit (pdf), but I’m not sure if I have enough patience or dominoes.

Do you have a favorite source for math puzzles, games, or activities?

Posted in K-12 Mathematics, Math Education, Recreational Mathematics | Tagged , , , , , , | 1 Comment