On Teaching Analysis

Posted on March 5, 2014 by evelynjlamb

Timothy Gowers, University of Cambridge mathematician and Fields Medalist, is teaching an analysis class this term, and fortunately for me, he’s blogging about it. Analysis IA is part of the first-year math major sequence at the University of Cambridge, and it is a rigorous approach to calculus at the undergraduate level. I am teaching a similar analysis class this semester, and although Gowers says that his posts are for his students, they’ve been useful for me as well. I have taught this class before, but it’s always good to see how someone with much more experience than I have approaches the subject.

In his first post about the class, Gowers gives a big picture overview of the course:

“One of the messages I want to get across is that in a sense the entire course is built on one axiom, namely the least upper bound axiom for the real numbers. I don’t really mean that, but it would be correct to say that it is built on one new axiom, together with other properties of the real numbers that you are so familiar with that you hardly give them a second’s thought.

If I want to say that more precisely, then I will say that the course is built on the following assumption: there is, up to isomorphism, exactly one complete ordered field. If the phrase ‘complete ordered field’ is unfamiliar to you, it doesn’t matter, though I will try to explain what it means in a moment. Roughly speaking, this assumption is saying that there is exactly one mathematical structure that has all the arithmetical and order properties that you would expect of the real numbers, and also satisfies the least upper bound axiom. And that structure is the one we call the real numbers.”

I also like his section about the difference between abstract and concrete in mathematics. The emphasis is mine.

“Up to now, you will have been used to thinking of the real numbers as infinite decimals. In other words, the real number system is just out there, an object that you look at and prove things about. But at university level one takes the abstract approach. We start with a set of properties (the properties of ordered fields, together with the least upper bound axiom) and use those to deduce everything else. It’s important to understand that this is what is going on, or else you will be confused when your lecturers spend time proving things that appear to be completely obvious, such as that the sequence 1/n converges to 0. Isn’t that obvious? Well, yes it is if you think of a real number as one of those things with a decimal expansion. But it takes quite a lot of work to prove, using just the properties of a complete ordered field, that every real number has a decimal expansion, and rather than rely on all that work it is much easier to prove directly that 1/n converges to 0.”

I sometimes struggle to articulate the big picture to my students effectively, and Gowers is great at making that broad vision clear.

The rest of the posts also have some similar gems. From How to work out proofs in Analysis I:

“For some reason, Analysis I contains a number of proofs that experienced mathematicians find easy but many beginners find very hard. I want to try in this post to explain why the experienced mathematicians are right: in a rather precise sense many of these proofs really are easy, in the sense that if you just repeatedly do the obvious thing you will solve them. Others are mostly like that, with perhaps one smallish idea needed when the obvious steps run out. And even the hardest ones have easy parts to them.”

This post also talks about some of the common “moves” we use when we do basic analysis proofs in the context of teaching a computer to do them.

Gowers mentions that his colleague Vicky Neale wrote blog posts after each analysis class when she taught it last year. I have not perused them yet, but I look forward to getting some ideas from them. In the past, I have also found Terry Tao’s blog helpful for understanding and teaching analysis, particularly the measure theory notes.

I love teaching analysis, and I’m very glad that I get to benefit from Gowers’ (and Neale’s and Tao’s) experience, especially when I’m trying to explain how individual theorems fit into the subject as a whole.

Posted in Math Education | Tagged , , , , | 4 Comments

My Top Ten Issues in Mathematics Education

Posted on February 25, 2014 by bfinegold

What are your “top ten”?  Sue VanHattum, a full time faculty member at Contra Costa College inspired me with her wonderful top ten list .

Below are my top ten Issues in Mathematics Education.  While this is my opinion, I do highly encourage you to check our Ms. VanHattum’s post as well as her blog Math Mamma writes…

10) Math IS, by its very nature, FUN!  A coworker of mine told me “I am never bored”.   He did not mean to say that he was constantly entertained. The cop-out statement  “Math is boring” or the equally ridiculous “Let’s MAKE math fun!” betray a general trend in society to dismiss what we don’t understand as simply being unappealing.  I was tempted to write about this earlier this year when the NY Times article “Who Says Math Has to be Boring?” and response piece in Slate “Math Has to Be at Least a Little Boring“came out, but I didn’t want to bore my readership.

9) Discovering and uncovering content should take precedence over covering and recovering content.  A quote from RL Moore “He who is taught the least learns the most.”  Or from Paul Halmos “The best way to learn is do; the worst way to teach is talk.”  Also see a great recent post from Grant Wiggins concerning when/if lecturing is an effective a teaching technique.

If you know you want to shut up more, but you have trouble (like me) not filling the dead space with your own voice sometimes, try Bob Kaplan’s advice for becoming invisible .  Read about how to help students become “productively stuck” at Math For Love.  Or for more information on Inquiry Based Learning, check the IBL blog. Lastly, I’ll share what I told an administrator who told me to “cover” more material while I was teaching High School.  I pointed to my desk (which was COVERED in student work) and I told him that the textbook was under there already!

8) Underrepresented groups in mathematics will remain underrepresented (especially in academia) unless measures are taken to recruit and retain them.  I am familiar with programs like EDGE, SK Days, Women in Number Theory, MSRI Connections for Women, etc, exist, but many women don’t know about them or aren’t actively encouraged by their departments to get involved in the mathematical community.   We can bring new perspectives into our field by providing role models for those who are traditionally underrepresented in our field, making the academic workplace more family friendly, and by breaking down stereotypes.  See Adriana Salerno’s recent post in PhD+Epsilon on the subtle ways in which women can be discriminated against.

7) Mathematics Educators deserve respect and more autonomy.  Without the freedom to teach as they see fit, educators cannot be experimental and take risks in their approaches.  Departments and school systems should reward creative teaching styles by having regular teaching observations of junior faculty by qualified individuals who can supply meaningful feedback.  These observations should be formal, made regularly by a small group of individuals, and play a greater role in advancement than test scores and/or student reviews.  As a postdoc, I would have loved more observations of my classroom.  If some courses need to have uniformity in curriculum, the instructors should be given a concise outline (such as the Common Core) of ideas to be studied.

6) Mathematics Educators deserve opportunities to further their own content knowledge for teaching.  Opportunities for ongoing professional development that truly connects research in education to implementation in the classroom are scant.  Both university and K-12 teachers tend to model their teaching after what they experienced as students regardless of whether it was truly effective.  Having reading groups on math ed papers is an activity done at some universities like the University of Arizona.

5) Mathematics (not the instructor) IS the Authority – this one is stolen directly from Ms. VanHattum’s list.  Part of the beauty of mathematics is that `proof by intimidation’ is not a valid method of proof.

4) All students can be trusted to learn mathematics, there is no Math Gene, and math courses should NOT be mandatory.  This does not mean that teachers should stop trying to inspire and excite students, but it does bear repeating that everyone can do math.  Many teachers unknowingly perpetuate the Math Gene myth by saying things like “Well, you COULD subtract ‘x’ from both sides… but that wouldn’t be very smart now would it?”.  Along this line of reasoning, we should trust students to take relevant courses. They are grown-ups and eventually they will figure out (perhaps with some advice) what skills they need to succeed in their area of interest.  I’m sure many people will disagree with this last point.

3) Students don’t realize that Math is fiddling around, turning your drawing upside down, looking at it through the paper, being befuddled, being sure that you are a genius….being sure that you are a moron, waking up and realizing that you didn’t actually prove the Beal Conjecture in your sleep, waking up and realizing that that lemma you thought was wrong in your thesis is actually right!  In other words, mathematics is joyful and unexpected.  See Math With Bad Drawings and Math Ed Matters “Be Predictably Unpredicatble”.  Share your own mathematical learning experiences with students.

2) Students don’t know that Math comes in many flavors. It’s hard to stay abreast of all the developments in your own field, much less others, but being curious is leading by example.  If you go to a colloquium that isn’t in your area it may pay off.  I’m often surprised at the number of ways there are to look at one problem like linear regression: as a machine learning problem where the data is a training set, as a geometric problem to be solved using singular value decomposition, as a parameter estimation problem involving Fisher’s matrix, as a classical minimization problem. Help students explore one toy problem from many perspectives.

1) Teachers should ask deep questions about basic ideas and be ready for questions or answers that might be better or different from what was anticipated. Sometimes when I go where students take me, it doesn’t align with my initial ideas.  This is both vexing and exciting!  It requires careful thought about definitions, purpose, and motivation behind concepts, not just examples and theorems.  Anyway, in the blog Math for Love, Dan Finkel talks about the rewards in pursuing a student-proposed problem:“A dollar that cost a dollar”.  I imagine that the ultimate goal in being an adviser is to have a PhD student who was “depending on you” become one who just totally blows you away with his/her conjectures and proofs. Of course, this has the potential for being simultaneously invigorating (“Yay!  I’m an awesome mentor”) and depressing (“Was I ever that creative?”).  So I put this at number one because I think that it is the item on this list that is potentially the most challenging and under-appreciated.

Wow!  Writing a top ten list is hard.

 

Posted in Issues in Higher Education, K-12 Mathematics, Math Education, people in math, women in math | 9 Comments

Why Should We Fund Math Research?

Posted on February 19, 2014 by evelynjlamb
Money. How do we convince people that math should get some? Image: Nick Eres, via flickr.

Money. How do we convince people that math should get some? Image: Nick Eres, via flickr.

As my co-blogger Brie Finegold mentioned last month, Cathy O’Neil of mathbabe.org has been writing about how MOOCs might change the face of math departments and, ultimately, how math research gets funded. O’Neil is concerned that without calculus classes to teach, math research funding could dry up unless we do a better job convincing the public and funding agencies that it is important. She writes,

“I’d like to argue for math research as a public good which deserves to be publicly funded. But although I’m sure that we need to make that case, the more I think about it the less sure I am how to make that case. I’d like your help.”

She gives the reasons she has come up with:

“1) Continuing math research is important because incredibly useful concepts like cryptography and calculus and image and signal processing have and continue to come from mathematics and are helping people solve real-world problems….
2) Continuing math research is important because it is beautiful. It is an art form, and more than that, an ancient and collaborative art form, performed by an entire community. Seen in this light it is one of the crowning achievements of our civilization….
3) Continuing math research is important because it trains people to think abstractly and to have a skeptical mindset….”

But she explains why she doesn’t find any of these arguments very compelling, and asks us to help her make the argument for funding math, even when mathematicians are no longer needed to teach calculus to future engineers. The comments section has quite a few interesting conversations about why and how to fund math, and O’Neil’s post has spawned at least one other blog post, by DJ Bruce.

Of course, as an early-career mathematician, the continued funding of my profession is important to me! I am not as convinced as she is that MOOCs threaten my job, but I think it’s good to think about how to make the case for math funding. I study math basically because of reason 2, but like O’Neil, I don’t think that alone makes a terribly compelling argument for funding math at the level of other sciences. I do think the combination of the aesthetic/creative and the practical makes mathematics very special, and I think that without mathematics, our search for truth in other sciences would be more difficult. I’m not sure the best way to sell the idea to the public and funding agencies, though. Can you do better? If you’d like to leave a comment on O’Neil’s post, you can do so here.

On a lighter note, Shecky Riemann recently posted an interview with O’Neil on his blog (cross-posted on her blog). My favorite line, about her daily blogging routine, was “…getting my daily blog on is kind of like having an awesome poop.” And that’s why I love reading Cathy O’Neil. She asks hard, provocative questions, and then she makes you think about poop.

Posted in Applied Math, Issues in Higher Education, people in math, Theoretical Mathematics | Tagged , , , , , | 1 Comment

Mistakes Are Interesting

Posted on February 11, 2014 by evelynjlamb

I just finished grading my first midterms of the semester, and I’m learning a lot about how my students think through the mistakes they made. (With apologies to Tolstoy, I’m definitely experiencing a bit of “correct solutions are all alike; every incorrect one is incorrect in its own way.”)

Find x. Image: Aaron Rotenberg, via Wikimedia Commons.

Find x. Image: Aaron Rotenberg, via Wikimedia Commons.

Last semester, the most frustrating (at least to me) mistake my students made on their first midterm was saying that if a set was open, then it wasn’t closed, and vice versa. They sometimes even came to the conclusion that Rd was neither open nor closed because it was both open and closed! That mistake taught me a lot about how language was influencing my students’ understanding of mathematical definitions, and I wrote about it last September on my other blog. This semester, my students largely avoided that mistake (maybe they read my post about it??), but they have been making other mistakes that I did not expect.

Michael Pershan is a math educator who collects and shares interesting Math Mistakes so he and other teachers can try to figure out what their students are thinking. One recent post came about because a lot of his students were writing things like three and a half fourths, instead of simplifying fractions the normal way. I also enjoy his continuing crusade to get people excited about exponent mistakes.

Pershan has the mistakes tagged based on where they fit into the common core standards, which is probably a helpful way for math teachers to see some real examples of mistakes their students might make, along with some possible reasons why. If you have an interesting mistake to share (elementary through high school level math), send it his way.

Mistakes can help us understand human thinking, but they can also show us how human thinking differs from the way computers calculate. Over at the aperiodical, Christian Perfect and David Cushing noticed a mistake in a Wolfram Alpha regression that can help us understand what computers are doing when they compute. And Patrick Honner had an interesting discussion with his class when the free online graphing calculator Desmos didn’t handle a removable discontinuity very well.

Ideally, I would not be learning so much about my students’ thinking on tests via their mistakes. It would be nice to be able to diagnose misunderstandings earlier. My weekly student problem sessions, group work in class, and one-on-one talks with students in my office give me a glimpse into their thinking, but I still don’t catch everything I’d like to before test time.

My teaching is of course a work in progress, and I am trying to figure out better ways to structure my classes and conduct assessments of student learning. The idea of using standards-based grading intrigues me, but I’m not quite ready to take the plunge. Joshua Bowman, who blogs at thalestriangles, has some reflections on what worked for him in standards-based grading. I’ve also been reading the standards-based grading posts at Bret Benesh’s and Kate Owen’s blogs. All of these blogs have given me a lot to think about as I reflect on how I want to organize my classes.

Posted in K-12 Mathematics, Math Education | Tagged , , , , , , , , | 3 Comments

How to get your friend to like math: A multipronged approach

Posted on February 5, 2014 by bfinegold

In Math with Bad Drawings, the author Ben Orlin calls the query in my title the most adorable ever, and I have to agree.  Now math is so awesome that it’s hard to believe that we actually have to develop any sort of strategy to entice our otherwise amazing friends (who don’t seem interested) to enjoy it.  Post-school, when it is no longer compulsory, we really do have a better chance at changing perceptions. So let’s think of some ways of luring friends ever closer to your world.

Introduce them to the concept of mathematical taste.  In other words, if you don’t like a certain genre of music, and that’s the only one that your music teacher ever played, then of course you would dislike “music”.  It’s never too late to develop a taste for mathematics, a hunger even, as described by Caroline Herschel in this poem, who at 31 started learning math from her brother only to become an astronomer.

Have them read Mandy Brown’s post on the pastry box, a forum in which 30 people who do interesting tech-related activities blog about themselves.  Mandy thought she would always be a language person, and not a math person… until she ended up majoring in physics!

Show them some amazing videos from George Hart at the Simon’s Foundation Site.  His most recent, posted just a few days ago is about Permutahedrons and Change Ringing (ringing church bells in beautiful patterns).

Force them to do something like make a Mobius band and cut it in half (always a conversation starter and crown pleaser).  Maybe if they are having fun, they’ll want to explore some recreational mathematics.  Here’s a brand new Recreational Mathematics Magazine.

Point out that it’s free!  One last thing that has always attracted me to mathematics from a philosophical point of view is its egalitarian nature.  In a world filled with expensive hobbies that require lots of equipment, travel, or expensive training, one can pick up a pencil (or a box of brightly colored pens if you prefer) and give math a try for free!

Posted in Math Education, Mathematics and the Arts, Recreational Mathematics | Tagged , , , , | 2 Comments

Job Security Calculus: Reasoning about our futures

Posted on January 31, 2014 by bfinegold

Most aGraphic from http://himalayawake.com/mooc/cademics have a love/hate relationship to teaching, and especially teaching Calculus.  Prior to the first exam of the semester, it seems that everyone in the class is there for learning’s sake, discussing ideas, engaging in problem-solving.  But we worry that we are providing too detailed feedback (that those more jaded might argue some students don’t even read).  Or that we spent too long creating the perfect exam when a not-so-perfect exam will do and afford us more time for research.  There are some who see their teaching as the perfect complement to research since it reminds us as we watch students stumble through our courses how we too are stumbling, just on the brink of discovery.  We are  warned nonetheless by our seniors of “liking teaching too much”.

But Catherine O’Neil, the author of MathBabe is worried about those Calculus classes disappearing.  As MOOC’s take over the function of teaching Calculus to the masses, there will be less need for Calculus Instructors, and therefore less need for Research Mathematicians at all but the most elite institutions.

While finding an academic job is already pretty difficult, she thinks it’s only going to get harder.  Dr. O’Neil writes:

“But for my younger friends who are interested in going to grad school now, I’m not writing them letters of recommendation before having this talk, because they’ll be looking around for tenured positions in about 10 years, and that’s the time scale at which I think math departments will be shrinking instead of expanding.”

What do others think about the future of research mathematics?  With 60 comments posted after her entry, its clear that many people have some opinion on the matter.  An early career mathematician, Kaisa Taipale, who is visiting at Cornell and got her PhD the same year I did (2010) writes on the Limit Institute Blog  about a recent panel she attended at the JMM about MOOCS:

“The economics, one of my favorite puzzles, recurred several times in discussion. Robert Ghrist and Tina Garrett both said that making a MOOC or a SPOC was not cheap or a real cost-saving measure. It comes out of tenured faculty time and perhaps special pots of administration money. I asked about the position of postdocs, graduate students, and others who might participate in online education initiatives …. There was some discussion of the fact that universities or colleges might hire adjuncts to do online courses in particular, which did not thrill me. Time to get into management I guess. There was universal acknowledgement that intellectual property and copyright rules have not yet been standardized. Patricia Hersh asked about the economics of asking recent PhDs to produce high-quality math materials for K-12 teachers. Hmmm… I have heard of no such official effort, and the economics are indeed interesting.”

By the way, The Limit Institute has a nifty mission: “The Limit Institute for Mathematics, Innovation, and Technology (LIMIT) is a loose affiliation of mathematicians at all levels of training and employment. We are interested in how technology is changing how we carry out math research, teach math, and even understand what mathematics is.” And it doesn’t hurt that they quote a Paul Simon lyric on their homepage.

For the cynics (like myself), the answer may be to seek jobs outside academia.  Izabella Laba wrote a post on her blog The Accidental Mathematician remarking on the lack of advice for those seeking non-academic jobs, especially on the AMS website.  She is seeking good sources as she will be helping to update the AMS site.

To me, the most significant point is that we should be thinking about these issues as a community and deciding how to best face them.  While the everyday pull of research, teaching, grant deadlines, and committee meetings, we may look up and find that administrators, businessmen, and bureaucrats  have made  all the decisions on behalf of mathematicians.  While Dr. O’Neil thinks relying on billionaires is not the right way to go (see her post ), there may be other alternatives.  How do you see MOOCs as changing the landscape, if at all?

 
Graphic from http://himalayawake.com/mooc/
Posted in Applied Math, Issues in Higher Education, Math Education, people in math, Theoretical Mathematics | Tagged , , , , | Comments Off on Job Security Calculus: Reasoning about our futures

Heads I Win, Tails You Lose

Posted on January 27, 2014 by evelynjlamb

Last semester, my university put on a production of Tom Stoppard’s play Rosencrantz and Guildenstern Are Dead that got me thinking about the likelihood of flipping a lot of heads in a row. I wrote about it on my other blog, Roots of Unity.

What are the odds? Image: Evelyn Lamb

What are the odds? Image: Evelyn Lamb

Around the time I saw the play, I read two blog posts about long runs of heads. Ben Orlin of Math with Bad Drawings wrote about The Swindler’s Coin. It is an imaginary dialogue between a teacher and student in which the teacher, who tells the student that he is flipping a fair coin, gets 30 heads in a row. The student, who believed that the coin was fair, finally accuses the teacher of using a swindler’s coin, and the teacher admits it. The post points out the naiveté with which we often teach probability.

“Some probability texts ask a similar question: ‘If a fair coin is tossed 50 times, and comes up heads each time, what is the probability that it comes up heads on the 51st toss?’ The ‘correct’ answer is ½. A fair coin always has a probability ½ of coming up heads, because that’s how we define ‘fair.’

“But guess what? If a coin comes up heads 50 times in a row—a 1-in-a-quadrillion event—then that ain’t no fair coin. The question could be paraphrased: ‘If I tell you a coin is fair, and then overwhelming evidence accumulates to the contrary, would you still believe me?’ And the ‘correct’ answer would be: ‘Yes, because I never reconsider my assumptions.'”

Later that week, John Cook wrote about different “levels of uncertainty” on his blog The Endeavour.

“The other day I heard someone say something like the following:

“‘I can’t believe how people don’t understand probability. They don’t realize that if a coin comes up heads 20 times, on the next flip there’s still a 50-50 chance of it coming up tails.’

“But if I saw a coin come up heads 20 times, I’d suspect it would come up heads the next time.

“There are two levels of uncertainty here. If the probability of a coin coming up heads is θ = 1/2 and the tosses are independent, then yes, the probability of a head is 1/2 each time, regardless of how many heads have shown before. The parameter θ models our uncertainty regarding which side will show after a toss of the coin. That’s the first level of uncertainty.

“But what about our uncertainty in the value of θ? Twenty flips showing the same side up should cause us to question whether θ really is 1/2. Maybe it’s a biased coin and θ is greater than 1/2. Or maybe it really is a fair coin and we’ve just seen a one-in-a-million event. (Such events do happen, but only one in a million times.) Our uncertainty regarding the value of θ is a second level of uncertainty.”

In Rosencrantz and Guildenstern Are Dead, the title characters flip heads more than 90 times in a row. I think Ben Orlin, John Cook, and I can all agree that they should probably take a second look at those coins!

Whenever I think about long runs of heads or tails, I remind myself that any individual string of n flips is exactly as likely (or unlikely) as any other individual one (if the coins are truly just as likely to come up heads as tails). But if someone says she flipped either 10 heads in a row or HTHHTTTTHT, you’re going to say the second one is more likely. Shecky Riemann wrote a post about a similar idea with rolls of a die. One interesting side note is that while any two individual strings of the same length are equally likely, true randomness looks different from human attempts at randomness. Justin Lanier mentioned this in a post on Math Munch, and Jim’s Random Notes fleshes this idea out a bit.

“Not only are people terrible at perceiving randomness, they’re also terrible at generating randomness. Asked to flip a coin 100 times and write down the results, many college students will ‘cheat’ and forego flipping the coin. They’ll just write down what they think is a random sequence. It’s usually easy to catch them because the idea of run of four or five tails just seems ‘not random.’ But getting five heads or five tails in a row is very common given 100 flips of a fair coin.”

He wrote a program to figure out how common different length runs are if you flip a coin 100 times. “If you play with the program a bit, you’ll find that runs of six tails happen more than half the time, runs of seven happen about a third of the time, and you’re twice as likely to get a run of 10 than not get a run of four.” I was definitely surprised about that!

Finally, Ask a Mathematician/Ask a Physicist explores whether “If you flip a coin forever, are you guaranteed to eventually flip an equal number of heads and tails?” The physicist takes a probabilistic look at the question and concludes that the answer is yes, but I would argue that with a literal interpretation of the question, the answer is no. I can easily invent infinite series of +1’s and -1’s such that no partial sum is 0.

Posted in Mathematics and the Arts, Statistics | Tagged , , | 2 Comments

The Revolution Will Be 3D Printed

Posted on January 14, 2014 by evelynjlamb

“What would you print if you had a 3D printer in your home?” James Madison University math professor Laura Taalman is printing a thing a day and blogging about it at MakerHome. Her family has a MakerBot Replicator 2 and an Afinia H-Series, and each post includes an .stl file of the project and some information about which printer they used, how long it took, and other pertinent details in case you want to play along at home.

Ducks and blocks illustrating different resolution options on the Afinia 3D printer. Image: Laura Taalman. Used with permission.

Ducks and blocks illustrating different resolution options on the Afinia 3D printer. Image: Laura Taalman. Used with permission.

Taalman’s project started last August, I assume to coincide with the beginning of school. Her university JMU has embraced 3D printing. They have a MakerLab where math students can learn about 3D printing (their blog is here) and a 3D printer-equipped general education classroom with its own website and blogs.

I’m impressed with the wide range of things Taalman prints at home. In addition to cool math stuff (more on that in a moment), she makes toys such as Daleks and polyhedral bears and useful household objects such as a coin-operated bottle opener, dietary restriction toothpicks, and a strainer funnel for hibiscus tea. She even 3D printed a grate to keep fingers safe from the 3D printer’s fan. (I like to think that if I had a 3D printer, my headboard would now be attached to my bed instead of propped against the wall because I don’t have the right sized bolts.)

The math-based designs are great. Some of my favorites are an icosahedron into which jawbreakers were dropped while it printed, a chain of interlocking rhombic dodecahedra, rocking knots that only touch the table in two places, a Hilbert curve, a tetrahedron puzzle, an icosiodecahedron, and a stereographic projection model. She even printed a knot without knowing what knot it is, or even whether it’s the unknot!

Is this the unknot? Image: Laura Taalman. Used with permission.

Is this the unknot? Maybe one of Laura Taalman’s students will figure out. Image: Laura Taalman. Used with permission.

Some things, like the Menger sponge coasters, and cosine-based cookie plate, are part math and part practical. Why a cosine-based cookie plate? Taalman writes, “I need a small cookie plate for a reception.  And maybe PLA [polylactic acid, a plastic used in some 3D printing] isn’t FDA-approved but frankly I think most people would probably eat a cookie that was sitting on the table with no plate at all, so a PLA plate is probably okay.  Just in case, however, here is a plate the minimizes cookie/PLA contact, because of the waviness of the plate – based off the cosine curve.”

I don’t have a 3D printer yet, but I am increasingly impressed with how professional their outputs look and how easy they seem to be to use. From Taalman’s blog, it looks like she has occasional frustrations with the printers, and I’m sure there’s a learning curve for figuring out what shapes work and what settings to use, but projects like hers make me think that maybe I could do this. The ability to print out exotic mathematical shapes could certainly make a big difference in a class like multivariable calculus or complex analysis in which concrete visualizations can be incredibly helpful. I don’t know if I’ll take the plunge anytime soon, but it’s very tempting. There are so many freely available projects already and endless ways to tweak them or make entirely new things.

3D prints of Henry Segerman's stereographic projection models. Image: Laura Taalman. Used with permission.

3D prints of Henry Segerman’s stereographic projection models. Image: Laura Taalman. Used with permission.

I first got interested in 3D-printed math when Saul Schleimer brought a 3D printed puzzle he and Henry Segerman had designed to a conference I attended, and I learned about Taalman’s blog from Segerman. I recently ordered a model of the hyperbolic plane from his Shapeways store to use in an upcoming talk. I can hardly wait for it to get here!

Posted in Math Education, Recreational Mathematics | Tagged , , , , , , , , , , | Comments Off on The Revolution Will Be 3D Printed

The best math in life is free? Help gather the data!

Posted on January 8, 2014 by bfinegold

Over at the Secret Blogging Seminar , Scott Morrison is championing a new project to analyze this year’s open-accessmathematics publications and draw attention to freely accessible papers.  The Mathematics Literature Project  is looking for your help in categorizing published articles according to whether they are freely accessible on arXiv, are freely accessible somewhere else, or are only accessible for a fee.  After enough data are collected, one may be able to correlate accessibility and quality as measured by number of citations.  There is a handy tutorial that will show you how to add information to the wiki by combing the internet for freely accessible versions of papers in well-known journals.    The color-coded progress bars displayed next to each Journal title on the wiki indicate how many articles have been categorized so far and how.  To see the key to the color-coding, you need to click on the bar itself.

In case you are interested in looking at some journals not listed on the wiki, the tables of contents of various journals are available through the site JournalTOCs.

Lastly, while reading Data Scientist Michale Li’s The Mathematics of Gamification post about how Foursquare uses Bayesian statistics to determine the quality of updates proposed by its users, I started thinking about how such a strategy might apply to peer review of academic papers in the future.  Foursquare uses “honeypots” to judge the quality of super-users’ updates.  Perhaps there could be certain academic papers that were “honey-pots” (i.e. were peer-reviewed in a traditional way and determined to be of very high quality).  These could help determine super-reviewers’ ratings.  Ratings could then also be informed in real-time by conditional probabilities.  In other words, a user’s rating could be informed by knowing the probability that a paper is “good” given that the reviewer deemed it “mediocre”.  What questions do YOU think that this data base might provoke or help answer?

Posted in Publishing in Math, Statistics | 1 Comment

Winter Break Reading: Baking and Math

Posted on December 23, 2013 by evelynjlamb
Sierpinski cookies. Oh yes. Image: Lenore Edman, via Flickr.

Sierpinski cookies. Image: Lenore Edman, via Flickr.

If you, like me, like both food and math, then maybe you should check out Yen Duong’s blog Baking and Math. Duong is a graduate student studying geometric group theory at the University of Illinois at Chicago. Her blog mostly consists of expository math posts and interesting recipes, but there are also some more reflective posts about being a mathematician and more specifically a woman in math. The blog is about a year old now, and her blogiversary post from November rounds up some of the highlights.

Some of my favorite posts:

On failure and coconut chocolate chip cookies. “I failed at these cookies. I fail at math sometimes. I am not a failure of a person, and while I enjoy baking and math, being great at either of them does not define me as a person.” I am writing this post after grading my final exams. The exam was hard, and I hope any of my students who are disappointed in their performance on the final know that they are not their grade on a math test! (In a related vein, Francis Su’s blog post about the lesson of grace in teaching really helped me think about how I communicate to my students that I care about them as people.)

A reaction to being a woman in math. “When I’m a super-minority (as in the only person with a particular characteristic in an otherwise homogenous situation), I feel a burden of responsibility to represent whatever that minority is: woman, mathematician, non-white person.” I definitely feel that burden sometimes when I am the only woman in a group of mathematicians!

On inaccessibility and why we do math. When an explanation of math is unclear, people are very ready to blame themselves rather than the source of the explanation, and it’s not always justified. I found myself nodding along as I read this post. Sometimes people tell me that my blog posts are good but they’re way over their heads. I think it’s meant as a compliment, but my goal is to make the posts accessible to most people who care to read them!

The three-part series about the curve complex, culminating in a proof that it is connected!

Chia chews. Duong doesn’t just make easy recipes like chocolate chip cookies. She also tries more daring stuff like making her own energy chews for workouts and running mini-triathlons!

I’d like to see baking and math combined a little more on the blog. I’m holding out hope that she will bake a model of the curve complex, mostly because I want to find out what locally infinite confections taste like! OK, that was just a ridiculous geometric group theory joke. I think the groanworthy puns in her recipes are going to my head! I’d better step away from the computer and make myself some banana pudding, even though it’s a “high-whisk investment.”

If I had all these things, I would make banana pudding. Image: Yen Duong.

If I had all these things, I would make banana pudding. Image: Yen Duong.

Posted in Math Education, Theoretical Mathematics, women in math | Tagged , | 1 Comment