Bad Statistics: Ignore or Call Out?

Andrew Gelman has been wondering how much time he should spend criticizing crappy research, and so am I. He wrote the post after a discussion with Jeff Leek of Simply Statistics about replication and criticism. Harsh criticism of preliminary studies could discourage new research, which would definitely be a bad thing, and Leek notes that shaming people who do research that gets retracted is probably not going to help science be self-correcting.

3-D pie charts are often misleading. Image: Smallman12q, via Wikimedia Commons.

In an earlier post on the topic, Gelman says that he is not against the publication of these early, possibly incorrect results. But if a bad study, especially about health, gets a lot of publicity, it could be harmful to people who read about it and take it seriously. In the later post, Gelman writes,

“A key decision point is what to do when we encounter bad research that gets publicity. Should we hype it up (the “Psychological Science” strategy), slam it (which is often what I do), ignore it (Jeff’s suggestion), or do further research to contextualize it (as Dan Kahan sometimes does)?”

More broadly, I’ve been wondering how much time should we spend criticizing bad science and math journalism or bad behavior in general. If misinformation is reported far and wide, it might be important to do some debunking, but in many cases, responding to bad things will just give the bad things more attention. (For example, I don’t think we should give Westboro more media coverage.)

Recently, there’s been a little kerfuffle about the fact that physicist Lawrence Krauss, among others, appears in a geocentric documentary made by a Holocaust denier. Krauss says that he doesn’t know how he got in the documentary and wants us to just ignore it. “Many people have suggested I litigate,” he writes. “But this approach seems to me to be completely wrong because it would elevate the profile of something that shouldn’t even rise to the level of popular discussion.” And in this case, I agree. The articles criticizing this documentary have given it tons of free publicity. Without them, it would have disappeared into the ether.

Right now I’m particularly fed up with news stories about how ignorant people are (as Cathy O’Neil writes, these are the stories about how many people think the sun goes around the earth), and I want to debunk them. After seeing a couple articles with a specific claim I found a little hard to believe, I got a hold of the poll data. And surprise, surprise: the articles make people out to be more ignorant than the poll seems to suggest (and the poll itself seems much less a representative sample of some society than a bunch of people who responded to an online poll that was only open for a few hours). Some misleading information about this poll is published in a few places, but it hasn’t gone viral. If I wrote about it, more people would see the original misleading information and might remember it instead of my correction. I would get some satisfaction from criticizing the other articles, but I don’t think it would help anything. Besides, I like to make people happy with math, and writing about something true and interesting is probably a better way to do it than taking something else down.

I really liked the end of Gelman’s post:

“A few months ago after I published an article criticizing some low-quality published research, I received the following email:

‘There are two kinds of people in science: bumblers and pointers. Bumblers are the people who get up every morning and make mistakes, trying to find truth but mainly tripping over their own feet, occasionally getting it right but typically getting it wrong. Pointers are the people who stand on the sidelines, point at them, and say “You bumbled, you bumbled.” These are our only choices in life.’

The sad thing is, this email came from a psychology professor! Pretty sad to think that he thought those were our two choices in life. I hope he doesn’t teach this to his students. I like to do both, indeed at the same time: When I do research (“bumble”), I aim criticism at myself, poking holes in everything I do (“pointing”). And when I criticize (“pointing”), I do so in the spirit of trying to find truth (“bumbling”).”

I’m going to take the easy way out and agree that we need balance. Personally, I don’t think I’m suited for doing a lot of pointing. Occasionally I write about something I think is bad and why, but mostly I’m going to keep writing about stuff I like and hope that my good stuff distracts from other crappy stuff. (As John D. Cook writes, quality over quantity. I found that post via another post of Gelman’s.) But there are other people who do a great job at criticizing the crappy stuff. This StatsChat post from Thomas Lumley about whether Generation Y spends a lot of money on fancy food cracked me up.

Of course, one reason I don’t do as much pointing is that I write more about math and less about statistics and how it’s used in other sciences. I think there’s more need and opportunity for pointing in those fields. When done well, I think pointing out bad statistical practice and the bad journalism it sometimes spawns might help journalists and readers approach scientific studies with the appropriate amount of skepticism and ask the right questions about them. A girl can dream.

Posted in Statistics | Tagged , , , , , , | 2 Comments

Mathemagical Thinking

So maybe you’ve seen the Flash Mind Reader.  If not, go ahead and try it!  I wouldn’t dream of depriving you (especially as this is year’s MAM theme is mathematics, magic, and mystery awareness).  What you are asked to do is to think of ANY two digit number, add the digits together, subtract that difference from the original number, and then look up your number in a table of symbols.  Then the computer will read your mind and produce the correct symbol.  So the mystery is – why does it work?  We know it’s a trick, and it can’t be that hard since the computer did it.  So it’s bound to bug you until you figure it out, and there are a number of people who post things like “Flash mind reader revealed!!” and then proceed to expound on the method used to do this trick.  They make tables, they do a lot of writing, all to express some simple algebra.

Certainly this is the kind of magic that much of the public associates with math – fancy tricks with numbers.  And it does bear a passing resemblance to another kind of magic with which many of us are familiar.  When you’re turning a problem (a mystery) around in your head over and over again because you just can’t let it go, using symmetry to simplify and transform the ideas, you sometimes feel in a tancelike state.  This kind of sentiment is echoed in Vi Hart’s most recent post about creating Art Code.  She writes “One thing led to another and soon I had a simple animation I called Lost Memories of Desert Sand, and couldn’t stop staring.”  One might say that “magic” allows our thoughts to fit together just so and create something beautiful.  The gasp that follows many math tricks is also the gasp that sometimes follows a good presentation of a proof.  There’s a sense that magic has just been performed – you were following each movement of the performer ever so closely when all of a sudden they finished the proof – why yes they did!  And it was so clever!  And it made sense, right?  Or did it?  Wait a minute, what about that part over there?  There’s a sense of skepticism, excitement, and awe that accompanies the practice of mathematics. had a great post on Magic and Symmetry in Mathematics that speaks to this type of “magic” in math. One of this year’s Sloan Research Fellows, Dr. Ivan Loseu says “Any scientific discovery involves some kind of magic,” That is, various pieces that may seem to be completely unrelated eventually start to fit together through the fruits of one’s labor. “Since pure math is pure, all this magic is much more clearly seen.”

Ready for a fun video from Tadashi Tokieda, whose work I learned about from a guest post at Scientific American about this year’s recent Gathering for Gardener?

Dr, Tokieda from University of Cambridge likes to play with “Toy Models” that demonstrate certain unexpected, and one might say “magical” properties.  Check it out, and you will probably find yourself spinning little tubes around and saying “paf, paf, paf!”.

So go ahead and work your magic this weekend!

Posted in Mathematics and Computing, Mathematics and the Arts, people in math, Recreational Mathematics | Tagged , , , , , , , | 2 Comments

Awesome K-12 Math Teachers Exist! And they have blogs.

I sometimes get tired of hearing about how “teachers (meaning K-12 teachers) just don’t understand this or that, or won’t try doing something new,” or are deficient in some way or another.  We often advise teachers to let go of the “deficit model” and trade in for the “abundance model” when thinking about their students, so why not encourage society to do the same for teachers.  In a sea of people bemoaning the state of our education system, we should consider these dedicated individuals are resources to be celebrated.  Disclaimer: I haven’t had either of these teachers myself, but based on their blogs, I am guessing they are awesome.  There are many wonderful teaching blogs out there, but these are a few that have recently caught my eye.

Fawn Nguyen’s blog, Finding Ways to Nguyen Kids Over, is full of ideas that could be used in both college and K-12 classrooms.

fawn nguyen math talks
One students strategy for finding the equation that describes the pattern: ” I see the center column as (n+1). Then there are 4 identical groups around the center, each one is a Gauss addition. My equation is C = (n+1) + 4[(n2+n)/2].”

My favorite part of her website is her Visual Patterns catalog, which is a whole bank of visual patterns that help middle schoolers connect algebraic expressions and geometric patterns.  This was especially nice from my point of view because she includes her way of doing “math talks” with these visual patterns, a simple but effective way of letting the kids think individually, share their ideas, and get feedback from classmates.In addition to sharing the mathematics that her class works on, Ms. Nguyen writes post like “I can’t afford not to” addressing the concern that there isn’t enough time to devote to activities outside of the typical curriculum like she does.  She shares some of the reflection of her students as evidence that Math Talks and other less conventional activities that she does in class are effective.  My favorite quote from a kid was “I thought it was clever because towards the end it wasn’t the rule, it was your rule.”


From a Math Munch post on paper representations of proofs of the Pythagorean Theorem. This is taken from an 1847 manuscript by Oliver Bryne.

The teachers at Math Munch: Anna Weltman, Justin Lanier, and Paul Salomon, who all taught or teach at Saint Ann’s School in Brooklyn.

I’ve liked this blog for a while and mentioned it before.  Recently, it featured a game with which, I am a bit ashamed to say, I became somewhat obsessed..2048.  If you haven’t already played this game, maybe it’s best that you continue to avoid it.  But if you are like me, you have not only played many times, but you play to win as quickly as possible.


Justin’s preparations for his math circle workshop

It seems that Justin Lanier is currently working at the Princeton Learning Cooperative and has his own blog “I choose math” which features most recently his foray into Celtic knot drawings at a math circle.




From Mike’s blog: “For example, from those two sources, and lots of trial and error, we were able to print out a hollowed out cube that illustrates the “Prince Rupert Problem” – a cube is actually able to pass through a second cube of the same size”

We shouldn’t forget that our parents are also our teachers, as is so obviously the case for kids who are homeschooled.  Take for instance, Mike, curator of

  • Mike has a day job, but also makes videos in which he explains math to his sons in front of what appears to be a room plastered entirely with whiteboard.  My favorite post is the one he made about Gauss and finding the expected value of the number of ways to write an integer as the sum of two squares.  I could easily see using this problem in a number theory, probability, or real analysis course!  He also had a great post about how his family was inspired by Laura Taalman’s 3D Printing Blog that was featured recently by my co-editor, Evelyn Lamb.

Lastly, I like this idea, that Pi Day is a great day to thank you favorite math teacher, from a writer and “radical homemaker” Alicia.  Here she describes her math teacher:

“Listening to him talk about math was like entering an alternate universe. This place was full of excitement, creativity + experimentation…nothing like the black and white worksheets I was used to. Instead of boring work like solving for X, we created 3D graphs, puzzling out shapes likes ice cream cones and clowns by using multivariable equations. Once everyone had calculated a perfect design, we printed them out and hung them in the hallway: a mathematical gallery sprang up to usher us to class.

Watching someone who passionately loves their subject matter talk about French literature, statistics or anthropology is enlightening. The material almost doesn’t matter when you can watch someone’s eyes light up and their every animated gesture convey their fascination. Pi Day, or 3/14, is the perfect day to nerd out and send a huge thanks to teachers who have changed our lives.”

Posted in K-12 Mathematics, Math Education, people in math, Theoretical Mathematics | Tagged , , , , , , , | Leave a comment

Geometry and the Imagination

If you like geometric group theory or amazing pictures (but especially geometric group theory), you might want to start reading Geometry and the Imagination, written by University of Chicago mathematician Danny Calegari. I’ve been following it for a while, but I got inspired to write about it here by a recent post on some new software he wrote, kleinian. Logically enough, it is a tool for visualizing Kleinian groups. And isn’t this visualization beautiful?

The universal cover of a genus 3 handlebody, visualized using the program kleinian. Image: Danny Calegari.

If I had to sum up the blog in a sentence, I’d say that Geometry and the Imagination contains expository posts about hyperbolic geometry and geometric group theory written for a mathematically sophisticated audience, with a few flights of fancy (like this post on solving the Rubik’s cube) thrown in. Like many research-oriented blogs, it delves into many of the technical details of theorems and proofs but with a lot more helpful “big picture” signposts than most research articles have. For me, at least, as a grad student and early-career researcher, seeing the forest for the trees has been the most difficult part of starting to do research, so I am a fan of blogs with signposts.

A few years ago, Calegari’s student Alden Walker wrote a series of posts containing notes from Calegari’s hyperbolic geometry course. (Calegari and Walker were both at Caltech at the time. Walker is now a postdoc at the University of Chicago.) As I mentioned in my last post, I find notes like this very helpful because even when I know a subject, I haven’t always thought deeply about what order to present it in, what examples to use, and so on.

One fun post is on Kenyon’s squarespirals:

“The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the squares rotate, they change size in such a way that the new (skewed, resized) squares still tile the same region. I thought it might be fun to try to guess how the image was constructed, and to produce my own version of his image.”

In the rest of the post, Calegari moves from square tilings of rectangles to square tilings of a torus to arrive at an image that is quite similar to Kenyon’s original. Aesthetically, I prefer this intermediate gif, of rotating squares that do not spiral, so I’m including it instead.

Squares that rotate but do not spiral. Image: Danny Calegari.

If heavier math is your thing, Calegari has a three-part series on Ian Agol’s proof of the virtual Haken conjecture and a post on his and Walker’s result that random groups contain surface subgroups.

But my favorite post is Random Turtles in the Hyperbolic Plane. With a title like that, how could it not be? In it, he riffs on his daughter’s Logo programming project. She programmed a turtle to take a random walk in the Euclidean plane, so he looks at what happens when turtles take random walks in the hyperbolic plane. Hyperbolic turtles display some very interesting behavior, with a “phase transition” from a beeline to the boundary to something that really “looks like” a random walk, depending on the how size of each step compares to the size of the turn the turtle makes each time. It’s a surprisingly rich line of inquiry!

The walk of a random turtle in the hyperbolic plane. Image: Danny Calegari.

The walk of a random turtle in the hyperbolic plane. Image: Danny Calegari.

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On Teaching Analysis

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

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 articleWho 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 | 6 Comments

Why Should We Fund Math Research?

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 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 , , , , , | Leave a comment

Mistakes Are Interesting

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 , , , , , , , , | 1 Comment

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

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 , , , , | 1 Comment

Job Security Calculus: Reasoning about our futures

Most aGraphic from 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?

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