More Graph Isomorphism Drama

That plucky graph isomorphism problem is at it again! In November 2015, University of Chicago computer scientist Laszlo Babai announced an algorithm to determine whether two graphs are isomorphic in quasipolynomial time, and there was much rejoicing. (My co-blogger Anna Haensch covered it here at the time.)

But earlier this month, University of Göttingen and CNRS mathematician Harald Helfgott posted on his blog that he had found an error in the proof, and it looked to be a major one. Babai’s algorithm was still a big improvement over earlier algorithms but only ran at subexponential rather than quasipolynomial time. (Complexity jargon got your head spinning? Check out Jeremy Kun’s primers on Big-O notation and P vs NP. There’s also a Wikipedia page on time complexity. He attended Babai’s talk announcing the result in November 2015 and posted about it then, with a few updates since.)

Luckily, Babai has fixed the problem, so people with potentially isomorphic graphs they’d like to check in quasipolynomial time can rejoice once again.

I’ve been keeping up with graph isomorphism news by reading Erica Klarreich’s posts about it on the Quanta Magazine Abstractions blog. She explained the error Helfgott found on January 5th and posted an update on the fix nine days later. I am especially fond of her analogy for quasipolynomial time: “Very roughly speaking, his algorithm carries the graph isomorphism problem almost all the way across the gulf between the problems that can’t be solved efficiently and the ones that can — it’s now splashing around in the shallow water off the coast of the efficiently-solvable problems, whose running time is what computer scientists call ‘polynomial.’”

On Saturday, Helfgott gave a Bourbaki lecture on graph isomorphism. Francophones can watch it on Youtube. (Others can also watch it on Youtube but will understand less of it.) I’ll also be keeping an eye on the Gödel’s Lost Letter and P=NP blog. They’ve been covering Babai’s work on the graph isomorphism problem since he announced it, and if the weather cooperated, they just went to a conference where Babai gave a distinguished lecture on the topic.

Update, January 17, 2017: As he reports on his blog, Helfgott’s Bourbaki article is now on arXiv as well (in French).

Posted in Mathematics and Computing | Tagged , , , , | Leave a comment

Best And Worst Of The Year

200508_firework_of_lake_of_annecy_festival_280

We made it through 2016, and now it’s that time when we reflect on a year gone by.

Best of 2016

There were several cool breakthroughs in math this year. My personal favorite involved the famous question of how to optimally stack higher dimensional spheres in space. This year Maryna Viazovska made a critical breakthrough, solving the 8-dimensional case, and several weeks later the 24-dimensional case tumbled too. This breakthrough is an important one because of its applications to coding theory and data transmission. When the result was announced Quanta published a very thorough history of the sphere packing problem that led to the breakthrough.

This year we also found some interesting (and huge!) new primes. The world record for longest known prime is now 22,338,618 digits. This bad-boy is a Mersenne Prime. In September there was also a new world record set for the largest twin primes. If we printed out all the new prime goodness we found this year it would take about 20 reams of printer paper.

My favorite math in pop culture this year was The Man Who Knew Infinity, the film about Ramanujan and Hardy. If you haven’t seen it yet, I urge you to. Several great books about math also came out this year, including Cathy O’Neil’s Weapons of Math Destruction about the dangers of data science, and Margot Lee Shetterly’s Hidden Figures about a group of African American woman mathematicians who contributed to the space race. I just received the latter as a gift for christmas, so you can expect a review of that in the next few weeks.

Worst of 2016

The real computational dunce cap of the year definitely goes to Facebook and their biased newsfeed algorithms that proliferated fake news during an historic and incredibly tense election. Cathy O’Neil did a nice job covering news of all things algorithmic this year before, during, and after the election. In general, this also reminds of the trouble we’ve had with bias in algorithms this year. For example, that algorithm that was supposed to help the legal system by predicting criminal behavior and instead has just contributed to our already incredibly racist justice system. I guess this was the year to remember that algorithms are run by computers, but written by humans.

On the theme of politics, it was a weird and bad year for polling too. I suppose we learned the value of 2 percentage points, and learning is a good thing, but I suspect we also had a false sense of reality going into the elections and that was a bad thing. While the speed with which we can consume infographics and data makes is quicker to digest numbers, it also leaves us with a pretty poor understanding of what’s going on in the margins. The lesson we learned here is that numbers need context.

And finally, the absolute worst of the worst this year (and perhaps a partial solution to the problem of the previous paragraph) was this craziness about the myth of algebra that just won’t seem to quit. I’m talking, of course, about Andrew Hacker and his infamous call to arms against mandatory high school algebra. This year he wrote a book on the subject, and I will concede that he makes a few good points about numeracy and problem solving. But he also makes dozens of horrible points about some made up algebra straw man that forces you to compute azimuths. So, I’m sorry Hacker, I just can’t. We need Algebra. So much Algebra.

Have a happy new year! And to all of you who are traveling to the JMM in Atlanta, have safe and speedy travels and stay tuned for our 2017 Joint Meetings Blog.

Posted in Events | Leave a comment

Mona Chalabi’s Datasketches

Hand-drawn data visualizations about farts and penises! If that has you hooked, no need to read any further. Just surf over to Mona Chalabi’s Instagram account and enjoy.

I first encountered Chalabi through her “Dear Mona” column at FiveThirtyEight, which has since moved to New York Magazine. There she answers people’s questions with both statistics and compassion. Since then, I’ve also been impressed with her work as data editor at Guardian US, including this November 9 column about why we should be treating polls with more skepticism. In a slightly less obviously mathematical vein, if you have a vagina or know someone who does, check out the Guardian’s Vagina Dispatches, a series of four fantastic and fearless videos by Chalabi and Mae Ryan that delve into emotional, cultural and health aspects of owning and operating that particular body part, of course using statistics to support their work.

Currently I’m obsessed with Chalabi’s “datasketches,” hand-drawn illustrations that visualize data in creative, accessible, and entertaining ways. One of the things that makes Chalabi’s visualizations so appealing is that she doesn’t shy away from taboo subjects: sex, pubic hair, periods, nose picking, death, you name it. They’re not exactly NSFW, but if you don’t want your boss to glance over and see a cartoon penis on your screen, you might want to save them for when you get home.

In a really nice Q&A with DigitalArts, Chalabi says she had grown frustrated with the inaccessible, academic way in which organizations tend to present important data and intellectual elitism in data visualization by journalists. She wanted to present things more clearly and accessibly and in a way in which people could feel free to ask questions about the data, which they do in the comments on Instagram. She always cites the source of the data she presents so if people are skeptical or think it has an agenda, they can explore it for themselves.

Chalabi’s datasketches were shortlisted for an Information is Beautiful prize and commended by the Royal Statistical Society this year. The citation says, “she has demonstrated that it is not always necessary to have sophisticated graphics packages or specialist programming knowledge in order to visualise data and tell a story.” In fact, one of the most affecting images is one of her simplest: the average size of a parking space versus a solitary confinement cell.

Sources: Texas Accessibility Standards, The Center for Investigative Reporting #datasketch

A photo posted by Mona Chalabi (@mona_chalabi) on Apr 23, 2016 at 7:44am PDT

Posted in Statistics | Tagged , , | Leave a comment

The Lure Of The Rubik’s Cube

The 3x3x3 Rubik's Cube, can you solve it?

The 3x3x3 Rubik’s Cube, can you solve it?

Who among us has not lost at least one afternoon of their life to that most seductive of toys: The Rubik’s Cube? Originally invented by the Hungarian architect Erno Rubik in 1974, this cube – although apparently not its patents – have stood the test of time.

The beauty of the Rubik’s Cube, much like the beauty of mathematics, is that it seems totally impossible at first. But as soon as you learn the solution, it becomes totally trivial. The problem is to take this jumbled up cube, and perform a series of permutations (by twisting across various axes) to get each face to display a single color. For a 3x3x3 cube there are 4.3252×1019 possible permutations to chose from. That’s quite a lot. But even so, computations taking 35-CPU years by a bank of computers at Google show that the worst possible jumbling of the cube can always be solved in 20 or fewer moves. This maximum number of moves to solve a Rubik’s cube is known as God’s Number.

So this means that for any jumbling, you’re always only 20 moves away from a solved cube. Now you see where things start to get tantalizing. Of course you may not solve the cube perfectly, that is, you might use an algorithm that ends up taking more than God’s Number. But just knowing the solution is so close at hand is already fun. The difficulty then is in coming up with an algorithm to solve the cube, and most methods do this by breaking down the algorithm in to several sets of moves, or “macros.” And these can be best thought of as operations in group theory. We can think of permutations of the cube as elements of a group, R, whose binary operation is concatenation of moves. Then building the macros to solve the cube can be thought of in terms of commutators and conjugates, see this great explainer for the full story.

So, if you are looking for a holiday gift to occupy please your mathematical loved ones: look no further! Math’s Gear will meet all of your Rubik’s related needs with competition grade speed cubes of all dimensions. They even have the really fun looking — but I’ll admit, slightly intimidating — Skewb puzzle cube. And what better to accompany this gift of procrastination than a paper on group theory to go with it, or for the les mathematically inclined, a plain english explanation of the macros and algorithm.

Just this week there was a new record set in the 3x3x3 Rubik’s cube by Feliks Zemdegs, a 20 year old Rubik’s cube speedsolver from Australia. This maniac can solve the cube in an insane 4.73 seconds. That’s faster than you can say “Hey Feliks, can you solve this Rubik’s Cube in under 5 seconds?” And remember, the maximum number of moves to solve the cube is 20. Suffice to say, I don’t think there’s anything I can do 20 times in the span of 4.7 seconds. You can watch him break the record in the video below, and it will make you feel really happy. (h/t to Matt Parker @standupmaths for sharing these links about Zemdegs.)

Recently the German semiconductor giant Infineon built a robot that can solve a 3x3x3 Rubik’s Cube in just 0.637 seconds. That’s so fast. That’s faster than you can say “Rubik’s Cube,” it’s faster than you can say “Ru-.” Actually, if the robot threw the Rubik’s Cube really hard at your shin, in .637 seconds the sensation of pain wouldn’t even have made it to your brain yet. The video below shows it in real time and then in slow motion, and it’s pretty incredible to watch.

In you’re looking for a mathematical gift that isn’t Rubik’s related, personally I’m hoping that someone gifts me an Otrio board this year. It’s a strategy and visual perception board game that’s sort of like Set — another great mathematical game — and Tic-Tac-Toe put together. What gifts are you hoping for? Let me know @extremefriday.

Posted in Recreational Mathematics | Tagged , | Leave a comment

The Pseudocontext 2016 Deserves

2016 has been the year of the lolsob. I have my reasons for feeling that way, and I’m guessing you might too. In that light, I’ve especially started looking forward to Dan Meyer’s “pseudocontext Saturday” postsIn each one, he finds a picture from a math book and challenges readers to figure out what math concept is being illustrated or tested with each one. Is a rock-climbing kid illustrating a question about types of quadrilaterals or counting by tens? Does a picture of a dartboard accompany a question about probability, circle sector areas, sequences of numbers, binomials, or the quadratic formula? With connections this tenuous, even if you get the question right, you lose.

Image: Sam Wolff, via Flickr.

Image: Sam Wolff, via Flickr.

What is pseudocontext? Meyer writes, “We create a pseudocontext when at least one of two conditions are met. First, given a context, the assigned question isn’t a question most human beings would ask about it. Second, given that question, the assigned method isn’t a method most human beings would use to find it.” (For my money, the all-time prize for pseudocontext will always be this question from the New York Regents Exam shared by Patrick Honner, though as he states, the story is so flimsy it’s not even pseudocontext.)

Pseudocontext Saturdays don’t just give us an opportunity to lolsob about the bizarre and irrelevant “real-world” questions math textbooks often ask. Commenters can also suggest better questions to ask that go with the picture or that explore the concept the picture was trying to ask about. Felicitously, as I was working on this post, I read Dana Ernst’s post about students generating examples on the MAA blog Teaching Tidbits. That post isn’t about students asking real-world questions necessarily, but it makes me wonder if it’s possible (or desirable) to get students in on the pseudocontext joke: 10 points to Gryffindor for the best math question that would actually relate to the picture in question!

If you’re not already reading Meyer’s blog, there’s a lot more there to enjoy beyond pseudocontext. Meyer is a former high school math teacher who now works for the online graphing calculator Desmos. Though I haven’t spent much time talking math with high schoolers, I appreciate the thought and energy he’s put into figuring out what will reach students the most effectively and how to spur them to ask the questions we want them to be asking about math. As a bonus, his blog is also one of the few places where you can really read the comments. He encourages people to participate and have real conversations in the comments section, often highlighting selected comments in his posts. How refreshing!

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

New PBS Show All About Math

Last week PBS launched a new show on YouTube all about math called Infinite Series. The first three episodes are up and they’re a ton of fun. The show is hosted by Kelsey Houston-Edwards, who is a graduate student at Cornell studying probability theory and the 2016 AMS-AAAS Mass Media Fellow.

In the first installation of the series, Houston-Edwards contemplates the sphere-packing problem, something that we talked about over here a few months ago. Aided by really cool animation and sound effects, she helps us to visualize spheres in higher dimensions and get a sense of how they might be packed. I especially liked her explanation of what happens when you pack spheres in more than 9 dimensions in a box. My mind — much like the sides of that box — was blown!

The second episode takes a somewhat of a philosophical turn. In it Houston-Edwards, who got her bachelor’s degree is in the interdisciplinary study of mathematics and philosophy, asks “Are prime numbers made up?” She delves into some of those tricky questions about whether math was invented, discovered or just…is. A question that certainly vexes those among us who dabble in math and patent law. Houston-Edwards says we can expect a few more episodes of this flavor.

Today I got a chance to catch up with Houston-Edwards to ask her about what’s headed our way in the next few episodes. “There are a couple of episodes like that which just came from personal knowledge, stuff that I just happen to know quite a bit about,” she says, “but the cool part about it, now that it’s aired, people are coming up to me like ‘Oh! You should make an episode about this!’ And that part’s really cool.” And given that she’s the one dreaming up all the ideas of the show I asked her if she was excited for all this feedback. She said, “Totally! I am more than happy to hear any ideas!” So feel free to pitch her all of your most strange and pressing math questions.

Kelsey Houston-Edwards, host of the new PBS online show about math.

Kelsey Houston-Edwards, host of the new PBS web series Infinite Series.

The most recent episode gives a very approachable treatment of the pigeon hole principle by answering that question that I know we all are wondering, “How many humans have the same number of body hairs?” Spoiler: tons and tons.

We can expect a new episode of Infinite Series every Thursday. If you’re interested in becoming a blogger or hosting a YouTube show of your own, a great place to start is with the AAAS Mass Media Fellowship. Evelyn and I are also both proud alumni of the program, and to learn more you can read about my experience at NPR or Evelyn’s experience at Scientific America. The fellowship program is accepting applications now until January 15th.

You can find Kelsey on Twitter @KelseyAHE. And while you’re there, you can find me too, @extremefriday, and let me know what else you’d like to see on this blog.

Posted in Math Communication | Tagged , , , , | Leave a comment

What Should Mathematicians Do Now?

Mathematicians sometimes pretend we are above the everyday vicissitudes of life, preferring to inhabit a realm of abstraction and perfection, but that’s a lie. We live here too. We are voters, citizens, residents, and teachers. What happens in our country matters. I’m sure Anna and I will eventually get back to writing about other parts of the math blogosphere, but the election is still big news, and we as mathematicians need to ask ourselves what to do next.

I know our readers are not a monolith, but a large number of you are mathematicians at universities in the US. I’ve written this post with that in mind, though much of it will be relevant to people in other careers as well. I am also aware that though I did not support Trump, some of my readers probably did. I am not arguing with you about that. I trust that in spite of that difference, we have similar standards for how to treat others, and we are in favor of a strong, healthy culture of math and science research.

So what are mathematicians to do? Many of the actions we take are the same actions any citizens should take right now: talk to our representatives about issues that are important to us, donate to groups that need our help, reach out to friends and family who are feeling scared, and take care of ourselves so we can continue those other actions long-term. But I think there are a few ways to take action that relate specifically to mathematicians and the jobs they do.

1. Keep students safe

In the wake of Trump’s election, many people feel scared. Trump’s rhetoric energized some people who are racist, sexist, Islamaphobic, homophobic, and transphobic. Since the election, there have been numerous reports of hate crimes targeting people of color, religious minorities, and LGBTQIA+ people. Professors should be doing everything they can to make sure their classrooms and campuses are safe.

It’s tempting to think that math classrooms should be politics-free, but the right response to the election is probably not business as usual. Many educators have written about how they’ve talked with their classes since the election. I especially appreciate Jose Vilson’s post: Politics are always at play in our classrooms. We also need to continue promoting diversity in mathematics. One way of doing that is to cut back on the hero-worship of dead white men. Astrophysicist Chanda Hsu Prescod-Weinstein has a list of resources for decolonizing science that can help us do just that. I’ve also written posts with resources about black mathematicians, Hispanic/Latinx mathematicians, and women in math.

One group likely to be at risk in the next administration is undocumented immigrants. If you are concerned about undocumented students, you might consider joining the hundreds of other professors who have signed this petition to extend the Deferred Action for Childhood Arrivals (DACA) program. DACA allows undocumented people who came to the US as children to obtain work permits and remain in the country.

2. Fight misinformation

As Anna mentioned in her last post, there is evidence that misinformation (“fake news”) may have affected the outcome of the election, thanks to the Facebook algorithm bubble. Since then, a lot has been written about how important the phenomenon was to this election and what we need to do to stop it. Cathy O’Neil’s book Weapons of Math Destruction feels especially prescient right now. (Read my review of it here.) Her blog mathbabe.org is one of my go-to resources, and she is part of a New York Times debate about how to best stop the fake news problem. Here are some other things I’ve read recently about fake news and the election:

This Analysis Shows How Fake Election News Stories Outperformed Read News On Facebook by Craig Silverman
Fake News Is Not the Only Problem by Gilad Lotan
The “They Had Their Minds Made Up Anyway” Excuse by Mike Caulfield
Factiness by Nathan Jurgenson
Post-Truth Antidote: Our Roles in Virtuous Spirals of Trust in Science by Hilda Bastian

Fighting misinformation is an area in which I think mathematicians are especially, though certainly not uniquely, equipped to take action. When we write proofs, we are trying to construct watertight arguments using pure logic. Ideally, we attempt to poke holes in our own work until we can ensure that it is impenetrable.

We need to use those skills when we read the news or the outrageous videos our friends share on Facebook, whether we agree or disagree with the conclusions of those stories or videos. Apply the same skepticism to the stories you want to believe are true as the ones you reject. Check Snopes, try to find the numbers instead of taking someone else’s word for it, listen to the full context of the quote, see how other sites are spinning it. Settle for an answer of “it’s complicated” if it is.

An example: in the past few days, a growing number of people have been calling for an audit of the vote in Wisconsin, Michigan, and Pennsylvania (update: as I’m posting this, the audit is looking more and more likely). Those of us who wanted a different outcome could latch on to the story that statistical anomalies make the election look “rigged.” There are a lot of numbers floating around in that article, and it sounds truthy. But J. Alex Halderman, one of the computer scientists urging Clinton to call for a recount, is more measured. “Were this year’s deviations from pre-election polls the results of a cyberattack? Probably not. I believe the most likely explanation is that the polls were systematically wrong, rather than that the election was hacked.” Zeynep Tufekci, a sociologist who studies our relationship to technology, wrote about voting machine vulnerability before the election. Her message is that it’s not likely that it affected this election, but we should be auditing the vote regularly and making sure we leave a paper trail. Halderman’s and Tufekci’s messages aren’t as sexy as “rigged election!” but we need to fight the urge to jump to the sexiest conclusions without sufficient evidence.

How else can we fight misinformation? By supporting real journalism. I recently subscribed to the Washington Post because I’ve found a lot of value in their coverage of Trump’s appointments and financial dealings, but there are many other media outlets that you might find equally or more valuable. The media certainly made mistakes in its coverage of the election, but we still need to support journalism. As subscribers, we should also hold media outlets accountable when they screw up.

We should probably also read more media we disagree with. Yen Duong of Baking and Math recommends the National Review. I recently read “You are still crying wolf” by Scott Alexander of Slate Star Codex. I don’t agree completely with his thesis in that post, but thinking about why instead of dismissing it outright has helped me think about where my preconceived notions come from and how to engage in this conversation.

3. Support climate change research

This is more specific than the above suggestions, but a Trump advisor recently suggested that we should defund NASA’s climate change research. Climate change is likely the most pressing issue of our time. We have to keep studying it and try to find ways to mitigate the damage it is causing.

4. Read history

I hope the people who are warning us that the US is falling into authoritarianism/fascism/kleptocracy are wrong. Or that their warnings help us avoid those dire predictions. But it has happened before, and it can happen again. I think mathematicians would do well to read up on the history of math in Göttingen in the 1930s, perhaps in this Notices article from 1995 by Saunders Mac Lane.

Finally, I’ll leave you with this post by Matilde of the blog Listening to Golem about the moral responsibilities of mathematics and science: “Pack all the tools you need in your bag: network theory, bayesian analysis, probability, differential equations, cryptography, computing, game theory, neural networks. We need them all and we need them now. Get down to work for the sake of our future.”

Posted in History of Mathematics, people in math | Tagged , , , , | 7 Comments

As The Dust Settles, Let’s Check The Numbers

I really didn’t want to write about the election. But probably, much like you, it’s all I can think about right now. News media is completely saturated with it and the blogs are churning out a steady stream of predictions and post mortem.

Even among those with college degrees and 2016 election was a divisive one.

Even among those with college degrees and 2016 election was a divisive one.

When we consider voters with post-graduate degrees, then we really start to see the ivory tower effect.

When we consider voters with post-graduate degrees, then we really start to see the ivory tower effect.

As an academic, I’m really troubled, but also fascinated by what this election and the reactions to it on college campuses tells us about the state of higher ed. Many of us wrestled with what to do last Wednesday when we stood in front of a room full of wide-eyed millennials. As Beth wrote over on the blog PhD Plus Epsilon, it was tough. The responses on college campuses have been extreme, and they tell a story perhaps different from the one we imagined.

On the one hand, the popular notion is of the ivory tower as a liberal bastion, and yet news anecdotes are giving the impression of college campuses which are massively divided, even in the least purple of states. But after the dust begins to settle, we can begin to try and understand more by looking at the numbers.

Several data sets published by the researches at The Chronicle analyze the voting outcomes over the past several elections across different swaths of the academic ecosystem. And it appears that our students (being people, I guess technically, without college degrees) have a much greater polarity that we do on the faculty. From these numbers, the whole liberal bastion ivory tower business doesn’t even seem to apply to students in the universities. For reasons I won’t delve into right here and now, I consider this a bit strange.

This data, gathered by the Chronicle of Higher Ed, shows a tendency towards the left in counties housing flagship universities.

This data, gathered by the Chronicle of Higher Ed, shows a tendency towards the left in counties housing flagship universities.

As the Chronicle of Higher Ed reported this week, college towns tend to be more liberal than the states they inhabit. In their research, counties that housed flagship universities tended to view the republican candidate less favorably that the state as a whole. Wisconsin-Madison was a particularly good example. The republican candidate won the state by about 1% but lost Dane County (home of University of Wisconsin – Madison) by about 48%. So the take-home here is that college towns lean left, which isn’t really a surprise. But then we also need to keep in mind that those left leanings are coming largely from the university affiliates and residents of the counties, not students themselves, since they typically aren’t registered to vote in the same place they go to college.

But in any case, I don’t exactly take heart at the sight of this data, because those numbers aren’t telling us college campuses are unified, as much as they are telling us that we as universities are alienated from our surroundings, and we as faculty are alienated from our students, which doesn’t feel so great. The Chronicle team generated several other data sets to explore the demographics of the vote across academic cross sections, also considering race and gender.

There are still plenty of lingering questions about what all of this means for us, as educators and academics. The Chronicle of Higher Ed is keeping us up to date with a lot of this in their series A Stunning Upset (my apologies, many of their articles are behind a paywall). Specific questions that I am concerned about, include what this all means for federal funding for research and, more broadly, for universities? Will the changing priorities of the government be reflected in changing priorities of institutions? And what’s in store for students who are still in the process of financing their educations?

I guess we wait and see.

I should also mention, as a mathematician, there is also a lot of interesting conversation going on about the efficacy of polling, and how biased algorithms might have shaped the outcome of the election. Blogger extraordinaire, Cathy O’Neil, has done some particularly great work in the past week discussing some of the data driven pitfalls of utter chaos 2016. Among other things, this election and the journalism surrounding it has reminded me how important it is to understand where numbers come from. An infographic with a needle swinging side-to-side is all well and good when it’s swinging in the proper direction, but when it starts to lean the other way, suddenly I’m forced to ask myself, “Wait, what am I even looking at right now?”

Posted in Data Science, Events, Issues in Higher Education | Tagged , , , | Leave a comment

Which One Doesn’t Belong?

1, 2, 4,…. What’s the next number in the sequence? I was a rule-follower as a kid, so I always got the “right” answer on questions like that, but they still bugged me. Sure, 8 would be predictable, but why couldn’t it be 7, 9, or 34 million, for that matter? It seemed like we were making an awful lot of assumptions about how sequences were going to behave without much evidence. Pattern recognition is an important part of doing math, but so is the skepticism that made me feel uneasy when I predicted what a sequence would do based on just a few beginning terms. Owen Elton describes why any answer would be “correct” using one of those awful Facebook “only 1 in a thousand will get it” math riddles that pops up every now and then.

Christopher Danielson’s book Which One Doesn’t Belong and Mary Bourassa’s blog of the same name would have been great for me as a kid. Each page in the book is a set of four shapes, and you have to say which one doesn’t belong. But any answer can be “right.” Each prompt can start a discussion of what traits the shapes/numbers/graphs have in common and do not. Instead of learning the one right answer and moving on, kids can discuss which answers jumped out at them and why. They can have open-ended conversations about math rather than just trying to find the one right answer.

I’ve seen posts about #wodb all over the #MTBoS, so I won’t even try to link to everyone who’s talked about using these prompts in the classroom, but I do want to mention Tracy Zager, who has a thoughtful post about using “which one doesn’t belong” in a second-grade classroom and the way open-ended math discussions can get both students and teachers thinking about what math words mean.

Danielson also writes the blog Talking Math with Your Kids, which aims to foster mathematical reasoning skills in early childhood by helping parents have low-stress conversations about math with their kids. Yes, please!

Helping parents have low-stress conversations about math with their kids is the aim of Bedtime Math, an app and blog. Each day it gives parents a fun prompt and some questions to start the discussion. I also love reading Malke Rosenfeld (currently blogging at Math in Unexpected Spaces) and Mike Lawler of Mike’s Math Page, who talk to their kids about math a lot. (I got nerdsniped yesterday by a fun area question from Lawler’s blog.)

I don’t have kids, so I’m mostly a bystander in talking math with kids, but I do have two young goddaughters. When we get together, we often count things together, and I hope as they grow up, I can keep talking with them about math in ways that are age-appropriate and fun. Reading blogs like Danielson’s, Zager’s, Rosenfeld’s, and Lawler’s and following the #tmwyk hashtag on Twitter are helpful for me when I’m thinking about how to talk with my goddaughters about math. I’m also partial to the #wodb hashtag. It’s just fun to see the cool mathematical “which one doesn’t belong” pictures created by both students and teachers. I’m hoping that in a few years, my goddaughters and I will be making some of them for ourselves.

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

Math Games Might Be Sort Of Good For Your Brain

Good news, all that time you spent playing World of Warcraft might have made you smarter. A study out of Stanford just showed that playing video games just 10 minutes each day can make you better at math. The study involved two cohorts of third grades, one group was taught math in the standard manner, and the other was given 10 minutes each day to play the game Wuzzit Trouble on school issued iPads. The video game playing group demonstrated significant improvement in what the researchers call “number sense,” including an improved ability to “apply their number sense to an unconventional problem.”

RoTopo may not give you number sense, but make you understand how cubes work.

RoTopo may not give you number sense, but make you understand how cubes work.

But wait, whoa, not so fast. Before you pick up that controller, I should note that the study was based on a very particular game. Certainly other similar games might see a similar effect, but I suspect Pokemon does very little to improve one’s number sense. And according to people who know a lot about this sort of thing, most math video games marketed as educational tools aren’t even all that impressive.

Incidentally, this study was well-timed to coincide with the American Academy of Pediatric (AAP) revised guidelines for kids and screentime, namely, that screens aren’t as bad for little brains as they initially thought.

I don't mean to brag, but...

I don’t mean to brag, but…

But for the full-grown brains of grown-up mathematicians, the internet is a cornucopia of highly educational addictive math games. In a recent post on Math Munch I learned about several online games exploring geometry and dimension. Rotopo is really fun, and has really soothing background music. The game doesn’t necessarily address math explicitly, but there is a strategy and spatial sense that you gain from the geometry of the game. If you need to procrastinate, then I highly recommend it.

Another game that recently came to my attention (tip of the hat to TJ Hitchman) is Euclidia. Using points, lines and circles (i.e. pencil, straightedge, and compass) you move through a series of exercises to construct equilateral triangles, perpendicular bisectors and increasingly difficult geometric constructions. I didn’t think it would be as fun as it is, and yet I started playing and couldn’t stop. The premise is simple, and yet it’s kind of a rush to actually do the constructions. And I would imagine even cooler if you were doing them for the first time. If I were teaching Euclidean geometry I would love to find a way to integrate this game into the course.

Apps are also a great place to get your math video fix. For the commuting and smart-phone wielding mathematician, The Aperiodical gave a good roundup of worthwhile games. Evelyn Lamb also wrote about games for understanding hyperbolic space for Scientific American.

And I just finally beat 2048, the most annoyingly addictive game to come out since snood (remember that?). So I’m feeling pretty good.

Posted in Math Education, Recreational Mathematics | Tagged , , , , | 1 Comment