Summer Odds and Ends

I promise I’ll start blogging again. But as followers of this blog might know, I like to take the summer off–both from teaching and blogging. I never take a break from math, though. Here are some fun things I’ve seen recently. Consider it my own little math carnival :-).

I love this comic, especially as I start my stat grad class this semester @ JHU. After this class, I’ll be half-way done with my masters. It’s a long road! [ht: Tim Chase]

Speaking of statistics, my brother also sent me this great list of lottery probabilities. Could be very useful in the classroom.

These math dice. Honestly I don’t know what I’d do with them, but you have to admit they’re awesome. [ht: Tim Chase]

These two articles about Khan academy and the other about edX I found very interesting. File all of them under ‘flipping the classroom.’ I’m still working up the strength to do a LITTLE flipping with my classroom. My dad forwarded these links to me. He has special interest in all things related to MIT (like Khan, and like edX) since it’s his alma mater.

I’ll be teaching BC Calculus for the first time this semester and we’re using a new book, so I read that this summer. Not much to say, except that I did actually enjoy reading it.

I also started a fabulous book, Fearless Symmetry by Avner Ash and Robert Gross. I have a bookmark in it half way through. But I already recommend it highly to anyone who has already had some college math courses. I just took a graduate course in Abstract Algebra recently and it has been a great way to tie the ‘big ideas’ in math together with what I just learned. The content is very deep but the tone is conversational and non-threatening. (My dad, who bought me the book, warns me that it gets painfully deep toward the end, however. That’s to be expected though, since the authors attempt to explain Wiles’ proof of Fermat’s Last Theorem!)

I had this paper on a juggling zeta function (!) sent to me by the author, Dr. Dominic Klyve (Central Washington University). I read it, and I pretended to understand all of it. I love the intersection of math and juggling, and I’m always on the look out for new developments in the field.

And most recently, I’ve been having a very active conversation with my math friends about the following problem posted to NCTM’s facebook page:

Feel free to go over to their facebook page and join the conversation. It’s still happening right now. There’s a lot to say about this problem, so I may devote more time to this problem later (and problems like it). At the very least, you should try doing the problem yourself!

I also highly recommend this post from Bon at Math Four on why math course prerequisites are over-rated. It goes along with something we all know: learning math isn’t as ‘linear’ an experience as we make it sometimes seem in our American classrooms.

And of course, if you haven’t yet checked out the 90th Carnival of Mathematics posted over at Walking Randomly (love the name!), you must do so. As usual, it’s a thorough summary of recent quality posts from the math blogging community.

Okay, that’s all for now. Thanks for letting me take a little random walk!

87th Carnival of Mathematics

The 87th Carnival of Mathematics has arrived!! Here’s a simple computation for you:

What is the sum of the squares of the first four prime numbers?

That’s right, it’s

Good job. Now, onto the carnival. This is my first carnival, so hopefully I’ll do all these posts justice. We had lots of great submissions, so I encourage you to read through this with a fine-toothed comb. Enjoy!


Here’s a post (rant) from Andrew Taylor regarding the coverage from the BBC and the Guardian on the Supermoon that occurred in March 2011. NASA reports the moon as being 14% larger and 30% brighter, but Andrew disagrees. Go check out the post, and join the conversation.

Have you ever heard someone abuse the phrase “exponentially better”? I know I have. One incorrect usage occurs when someone makes the claim that something is “exponentially better” based on only two data points. Rebecka Peterson has some words for you here, if you’re the kind of person who says this!

Physics and Science-flavored

Frederick Koh submitted Problem 19: Mechanics of Two Separate Particles Projected Vertically From Different Heights to the carnival. It’s a fun projectile motion question which would be appropriate for a Precalculus classroom (or Calculus). I like the problem, and I think my students would like it too.

John D. Cook highlights a question you’ve probably heard before: Should you walk or run in the rain? An active discussion is going on in the comments section. It’s been discussed in many other places too, including twice on Mythbusters. (I feel like I read an article in an MAA or NCTM magazine on this topic once, as well. Anyone remember that?)

Murray Bourne submitted this awesome post about modeling fish stocks. Murray says his post is an “attempt to make mathematical modeling a bit less scary than in most textbooks.” I think he achieves his goal in this thorough development of a mathematical model for sustainable fisheries (see the graph above for one of his later examples of a stable solution under lots of interesting constraints). If I taught differential equations, I would  absolutely use his examples.

Last week I highlighted this new physics blog, but I wanted to point you there again: Go check out Five Minute Physics! A few more videos have been posted, and also a link to this great video about the physics of a dropping Slinky (see above).

Statistics, Probability, & Combinatorics

Mr. Gregg analyzes European football using the Poisson distribution in his post, The Table Never Lies. I liked how much real world data he brought to the discussion. And I also liked that he admitted when his model worked and when it didn’t–he lets you in on his own mathematical thought process. As you read this post, you too will find yourself thinking out loud with Mr. Gregg.

Card Colm has written this excellent post that will help you wrap your mind around the number of arrangements of cards in a deck. It’s a simple high school-level topic, but he really puts it into perspective:

the number of possible ways to order or permute just the hearts is 13!=6,227,020,800. That’s about what the world population was in 2002. So back then if somebody could have made a list of all possible ways to arrange those 13 cards in a row, there would have been enough people on the planet for everyone to get one such permutation.

I think it’s good to remind ourselves that whenever we shuffle the deck, we can be almost certain that our arrangement has never been created before (since  52!\approx 8\times 10^{67}  arrangements are possible). Wow!

Alex is looking for “random” numbers by simply asking people. Go contribute your own “random” number here. Can’t wait to see the results!

Quick! Think of an example of a real-world bimodal distribution! Maybe you have a ready example if you teach stat, but here’s a really nice example from Michael Lugo: Book prices. Before you read his post, you should make a guess as to why the book prices he looked at are bimodal (see histogram above).

Philosophy and History of Math

Mike Thayer just attended the NCTM conference in Philadelphia and brings us a thoughtful reaction in his post, The Learning of Mathematics in the 21st Century. Mike wrote this post because he had been left with “an ambivalent feeling” after the conference. He wants to “engage others in mathematics education in discussions about ways to improve what we do outside of the frameworks that are being imposed on us by those outside of our field.” As a secondary educator, I agree with Mike completely and really enjoyed his post. Mike isn’t satisfied with where education is going. In his post, he writes, “We are leaping ahead into the unknown with new educational models, and we never took the time to get the old ones right.”

Edmund Harriss asks Have we ever lost mathematics? He gives a nice recap of foundational crises throughout the history of mathematics, and wonders, ultimately, if we’ve actually lost any mathematics. There’s also a short discussion in the comments section which I recommend to you.

Peter Woit reflects on 25 Years of Topological Quantum Field Theory. Maybe if you have degree in math and physics you might appreciate this post. It went over my head a bit, I’m afraid!

Book Reviews

In this post, Matt reviews a 2012 book release, Who’s #1, by Amy N. Langville and Carl D. Meyer. The book discusses the ranking systems used by popular websites like Amazon or Netflix. His review is thorough and balanced–Matt has good things to say about the book, but also delivers a bit of criticism for their treatment of Arrow’s Impossibility Theorem. Thanks for this contribution, Matt! [edit: Thanks MATT!]

Shecky R reviews of David Berlinski’s 2011 book, One, Two Three…Absolutely Elementary mathematics in his Brief Berlinski Book Blurb. I’m not sure his review is an *endorsement*. It sounds like a book that only a small eclectic crowd will enjoy.


Peter Rowlett submitted this post about linear programming and provides a link to an interactive problems solving environment.

Peter Rowlett also weighs in on the recent news about a German high school boy who has (reportedly) solved an open problem. Many news sources have picked up on this, and I’ve only followed the news from a distance. So I was grateful for Peter’s comments–he questions the validity of the news in his recent post “Has schoolboy genius solved problems that baffled mathematicians for centuries?” His comments in another recent post are perhaps even more important though–Peter encourages us to think of ways we can remind our students that lots of open problems still exist, and “Mathematics is an evolving, alive subject to which you could contribute.”

Jess Hawke IS *Heptagrin Girl*

Here’s a fun-loving post about Heptagrins, and all the crazy craft projects you can do with them. Don’t know what a Heptagrin is? Neither did I. But go check out Jess Hawke’s post and she’ll tell you all about them!

Any Lewis Carroll lovers out there? Julia Collins submitted a post entitled “A Night in Wonderland” about a Lewis Carroll-themed night at the National Museum of Scotland. She writes, “Other people might be interested in the ideas we had and also hearing about what a snark is and why it’s still important.” When you check out this post, you’ll not only learn about snarks but also about creating projective planes with your sewing machine. Cool!

Mike Croucher over at Walking Randomly gives a shout out to the free software Octave, which is a MATLAB replacement. Check out his post, here. MATLAB is ridiculously expensive, and so the world needs an alternative like Octave. He provides links to the Kickstarter campaign–and Mike has backed the project himself. I too believe in Octave. I’ve used it a few times for my grad work and I’ve been very grateful for a free alternative to MATLAB.

The End 

Okay, that’s it for the 87th Carnival of Mathematics. Hope you enjoyed all the posts! Sorry it took me a couple days to post it–there was a lot to digest :-).

If you missed the previous carnival (#86), you can find it here. The next carnival (#88) will be hosted by Christian at For a complete listing of all the carnivals, and more information & FAQ about the carnivals, follow this link.



This article about the saddle-shape of Pringles is a joy to read [ht: Prisca Chase]. I’ll give you an excerpt, but I encourage you to read the whole thing. It’s both mathematically stimulating and extremely funny:

Saddle up for maximum snack satisfaction (mathematically speaking)

Stephanie V.W. Lucianovic

My husband is a calculus professor and one who brings food items into the classroom with surprising regularity. No, he doesn’t bring pies on Pi day – though he can recite the string up to a couple dozen digits – but he does bring Pringles. As a teaching aid.

This afternoon when I walked into his study, I nearly tripped over a plastic Safeway bag filled with six red cans of Pringles. “Is it Pringles Day already?” I asked, nudging the bag. Pringles Day is the day Dr. Mathra lectures on the classification of critical points in multivariable calculus, and he uses the saddle-shaped Pringles to illustrate his points.

After class, the students get to eat his illustrations. It’s their favorite day.


Later in the article, the fact that a Pringle can’t be made from a sheet of paper is mentioned. For a normal sheet of paper, this is true. But you can fold paper in such a way as to approximate a hyperbolic parabaloid. I’ve mentioned this before here and here. So go try it!

Math paper contains no math

I feel like I need to post about this too–just to get the word out.

At first I thought it was funny, but now it just makes me angry. I first heard about this paper thanks to my brother, Tim Chase, who shared this news via Retraction Watch. Then today I learned a bit more information by way of Alexander Bogomolny and his blog.

Okay, what’s going on? Authors M. Sivasubramanian and S. Kalimuthu have published this completely nonsensical math paper, and here’s what Retraction Watch had to say:

Have a seat, this one’s a howler.

According to a retraction notice for “Computer application in mathematics,” published in Computers & Mathematics with Applications:

This article has been retracted at the request of the Publisher, as the article contains no scientific content and was accepted because of an administrative error. Apologies are offered to readers of the journal that this was not detected during the submission process.


Go read the whole paper in full text available here. At the very least, this paper has been retracted. That’s good.

But sadly, S. Kalimuthu and his coauthors are responsible for many other terrible papers too (seriously, go check them out!). How does this happen? Can anyone explain it? And why hasn’t he been stopped?

In one paper in particular, he has completely plagiarized Alexander Bogomolny’s site–as one commenter noticed. Check out Alexander’s blog CTK Insights for his coverage.

Like I said, this man needs to be stopped.


When cars collide

[Another guest column from Dr. Gene Chase.]

Suppose two equally weighted cars collide in a head-on collision, each traveling at 50 miles per hour.  Do you think that the impact for one car will be more severe on the car and driver than the impact of that car’s hitting a brick wall?

To be fair, we have to assume that neither the cars nor the wall compress at all.  If the wall is as soft as a pillow, I’ll take the wall every time.

Marilyn vos Savant’s recent column in Parade Magazine says that hitting an oncoming car in that way is no more severe than hitting a solid wall.   They both stop dead, whether the wall or the other car causes it.

Each experiences a momentum change that is the same as if they hit a wall, not twice as much. That’s clear when I think of it now, using the law that momentum = impulse (that is, mass * velocity = force * time) but I’ve been mistaken when I’ve only thought about it casually, thinking it must be a 100 mph impact..

If a bike hits a car head-on, the situation is different, because the “bike-car” combination will continue to move in the direction of the car, so my intuition is correct in that case:  The bike driver fares worse than the car driver.  Comments at Marilyn vos Savant’s blog say as much.

I used to think that car bumpers that collapse at the slightest impact were poorly made.  In fact, if momentum is constant, extending the time of impact will decrease the force, to keep force * time constant.

Give me “cheap” bumpers and a wall made of pillows every time.

Minnesota Senator loves math!

I really enjoy reading J. Michael Shaughnessy’s column. He’s the president of the NCTM and always has interesting, timely things to say about math and math education. Here’s an excerpt from this week’s column, where he recounts his recent conversation with Senator Al Franken (D-Minn) as he eagerly shared a proof with President Shaughnessy. Go check it out!


Seen Any Good Proofs Lately? Raising the Social Currency of Mathematics

We all probably have had a friend or acquaintance, or even a perfect stranger, raving about a book she has just read, or a movie he has recently seen, and then saying, “Oh, you must read this book!” or, “You must see that film!” But how many of us have had this kind of experience in a social occasion where the person exclaimed, “Oh, you must see this proof!” So it was indeed refreshing to meet someone who really likes mathematics, as I did several weeks ago, in what might seem like a very unlikely setting—the Hart Senate Office Building in Washington, D.C.

On Wednesday mornings when Congress is in session, Senator Al Franken (D-Minn.) holds a breakfast gathering in his office for his constituents. Visitors to the breakfast consist primarily of people from Minnesota, but I received an invitation from a mathematics teacher who is spending the year working on the senator’s staff. A famous hearty porridge is served up at these breakfasts, and once guests have begun to circulate, Senator Franken drops in and greets everyone. I had been misinformed and thought that the Senator had been a mathematics major in college. When I asked him about this, he said that the rumor was false, but he agreed that his good grades in math had probably helped him get admitted to college.

After breakfast, the visitors were escorted to a terrace area in the hallway outside the office, where the senator spoke for a few minutes about events being debated in Congress and answered questions. Guests then lined up to have their pictures taken with the senator. I was at the end of the line, and as I shook his hand and introduced myself as the president of NCTM, he said, “Let me show you my geometric proof of the Pythagorean theorem!” Senator Franken then proceeded to grab scratch paper and a pen from one of his staffers and plopped down cross-legged on the hallway carpet. As I sat next to him, he began to sketch out his proof. He explained what he was doing, and why it worked, and I paraphrased each move he made so that it was clear to both of us how he was thinking and what he was doing.





Cooking math


This article by Samuel Arbesman came through today on

Are There Fundamental Laws of Cooking?

Cooking is a field that has in recent years seen a shift from the artistic to the scientific. While there are certainly still subjective and somewhat impenetrable qualities to one’s cuisine — de gustibus non est disputandum — there is an increasing rigor in the kitchen. From molecular gastronomy to Modernist Cuisine, there is a rapid growth in the science of cooking.

And mathematics is also becoming part of this. For example, Michael Ruhlman has explored how certain ingredient ratios can allow one to be more creative while cooking. Therefore, it should come as no surprise that we can go further, and even use a bit of network science, when it comes to thinking about food.

Yong-Yeol Ahn and his colleagues, in a recent paper titled Flavor network and the principles of food pairing, explored the components of cooking ingredients in different regional cuisines. In doing so, they were able to rigorously examine a recent claim: the food pairing hypothesis. The food pairing hypothesis is the idea that foods that go best together contain similar molecular components. While this sounds elegant, Ahn and his collaborators set out to determine whether or not this is true.

Using recipes from such websites as Epicurious, the researchers examined more than 50,000 recipes. They combined these recipe data with information about the chemical components in each of the ingredients, in order to create a network map of related ingredients. For example, shrimp and parmesan are connected in the network, because they contain the same flavor compounds, such as 1-penten-3-ol. A large flavor network of different ingredients is [above].


He later gives a reference to George Hart’s “Incompatible Food Triad” problem and the associated website:

An example solution would be three pizza toppings — A, B, and C — such that a pizza with A and B is good, and a pizza with A and C is good, and a pizza with B and C is good, but a pizza with A, B, and C is bad. Or you might find three different spices or other ingredients which do not go together in some recipe yet any pair of them is fine.

Has any of this ever crossed your mind? Me neither.




LEGO math

This article was just posted to today and is an interesting summary of some research from 2002–but it is new to me. Here’s an excerpt from Samuel Arbesman’s article:

Most objects are made up of smaller parts, combined in complicated and diverse ways… In the wonderfully titled paper Scaling of Differentiation in Networks: Nervous Systems, Organisms, Ant Colonies, Ecosystems, Businesses, Universities, Cities, Electronic Circuits, and Legos,Mark Changizi and his colleagues set out to understand this concept. They found that in every single one of the systems in the wildly interdisciplinary list of the subtitle there was an increase in the number of types of components as the total number of pieces grew. The larger something is, the more types of building blocks it uses.

And this includes, of course, Lego bricks. Using a dataset of 389 Lego sets (this was done back in 2002, so if anyone can download the data easily, I would love to see if the results hold up with a richer dataset), they examined the number of distinct types of pieces in a set versus the total number of pieces in that set (examples of sets include “Air Patrol”, “Spy Boat”, and “Cargo Crane”, and a master list of Lego piece types is here).

They found that the number of piece types to total number of pieces could be fit nicely to a power law. Here it is on a log-log scale:

This curve demonstrates that as the number of pieces in a set grows, so do the number of piece types. However, the number of piece types grows sublinearly: while a larger set uses more piece types, as sets becomes larger, they use progressively fewer additional piece types (so larger sets actually use fewer types per piece). This is similar to other sublinear curves, where larger animals use less energy per cell for metabolism or larger cities actually need fewer gas stations per capita. Essentially, larger sets become more efficient, using the same pieces that smaller sets do, but in a more complex and diverse way.



Now, just for fun, here’s a video of a  great LEGO contraption (HT: Tim Chase).

Also, just for fun, here’s a photo of a 4-foot Lego 737 that my friend Matthew and I built. We’ve actually finished most of it, I just don’t have a recent picture. (Notice in this photo the roof, tail, and wings are missing.) But this gives you a taste:







Teaching math with applications

Wow, if the title of this post didn’t grab you, I don’t know what will. Pretty riveting, right? Has anyone ever thought of teaching math with applications? </end sarcasm>

This is the basic thesis of a recent article from the NY Times, “How to Fix Our Math Education” by Sol Garfunkel and David Mumford:

THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.

All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.

Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.

For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a “group of transformations” or a “complex number”? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.



To be fair, the real thesis–if you read further in the article–is that we should primarily teach applications and math can swoop in and rescue us if and when it’s needed:

Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.

If you haven’t gathered already, I don’t agree at all with this thesis. It’s my opinion that math should be taught as math, respected as its own field of study, and a valuable part of a high school liberal arts curriculum. Students should value math for its inherent, abstract beauty. Applications are of course a must in any course. But I find that in the text resources I’ve used, the applications are often contrived. Extremely contrived. Doing math should feel like playing a game, like working on a puzzle, or like arguing.

In fact, when high school students ask why are we learning this?, I FIRST respond with the things I just said: It’s part of a liberal education; it will make you a well-rounded, intelligent person who can hold conversations with other smart people in other fields; and it’s fun. I mention SECOND what applications exist for the math we’re learning. For high school students, if we’re honest, most of them will never need any of the math we’re teaching. Seriously. If you’re not working in a math or science field, when was the last time you had to factor a polynomial?

The authors go on to say “Science and math were originally discovered together, and they are best learned together now.” But this is not universally true. In many cases, the ‘useless’ math was developed first (think of number theory for instance) and then only later were applications discovered (think of the RSA or El Gamal public key encryption schemes).

So why learn math? There was a nice post about this yesterday on one of my new favorite blogs, dy/dan, titled “Cornered By The Real World.” He highlights this great article by Samuel Otten in this August’s Math Teacher magazine. I highly recommend reading the whole article. As a taste, I’ll include the same snippet that Dan shared:

I believe that thinking and acting as if the justification for teaching and learning mathematics is found solely in everyday applications can be dangerous. Mathematics does not exist only to serve other professions, nor is it merely a collection of algorithms and procedures for dealing with real-world situations. Such a mind-set essentially paints our discipline into a weak and lonely corner and leaves undefended many of its greatest aspects.

I could say more about this, but I’ll spare you. I’m passionate about making math a subject worth learning all on its own. If I say more, I’ll start to sound like Paul Lockhart.