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!

Rants

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.

Uncategorized…

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 checkmyworking.com. For a complete listing of all the carnivals, and more information & FAQ about the carnivals, follow this link.

Cheers!

New Physics Blog

Shout out to Chase Martin, who has just started a great physics blog, Five Minute Physics. My friend Chase and I have a lot in common:

  • Our names
  • Our love for juggling
  • Our love for math & physics
  • Our love for teaching
  • Our blogs

Chase is awesome, and you’ll love his fun-loving lecture style in these videos. His goal is ambitious: to put the entire lecture content of his high school physics course on youtube. Wow! File this under ‘flipping the classroom.’

Here are a few of his first videos for your enjoyment.

Go check out his website for more!

 

PS: If you haven’t checked out Minute Physics yet, it’s also a great youtube channel with fun entertaining videos!

More on Microsoft Equation Editor

As some of you know, I recently posted about Microsoft Equation Editor (here) and the way it’s been totally upgraded. I’ve been using Microsoft’s Equation Editor more and more, and I’ve learned a lot of new things, but I also still have questions (for instance, how do you force it to do display or in-line mode?).

Before, when I had questions, it seemed like Microsoft had no answers. I searched their website and found minimal help. I found help from third-parties, like this wonderful cheat-sheet which I still highly recommend. But today when I went searching for some more answers, I found this page on Microsoft’s website, which I swear wasn’t online two months ago.

The most interesting thing is that they mention their use of Unicode Nearly Plain-text Encoding of Mathematics and they claim that the Microsoft Equation editor adheres to the standards set forth in Unicode Technical Note 28.  I’ve now completely read this Unicode guide and it was very helpful.

I think I can finally use the new Microsoft Equation Editor without ever leaving the keyboard.

In particular, here are a few things I learned how to do. Hopefully this will save you the time of having to read through it all yourself:

Tips & Tricks with the new Microsoft Equation Editor

To start with, here are a handful of things I didn’t know how to do without visiting the toolbar. Now I can do them just by typing.

Boxed formula:   \rect(a/b) produces

boxed formula

Matrix:   (\matrix(a&b@&c&d))   produces

matrix

Radicals:  \sqrt(5&a^2)    produces

radical

Equation arrays are something I found hard to do in Microsoft Equation Editor. In their documentation, I learned you can type “Shift+Enter” to keep the next line as part of the same equation array. But here’s the more finely-grained method:

\eqarray(x+1&=2@1+2+3+y&=z@3/x&=6)

resolves to this:

equation array

A more complicated example of alignment, and a description of how it is interpreted comes from the Unicode page:

3.19 Equation Arrays
To align one equation relative to another vertically, one can use an equation array, such as

system of equationswhich has the linear format █(10&x+&3&y=2@3&x+&13&y=4), where █ is U+2588. Here the meaning of the ampersands alternate between align and spacer, with an implied spacer at the start of the line. So every odd & is an alignment point and every even & is a place where space may be added to align the equations. This convention is used in AmSTeX.

Instead of █, one can type \eqarray in Microsoft office. Also, to include a numbered equation is simple:  E=mc^2#(30).

Another nice thing I learned is how to quickly include text in your equations, without having to visit the toolbar (in retrospect, it’s somewhat obvious):

 “rate”=”distance”/”time”

resolves to

\text{rate}=\frac{\text{distance}}{\text{time}}

Like I said, one unresolved issue I still have is how to force math to be displayed in ‘in-line’ or ‘display’ mode. This is very easy in \LaTeX with the use of $ or $$. Section 3.20 of the Unicode notes isn’t very satisfying:

Note that although there’s no way to specify display versus inline  modes (TeX ‘s $ versus $$), a useful convention for systems that mark math zones is that a paragraph a paragraph consisting of a math zone is in display mode.  If any part of the paragraph isn’t in a math zone including a possible terminating period, then inline rendering is used.

So there you have it–more of what I’ve learned about the Microsoft Equation Editor. Please do share if you have other useful information.

Microsoft Office Equation Editor

Even though I’d love to say I use \LaTeX for everything, I actually only use it for my grad school assignments. I don’t use it for all my worksheets and assessments. There is a teacher in our math department who does use \LaTeX for everything, but it’s not me.

That being said, Microsoft has made a significant upgrade to its equation editor with the release of Office 2007 (I know, pretty stale news–but my school just upgraded this past year) and \LaTeX lovers will love it if they haven’t tried it yet. The old Microsoft Equation 3.0 which shipped with earlier Office products had a few shortcuts, but it was still pretty hard to type equations without using the toolbar. Color-coding was problematic, and equation objects didn’t respond to font-size changes or other formatting properties. Animations in powerpoint were also difficult.

The new equation editor is much better for the following reasons:

1. The shortcuts are amazing, and most simple \LaTeX commands work. For a complete list of shortcuts go here for a great pdf cheat sheet. You can even add your own custom commands if you go into your options to Proofing > AutoCorrect Options and click on the “Math AutoCorrect” tab. Also, pressing Alt+= will immediately launch the editor. So inserting an equation is fast and you never need to leave the keyboard.

2. Most calculator-style syntax is accepted as well. So typing 3^x [space] / 4^y [space][space] results in \frac{3^x}{3^y}, without any extra effort. Tapping the spacebar will automatically convert your calculator syntax into pretty display math. For a more complicated example, consider this:

\lim_{n\to\infty} \frac{(2n+1)(3n-2)}{4n^2}=\frac{3}{2}

produced by typing “lim_(n\to\infty)[space]((2n+1)(3n-2))/(4n^2)[space]=3/2[space].”

3. As hinted above, the new equation editor responds to all the normal font formatting options in Microsoft Office. You can color your formulas, you can change the font size, and you can apply any other text effect like shadow/glow/outline/etc. [edit: Though you can change all those things, no, you cannot change the font face. There are a limited number of fonts available for use, and the only one I know of is the default, Cambria Math–if you know of another one, please share!]

4. In powerpoint, animations are quite a bit easier, since you can do all the equations in-line as part of the text, rather than juggling scads of different text and equation objects.

For more on Microsoft’s new  Equation Editor, please check out my more recent post here!

Bedtime Math

 

 

This math website has a great idea: If we can read bedtime stories to our children each night to increase literacy, shouldn’t also be reasonable to do a math problem before bed each night too? My sister and brother-in-law already do this, and it’s great fun. But this website does all the work for you and you can even sign up for a daily problem in your email. All of this is thanks to website creator, Laura Bilodeau Overdeck. Check it out here.

When my little daughter is a bit older, I’ll definitely be giving her some daily math problems.

 

[HT: Tim Chase, via Geekdad]

The Manga Guide to Calculus

This summer I finally finished reading the Manga Guide to Calculus by Hiroyuki Kojima and Shin Togami. Here are my two cents:

The Manga Guide to Calculus is chocked full of great mathematics and lots of quality comic art (the author went to great lengths to ensure it was authentic manga, with illustrations by popular artist Shin Togami).

That being said, I don’t think anyone could ever learn Calculus using this book. In fact, I think Kojima must know that. He never claims this can be used as a textbook replacement. The math isn’t presented in a very systematic way, and there are very few real exercises for the reader. Right from the beginning he puts heavy emphasis on linear approximation. He takes a very different approach to presenting Calculus than a math book would. It is a story most of all. Kojima, in his preface, says its a great book for those who already have Calculus knowledge–both for those who love Calculus and for those who have been “hurt by it.” I tend to agree.

As for the story, well, it’s a bit contrived. But what story that tries to smuggle in some math doesn’t seem a little contrived? Sometimes it’s a bit of a stretch and the story suffers. You should still give it a chance, though.

So to those looking for a Calculus textbook, you need to look elsewhere. For instance, I was looking for things I might be able to use in the Calculus class I teach, but couldn’t find much usable content. But for those interested in math and are looking for a fun read, I would recommend picking it up.

Khan Academy

I’ve mentioned Khan Academy before, asking “could your math teacher could be replaced by video?

Here’s Salman Khan’s TED talk from just a few days ago. I simply love this idea. I’ve been using catchupmath.com with a few students, but this seems even larger in scope. It’s extensive, not just limited to math, free and open, and very powerful. I’d really like to try this.

If you haven’t been to Khan Academy yet, you need to go there now. Check out the videos, do some practice problems. I did a bunch, and it was a bit addictive. You even get points and badges!

I’m going to really think about this: can lecture happen outside of class and practice problems happen IN class?

MathOverflow

I’m sure most of the mathematical community already knows all about  mathoverflow.net, but just in case, thought I’d post a link here. It was news to me…perhaps just because I’m not really in the collegiate academic math community. (Though the wikipedia article says it was created in 2009, so it’s still in its infancy.) But it’s definitely a resource I’ll be using as I do my own mathematical investigations and my grad coursework. I’ve enjoyed the original stackoverflow.com for looking up answers to questions, and in two instances so far, asking my own. StackOverflow and sites like them fill a vital role in the online community–quick, thorough answers to pointed questions.

Geogebra has new skills

A new version of Geogebra has been released, in beta. It’s called Geogebra 5.0, and you can see the news about it here. Or, here’s a direct link to launch it right away. Thanks to The Cheap Researcher for the lead on this. As readers of this blog may already know, I love Geogebra!

One of the main highlights is that Geogebra now supports 3D manipulations. Awesome! However, don’t get too excited–it doesn’t let you graph anything except planes. No surfaces. It will do geometric constructions, like spheres and prisms. Using parametric equations and the locus feature, you can coax it into rendering spirals or other space curves. [edit: I figured this was possible, but it actually wasn’t. Not sure why.]

Another highlight, which I find even more exciting, is that Geogebra now has a built in CAS. Here’s a screen shot of me playing around with a few of its features. It also has a ways to go, especially for those who are used to more robust systems like Mathematica/Maple/Derive/TI-89. But this is a great step in the right direction, and 10 points for the open-source camp!

Notice that it can work with polynomials in ways you would expect, it can symbolically integrate and derive (simple things), perform partial fraction decomposition, evaluate limits, and find roots. Here are a few more things it can do. Strangely, it had problems finding the complex roots of a quadratic (easy), but not a cubic (hard). Just take a look at my screen shot. Seeing that it did okay finding the complex roots, I wondered if it could also plot them for me. I started by entering (copying and pasting) the complex zeros as points in Geogebra, which worked. But then I discovered the new ComplexRoot[] function which approximates the roots and plots them on the coordinate plane all at once. Cool! Here’s the screenshot:

The seven complex roots of f(z)=z^7+5z^4-z^2+z-15

As you can see, I asked for the roots of a 7th degree polynomial. Since the polynomial had real coefficients, notice that every zero’s conjugate is also a zero, as we’d expect. And we also expect that at least one solution of an odd-degreed polynomial will be real (notice this one has only one real root, approximately 1.22).

That’s all I’ve discovered so far. I’ll let you know if I come across anything else exciting. Keep in mind that this is beta, so the final release will likely have all the bugs worked out and more features.

Could your math teacher be replaced by video?

Before I get to the titular topic, let me share some links. I’ve been meaning to post links to a couple of online resources that are astonishingly thorough. I strongly encourage you to check all these out.

  • Drexel Math Forum — This site has been around for years, I’m just getting around to posting about it now. But if you’ve never been there, I highly recommend it. Almost any math question high school students could asked has been answered and cataloged on this site (including misconceptions about asymptotes like I posted about the other day).
  • Interact  Math — When you first link to this page you’ll be unimpressed. But select a book from the drop down menu and then pick a chapter and set of exercises. Then, click on an exercise and prepare to take an interactive tour of that problem. The site let’s you graph lines, type math equations, do multiple choice problems, and more. If you have trouble with the problem, it will interactively walk you through each step, asking you simpler questions along the way. What a fantastic resource! Unfortunately, almost none of our books are on the drop down list. That doesn’t keep it from being useful. Just find problems similar to what you’re struggling with and try those.
  • Khan Academy — A nonprofit organization started by Sal Khan, this site has 1800+ youtube instructional videos, nicely organized by course and topic. You can go learn everything from basic arithmetic to college level Calculus (and Differential Equations, Linear Algebra, Statistics, Biology, Chemistry, Physics, Economics…). Sal’s mission is to provide a world class education to anyone in the world for free. It’s very exciting to see how this site will grow, and possibly change how we do education.

Math Teaching by Video

Some of these sites, especially the Khan Academy, make me wonder how long our modern American school system will remain in its present form.  Will we always have a teacher in the front of the math classroom delivering instruction?

I’m not afraid of the idea that we (teachers) could be partially replaced by video lessons. It’s actually a pretty good idea. The very best instructional practices could be incorporated into a flawlessly edited video. Teachers wouldn’t make frustrating, careless mistakes, students could replay the videos at any time, and substitute teachers could easily run the class. Every school, even the poorest and most marginalized would be able to deliver top-notch, world class instruction.

And what would teachers do, then? Qualified teachers could turn their efforts toward more of “coaching” and “discussion leading” role, concentrating on one-on-one sessions, remediation, reteaching, providing feedback, grading, seminars, open forums, field trips, and inquiry-based instruction that supplements the more formal video presentations. And don’t forget blogging! 🙂 So much of a teacher’s time is currently spent preparing lessons and teaching them that they have very little time for all those other (more?) important aspects of teaching. All this time devoted to preparation is being spent by teachers everywhere. Imagine the possibilities if we devoted the bulk of our time to these other aspects instead of preparing instruction. Sounds really great to me.