We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image.

Ah, mathematics. That single word, mathematics, can divide - and strike fear in the hearts of - individuals like no other term.

I just read two very different science books, Daniel Kahneman's Thinking, Fast and Slow and Scott Aaronson's Quantum Computing since Democritus. Not much to connect the two except both deal to some extent about probability and computation and I want to write a blog post for each chapter, for much I disagree with both authors. But that's what makes them so much fun, so rare to find science-oriented books both worth reading that have the guts to say things that one can disagree with.

In full disclosure, Scott and I agree that he would post about my book if I wrote about his but what a deal. Scott's book is a pleasure to read. He weaves the story of logic, computation and quantum computing into a wonderful tour. You can get an idea of Scott's style by how he explains how he will explain quantum.

The second way to teach quantum mechanics eschews a blow-by-blow account of its discovery, and instead starts directly from the conceptual core - namely, a certain generalization of the laws of probability to allow minu signs (and more generally, complex numbers). Once you understand that core, you can then sprinkle in physics to taste, and calculate the spectrum of whatever atom you want.

He approaches the whole book by this philosophy.  Every now and then he moves into technical details that are best skipped--either you already know it or will get lost trying to follow. But no problem, the story remains. You need to appreciate Scott's sense of humor and his philosophical tendencies, and he does get way too philosophical near the end, particularly a strange attack on Bayesian that involves God flipping a coin. At the end of the book Scott contemplates whether computer science should have been part of a physics department but after one reads this book the real question is whether physics should be part of a CS department.

Kahneman gives a readable tour of behavioral economics with a variety of examples, though I don't agree with his interpretation of many of them. His fast and slow refers to decisions we make instinctively and quickly (like judging a person based on first impressions) versus more slow and deliberative (like multiplying numbers). There is a computer science analogy, in that his fast refers to what we can do with machine learning, simple trained models to make quick judgments that occasionally gets things wrong. I'm not a huge fan of behavioral economics, but it is useful in life to know the probability mistakes people make so you can avoid making them yourself. The wikipedia article has a nice summary of the effects mentioned in the book.

While these two books cover completely different areas, the themes of probability and computation pervade both of them. One simply cannot truly understand physics, economics, psychology and for that matter biology unless one realizes the computational underpinnings of all of them.

A well written note by Frenkel that attempts to explain Deligne’s work that won him this year’s Abel Prize.

When watching Jeopardy with Darling if I get a question correct that is NOT in my usual store of knowledge (that is NOT Ramsey Theory, NOT Vice Presidents, NOT Satires of Bob Dylan) Darling asks me How did you know that? I usually reply I do not know how I knew that. Recently I DID know and I'll get to that later, but for now the question arises: Do you know how you know what you know?

1. As an undergrad I learned mostly from taking courses. Hence I could say things like I Know Group theory from a course I had in Abstract Algebra in the Fall of 1978 (Side Note- I know why I should care about groups from reading the algorithm for graph isom for graphs of bounded degree---in 1988). I learned a few things on my own- I learned that a graph is Eulerian iff every vertex has even degree from a Martin Gardner article. But since most of my knowledge was from courses I knew how I knew what I knew.
   
2. As a grad students I still took courses but more routes to knowledge emerged. Papers! I could say things like
   I know the oracle constructions about P vs NP because I read the Baker Gill Solovay paper on October 23, 1981. It helps that Oct 23 is Weird Al's birthday. But even here things get a bit murky- someone TOLD ME about the paper which lead me to read it, but I don't recall who. So one more route to knowledge emerged- people telling you stuff in the hallways.
   
3. I saw Anil Nerode give a talk on Recursive Mathematics and that day went to the library (ask your grandmother what a library is) and read some articles on it. This was well timed- I knew enough recursion theory and combinatorics to read up on recursive combinatorics. In this case I know exactly how I know what I know. Might be the last time.
   
4. As a professor I read papers, hear talks, hear things in hallways, and learn stuff. Its getting harder to know how I know things, but to some extend I still could. Until...
   
5. THE WEB. The Web is the main reason I don't know how I know things. I sometimes tell Darling I read it on the web which is (a) prob true, and (b) prob not very insightful.
   

So- do you know how you know what you know?

On Jeopardy recently the final Jeopardy question was as follows.

TOPIC: Island Countries.

ANSWER: No longer Western, this one-word nation has moved to the west side of the international Date Line to join Asia and Australia.

BILL: What is SAMOA!?

Darling wondered how I know that:

DARLING: How did you know that? Is there a Ramsey Theorist in Samoa?

BILL: Not that I know if, but that's a good guess as to how I knew that. Actually Lance had a blog post Those Happy Samoans about Samoa going over the international dateline and losing the advantage of having more time to work on their conference submissions.

DARLING: Too bad there isn't a Ramsey Theorist there to take advantage of that!

Thanks Lance!- In this one case I know how I know what I know!

Jo Boaler has kindly asked me to spread the word about her free, upcoming course How to Learn Math. It sounds intriguing; in fact, I signed up and hope to be able to attend (to have the time).

Here's her description of it:

The course is a short intervention designed to change students' relationships with math. I have taught this intervention successfully in the past (in classrooms); it caused students to re-engage successfully with math, taking a new approach to the subject and their learning.

In the 2013-2014 school year the course will be offered to learners of math but in July of 2013 I will release a version of the course designed for teachers and other helpers of math learners, such as parents. In the teacher/parent version I will share the ideas I will present to students and hold a conversation with teachers and parents about the ideas. There will also be sessions giving teachers/parents particular strategies for achieving changes in students and opportunities for participants to work together on ideas through the forum pages.

Concepts

1. Knocking down the myths about math.
Math is not about speed, memorization or learning lots of rules. There is no such thing as “math people” and non-math people. Girls are equally capable of the highest achievement. This session will include interviews with students.

2. Math and Mindset.
Participants will be encouraged to develop a growth mindset, they will see evidence of how mindset changes students’ learning trajectories, and learn how it can be developed.

3. Teaching Math for a Growth Mindset.
This session will give strategies to teachers and parents for helping students develop a growth mindset and will include an interview with Carol Dweck.

4. Mistakes, Challenges & Persistence.
What is math persistence? Why are mistakes so important? How is math linked to creativity? This session will focus on the importance of mistakes, struggles and persistence.

5. Conceptual Learning. Part I. Number Sense.
Math is a conceptual subject– we will see evidence of the importance of conceptual thinking and participants will be given number problems that can be solved in many ways and represented visually.

6. Conceptual Learning. Part II. Connections, Representations, Questions.
In this session we will look at and solve math problems at many different grade levels and see the difference in approaching them procedurally and conceptually. Interviews with successful users of math in different, interesting jobs (film maker, inventor of self-driving cars etc) will show the importance of conceptual math.

7. Appreciating Algebra.
Participants will be asked to engage in problems illustrating the beautiful simplicity of a subject with which they may have had terrible experiences.

8. Going From This Course to a New Mathematical Future.
This session will review where you are, what you can do and the strategies you can use to be really successful.

Earlier this week Georgia Tech announced the Online Masters of Science in Computer Science, a MOOCs-based degree with a total tuition of about $7000. This degree came out of a collaboration between Sebastian Thrun of Udacity and my dean Zvi Galil with some significant financial support from AT&T. We've spent several months getting faculty input and buy-in to the program and we're very excited about taking a new leading role in the MOOCs revolution.

We will roll out slowly, with a smaller scale courses to corporate affiliates to work out the kinks and the plan to go to the general public in fall 2014. Read the FAQ to get more information about the program.

It's been fun watching the development of this degree, in particular hearing Sebastian talk about his MOOC 2.0 plans to scale courses with a small amount of expense that we pull from the tuition. No doubt we will have challenges in making this degree truly work at a large scale but I'm truly bullish that we'll a self-sustaining quality Masters program that will reach tens if not hundreds of thousands of students.

Here we go.

You probably already know about the two packages that you can use to typeset Fitch-style natural deducation proofs in LaTeX.  Here's another, which you may be interested in if you use Barker-Plummer, Barwise, and Etchemendy's popular logic text Language, Proof, and Logic. It makes proofs like this:

I've taken Etch's original style file and Dave's documentation, put it together in standard docstrip format, cleaned up the code a bit and added a few features.  You can download the beta from

https://github.com/rzach/lplfitch

I've also attached the documentation here.

Please file any problem reports on github, if you could, or email me directly. (The comment system here is unreliable.)  I'm hoping to put it on CTAN in a month.

Math Teachers at Play carnival is posted at Denise's blog. As usual, it is FULL of "yummy" math posts! Go check it out.

I especially enjoyed using the "parrot-talk" in math class, and Fawn taking a textbook problem and making it into an open-ended activity -- her students struggled and learned a lot!

I'm just posting this flyer that was sent to me about the Elements of Mathematics course. This is not paid advertising; I am doing this for them for free, because I believe it is not easy for them to find their "target audience" - gifted middle school children that are very interested in mathematics.

Problem: On Mothers day (May 12 this year) restaurants are very crowded because many people take their mothers, grandmothers, great-grandmothers, etc out to lunch. (Grandparents day is in September but I think most people ignore that and honor their grandmothers on mothers day and their grandfathers on fathers day.)

My solution: Take mom out to lunch the FOLLOWING week. Some of my friends tell me NO- you can't just MOVE Mothers day- what are you--- The Master of Space and Time? The key is that my mom AGREES with me and in fact raised me with these values: (1) Never do X when everyone else is doing X, its too crowed, and (2) Learn the polynomial VDW theorem.

While this solution may work for me, it may not work for everyone. Here are some options to alleviate the restaurant crunch:

1. Declare the second WEEKEND in May to be MOTHERS WEEKEND. People take their moms out to lunch SATURDAY or SUNDAY. This would split the restaurant load in half.
   
2. Declare May MOTHERS MONTH. People take their moms out to lunch ONE Sunday in May. This would split the restaurant load by 4.
   
3. Declare May MOTHERS MONTH. People take their moms out to lunch ONE Saturday OR Sunday in May. This would split the restaurant load by 8.
   
4. Declare May MOTHERS MONTH. People take their moms out to breakfast OR lunch OR Dinner ONE Saturday OR Sunday in May. This would split the restaurant load by 24.

How would people DECIDE which day to do:

1. The last day of April have mom either (depending on which of the above schemes) flip a coin, role a 4-sided die, or role an 8-sided die or role two 12-sided dice to determine which day to be taken to lunch. Fortunately, due to the Dungeons-and-Dragons craze that girls got into about 40 years ago, most mothers have these dice. But in case she does not, here is a nice MATH PROBLEM (I am sure already solved): USE fair coins and fair 6-sided dice to simulate other random choices fairly. In our case 4-sided, 8-sided, and 24-sided. Which random choice can be simulated? Which can't?
   
2. Say we do the Saturday/Sunday/breakfast/lunch/dinner solution. Everyone with last name beginning with A goes to breakfast on the first Saturday. Everyone with last name beginning with B goes to lunch on the first Saturday. etc. There are only 24 lunches and 26 letters, so merge P and Q, and merge Y and Z.
   

How likely is any of this to come about? It would need to evolve naturally as a social custom. It also would have to not be that hard to implement. As such the 24-meal-plan probably won't catch on. Also, if Mother's Day become Mothers one-of-24-meals-day it may lose something. Hence the 2-meal-plan solution is probably the best.

However, the entire tradition of taking mom out to lunch on mothers day may fade. The origin is that mom cooks for the family most days, so this ONE day they take her out. Nice! But more and more households share responsibilities (NOTE- I have no facts or stats to back this up but it has a certain truthiness about it) hence the notion of taking mom out to lunch may seem more and more odd over time. Then again, its still nice being taken out to lunch.

Get *25% off* of ALL Math Mammoth downloads and CDs at Kagi store with coupon code MAY2013.

The sale starts NOW and runs till the end of May (May 31).

This includes all Math Mammoth & Make It Real Learning downloads CD products at Kagi, including the already discounted bundles!

You can go to MathMammoth.com first, then find the links to Kagi's order pages there. Or, you can use these direct links to the order pages:

• Light Blue series (complete curriculum)
• Blue series
• Golden and Green Series
• Make It Real Learning activity workbooks.
• Bundles (CDs or downloads).

From today, May 13, till May 31! Use coupon code MAY2013, and get 25% off!

Another report on how students don’t learn as well when they attempt to multi-task.

Here is a neat factoring game I found online: Factorization Forest

You pick among six different types of seeds given. Then, you have to factorize a number to its prime factors, and the game then lets you grow a tree and place it into your “forest”. You can change the size of the tree and move it around.

Then just grow another tree by factoring another number! You can do it for as long as you want. Great fun! My daughter really liked it, and has made several "forests." I basically replaced the practice problems about prime factorization in  her math book with this game.

Here are four screenshots from the game:

Back around 1980, I used to write computer games for the Apple II. Plotting a point on the Apple II screen required dividing by 7, a lengthy process for the 6502 microprocessor. Asking around, we learned how to make division by 7 much faster--lookup tables.

As computer gaming got more intense in the decades that followed, we first had graphics cards designed to speed up the process and later Graphics Processing Units or GPUs, dedicated processors devoted to graphics.

Around the turn of the century, people started using GPUs for more than just graphics. GPUs did certain kinds of vector manipulation quickly and one could use these for a variety of mostly scientific computing. But GPUs weren't really well designed for other purposes. Following the cupholder principle, GPUs began to evolve to allow easier to access APIs from more common programming languages becoming General Purpose GPU or GPGPUs. Several systems researchers at Georgia Tech and elsewhere are now redesigning chip layouts to make the best most efficient uses of CPUs and GPGPUs.

The theory community hasn't seem to catch on yet. There should be some nice theoretical model that captures the vector and other operations of a GPGPU and then we should search for algorithms that make the best use of the hardware. The theory world writes algorithms for non-existent quantum computers but not for the machines that we currently use.

You've probably heard me talk about the Supercharged Science curriculum before - it is a great science curriculum, and I use it with my own kids.

Today I have something to tell you I think you will appreciate! Aurora, the owner, is doing a free online science class TOMORROW, Wednesday.

You can reserve a free spot by clicking here:
www.sciencelearningspace.com/members/go.php?r=3095&i=l25

She's done these classes several times before (as you may know :) ), and my kids have really enjoyed them. She's warm and engaging - and knows her science.

Kids not only learn solid academics, but get to do hands-on activities during the class. (How? You prepare the materials before the class, and during it, she gives you 5-10 minutes to do a certain activity, then discusses it.) 

This week's topic is Rocketry and Spaceflight. This is really appropriate because Aurora actually used to work for NASA, and is a real rocket scientist!

If you've never experienced Aurora in action, you need to sign up for this class and try it out!

It is also great for parents who don't like teaching science -- because she does it for you, and kids stay engaged!

Reserve your free spot now by clicking here:
www.sciencelearningspace.com/members/go.php?r=3095&i=l25

Remember, it's tomorrow. Enjoy!

My darling sometimes watches TV in the middle of the night when she can't sleep.So I found myself watching (actually listening) to the quiz show
Are You Smarter than a Fifth Grader? They asked the following Math Question:

What number do you need to add to 3 to get a double fact?

I had never heard the term double fact! I really didn't know and there was no way toderive it! I don't recall what my guess was but it was incorrect.See herefor what they are.

Is this a common term? If you Google

"Double fact" math

You get roughly 6,000 hits. (Down from 17,000 a few months ago when I first sketched out this post.)Is that enough hits to be a real term? Is number-of-hits a good measure?

Are there other math terms that are being taught in elementaryschool that are not that well known to people like us? (Though if you have children perhaps you know them.)Note that no matter how much math you know, there may be terms you don't know and can't derive (though you can make an intelligent guess).

My name is Bill Gasarch, and I am NOT smarter than a fifth grader.

If you have taught math any length of time, I'm sure you have encountered the question, "Where do I ever need this?"

I've updated an article of mine on this topic. It specifically uses the example of where students might need the idea or concept of square root.

Where do you need math, square roots, or algebra?

The article also lists several resources that are designed to help students understand where math is needed in real-life.

Enjoy!

Educents is a fairly new site that runs special deals for educational products.

You can now get certain Math Mammoth bundles on Educents for 30% off! This deal runs till Monday, May 13.

The bundles included in the deal (as downloads) are the Light Blue Series, Blue Series, and All Inclusive.

Following the coloring theme from Bill's last post, a few years ago I asked you readers for natural examples of maps that were and were not three colorable. Chris Bogart gave a nice non-trivial example of a three-colorable country, Armenia.




But I also wanted a natural example that was four-colorable even though every interior region had an even number of neighbors. In my book I ended up making up my own fake country map.

Kenneth Appel, of Appel-Haken Four Color Theorem Fame, died recently. See here for an obit.

In 1972 I read that the four-color theorem was an open problem. From what I read it seemed like there was some progress on it (e.g., results like `if its false the graph has to be yah-big) but it seemed to be years away from being solved. I assumed that a new idea was needed to solve it.

Then, in 1976, it was SOLVED by Appel-Haken. From what I read it wasn't so much a new idea but very clever use of old ideas and a computer program. I also heard that it was just at the brink of what computers could do at the time, and that it would have taken 1200 grad student hours. (There is a good description of the proof on Wikipedia here.)

At the time I heard there were objections to the proof. Later when I read some of them they didn't seem like real objections. They boiled down to either

1. I wish there was a shorter proof. This is true, but not a reason to object.
   
2. It can't be hand checked. I trust a computer-checker MORE THAN a human checker
   
3. We don't know WHY its true. This is a more reasonable objection- but we do know how they got it down to a finite number of cases, so I'm happy with that.
   

In 1996 Robertson, Sanders, Seymour, Thomas was obtained a simpler proof. In 2005 Werner and Gontheir formalized the proof inside Coq- a proof assistant. To quote Wikipedia This removed the need to trust the various computer programs used to verify particular cases;it is only necessary to trust the Coq Kernel At this point I doubt anyone seriously doubts that the theorem has been proven.

There have been more computer-assisted proofs since then. See here for a list of some of them. That article also claims that such proofs are controversial and not always accepted. Is this really true? I thought the controversy was gone except for the topic of the next paragraph.

A famous computer assisted proof (or perhaps ``proof'') is the Kepler Conjecture. In 1998 Thomas Hale claims to have proven it. The proof involved rather complex computer calculations. The referees say they are 99% sure its true. Here's hoping an easier proof is found.

Computer assisted proofs may become more common. I just hope we still know WHY things are true.

Was Appel-Haken the first use of computer assisted proofs? I doubt it, but it was likely the first one to have an impact. It was important to know that this kind of proof could be done.

Is there a much shorter proof? A combinatorist once told me that since the function

f(n) = max size of proofs of statements of length n

grows faster than any computable functions, there have to be some statements that have very long proofs; and perhaps the four color theorem is one of them.

30 Apr 2013

In this Newsletter:

1. Math resource: MathGraph32
2. Prime numbered cicadas
3. Relieving test anxiety
4. Math puzzle
5. IntMath Poll
6. Final thought:

Once again I apologize for the long gap since the last IntMath Newsletter. I’m involved in several large projects (including developing online modules for a math course and 2 academic integrity courses). Once those are done, hopefully I’ll be able to get back to a regular schedule of writing.

1. Math resource: MathGraph32

MathGraph32 is a great free tool for exploring 2D and 3D math concepts. MathGraph32

2. Prime numbered cicadas

The 17-year cicada is due to emerge in north-eastern parts of USA in Spring 2013. What is their connection to prime numbers? Prime numbered cicadas

3. Relieving test anxiety

Annie Murphy Paul writes the Brilliant Report for Time magazine, It’s an interesting collection of research about the brain and how to squeeze more out of it.

A recent article, How to Eliminate Test Anxiety gives some good pointers, which as she says, are reasonably simple, inexpensive and, as recent studies show, effective."

Here’s some short quotes from her list of suggestions:

1. Unload on paper. Spend ten minutes writing about your thoughts and feelings immediately before taking a test.

The practice, called "expressive writing," is used by psychologists to reduce negative thoughts in people with depression. They tried the intervention on college students placed in a testing situation in Beilock’s lab, and in an actual Chicago school, where ninth-grade students engaged in the writing exercise before their first high school final. In both cases, students’ test scores “significantly improved,” according to an article they published last year in the journal Science.

2. Affirm your values. Apprehension over tests can be especially common among minority and female students. That’s because the prospect of evaluation poses for them what psychologists call "stereotype threat"—the possibility that a poor performance will confirm negative assumptions about the group to which they belong ([this posits] that girls can’t excel in math and science; blacks and Latinos aren’t college material).

3. Engage in relaxation exercises. Younger kids aren’t immune from test anxiety. As early as first and second grade, researchers see evidence of anxiety about testing. Their worries tend to manifest in non-verbal signs that adults may miss, [like] stomachaches, difficulty sleeping, and a persistent urge to leave the classroom to go to the bathroom.

See more details on these anxiety-reduction techniques here:

From The Brilliant Report: How To Eliminate Test Anxiety

4. Math puzzles

The puzzle in the last IntMath Newsletter was about expressing the number one using nines, a minus sign and dots.

Correct answers were given by dalcde, Christopher, Bonnie, Andrzej, Dineth, Nicos and Thomas.

Math symbols were a challenge: I knew it was going to be tricky to type in the answers for this puzzle. I started to write some pointers about how to do it and stopped, because that would have given the answers away! (There is a "Preview" button on the response box for the blog. You can use it to make sure your math looks OK before posting.)

1. Superscripts (powers): You can create a superscript by typing it like this in the "Respond" box of most blogs:

   9^9-9 = 1

   It will look like this:

   9^9-9 = 1

2. Dot above a number: This one is a bit trickier, as there is no HTML solution that works nicely in all browsers. The best way is to create an image using a tool like Codecogs Equation Editor. It has a visual interface, but requires some knowledge of LaTex. You need to enter:

   0.\dot{9} = 1

   At the bottom of the equation editor page there is an "embed" box with code. Copy that code into the "respond" box in most blogs and it will look like:

   

Note 1: I included the "0" before the decimal point in my answer above. A lot of people don’t notice the dot, and misread decimal numbers. (Yes, I know the question specified "one nine" only, but it’s worth mentioning.)

Note 2: I was aware the question was somewhat country-specific. The Europeans write a decimal number using a comma, not a dot.

And as Andrzej pointed out in his response regarding the recurring part:

There are three symbols for recurring fractions: dash, dot (UK and USA) and () in Poland. So the solution is slightly tricky; it depends on nationality.

It would be really nice if we had consistent math notation around the world, especially when for something as simple as writing numbers!

New puzzle: 216 cubes of side length 1 cm are arranged to make a cube with side length 6 cm.

A sphere of diameter 6 cm is inscribed in the large cube such that the center of the sphere is the center of the cube. How many complete unit cubes is contained in the sphere?

You can leave your responses here.

5. Final thought: Watching

Donald Trump is a real estate millionaire who became quite famous as a result of his role in the TV series The Apprentice. He once said:

Work hard. Someone’s always watching.

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: Prime cicadas appeared first on squareCircleZ.

MathGraph32 is a great free tool for exploring 2D and 3D math concepts.

(It’s been around for a while, but I only recently discovered it.)

There is a download version as well as an online Java-based version.

According to the site, MathGraph32 is an:

Open source cross-platform software of geometry, analysis and simulation.

It was developed by French-speaking mathematician, Yves Biton. Some of the examples are in French, but it’s quite easy to see what is going on. (There is an English version of the program).

Screen Shots

Here are some examples from MathGraph32 (images by them).

You can explore several examples (some for “teaching purposes”), and there are tutorials that help explain the use of the applet.

It’s well worth checking out!

The link again: MathGraph32

The post MathGraph32 appeared first on squareCircleZ.

I keep a list of ideas for blog posts, but some will never turn into posts. So here are a few random thoughts from that list.

• Some people like to write prose, some people like to write lists, like Bill's last post. Bill will often send me an email that's a list of items. I prefer the prose and usually avoid the lists with today being the "exception that proves the rule" (an expression that I never understood).
  
  I do have to admit that lists are very efficient, when I can respond to Bill like 
1. yes
2. no
3. Friday
4. Did you really expect happy comments on that post?
• Marissa Mayer has banned working at home for Yahoo employees. Lots of academics work at home when they aren't teaching. I didn't have any more deep insights here so it didn't become a post.
• I have a note to write a blog post on "confusing university names". I wonder what was confusing me.

I just wanted to share this testimonial that Lisa K. sent in a few days ago. I find this situation is fairly common among children who come from public school and start Math Mammoth - they are behind in math, sometimes very much so (many grade levels). Yet, it's not a time to despair, because it IS possible to catch up!

It is often necessary to go back and make sure all the basics are mastered. Then, the child often progresses at a much faster pace, and within a few years they usually attain grade level.

The idea is not to have a race to keep up with public school, but to gain a solid  understanding of math for life.

Here's Lisa's comment. Her daughter still has some more catching up to do, but she's progressing at a good pace!

Maria,

I just have to thank you for your wonderful math program. I emailed you late last summer about my 6th grade daughter who was terribly behind in math. You suggested I start her back in the grade 1 book to give her a fresh and solid start. I did and she has done great! She has completed both first grade books, both second grade books and has now started the multiplication "theme" books (I decided to just do multiplication because most of the other things in the 3rd grade book she already knows). I can't believe she's flown through 2 grade levels and is starting a third in less than one school year. She still does not love math, but she is doing well and has built some confidence. Your method for learning multiplication tables is terrific. She's learned 2, 4, 3, 5, 10, 11 in about 3 weeks!

I love that you focus, not just on learning the facts, but on understanding the concepts. Not just "this works", but "this is why this works". Many programs don't do that; it's just rote memorization which is partly why Grace got so behind. She never understood why she was carrying (for instance), she just did it. Now that it makes sense, it's really sticking with her.

I'm using your program with my other younger children, too, and am just as happy. Also, it's so easy for me to use. Almost no prep time and free worksheets when they need a little extra help or some review!

Thank you again and again for your wonderful program!
Blessings,
Lisa

If you're in Austin, you probably know this already. If you're not, it's probably too late. But this is what I'll be doing this weekend:

Friday, 26 April 2012
Thomas Uebel, University of Manchester, “The Logic of Science and the Pragmatics of Science: The Challenge of Complementarity.”
Christopher French, University of British Columbia, “Carnap, Jeffrey and Explication of Radical Probabilism.”
Sebastian Lutz, Ludwig-Maximilians-Universität München, “The Criteria for the Empirical Significance of Terms.”
Saturday, 27 April 2012
Sahotra Sarkar, University of Texas, “Nagel on Reduction.”
Michael Stoeltzner, University of South Carolina, “Could Mathematical Physics serve as a Model for Formal Epistemology?”
Flavia Padovani, Drexel University, “Reichenbach On Causality in 1923: One Word, Many Concepts”
Richard Zach, University of Calgary, “Carnap on Logic.”

Thanks to Sahotra Sarkar for putting this together!

You can now wear a device, Memoto that will take a picture every 30 seconds. Do you really have a Kodak Moment every 30 seconds? No, but this way when you do have one it will be captured (unless it happens at just the wrong time.) There is another one called Autographer which claims to be a "smart" camera that will take tons of pictures of your day. A photographer wore one and had his team all wearing them during a photo shoot.

Current historians who study ancient civilizations have the problem of very little being preserved. They go on whatever, perhaps by chance, happened to survive. King Tut is studied NOT because he was an important king, but because we have LOTS of his stuff. (What if America is destroyed and all that is left is the Gerald Ford Library?}

Future Historians may have the problem of having too much written down, recorded, emailed, blogged, tweeted. Or they may have the problem current historians have if some of these technologies go out-of-date so a lot is lost. Floppy's are already unreadable, Video Tapes and VCR's are degrading, beta-tapes--- as your grandparents what they were. They may also have the Galaxy Quest problem of mistaking our TV shows for historical documents (oh that poor Gilligan!).

What do we and don't we preserve?

1. Almost every TV show and Movie produced since DVDs came out, no matter how bad, is on DVD. Is there a directors cuts of Dude, where's my car?? Will future generations need that?
   
2. News shows are not preserved. Too bad- I might want a complete set of THE DAILY SHOW and THE COLBERT REPORT. Future historians may have the problem of mistaking these shows for satire.
   
3. Quiz shows are not preserved. Too bad- I liked BEAT THE GEEKS but all I have are some video tapes I recorded when it was on.
   
4. Sporting events- This is borderline since there are DVDs of past Superbowl's, World Series, etc, but not of ordinary games.
   
5. Many TV movies don't make it to DVD. Is that a loss?
   
6. Much TV from before the video tape age is lost. Some survives. It is somewhat arbitrary. I Love Lucy was done on some sort of medium that survived, others did not.
   
7. A very odd case: Dr. Who. There are rumors that some fans audiotaped the early episodes and that the only form in which they are preserved. See here, here.
   
8. Commercials. Aside from arbitrary things people happen to have taped, most commercials will not survive! What will future generations do without all of those insurance companies claiming that they are cheaper than the other ones?
   
9. What if the only knowledge of complexity theory that survives is Lance's book. That would be okay!
   
10. What if the only knowledge of complexity theory that survives are the Bill/Lance Podcasts. Hmm...
    

Quasi related to the theme of today's post (today's post has a theme?) is the question Who was the first married sitcom couple to be shown sleeping in the same bed? The funny answers are The Flintstones and The Munsters, indicating that America was so prudish they couldn't show real humans in bed. The first real-human couple is then said to be the Brady Bunch. But actually a couple named Mary Kay and Johnny from a sitcom in 1947 are said to be the first. The networks were less prudish because the couple really was married (also, TV was SO new back then). But here is my point- The show DOES NOT EXIST ON VIDEO TAPE OR DVD OR ANY MEDIUM. So the answer is only people's memories which could be faulty. We are already losing access to some TV trivia! Incidentally, are there any happily married couples on TV anymore? (The FBI agent on White Collar comes to mine, but nobody else.) (Added later: Our knowledge of TV trivia is actually increasing! see here.


What about journal articles? Do we do the future a DISservice by publishing too much and making it harder to find stuff? Or will they have the tools to find what they want? Today we have Google Scholar and some other tools- but are we putting all of our eggs in one basket? Will all of our papers survive or only the good ones? Or is every paper a gem worthy of preserving for future generations? While I doubt this is true, you never know when someone is going to need the Canonical a-ary Ramsey Theorem to prove something in geometry.

You've surely heard of the acronym PEMDAS for the order of operations (Please Excuse My Dear Aunt Sally) - standing for Parentheses, Exponents, Multiplication & Division, Addition & Subtraction.

There's a new game for order of operations called My Dear Aunt Sally. You can play it free online, or purchase an inexpensive app for your tablet.

It's a very good game, and takes some thinking! You need to place the given numbers into two expressions so that the operations make the two expressions have the same value.

Here are some screenshots. The first one is the easiest level. The addition on the top has to have the same value as the multiplication/addition expression on the bottom.


 It gets harder if you choose to include exponents:

You can also choose to use fractions, so it becomes harder yet:

Some 3-d geometry and knife, and your breakfast bagel will turn into a two-twist Möbius strip!


Bagel by George Hart

Here is the newest worksheet generator that I have added to HomeschoolMath.net:

Area and perimeter of rectangles/squares 

It is very versatile and makes many different kinds of problems related to the area and perimeter of rectangles and squares.

You can make:

• problems for the area and perimeter of rectangles and squares, with grid images or normal images
• word problems, including ones asking for a side length when area or perimeter is given
• problems with irregular rectangular shapes
• problems to practice distributive property with two-part rectangular areas (required by Common Core Standards in 3rd grade).

You can make worksheets with just one type of problems from that list, or mix them up however you want. For example:
 


I especially made this worksheet generator because I feel many children may have some difficulty with the new Common Core Standards requirement for 3rd grade, that is 3.MD.7.c

"Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning."

You can generate worksheets for this exact standard! They can look like this:



or you can ask children to draw the two-part rectangle.


Here are some quick links for you to try the generator:

• Draw a rectangle with given area, or find the area of a given rectangle (grid image; about 2nd grade)
• Draw a rectangle with given area, or find the area of a given rectangle (grid image; 3rd grade)
• Find the area and perimeter of the given rectangle (grid image)
• Find area, perimeter, or the missing side length (rectangle image; 3rd - 4th grade)
• Find area, perimeter, or the missing side length (word problem or rectangle image; 3rd - 4th grade)
• Fill in a number sentence for the two-part rectangle, thinking of one rectangle or two (distributive property) (3rd grade Common Core standards; in your browser options, make sure background colors get printed)
• Draw a two-part rectangle to match the given number sentence for its total area (distributive property) (3rd grade Common Core standards; in your browser options, make sure background colors get printed)
• Draw a rectangle OR write its area using distributive property (3rd grade Common Core standards, mixed practice)
• Find the area and perimeter of irregular rectangular shapes (grades 4-5)
• A variety of problems about area and perimeter (grades 4-5; in your browser options, make sure background colors get printed)

Today is the official publication date of my first book The Golden Ticket: P, NP and the Search for the Impossible by Princeton University Press, though the book has been available on Amazon and in some bookstores for a couple of weeks now. This book takes a non-technical tour of our favorite open question through a series of stories and examples, covering P, NP, NP-complete, the "beautiful world" if P = NP, and how to deal with hard problems since we surely don't live in that world and a bit of history, cryptography and quantum. How "non-technical": I never actually define P and NP and avoid formulas and terminology except as needed to describe circuits and Cook's theorem.

I spent three years on this book project, building on my CACM survey. Thanks to all of you readers for your support, your suggestions for maps (some of which I used), titles and epigraphs (none of which I used).

You can keep up with all the happenings of the book on its own twitter and website.The Golden Ticket has already received some nice reviews. You can also check out an article I wrote for the Daily Dot and podcasts interview at  Wild About Math and the New Books Network. There are Japanese and Chinese translations in the works.

Hope you enjoy the book, recommend it all all your friends and help preach the gospel of P v NP.

Recall that on April Fools Day I had a post about R(5) being discovered via a collaboration of Math and History. Many readers emailed me asking how many people were fooled (apparently they weren't!). I can't tell if it fooled ANY of my astute readers, though I hope it amused them. I CAN tell how many of my STUDENTS if fooled. I did an experiment on my class for which I CAN report the results.

1. All semester I gave would say things like there has been some progress on R(5), but I'll talk about that later
   
2. On March 28 I gave them my paper Ramsey Theory and the History of Pre-Christian England: An Example of Interdisciplinary Research and told them that I was going to do an experiment with the flipped classroom concept (this is Real educational thing)- they would read the paper, and on Tues Apr 2 I would give them a quiz on it (to make sure they read it) and we would discuss it.
   

I DID give them a quiz, but I asked them to NOT sign it so it was more of a survey. Here is a summary of the questions, the stats on it, and what to make of it. 17 people took the quiz (the class has 16 and some auditors.)

1. When did you Realize it was a hoax?
   
   1. When I first took this quiz: 5
      
   2. When I saw that Fifty shades of Grey was one of the references: 1
      
   3. When I went to the website that was supposed to have R(5) and it revealed the hoax: 3
      
   4. When I saw the name H.K. Donnut: 2
      
   5. When I read your April 1 blog: 1
      
   6. When I heard other students talk about it (though I kind of knew anyway): 4
      
   7. When I read the title: 1
      
   UPSHOT: I would count answers 1,2,3 as being fooled. So NINE were fooled. An earlier proofreader believed it because he wanted to so badly.
2. Was it okay to have you read a hoax paper? (NOTE: My wife thought NO.)
   
   1. This assignment was awesome!: 9
      
   2. Liked learning to not always belief a prof: 1 (When I tell him the Large Ramsey Theorem can't be prove in PA will he say That's bullshit man!):
      
   3. Assignment was okay (I could tell these people were not amused): 7
      
   4. Okay as far as it goes, but 10 pages! C'mon, that's too much: 1 (This is quite fair, but it was an easy read).
      
   UPSHOT: I all it a success!
3. Which of the names are Real and which did I make up?
   
   1. Eugene Wigner. REAL. Fake:4, Real:13. Name does sound funny.
      
   2. Herbert Scarf. REAL. Fake:9 Real:8. Real name that fooled the most people.
      
   3. Samuel Harrington. REAL. Fake:1 Real:16
      
   4. Dorwin Cartrwright. REAL. Fake: 6 Real: 11. I thought this would fool more people.
      
   5. Frank Harary. REAL. Fake: 3, Real: 14 (One thought I misspelled the real name. I didn't, but given my bad spelling, and the nature of the name, I can see why they thought so.)
      
   6. Charles Percy Snow. REAL. Fake: 5, Real: 12
      
   7. Jacob Fox. REAL. Fake:3, Real: 14. I've met him, he's real!
      
   8. Sandor Szalai. REAL. Fake: 4, Real: 13. Looks fake to me.
      
   9. Paul Erdos. REAL. Fake: 0, Real: 17. Seems like a mythical figure.
      
   10. Paul Turan. REAL. Fake: 3, Real: 14. One thought it was a play on Turing.
       
   11. Vera Sos. REAL. Fake: 3, Real: 14. I'm surprised people thought this name is real- I wouldn't have.
       
   12. Sir Woodson Kneading. FAKE- It's an anagram of Doris Kearns Goodwin, historian. Fake: 10, Real: 7. I'm amazed that 7 thought it was real.
       
   13. H.K. Donnut. FAKE- It's an anagram of Don Knuth. Fake: 13. Real: 4. Looks so fake it has to be Real? I was originally going to use Hal D.K. Donnut which is an anagram for Donald Knuth. I still don't know which looks more real.
       
   14. Moss Chill Beaches- FAKE- it's an anagram of Michael Beschloss, a presidential historian (He studies presidents, he is not, himself presidential). Fake: 13, Real: 4. My most fake looking name.
       
   15. Tim Andrer Grant- FAKE- It's an anagram of Martin Gardner. Fake: 5, Real:12. Tim is a reasonable first name, Grant is a reasonable last name, and Andrer--- well, middle names are sometimes weird.
       
   16. Alma Rho Grand- FAKE- It's an anagram of Ronald Graham. Fake: 14, Real: 3. My favorite fake name.
       
   17. D.H.J. Polymath- FAKE, but not my invention. It's used on the Polymath paper that proved the Density Hales Jewitt Theorem using elem methods (see here). Fake: 12, Real: 5. The people who thought it was real were either kidding OR read the question as a trick question Fake name that BILL MADE UP-- this is a fake name but BILL didn't make it up.
       
   18. Ana Writset- FAKE, It's an anagram of Ian Stewart. Fake: 4, Real: 13. Fake name that fooled the most people.
       
   19. Tee A. Cornet- FAKE- It's an anagram of Terence Tao. Fake: 8. Real:9. Is Tee anyone's first name?
       
   20. Andy Parrish- REAL. Fake: 1, Real: 16. I hope he's real, he's one of my proofreaders.
       
   21. Stephen Fenner- REAL. Fake: 1, Real: 16. I hope he's real- I taught him the construction of an r.e. minimal pair.
       
   22. Clyde Kruskal- REAL. Fake: 0, Real: 17. Real on a good day.
       
   One of the students who knew the Fake names were anagrams, and thought Dorwin Cartrwright was fake, spend an hour trying to find the anagram it was of. Can people spot fake names easily? I doubt it since many real names look fake, and many fake names look real. Along those lines, what do the stage names Clint Eastwood and Dolly Parton have in common? See here for the answer.
4. Speculate on how I came up with the false names. Most left this blank. Some said Anagrams, some mentioned anagram-programs on the web (I did use one), some said random-name-generator pointing out that Vera Sos and Sandor Szalai look like they were produced by a not-very-good random name generator.
   UPSHOT: As Blanch Nail Roam said You can fool some of the people all of the time, and all of the people some of the time, but you can't fool all of the people all of the time.
   

Google's logo for today (interactive, by the way) is a tribute to Leonhard Euler -- a very famous mathematician from the 1700s. Today is the 306th anniversary of his birth (he was born on April 15, 1707).

You'll hear about him when you study calculus. The constant e bears his name. Euler's identity, or the formula


is called the most famous formula of mathematics. It ties together the important numbers 0, 1, e, i, and Pi.

Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron.

A lot of the notation we use is attributed to him: for example, Euler was the first to write f(x) to denote the function f applied to the argument x. He also introduced the notation i for the imaginary unit and the Greek letter Σ (sigma) for summation.

Then, if you study graph theory, you'll immediately encounter the famous problem about the Seven Bridges of Königsberg. Euler solved that, as well.

He also proved many results in number theory. And on and on and on... he wrote volumes and volumes of works in mathematics, and is a fellow worth knowing something about, I feel!

TL;DR

Introduction

There's a popular argument used to explain why atoms are stable. It shows there is a lower bound on the energy level of an electron in the atom that makes it impossible for electrons to keep "falling" forever all the way down to the nucleus. You'll find it not only in popular science books but in courses and textbooks on quantum mechanics.

A rough version of the argument goes like this: the closer an electron falls towards the nucleus the lower its potential energy gets. But the more closely bound to the nucleus it is, the more accurately we know its position and hence, by Heisenberg's uncertainty principle (HUP), the less accurately we know its momentum. Increased variance in the momentum corresponds to an increase in kinetic energy. Eventually the decrease in potential energy as the electron falls is balanced by an increase in kinetic energy and the electron has reached a stable state.

The problem is, this argument is wrong. It's wrong related to the kind of heuristic reasoning about wavefunctions that I've talked about before.

Before showing it's wrong, let's make the argument a bit more rigorous.

Bounding wavefunctions

(assuming we're in a frame of reference where the expected values of p and x are zero).

If the wavefunction is spread out with a standard deviation of a then the electron is mostly around a distance a from the nucleus. So the second fact is that we can roughly approximate the expected value of 1/r as 1/a.

Combine these two facts and we get, roughly, that
I hope you can see that the right hand side, as a function of a, is bounded below. The graph of the right hand side as a function of a looks like:
It's now an exercise in calculus to find a lower bound on the expected energy. You can find the details in countless places on the web. Here a link to an example from MIT, which may have come directly from Feynman's Lectures on Physics.

The problem

The above discussion assumes that the wavefunction is basically a single packet confined around a distance a from the nucleus, something like that graphed above. But if a lower energy state can be found with a different wavefunction the electron will eventually find it, or an even lower energy state. In fact, by using a wavefunction with multiple peaks we will find that the Heisenberg uncertainty principle doesn't give a lower bound at all.

We'll use a wavefunction like this:
It has a packet around the origin just like before but it also has a sharp peak around r=l. As I'm showing ψ as a function of r this means we have a shell of radius l.

Let's say

where ψ₁ is normalized and peaked near the original and ψ₂ is our shell of radius l. Assume no overlap between ψ₁ and ψ₂.

In this case you can see that we can make
For a large enough shell, ψ₂ contributes little to the total expected potential energy, but ψ₁ can contribute an arbitrarily low amount because we can concentrate it in areas where 1/r is as large as we want. So we can make the total expected potential energy as low as we like. And yet we can also keep the estimate of the kinetic energy given by HUP as close to zero as we like. So contrary to the original argument, the Heisenberg uncertainty principle doesn't give us a lower bound on the energy at all. The argument is wrong.

But wait, we know there is a lowest energy state...

But wait, the Heisenberg uncertainty principle argument gives the right energy...

But wait, it's just a heuristic argument...

Math Teachers at Play carnival #61 is posted at Math Hombre.

It looks shock full of good posts - waters my "math mouth!" Unfortunately, it may take me till next week to get to reading all that but it really looks good. Go check it out!

Once again there is some money from ACM and from NSF for students to goto STOC, and I am the one to send the applications to. The link for info on how to apply is on the STOC webpage, but I give it here as well. Note that the deadline is April 16.

ALSO there is travel money for CCC: see here (though I am NOT the one to send applications for that one).

If you are a grad student going to STOC and CCC then you SHOULD apply for both. If you are a grad student going to STOC but NOT CCC then you SHOULD apply for STOC. If you are a grad student going to CCC but not STOC then you SHOULD apply for CCC. If you are a grad student going to neither then... well, I have no advice for you.

1. The application process is EASY.
   
2. Not that many have applied so you have a chance. But see next note.
   
3. THIS blog posting may make point 2 false.
   
4. For STOC it was only one applicant per advisor; however, we have RAISED it to TWO.
   
5. There is a priority for members of underrepresented groups. This DOES NOT just mean Women and minorities, it also means people from institutions that don't normally send students to STOC. However, you should APPLY even if you don't fit these descriptions.
   
6. We prefer people whose advisor don't have the money to send them, or are at least short on cold hard cash.
   
7. So what I really want you to is tell people who should want to go but either don't quite know what it is or don't have the money.
   
8. The above are only priorities. We want to support students as much as we can, so we will be spending money, not hoarding it. Even if you thing you have low priority for one of the reasons above, you should still apply. And again- APPLYING IS EASY.
   
9. The deadline is April 16, so get to it!
   

This also raises the question: SHOULD you goto STOC? YES

1. Even if you are a first year theory grad students its good to see whats out there.
   
2. Things you see may inspire you. My interest in Ramsey Theory came partially from seeing a STOC talk by Lipton or Chandra of Furst on bounds on Multiparty Comm Complexity that used the Gallai-Witt theorem
   
3. You get to meet people. Note that many of the people who proved basic theorems are still alive.
   
4. If you are work in complexity and also apply for the Student Travel Grant for CCC, and get both, you can goto BOTH!
   

Entertainment Weekly reported last month on the bankruptcy of the special effects company Rhythm and Hues and the troubled industry. I remarked five years ago that the special effects industry lost its ability to surprise us. Without the ability to innovate, and with most of the effects handled in software, the need for specialized talent and companies disappears.

We've always could take comfort that computation and its related efficiencies have led to more, often safer and higher paying jobs. But is that still true? The stock market has hit historic highs but the unemployment rate remains high. Are companies who pared down during the last recession realizing they don't need to hire as the economy comes back?

Erik Brynjolfsson and Andrew McAfee in their 2012 book Race Against the Machine: How the Digital Revolution is Accelerating Innovation, Driving Productivity, and Irreversibly Transforming Employment and the Economy, argue that computer technology has truly gotten us to the point where we need fewer people to perform the task at the middle of the economy. The world needs computer scientists and welders, but far less people doing mid-level professional work. On the other side, Henrik Christensen, a Georgia Tech roboticist, argues that technology will continue to produce far more jobs than it displaces.

My take: It's just too early to tell. The economy could completely turn around and near full employment. Or we can see a permanent loss in returning jobs. History doesn't seem like a good guide here.

There are so many issues tied to the current state of employment and this sense of technological efficiency replacing jobs: Is college really worth the cost? Should an undergraduate student major in a STEM field or follow their passion if it lies elsewhere? Do we need MOOCs to improve access to quality education and/or keep down education costs? Are academics the next group to meet the efficiency maker?

My oldest daughter starts college next year and I don't even know what advice to give her.

I found this nice post from a fellow Math Mammoth user:

10 Ways to Practice Math During a Break

Wanted to add two other ideas to her list:

• Use online math games - there are SO many! See a list here.
• Sign up to a free trial to one of the many online math practice curricula.
  
  There are many, and they typically offer a 2-week trial. Over the years, we have tried Mathletics, IXL, Dreambox, K5Learning, Time4Learning, Reflex, Math Whizz, and probably a few others that don't come to mind now. I have reviewed several of them.

I'm sure you can add to this!

Roger Ebert died on April 4, 2013. Margaret Thatcher died on April 8, 2013. I have heard the urban legend that celebrities die in threes.

1. Does anyone really believe this? Is it an urban legend that this is an urban legend?
   
2. YES- there was an episode of 30 Rock based on this.
   
3. NO- 30 Rock is FICTION.
   
4. Whenever I've seen ``examples'' of this at least one of the three isn't a celebrity. Are politicians really celebrities? For that matter, are Movie Critics really celebrities (Roger Ebert may qualify but very few others would.)
   
5. The notion of Celebrity is not well defined. People change it to make the rule-of-threes work. Can there be a rigorous definition of celebrity, perhaps based on the indegree of the I"VE HEARD OF THEM directed graph.
   
6. I predict that somebody marginally famous will die in the next few weeks and someone will say its the rule of threes. But will the person who says it be serious? And will the person who dies really be a celebrity (whatever that means)?
   
7. The paradox: there are so many famous people that I haven't heard of most of them.
   

I keep a list of old celebrities (defined as people that I have heard of-- websites of old celebrities have lots of people I never heard of) so that when they die I am NOT one of those saying I thought they were already dead. I noticed this when Dear Abby died: (1) young people asked Whose that? where as older people said I thought she was already dead. Margaret Thatcher and Roger Ebert people seemed to know they were alive.

The 2013 Knuth Prize will be awarded to Gary Miller at STOC (ACM Press Release). The Knuth prize is now given yearly for outstanding contributions to the foundations of computer science, jointly sponsored by ACM SIGACT and the IEEE TC on Mathematical Foundations of Computer Science.

In 1975, Gary Miller gave a polynomial-time algorithm for primality assuming that the extended Riemann Hypothesis is true. For a given n, if there is an x and j such that x^2j mod n is not 1 or -1 and x^2j+1 mod n ≡ 1 then n can't be prime. Gary showed that ERH implied that if n is composite there must be such an x ≤ O(log² n). Given ERH one could just search all possible x and j in time polynomial in the number of bits to describe n.

Rabin noted that one could choose x at random to get a probabilistic algorithm that was correct with high confidence without any ERH assumption, now called the Miller-Rabin test. In 2002, Agrawal, Kayal and Saxena showed a polynomial-time algorithm for primality without assumption, but the Miller-Rabin test is still much faster in practice.

This is just part of Miller's contribution. From the press release:

Miller also made significant contributions to the theory of isomorphism testing—the problem of telling whether two structures are the same except for the labeling of their components.  He showed the equivalence of many different isomorphism problems to the still-open problem of graph isomorphism, and identified many special cases that could be solved efficiently. These included the problem of testing isomorphism for a special case known as bounded-genus graphs, a result he obtained with John Reif in 1980.  In 1985, in another collaboration with Reif, Miller invented the concept of "parallel tree contraction." This is one of the most fundamental primitives in parallel algorithm design with wide applications to graph theoretical and algebraic problems.

In 1984, Miller moved into the area of scientific computing. He set up the theoretical foundations for mesh generation, and was the first to design meshing algorithms with near-optimal runtime guarantees.  His subsequent research led to his breakthrough 2010 results with Ioannis Koutis and Richard Peng that currently provide the fastest algorithms—in theory and practice—for solving "symmetric diagonally dominant" linear systems. These systems have important applications in image processing, network algorithms, engineering, and physical simulations.  

Come to STOC and see Gary Miller's Knuth prize lecture. The program has just been posted and if you need financial help to attend, the student travel grant deadline is April 16.

Math Mammoth Grade 5 revised version is now out!

If you are an old customer and would like the updated files, please use the contact form here and include your purchase information (email/name used, if at Kagi or Co-op). Currclick customers can log in to their account at Currclick and download it from there.

Download below:
5-A contents and samples
5-B contents and samples


Here's a summary of the changes between the old and the new. You can also find this information here.

The biggest changes in grade 5 are that the two optional chapters (percent and integers) have been removed, and the topics in geometry chapter are different. However, overall, the structure and contents of this grade is very similar to the old.

• Chapter 1 starts out like the old version, with some review topics and multi-digit multiplication. There is now more focus and practice problems on long division with two-digit divisors. One big change is that the lessons on problem solving and equations were moved to their own chapter (chapter 3 in the revised version). Another is that I have now included the topics of divisibility, factors and primes, and prime factorization (used to be in the old version grade 6).
   
• Chapter 2 (large numbers) is also very similar to the old version. The lesson A Little Bit of Millions was added (it used to be in 4th grade).
   
• Chapter 3 is about problem solving and simple equations (originally these lessons were in the 1st chapter).
   
• Chapter 4  (decimals) is very similar to the old version in its contents. The topic of writing decimals in expanded form was added to one of the lessons. Other changes are cosmetic - improved layout, images, scaffolding of concepts, and order of presentation (in some lessons).
   
• Chapter 5 (graphing and statistics) is very similar to the old version. I added one more lesson on patterns in the coordinate grid. This chapter is now Chapter 5 (used to be chapter 4), and is moved to the part B.
  
  It actually exceeds Common Core standards (line graphs, histograms, analyzing graphs, mean, and mode). However, I feel students will need this as a background before the statistics topics of 6th grade, which are somewhat advanced. For example, in 6th grade students will study interquartile range and/or mean absolute deviation, and will need to relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. I do not feel that you can read Common Core Standards as "include this and nothing else", because the standards are not all-inclusive: there are many topics they do not explicitly mention. In my opinion, for students to be able to make sense of mean absolute deviation and relate the choice of measures of center and variability to the shape of the data distribution in 6th grade, they need to analyze and study statistical graphs in easier manners in 5th grade.
   

• Chapter 6 is fraction addition and subtraction. It is, again, very similar to the earlier version in its contents. I have added a few new problems about how to spot calculation errors using "fraction number sensen" and about line plots.
   
• Chapter 7 is fraction multiplication and division. It is similar to the earlier version, but has some changes as well. The lesson Fraction Multiplication and Area is expanded to include problems where the student extends the sides of the given rectangle to get a square unit, and then write a multiplication for the area of the original rectangle. There are two new lesson Multiplication as Scaling/Resizing and Fractions are Divisions. Line plots are included in several of the lessons.
  Fraction division is only dealt in these special cases: sharing divisions (such as 4/6 ÷ 2), dividing unit fractions by whole numbers (1/5 ÷ 3) and dividing whole numbers by unit fractions (such as 5 ÷ (1/3)). These types of divisions can be solved with mental reasoning, without using the "rule". The general case of fraction division is left for 6th grade.
   
• Chapter 8 is geometry, and its topics are different from the older version. Now, it includes a review of angles, a review of area and perimeter, drawing circles, classifying triangles, classifying quadrilaterals, and volume of rectangular prisms. It used to include many lessons about the area of polygons, which will be moving to 6th grade.

As you may know, this is also the version that is aligned to the Common Core. It turns out that some may think the alignment is not "perfect", as I felt I needed to include a few things outside the standards. Like I've said before... I do not feel Common Core Standards are "all-inclusive". In other words, if something is not mentioned in the standards does not mean that you leave that out.

For example, rounding and estimating are not mentioned in the standards for 5th grade. However, since students learn and use bigger numbers in this grade than they did in 4th grade, I feel it is good to review and practice them one more time.

In number theory topics, CCS mention that in grade 4, students are to "gain familiarity with factors and multiples." Then in 6th grade, students find common factors and multiples. However, CCS do not mention in which grade students should study prime factorization. Obviously you cannot leave that out, and so I put it in 5th grade.

Then, problem solving or word problems are not mentioned in any of the numbered individual standards for 5th grade. I included a whole chapter on problem solving, and I feel obviously word problems are to be included in 5th grade mathematics. Common Core Standards mention as one of the recommended mathematical practices that students are to "Make sense of problems and persevere in solving them."

Yet one more: 5th grade standards do not include any statistics topics. However, I feel students will benefit from being introduced to some of the concepts (line graph, histogram, mean, and mode), and from being used to analyzing graphs in order to prepare them for 6th grade statistics topics, which are somewhat advanced. In 6th grade, students will study for example interquartile range and/or mean absolute deviation, and will need to relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

See also: Math Mammoth Common Core alignment, Grade 5 - a list of lessons and the corresponding Common Core Standards in Math Mammoth grade 5 (a PDF file).

All of the math and history in this post is elaborated on in my paper here.

Are there any interesting applications of PURE math to the Social Sciences or History? Scarf's application of the Brouwer fixed point theorem to Economics is one of many examples of applying (arguably) pure Mathematics to Economics. Cartwright, Harary, and others appear to have used graph theory to model social relationships; however, on closer inspection they just used the language of graph theory. In C.P. Snow's article, The Two Cultures, he speculates that there is a cultural divide between the sciences and the humanities, which may make such collaborations difficult. This points to the lack of interaction between the sciences and the humanities being a sociological problem in itself; however, we are not going to go there.

There was an application of Ramsey theory to sociology in the 1950's. In Jacob Fox's Lecture Notes in Combinatorics he tells the the following well known story:

In the 1950's, a Hungarian sociologist Sandor Szalai studied friendship relationships between children. He observed that in any group of around 20 children he was able to find four children who were mutual friends, or four children such that no two of them were friends. Before drawing any sociological conclusions, Szalai consulted three eminent mathematicians in Hungary at that time: Paul Erdos, Paul Turan, and Vera Sos. A brief discussion revealed that indeed this is a mathematical phenomenon rather than a sociological one. (Namely R(4)=18&leq20.)

This is more of an anti-application since math was used to prove there was NO interesting sociology.


Recently there was a REAL application of Ramsey Theory to History, and later of History to Ramsey Theory. We summarize the results; however, the reader should look at the link above for more details.

1. Sir Woodson Kneading, a scholar of pre-christian history of England noted that, from 600BC to 400BC, whenever 6 lords were in close proximity, war broke out (with one exception). Either 3,4, or 5 of them formed an alliance against the rest, or 3,4,5, or 6 hated each other and went to war. The exception: all six formed an alliance. Kneading hired a CS grad student, H.K. Donnut, to verify the data. (Note that they really used R(3)=6.)
   
2. Kneading noted that, between 400BC and 200BC, whenever 18 lords were in close proximity, war broke out. Either 4,...,17 of them formed an alliance against the rest, or 4,...,18 hated each other and went to war. Again, Donnut verified the data (Note that they really used R(4)=18.)
   
3. Kneading found more results of this sort. His resuls and speculations, when translated to mathematics, are Ramsey Theoretic.
   
4. Kneading wrote a 300 page book on this topic using the data that he colleced and Donnut verified (Donnut declined to be a co-author since, in his view, Kneading did the intellectual heavy lifting).
   
5. Alma Rho Grand, a combinatorist, saw Kneading's book and realized that Ramsey Theory would simplify the work tremendously. Grand and Kneading wrote an article of which Kneading said My paper with Alma says cleanly in 30 pages what I said clumsily in 300 pages.
   
6. Grand noticed that one of Kneading's examples had 48 lords in proximity but no war broke out. This was in an era where if 5 formed an alliance or if 5 hated each other then a war should happen. She verified that this configuration showed R(5)&geq49. It is already known that R(5) &leq 49. Hence she showed R(5)=49. (Note that R(5) was unknown before this time.)
   

This is a case where Ramsey Theory helped History and History helped Ramsey Theory. Hopefully there will be more.
~

Here's something for all of us puzzle lovers: logic imbalance problems invented by Paul Salomon (HT Denise). You need to order the shapes by their 'weight':



Think logically - or write down some inequalities and use algebra. Pretty cool. They are simple, yet captivating. A new, creative idea! Paul also recommends you start making your own imbalance puzzles, as a more 'puzzling' exercise.

I'm having difficulty in solving this question which involves calculating fractions - this question relates to finding an arc length.

140 divided by 360, multiplied by 2, multiplied by 22 divided by 7, multiplied by 12:

140 360

×

2

×

22 7

×

12

Solution:

You can either put everything in the calculator, multiplying the top numbers, then dividing by 360 and 7.

Or, you can simplify before you multiply. This process is actually quite handy!

For example, the first fraction 140/360 can be simplified into 14/36, and then further into 7/18 before you multiply.

We get

7 18

×

2

×

22 7

×

12

Now, the 7 in the numerator and the 7 in the denominator cancel out.

Why? Every time we have the same number in the numerator and the denominator, and the only other operation involved is multiplication (like in our example), that number cancels out. It becomes the same situation as if you multiply by 7 and divide by 7: the result is 1. As a shortcut, we can cancel out those numbers and write 1's in their places.

Now we get

1 18

×

2

×

22 1

×

12

Then, 22 and 18 have a common factor 2... so that 2 cancels out. You can think of it as being...

1 2 × 9

×

2

×

2 × 11 1

×

12

... or you can think of it as if  the fraction 22/18 was in there, which simplifies to 11/9.

1 9

×

2

×

11 1

×

12

One last simplification: 12 in the top and 9 in the bottom have a common factor 3... so, divide both 12 and 9 by that 3 and get:

1 3

×

2

×

11 1

×

4

Now it is easy to multiply mentally (regular fraction multiplication):

1 3

×

2

×

11 1

×

4

=

88 3

The South African version of Math Mammoth is now available for grades 1-3!

You can either purchase any of the grade levels 1-3 separately, or purchase them all in a discounted bundle (33% off).

The order page is here.


Please see more information here:

Math Mammoth Grade 1, South African version
Math Mammoth Grade 2, South African version
Math Mammoth Grade 3, South African version

I hope South African homeschoolers and parents can spread the word!

Let n_σ(w) is the number of σ's in w.

We often ask our students about languages like { w | n_a(w) = 2n_b(w) } (CFL but not REG). Lets formally define languages that are like that.

A COUNTING DESCRIPTION is a boolean combination of linear equations and inequalities involving n_σ(w). For example

(n_a(w) &leq 2n_b(w)+3) AND NOT( n_c(w) = n_b(w) ).

We denote a counting description by E. Let L(E) = { w : E(w) is true } A lang L is CD if there exists an E such that L=L(E). What to make of the class CD?

The following items came out of emails between myself and Eric Allender, Dave Barrington, and Neil Immerman.

1. CD is contained in 1-way log space and in uniform TC₀.
   
2. CD is incomparable to REG since (1) from above we see there are langs in CD that are not REG, and (2) ba^* is not in CD.
   
3. CD intersect REG is in uniform AC₀
   
4. Parikh's Theorem yields a large class of CD's that produce context free languages.
   
5. If E is a Boolean combination of threshold and mod statement, each about a single n_σ(w), then L(E) is regular
   
6. The following papers may help answer some of the questions one could ask: here and here.
   

Are the following decidable:

1. Given a counting description E, is L(E) regular?
   
2. Given a counting description E, is L(E) context free?
   

Richard Beigel has shown these problems, with an unbounded alphabet size, are NP-hard. see here. I am very curious about the case where the alphabet size is bounded. Are there other (NATURAL!) classes one could ask about? NOT context sensitive since CSL contain log space. NOT the class DSPACE((log n)^1/2) since that's not natural. Maybe some sublinear class would be interesting. We can also ask variants of these questions involving any combinations of the following variants:

1. Do not allow negation.
   
2. Do not allow intersection.
   
3. Do not allow inequalities.
   
4. Do not allow additive constants.
   
5. Do not allow multiplicative constants.
   
6. Only allow a bounded alphabet size.
   
7. Allow other types of equations, for example n_a(w) = n_b(w)².
   
8. Allow other primitives such ast n_a(w) == n_b(w) mod 9.
   
9. Have a non-uniform version of Counting Descriptions and bound the size of the formula as a function of n.
   

In problem 7 if you allow any poly then the problem is undecidable by the solution to Hilbert's tenth problem.

The great Hungarian combinatorialist Paul Erdős was born one hundred years ago today. The big celebration will happen in Budapest in July.

It's hard to say more about Erdős than I've already said in this blog so let's recap some of those highlights.

• What is your Erdős number?
• The Erdős-de Bruijn Theorem
• The Erdős distance problem solved
• Erdős-Szekeres monotone sequence theorem
• Erdős-Rado sunflower theorem
• Erdős-Turan conjecture
• Erdős and the "stupid question"
• Solving math problems for money
• Sum-Product Theorems
• Sauer's Lemma (which may or may not have been first conjectured by Erdős)

Why do scientists publish so much? There is the whole "publish or perish" thing but that doesn't explain the large increase in quantity of publications. With a focus on important publications and measures like H-indices, a simple publication in a conference often won't help someone's career. Yet we continue to publish as much as we can. Why?

When you play a slot machine and you win, you get lights, music, sounds of coins coming out of the machine. If you lose, nothing. Lots of positive feedback if you win with no negative feedback if you don't.

If you submit a paper and it gets accepted into a conference, you feel excited. Excited to see your name on the list of accepted papers. Excited to update your CV and to give that talk or poster that only those few who's paper was accepted get to give.

If your paper is rejected, nothing. You don't list rejected papers on your CV. Nobody will even know you submitted the paper. And you can just take that paper and submit it to another conference. Lots of positive feedback if your paper is accepted with no negative feedback if it isn't.

Some people might argue the analogy doesn't work since slot machines are arbitrary and random. Those people have never seen a program committee in action.

AP press 2050: The new pope was elected in just 2 hours using EasyPope, the software based on EasyChair, software designed to deciding which papers get into a conference. The new Pope was quoted as saying How did they manage in the old days actually Flying to Vatican to elect someone. This just seems silly.

(I do not know which is more unrealistic: That the Pope will be picked this way or that there will be an AP press in 2050.)

The recent Papal election was done the old fashioned way--- in person. Could they have done it over the web? Perhaps each Cardinal gives each cardinal (except themselves) a number between 1 and 10 for how good they would be, and a number between 1 and 3 for how confident they are in their vote.
Or a more complex system like e-harmony uses?

I doubt this will change anytime soon, and I doubt that it should. So what are the PROS of each way to meet? Some of this was covered in the Net vs Jet discussion.

PROS of meeting in person:

1. With current technology (and this may change) its easier to have a back and fourth in person. Even now this is possible with teleconferencing, though I wonder if this would work with the 115 cardinals.
   
2. You can read peoples faces and enthusiasm.
   
3. There are things that can come up and be discussed that you may not have thought of if you were just alone at your computer.
   
4. Technology fails sometimes.
   
5. Time limited- really has to end (Papal Elections can go on for a while; however, one of the reasons for the Papal Enclave is to FORCE them to get a Pope relatively soon.)
   
6. Builds connections. Recall the famous quote:
   

   As a society we are gaining efficiency but loosing connectivity
   

7. The meeting gives you a global view of the issues.
   

PROS of meeting just on line.

1. Money and Time are saved.
   
2. Environmental concerns of flying
   
3. Less likely to have Group Think set in.
   

So- which types of meetings are better for which events? And what is the criteria? Is the goal to have a better decision in the end? This is only one goal- the connections formed at the meeting are valuable also.

1. Papal Election. There are so many candidates and so many issues that I think in person is better. I also think it won't change--- NOT because the Vatican is Tech-Shy (the last Pope had a Twitter account) but for the reasons above.
   
2. The Maryland Math Competition. We used to meet four times a year, then two, and now its down to zero--- its all online now. We will go back to two meetings a year--- having someone explain a problem and its solution to you is much better than email. Note that for this a meeting is a time sink but not a money sink. A Memory--- my first year on the committee we met at the end for Pizza and Beer and they got me a non-alcoholic beer (since they knew I didn't drink). They were welcoming me into the club. By contrast I don't even know whose on the committee anymore---- just their email addresses.
   
3. Program Committees- The money and time involved in getting everyone to the same place is rather a lot so I suspect these will be mostly online. I know there are some exceptions, and some meetings of subsets of the committee. At one time the COLT meeting was held AT the STOC conference where everyone would be there. Even so, it seems like the advantages of in-person are out weighted by the time and money.
   
4. Conferences themselves. Could all be videotaped an available (some are) but somehow its hard for me to really GOTO an online talk.
   
5. Faculty meetings. Sometimes when we try to resolve an issue online the emails just go on forever. Best to just meet and get it over with.
   
6. Grad Admissions. Its now online. I miss the days when we would meet with paper folders in front of us and order a pizza.
   

We had two terrible losses this week. Not people but websites, Intrade and Google Reader.


The corporate auditors for the real-money Irish prediction markets site Intrade found improprieties with payments to the late founder John Delaney and have effectively shut down the site, probably for good. I never bet money on Intrade but they made their bet data easily available. I used Intrade data to power my electoral markets map, possibly the most accurate predictor of elections, at least until Nate Silver came along. Even then our map updated in real time based on new information reflected in market prices, while Nate had to wait for poll data. I also used Intrade generated graphs of prediction market prices in dozens of talks I've given over the past decade. There are other sites for prediction aggregation but painful to lose the granddaddy of them all.

Google announced it is closing down Google Reader, Google's newsreader on July 1. I find Reader invaluable for keeping up with the theory blogs, especially those that don't post that often (I'm looking at you Scott and Luca) as well as a various collection of other blogs and my daily Dilbert. Not to mention the 2241 people who subscribe to this blog on Google Reader. There are alternatives (I'm trying Feedly) and I will be more vigilant on sharing posts on Twitter and Google+ but the real question is to why do fewer people use Google Reader. Is the flood of stuff on the Internet gotten so large we don't even try to keep up with it?

The 2012 ACM Turing Award, the highest honor in computer science, will be given to MIT cryptographers Shafi Goldwasser and Silvio Micali. From the press release:

Working together, they pioneered the field of provable security, which laid the mathematical foundations that made modern cryptography possible. By formalizing the concept that cryptographic security had to be computational rather than absolute, they created mathematical structures that turned cryptography from an art into a science. Their work addresses important practical problems such as the protection of data from being viewed or modified, providing a secure means of communications and transactions over the Internet. Their advances led to the notion of interactive and probabalistic proofs and had a profound impact on computational complexity, an area that focuses on classifying computational problems according to their inherent difficulty.

Shafi and Silvio's paper Probabilistic Encrytion really did set the stage for modern cryptography. Their paper with Charlie Rackoff, The Knowledge Complexity of Interactive Proof Systems started my own research in that area. When I did my graduate work at MIT, I had many great discussions with Shafi and Silvio about cryptography and proof systems and I owe them much for my own research career.

Congrats to Shafi and Silvio!

Questions about Math Mammoth and the Common Core Standards alignment?

I made a new, hopefully comprehensive FAQ on this topic.

Please refer to that document, and hopefully it will answer everyone's questions.

A professor tells the class that he will use the highest grade to set the curve. The students all conspire to NOT take the exam, so the highest score is 0, so they should all get A's. If you were the prof what would you do?

This is NOT hypothetical. It happened- see here.

1. The prof gave all A's and didn't even mind it since the students learned to cooperate.
   
2. The prof then changed his grading scheme.
   
3. The article calls it a prisoners dilemma problem. I don't think thats right; however, it is the case that someone could have `defected'
   
4. Curving an exam based on the BEST student seems odd.
   

Denise just posted an article How to Recognize a Successful Homeschool Math Programon her blog.  I enjoyed that a lot and recommend you read it too, no matter what math curriculum you are using!


She summarizes it this way:

If you are wondering how well your homeschool math program is working, pay attention to your children.

• Do they understand that common sense applies to math?
• Can they give logical reasons for their answers?
• Even when they get confused, do they know that math is nothing to fear?

If so, then be assured: your children are already miles ahead of most of their peers. Their foundations are solid, and the details will eventually fall into place as you continue to play with mathematical ideas together.

She also notes her 'yardstick' for measuring math anxiety: if your child does not fear word problems, he/she is not suffering from math anxiety.



There was a time when my second daughter actually relished word problems and thought they were the BEST part of her math work (it was about 2nd- 3rd grade). Now she said she still enjoys them, but likes mental math problems best (she just started 5th).

When it comes to mental math, I sometimes give myself a little challenge (such as when making an answer key to my books): can I do this problem mentally instead of a calculator? It's not anything I fear - it's enjoyable in a sense.

I feel this is similar to when people do crossword puzzles, solve Sudoku, or even play Freecell: you actually enjoy the mental challenge, right? The same can happen with mental math, or with math in general:- it doesn't have to be something fearful, disgusting, or repulsive -- far from that! : )

I've just completed making two new worksheet generators for HomeschoolMath.net:

Classify triangles  - make worksheets for classifying triangles by their sides, angles, or both.

Classify quadrilaterals  - make worksheets for classifying (recognizing, idenfitying, naming) quadrilaterals. There are seven special types of quadrilaterals: square, rectangle, rhombus, parallelogram, trapezoid, kite, scalene, and these worksheets ask students to name the quadrilaterals among these seven types.

They look sort of like this:



Use the links above to set your options (image size, number of problems, etc.)

I had fun making the scripts, though also some challenges. But overall I enjoy such work - programming is similar to problem solving in math.

In this case the problems often were math, such as how to make a script that gives me a kite or a scalene quadrilateraly with varying dimensions. I was using php GD library to first create a bunch of images, and then made the worksheet script that simply chooses randomly among the pre-made images.

Enjoy!

I've really learned to enjoy opera (the art form not the browser) and while I've come to really enjoy Atlanta, the city only has a regional opera company. So I tried out the Metropolitan Opera HD broadcasts in a local movie theater here. I have to say the experience was quite amazing, the picture and sound was amazing. In many ways a better experience than watching the opera live at the Met, with close-ups and back stage interviews. It doesn't replicate the atmosphere of seeing an opera live but it does give people who don't have access to the Met a great opera experience.

Watching the opera made me think of an analogy to MOOCs. There is limited scaling we can do in a classroom or an opera house, but the Opera HD and MOOCs can scale tremendously with only moderate additional cost. A MOOC doesn't replicate the classroom experience but done well it can offer some advantages to a classroom.

So this seems like a win for the opera lover. But not every opera company benefits. I went to see the Parsifal on Met HD last Saturday instead of the Altanta Opera's Traviata. Even other great US opera companies like the Chicago Lyric might not get a huge audience if they tried broadcasting their operas in movie theaters. How does the analogy play out for universities and MOOCs?

I have updated the Basic operations worksheet generator at HomeschoolMath.net—with new features!

Now you can add a border around each problem, add additional workspace below the problem, and use a variable instead of an empty line.

Even more: instead of choosing only one operation, you can also choose addition and subtraction, or multiplication and division.

This worksheet generator makes printable worksheets both for whole numbers and integers, and for both horizontal and vertical form of operations. It is very versatile. You can make worksheets for...

• basic addition/subtraction facts
• mental addition and subtraction (for example, whole hundreds)
• multiplication tables, including missing factors
• basic division facts, including with remainders
• mental multiplication and division (for example, by powers of ten)
• column-form addition and subtraction, including subtraction with or without regrouping
• column-form multiplication (long multiplication)
• long division
• simple equations using either an empty line or a variable (choose missing addend/subtrahend/minuend/factor/dividend/divisor). This is usable from first grade up to pre-algebra/algebra 1!

Click here to use it! Below you'll see a screenshot of the generator.

I’ve been off Being Human (the original UK version) for a while now. I stopped watching after the first episode of season 3, in part because the American version came out and I got into that before it started circling the drain, and in part because Mitchell gets on my damned last nerve.

Now I’m back to watching it after I discovered all the original cast members leave by the end of the show. I’m so looking forward to seeing emo-goth pretty boy Mitchell get dusted— or maybe they’ll dial up his already epic levels of self-pity, narcicissm, and angst to the point that he combusts— that this was just what I needed to get back in the game.

Too bad that means Lenora Crichlow and Russell Tovey will be leaving eventually too. Their acting is the strongest reason I watch this show: when they cry, I cry. When Lenora gives her monologues, I cry a little. When Russell opens up about his feelings, I ball. It’s pathetic, I know, but the point is that they’re very good at getting across the underlying message of the show: that even monsters can be human, because humanity is about our relationships.

Anyhooo, now I’m back to watching Robson Green play a homeless(?) drifter werewolf dad forced into underground supernatural cage fights.

I know I'm late at blogging about this but here goes anyway. Curtis Cooper at the University of Central Missouri in Warrensburg has found the largest prime number yet.

(He hasn't found the largest prime as there is no such thing -- he's just found a new prime that is larger than any other primes people have found.)

It is a Mersenne prime, which means it is of the form 2^P − 1, where P itself is prime. The one Cooper found is 2^57,885,161 − 1, and it has 17 million digits!!

 So no, I'm not going to type it out here! Writing it in the form 2^57,885,161 − 1 is way handier, isn't it? Shows us how important exponents are. So, this new prime is 2 multiplied by itself 57,885,161 times, and then you subtract 1.

This is what Cooper himself says about the hunt for new primes:

"Every time I find one it is incredible," Cooper said. "I kind of consider it like climbing Mount Everest or finding a really rare diamond or landing somebody on the moon. It's an accomplishment. It's a scientific feat."

Read more:  Missouri researchers ID largest prime number yet. If you wish to see the number, you can download a text file of it here.

Math Teachers at Play blog carnival #59 is posted at Learners in Bloom. Looks good, as usual!

I want to especially check out the various science and math Youtube channels mentioned -- hope to find time for that.

A quick question I asked my daughter as posting this: is 59 a prime number? (She's reviewing factors and primes right now.) Just one example of how you can just weave math into everyday life. Another question I asked her just a few minutes ago at the lunch table was to find a prime between 90 and 100. And she found it!