If fractions make you pale, then scale, scale, scale! OR When kids invent algorithms

My goofy title is in response to the annoying rhyme you may have learned "Don't ask why, just invert and multiply".

In the course of writing a Professional Development module for teachers, and I've spent hours downloading papers about algorithms that kids invent for division of fractions.

Here's a summary of a not-so-common, but very useful method for dividing fractions and the representation that facilitates its discovery. I'll call this the scaling algorithm because it is born out of considering areas of rectangles and how the ratio of area to length of a side is preserved under scaling.  The algorithm is:

For example:

What representation would lead to this algorithm's discovery?  Ask the question in the context of area:

"What is the width of a rectangle whose area is 2 sevenths and whose length is 3 fourths?"

To answer this, let's make seven copies of the rectangle -- Why? Because integers are easier to deal with and we all know that 7x(2/7)= 2.  The question has now been reduced to "What is the width of a rectangle whose area is 2 and whose length is 7x 3 fourths?" 

Since 7x3=21, we have 21 fourths as one side of the rectangle.   We have exchanged ? x (3/4) = 2/7 for ?x (21/4) =2.  We may also notice that scaling each factor by the same number preserves the answer!

Let's scale again!!!  Take four copies of the big rectangle, which gives you a total area of 8, and a total side length of 21.
We have now replaced all fractions with integers and traded the original question in for the simpler sounding:
"What is the width of a rectangle whose area is 8 and whose length is 21?" At this point, we see that the answer is 8 divided by 21 or 8/21!  Woohoo!!!

Do kids invent this scaling algorithm?  Indeed they do if they are given the area context and their brains are not stuffed with "this is how you do it" algorithms.  Want to read a paper that talks about this?  Jaehoon Yim from South Korea has a 2009 paper that's fascinating -- Children’s strategies for division by fractions in the context of the area of a rectangle -- I'm not sure if you'll be able to access it for free.  But, the general gist is that ten and eleven year old students were able to "make the width equal to 1, make the area equal to 1, and change both area and width to natural numbers".  The strategy outlined above is of the last variety.  She even researched whether children could formalize their pictorial drawings to create a numeric algorithm -- and surprise, surprise -- they could!  Granted, these were students who were picked because they had a "positive attitude towards mathematics".

Incidentally, if the numbers are nice like 6/20 divided by 3/4, then they can use the area model to see that the answer is 2/5 since 6 divided 3 is 2 and 20 divided by 4 is 5.  In other words,

Here's a picture that shows this.  Can you see how?  The strategy is a little different from the one above as it starts from a unit square.

Six pointed stars -- just in time for the holidays!

So I'm on vacation in Maine where it was snowing today, and we took my little one for a walk in the snow.  A perfectly formed little snowflake fell on his eyelash, and we all just stood around and stared at him for a few minutes while it melted.  It was so beautiful!  I'd never been able to see that symmetry so well with my naked eye.  We took a picture, but it just didn't do it justice.
   Anyway, it reminded me of a book I'd seen once that would make a great gift.  The book contains snowflake photos taken and explanations by Caltech physicist Kenneth Libbrecht.  Apparently photographing snow is an expensive hobby, costing over a thousand dollars, not to mention you need to keep your camera warm!
   Then I just happened to see this wonderful video made by George Hart, Vi Hart's father and also one of the designers of many of MoMath's new exhibits.  The key in understanding what is obtained when you slice the Menger sponge in half along its diagonal is that we can view a star of David as being the union of three Rhombi, not just two equilateral triangles.  This different way of constructing the Star of David was exploited by an artist I recently saw at Tucson's 4th Avenue Street Fair.  Unfortunately, I can't find his site!  But the general idea can be seen in this basic picture I made.
It's just amazing to me how the things I see in everyday life sometimes link together in these nice ways... Anyway, there's a lot of six fold symmetry going on just in time for the holidays.

Learning to Teach by Teaching Teachers to Learn

I am currently helping twenty-seven K-8 teachers brush up on their math skillz.  The teachers I am working with are honestly a JOY to work with.  The first few days we had comments in the daily feedback that revealed discomfort and frustration with math and with the idea that we were asking the participants to NOT FOCUS ON THEIR TEACHING but to FOCUS ON THEIR OWN UNDERSTANDING.  Boy have they come around!!!!

Teacher participant (after seeing a colleague at the board):  "Are you going to ask for more people to share whether they did the problem differently?"

Me:  "Why yes, I was now that you mention it."

Teacher participant (in the process of standing up to come to the board):  "Oh, good because I did it a totally different way, and I want to show what I did!"

Another paraphrase of a student talking from her seat:
"Can I share something I thought of about the area model?  It's kind of like when we add 0 a bunch of times so that we could subtract negative numbers....because you can add 0 as many times as you want and still get 0.  It's like that, except... really I should just show you."

Marie's "1 and 3/4 times 1 and 2/3"

She proceeded to place her drawing under the document camera, which looked something like the figure to the right.  She said

 "we had 1 and 2/3 going on to the right forever and 1 and 3/4 going down forever, and the product is just the overlap...So we don't need to think ahead of time of how big a rectangle we need".  So since you can see 35 squares, each of which is 1/12, you know that the product of 1+3/4 and 1+2/3 is 35/12.

I thought that her invention was really slick.  She noticed that adding extraneous area actually made the concept shine through.  We had just finished talking about how "adding 0" and "multiplying by 1" could be useful.....She really internalized that..

Marie asks LOTS of questions.  And as the workshop has progressed, the number of excellent questions that are brought up AND answered by the participants themselves just keeps growing.  Granted, these teacher participants are being paid to attend, but I don't think that's all that is motivating them to ask such wonderful questions.  I think one reason is that they are not being graded.


Many mathematicians (including myself) are appalled at the low level of understanding and lack of interest in understanding shown by many elementary school teachers as well as those who aspire to be elementary school teachers.   But being appalled doesn't accomplish anything.

LEARNERS' RIGHTS and MY RESPONSIBILITIES AS AN INSTRUCTOR:


Many people talk about the "Rights of the Learner".  But we (my teaching partner and I) REALLY took that seriously.  Clearly it's important to create a classroom environment that is comfortable, but that doesn't mean "feel-good" or "dumbed-down".  I now attempt to do the following:


1) meet anger about and frustration with the material with a sense of humor as opposed to dismay or disapproval.   


2) acknowledge that learners may have previous experiences that we can learn from.  So we should solicit anonymous feedback, show that we have read it, and publicly address or discuss it.


3) encourage learners to reconcile old ideas (whether correct or incorrect) with new ones   Asking them to "just start from scratch" or "just forget about the algorithm" is unfair. 


4) keep in mind that learners want to please the teacher (especially if a grade is involved).  When there is no grade involved, students are more likely to express their frustrations honestly, but still hesitate for fear of "upsetting" the instructor.


5) be consistent in "walking the walk".  In other words, if I talk about the philosophy of "why something should be discovered or learned independently", I have to try to avoid things like writing on students' papers or directing them on what to write.  I have to invest in their thought process whole-heartedly regardless of my concern for "covering material".


 For instance, after actually discussing and debating about the abhorred-by-mathematicians "FOIL method" of multiplying binomials and how it was related to the distributive property, one student jokingly said "Well I prefer the SARAN WRAP method."  After we had discussed the method (its advantages and disadvantages) heatedly AND with humor, even though they knew I "didn't approve", students felt comfortable saying that they "used FOIL" to solve the problem.   To me, this is much more preferable because they are being HONEST about how they thought about the problem rather than trying to make me feel good.

That's all for now...So many thoughts are running through my head.  But here's a link about Finland's lack of grading which has means that in Finland The differences between weakest and strongest students are the smallest in the world!!!

Let's just say a lot's happened since my last entry.
For instance, I had a baby!  So what's it like teaching and doing mathematics with a baby?
Let me just tell you that my 8-month-old does not think that he should have to share me with my computer...

   After a particularly productive bit of time spent typing on my computer, feeling proud of myself, and ready to email my work out, I realized that I had been ignoring the little firefly who was screaming and wailing for me to pay attention to him.  So I picked him up, and as I turned around, he spit up ever-so-adeptly on my keyboard.  Of all the directions he could have chosen, the spit-up landed just so, and tt took me a second to realize that my computer was probably fried.  After turning it off, waiting a while, and turning it on only to find out that the "k" key now controlled the volume of the speakers, I went to try and get it repaired!  "Water-damaged?" they asked.  "Why, whatever do you mean?" I replied.

   Another morning, after staying up late to grade a katrillion exams, my kiddo woke up with a fever!  Teething had commenced, and he didn't really care if I still had the last day of classes, reviews, extra office hours, and more grading to contend with.  What good are exams unless you can chew on them?

   But really he's a pleasure--- in office hours, he plays in the pack 'n play quietly while my students do math at the board, and they are more likely to try and engage him then he is to distract them.  That's right, I take him to school with me about once a week.  And no, I don't leave him in daycare.  Either my husband, one of my friends, or myself takes care of our sweet baby at all times....

   But I'd like to hear if anyone else in mathematics has stories about raising children while working.  I'd especially like to hear about women who are doing that...  I don't meet too many, but maybe that's just because we're all too busy to be bumming around at coffee shops or chit-chatting in the hallways....or reading/writing obscure blogs :).....

Topology and the Moore Method--i.e. what I'll blog about for the next few weeks

  So I'd like to start blogging about this class I'm teaching.  I decided to use the Moore Method to teach my Topology course.  I'll start by giving a little background.  My first and favorite math courses were taught by Dr. Phil Tonne and Dr. Bill Mahavier at Emory University.  Dr. Tonne taught me Matrix Algebra and Calculus, and Dr. Mahavier taught my Real Analysis class.  I was a teenager when I took these classes and the format of the classes spurred me to work for hours at a time on math at the kitchen table with my mother occasionally putting food in front of me or telling me it was time for ballet class.  What I wouldn't give to get back to the level of focus and dogged determination I had at that time!
   Anyway, I got up to the board in those classes, sharing solutions to problems in Calculus class, and proofs in Real Analysis, and I still remember some of the proofs and ideas I learned in those classes after fifteen years.  Most of all, I remember the feeling of accomplishment and excitement that I got from doing some really hard problems myself...And I didn't really take very many notes either...I went on to take some time off from math in my Junior and Senior years of high school when I turned my attention to French, Latin, and Linguistics, then back to math in college, and again in graduate school.  In college, I had more amazing Texas-style classes with Dr. John Neuberger, who made me think that I might study Differential Equations simply because his class made it seem so great.  But ultimately I was drawn to Topology, the field of the illustrious Dr. R.L. Moore, after whom the method of teaching I am inspired by is named.
   So that's why I'm inspired to teach by the Moore Method.  It was so influential in my own pursuit of mathematics that I feel it might stoke the fire of others who are already somewhat interested in mathematics.  And since teaching this way means that I don't get to talk much in class since the students are doing most of the talking, it means that I should talk here instead :).
   Being already about two weeks into the course, I'll tell you where we are right now.  We have made it almost all the way through about nine pages of notes, which briefly cover the Topology of the real line, the general definition of a Topology,  exercises on what makes two topologies the same, the separation axioms and their relationships to one another, many theorems and false statements concerning boundary, limit points, and interiors of sets in topologies satisfying various separation axioms.
    The students present around two or three exercises, counter examples, or proofs per class meeting, but only about half the students (out of 20) have made it to the board so far.  There has been quite a bit of juggling of the roster as students drop and add.  I started with 28 students signed up.  After listening to my opening spiel the first day, about five of them dropped immediately.  So I suppose that this scenario could have been a lot worse if I'd started with only those five students.
   The logistics of the class are a little hard to stick to as I sometimes get excited and forget to give those who have not yet presented a chance to put something up.  Instead, I've forgotten and asked for volunteers, which is honestly not seeming like such a good idea since I'll get the same five or ten kids the whole semester I suspect.  I have been good about going to the back of the room and being part of the audience, but there is some reticence to ask questions amongst the students.  So I've been having to sort of drag it out of them by calling on people at random to summarize the work of their peers.
    I'm really wondering how this would work in my introductory Calculus course.  I feel like it would be worthwhile since I'm writing their first exam and have only a vague idea of what anyone REALLY knows.  But they might hate me... Too late to change the format of the class now.  One thing I've learned in teaching:  stick to one thing and make it work.

EQUATIONS are now a monthly staple in Wired

 A great teaching tool is now appearing every month in the media!  Starting back in May, mathematics writer Julie Rehmeyer started writing a new column in Wired Magazine.  An entire page features one equation along with a fabulous graphic (much better than the silly one I made to the right) to help bring it to life.  There are even sliders on the variables to help readers understand the range of possible scenarios modeled by the equation.

May: Carbon Emissions
June: Phantom Traffic Jams
July: 3-D Rendering
August: Power of Waves
September: Roller Coasters

It hasn't been that long since my last post

    It's been about 7 months, maybe 200 days or so, since my last post.  And while you may not have been counting the 4800 hours, you probably aware of the 5 million barrels or 210 million gallons of oil that were being dumped into the gulf.  This recent SEED slide show highlights the our deficiency as humans when it comes to comprehending these sorts of massive numbers.  Perhaps this is part of the reason we cannot stop buying drinks in disposable containers.  The images in the slide show come from Chris Jordan's 2009 book entitled Running The Numbers .  Of course, the idea of using technology and imagery to help us wrap our minds around the gargantuan nature of our world is nothing new.  The short video Powers of Ten from the 1960's highlights the wonders of the universe by expanding our field of view by one power of ten every 10 seconds.  In the video, we see a square that is 10^8 meters on a side framing the earth and a square that is 10^-8 meters per side framing a coil of DNA.

   Recently, while teaching my Math 124 Calculus Course, I came up with a little related rates program to help the students wrap their minds around the spreading of the oil in the gulf coast.  According to a NY times article from this summer, anywhere from 12 to 25 thousand barrels of oil per day were being dumped.  Wikipedia's site on oil slicks asserts that an oil slick is no thicker than about .002 millimeters.
There are 42 gallons of oil in a barrel.  So we will go with the round and reasonable number of about 1 million gallons per day.  A quick conversion gives us

(1 millions gallons/day) (3.79 liters/gallon) (10^6 mm^3/liter) = 3.79 (10^12) mm^3/day

Now we make a few fairly broad assumptions:

1. The oil spreads in a circular manner and always has uniform thickness. 
2. The oil is being spilled at a constant rate.

Most of us have experienced that as liquid spills out onto a surface, it spreads quickly at first and then slows down over time even if we do pour at a constant rate.  But how exactly is the rate of change of the radius of the spill related to the current radius of the spill?

We now answer that question with a little calculus.
Suppose V(t) denotes the volume of oil spilled at a time t where t is measured in days.  According to the assumptions we made,
V(t)=pi* r(t)^2 h where r(t) is the radius of the slick in millimeters at time t (in days) and h is the thickness of the slick.  So, using the chain rule, we have V'(t)= 2pi*h*r(t)*r'(t).
Because we know that the rate of change of volume of the oil spilled is constant at 3.79 (10^12) mm^3/day, we can determine the rate of change of the radius with respect to time in terms of the radius of the slick at a particular time.  So we see that r'(t)*r(t) is approximately 302(10^12)mm^2/day.  If we choose to measure r(t) in kilometers, then r'(t)*r(t) is approximately 302 km^2/day.
In other words, the rate of change of the radius of the slick is inversely proportional to the current radius.
So, when the slick is 1 km in radius, it is spreading at a rate of 302 kilometers per day.
But when the radius is 302 km, the radius is changing at a rate of only 1 kilometer per day.
By the way, 302 km is about 188 miles.  But the rate of spread is slowing, so how long would it take for the oil to reach shore if the spill occurred 100 miles off shore?  If the rate of the spill really does stay constant (i.e. no successful clean up) then we see can approximate r(t) as 25 t^(1/2).   When is 100=25t^(1/2)?  After about 16 days. So a major oil spill, even if it occurred 100 miles out into the ocean could reach the shore in a few weeks!
   I can only hope that the spread of information is as successful and uniform.  And I also make the observation that I waited about the same number of days to post as the number of kilometers that the oil slick would have grown in even one day.

P.S. How accurate is this little estimate?  Certainly we made many simplifications and overlooked all clean up efforts.  Looking at an interesting app from the NY times, I see that it did indeed take about three weeks for the spill to be noticed on the shore of Mississippi about 100 miles away from the source.

The Last Year of Grad School

  Nobody tells you about the emotional part of leaving grad school.  The time I've spent here has been longer than the time I've spent anywhere outside of the town I grew up in.  I've developed relationships here that will (I hope) last me a lifetime.  I have become a wife here, an adopted "auntie", and a teacher to many.  As I bike into my apartment complex, I wave left and right at familiar faces.  I see children who were babies when I started grad school here.  I stop to talk with former students who are themselves moving on to grad school, new jobs, teach for america, or travel abroad.  Getting ready to go isn't just about writing my thesis -- it's about saying goodbye to the people and places I've bonded with.
   Granted, I'm happy right now to be finishing the umpteenth grade and to be starting a new chapter in my life.  But I'm just warning you -- steel yourself against the fact that you will be processing a lot more than just your mathematics as you write up that thesis.   You'll be thinking about what  to keep and what to toss, you'll be spending time with friends, reconnecting with those you've lost touch with, visiting with those who are just near enough to drive to see and just far enough to not bump into.  You'll be wondering if anyone else will carry on the traditions you've started.  You'll see your surroundings with different eyes.  Try it now...It might be an interesting exercise.

Manifold Fashion

   This week Dai Fujiwara, a fashion designer, presented a collection based on Topology, specifically Fujiwara discussed his collection with Professor Bill Thurston, who is famous for his Geometrization Conjecture .  Here is a blog entry with pictures that discusses the role of geometry in this years fashion.  Another designer is also mentioned as incorporating geometric ideas in the NY Times.  Also see the YouTube interview of Thurston and Fujiwara .  

Census and Sensibility

    With the 2010 Census about to take place in the middle of this month during what is Spring Break for many college students, we might start to think about fairness.  As people receive so much junk mail, it's important to convey the importance of filling out the ten questions (which is the fewest number to ever appear on a census).  According to the Census2010 website $400 BILLION is distributed according to the numbers recorded in the census.  Just recently, a house committee decided to table a proposal for a bipartisan commission to help decide the redistricting that will occur when the census is over.  So how will redistricting be decided fairly by those people who have themselves benefitted from possibly unfairly drawn districts?

    Here we see the 12th congressional district that was approved in 1994, and which was controversial for being racially "gerry-mandered" .  Now, we have to ask ourselves if there is anything wrong with a district having an "bizarre" shape, whether this is somehow giving an unfair advantage.  Perhaps requiring that districts be "sensibly shaped" in some way is a reasonable way to ensure fairness.  Perhaps not.

Let's also take into consideration that "There's nothing Maryland can do about its bizarre shape," as economist Christopher Chambers said at the AAAS symposium on Fairness and Mathematics.  Professor Chambers is one of the authors of an academic article entitled "A Measure of Bizarreness"  that will soon be published in the Quarterly Journal of Political Science.  So however we may choose to answer the question of "How Bizarre is that shape?" it must take into consideration that the shape may already live in a somewhat bizarre larger one.  The way to measure this according the the authors is to use a path-based measure of convexity.  Given a district, its bizarreness is determined by the probability that it contains the shortest path within the state that joins two randomly selected points in the district.  The higher the probability, the lower the bizarreness.

Here's a January article from Slate magazine concerning this matter.

     

What's Massive about Media?

    Media is the plural of medium, and is a substrate through which information flows.  During the AAAS Mass Media Luncheon, speaker Dr. Jeffrey Kirsch , the Executive Director of the Fleet Science Center , asked us for our reactions to the phrase coined by Marshall McLuhan in the 60's: "The medium is the message."  Just the day before in the Counter-terrorism Symposium, Keith Devlin spoke about the difficulty of quantifying "information" and its reliability.  These discussions led me to think about information like light -- a pure and difficult to measure substance whose appearance is determined by the substance through which it passes.  Rather than thinking of information as being stuck in some sort of box to which an elite few hold the key, I recognize that information is difficult to confine.  Just as we created the light bulb, neon signage, fiber optics, and lasers to channel light which is typically free to move about, we design means by which we channel information so that it can be viewed, touched, heard, smelled, and sensed as we see fit.
    We control some of the media through which information passes: our own mental framework, a blog, a podcast, an imax movie, a newspaper article, our social interactions.  But we don't have nearly as much control over the information itself, which is floating everywhere around us in more or less dense and disorganized clouds.  So people who choose to be involved in media are attempting to corral information, this unruly light-like substance, so as to harness some of it's power and help others use it as a tool to brighten their lives.
     With this metaphor in mind, I see that there is power in heterogeneity. With a variety of media, what remains to be seen is which media will excel at fulfilling which roles.  While some media will replace others (like compact fluorescents are increasingly replacing incandescent bulbs), many will coexist or work in concert.
   Still there is the concern that people will be blinded by so much information, as if we are in a world covered in a thick blanket of pure white snow.  Well, I think that's what sunglasses are for!  In other words, people will have to squint for a while before they realize which media they need in order to function, and many members of the public are probably in that squinting phase right now.  So when they realize that they need a way to filter out the intensity in order to focus in on some of the details and survive, they will be scrambling to find good journalists.

Frenemies and Functiononmeters - Countering Terrorism

   According to Dr. Gordon Woo, a catastrophe risk consultant who has also studied natural disasters, "It is the social networks between terrorists which ultimately are their undoing." Approximately 1 out of every 20 of a terrorists friends is either someone being watched by a security agency or an informant.   He concludes that terrorist cells with six or more members have about an 80% chance of being caught.  So one method of decreasing terrorist activities may be to model their extended social networks and the internal mechanisms of their cells.  Colonel Steve Horton's research focuses on the question of whether you can use raw data to infer social structure.  As a starting point he looks at the records over the last 30 years of bridge game scores to determine relationships between players.  
    On a different note, Dr. Paul Tannenbaum attempts to link the Research and Development world to the Battle Field by creating "functionometers" for the devices used by soldiers.  His hope is that by partially ordering the various functions of a device by their importance to a given mission, he can create a tool which quickly diagnoses the situation for the user.  In other words, the tool would be like a gas gauge of functionality ranging from "There's no way you can complete this mission with this device" to "Go for it!".
   Considering that several of the speakers for this Symposia were sick, a lot of information was presented.  I would loved to have heard more about how mathematics, as one speaker put it, "helps win over the hearts and minds" of insurgents' communities.  There are also ways to think of the mental attitude that condones terrorism as an "infection" that can be modeled in the same ways as disease.  For information on this and other research done in this area, check out Consortium for Mathematical Methods in Counterterrorism which was founded by Professor Jonathan Farley, the organizer of the symposium.
    
  

Higher School Mathematics

   So, there's the high school mathematics we all know and then there's the mathematics that the students pictured here are doing -- let's call it higher school mathematics!  These students were participating in the AAAS high school poster session, and had the opportunity to attend a conference right along side experts in many areas of science.

    Justin Lozano focused on solving the rubik's cube and variations using a computer programming approach.  Yesterday afternoon, he was among several students whose posters were on display in the exhibit hall.  When we talked about how he might use group theory to think about the rubik's cube, he commented "It's interesting that you think of the problem from a different point of view."

   Aishwarya Vardhana's poster features her work using linear programming methods to create a program that would show a company how to optimize their use of green energies such as wind and solar energy.    And, last but not least...

Varuna Rao worked on facial recognition.  She created a database with photographs for reference, and then wrote a computer program that assigns a number to a photo based on certain measurements obtained from the photo and compares different photos by comparing their numbers.  She says that now that she's learned about mathlab, she could see redoing her project using that tool, and that in the future, she would like to use more than one number to compare photos to improve her current recognition rate of 45%.

    Good luck to all of these students!

Gangs and Statistical Mechanics?

   The Los Angeles Police Department may have a new ally who, while not as all-knowing as Charlie Epps from Numb3rs, will help reduce crime by predicting where and when it might occur.   This morning, Professor Andrea Bertozzi  spoke about models that she and her team of post-docs at UCLA are developing to model gang violence.  Using statistical mechanics, bifurcation theory, and partial differential equations, she aims to predict where and when hot-spots of gang-related activity will emerge.  These tools have a long history in the physical and biological sciences, and are similar to those tools used to study the behavior of swarms of insects, which is the subject of some of her past research.
Dr. Bertozzi's newest paper on modeling gang activity will soon be published in the Proceedings of National Academy of Sciences.
   In order to model crime occurrence, Dr. Bertozzi consults the LAPD as well as Anthropologist Jeff Brantingham , whose research shows that criminals tend to commit their crimes near their own homes in areas with which they are familiar.  The model is designed with this in mind, and consists of a grid with a "house" situated at each vertex, freely moving "burglars", and an "attractiveness" function that depends on both space and time.  Different factors determine the attractiveness of a house to a burglar -- these include how close the house is to the burglar's home, whether the house was recently burgled, whether any of its neighbors have been victimized recently, and constants like the presence of graffiti or the type of housing prevalent in the area.  Two behaviors emerge from the simulations:  one in which a police presence would simply displace the criminal activity and one in which a police presence would actually eliminate the problem.  The model compares favorably to data collected over the period of a year in a particular LA neighborhood, and Dr. Bertozzi sees this as a first step in applying mathematical modeling to other social science issues.
    Where are the gangs?  Gang-related violence modeling is done by post-doc Alethea Barbaro , who studies gang networks, patterns in tagging, and how rivalries arise and dissipate.  In response to a question concerning how this work might inform decisions concerning the balance between hiring more policemen and "cleaning up the streets" to reduce the "attractiveness", Dr. Bertozzi responded "Sounds like a good research proposal!  These questions are excellent and hopefully will employ people like us for years to come."
 

World of Mathcraft?!

  Imagine your avatar immersed in a complex world in which mathematics is the greatest weapon, in which you can "turn the world off", sit down to build a new virtual tool with other players, and pick up again when you are ready to test the workability of your new tool.  Mathematician Keith Devlin thinks that such a game is entirely feasible, and could be a new way to attract students to mathematics.  Unfortunately, he also thinks that to develop such a game might be an expensive undertaking -- on the order of $100 million.  "But if we think of this as a national initiative then compared to Apollo it's not so expensive."

    Why should this be a national initiative?  The slide above was an illustration that 98% of students' responses to mathematical questions are correct when the questions are posed in a context relevant to the students' everyday lives while the students correctly answered only 37% of the SAME questions in a paper and pencil exam.  So one argument for creating math-based games for learning is that they have the potential to replicate the physical experience of using mathematics in everyday life much more effectively than typical textbook problems. Linguist James Paul Gee looks at textbooks as analogous to the dry manuals that accompany games, pointing out that "If you play the game you cannot fail to understand the manual."  Furthermore, he distinguishes between quality games and "skill and drill", saying "If we keep doing skill and drill, the only class your kids will care about is Mandarin."  However, he acknowledges that many of the people playing technical games are upper class, and that for a game to be successful as an educational tool, there must be an accompanying social network to which players can rely for mentorship and camaraderie.  Some of the games mentioned that you can learn about and try on-line were:

• Portal -- a game that builds ones physical intuition
• Fold-it  (Zoran Popovic )-- a game that helps researchers better understand protein folding
• The products of Dream Box , a company whose CEO, Lou Gray, spoke in the session and whose products have been influenced by the research of math educators like Cathy Fosnot

But how can students really appreciate the mathematics behind the game itself?  Professor Frank Wattenberg has his students build simple games or models themselves in which they can see the relevance of exponential decay and differential equations up close and personal.  Similarily, Professor Brianno Coller teaches students the basics of control theory by having them learn to "steer" a virtual car -- you can see the simulations he uses on his website.  

     Commenting on the importance of understanding the nuts and bolts of mathematical models (including their limitations), Dr. Wattenberg commented that to update the old saying "There are three kinds of lies: lies, damn Lies, and statistics", many people would replace the word statistics with "modeling".  Perhaps when viewing games as a component of education, it is key to recognize that games themselves are models-- useful models that can enhance our understanding of reality.

Sea Ice, No See Ice

   We're on thin ice around the arctic, and the most well-respected mathematical models have underestimated the loss of sea ice shown by current measurements.   In particular, the photo above shows the loss of "multi-year" ice from March 2004 to March 2008.  Many people may wonder about whether we have reached a "tipping point" or point at which the loss in sea ice has passed the point of no return.  As Donald Perovich, a researcher with a US Army Laboratory, said during his presentation: "It's really a question of whether we will fall off the edge of the stage or fall down the stairs, bumping along the way"
    What's the difference between sea ice and the ice from your fridge?  The tiny channels of brine which carry algae and make modeling sea ice considerably different from modeling a liquid.  In addition to being different on the microscopic scale, sea ice covers vast expanses of water and melts because of effects from underneath (the ocean currents), above (the sun, snow, and atmosphere), and within (topography, permeability).  The "albedo" that I talked about in the last post is a ratio of reflected to incident sunlight, and the average albedo of the sea ice changes as the mosaic of ponds and ice evolves through the seasons.  First-year ice has a lower albedo and therefore absorbs more heat and melts quite differently from multi-year ice.   This creates a feedback loop that causes more ice to melt and lowers the albedo more....   Many of the presentations focused on finding better ways to model the albedo by better understanding pond-formation.
   But the view from the picture above is only one slice of the story.  I'll say more about ice volume and flesh out more of the picture about Sea Ice later tonight!

In Search of Mathematics at the AAAS Annual Meeting: A PREVIEW

Watching the ocean views, graffiti, and tract housing of So Cal pass by my window, I am headed to The American Association for the Advancement of Science Annual Meeting in San Diego. There I will join over 10,000 scientists, journalists, and members of the public as we attend symposia ranging in topic from Astrobiology to Zoology. My mission is to attend as many mathematical Symposia as possible at this years meeting, whose theme is "Bridging Science and Society". Here is a list of the symposia I plan on attending. Each is about three hours long with several speakers from all over the United States and beyond.

• Sea Ice in the Changing Climate: Modeling a Multiscale Nonlinear System
  
• First-person Solvers? Learning Mathematics in a Video Game
• Traffic, Crowds, and Society
  
• Real Numbers: Mathematical Technologies for Counterterrorism and Border Security
  
• TIMSS 2007: Exploring the Dramatic Improvements in Performance in Two States
  
• Moving Across Scales: Mathematics for Investigating Biological Hierarchies 
• Mathematics and the Analysis of Fairness in Political Processes
• Can Singapore Mathematics Enhance Student Learning in the United States?

Tune in tomorrow to learn about the "ice-albedo" feedback loop, and how we can model the melting of our polar ice caps. And yes, "albedo" and "albino" have the same root --meaning "white". The albedo of an object is a measure of how reflective it is. And I hopethat there will be some reflection by readers on my writing this week. Feel free to comment!

When Food-For-Thought becomes Food-For-Your-Family

    Many people, even mathematicians, feel that pure mathematics is useless. But I would argue that at the very least, pure mathematics has always been useful to those who love it-- in keeping us happy, mentally healthy people! Healthy happy people make better life partners, better parents, and citizens who are more likely to help maintain order and fairness. In this way mathematics is a lot like art or sports.
    So what happens when an artist or an athlete finally starts to make money with their art or sport? Well, it messes with our heads. We start to question our motivations. On those days when we just don't feel like thinking about math non-stop or when it feels like we are banging our heads against a wall, we feel guilty because we're not doing our job or we're not doing it well. We're goofing off, we're not productive! "Wait a minute," the brain says, "I used to goof of by DOING MATH!" The brain starts to freak out that its owner will never solve that thesis problem, will not be a good enough professional mathematician, that "gasp!" the brain will lie dormant while its owner teaches BUSINESS CALCULUS FOREVER!!! That may be every math graduate students biggest fear. So how do we recover the feeling of joy and amazement that we first had??
      My first love of pure mathematics grew out of my joy at solving puzzles as a child, an excitement at having to exercise my brain, the struggle followed by the success, and a feeling that I owned the solution. I never cared whether the solution to the puzzle would lead to a cure for a disease, a better battery, or a million dollars. I did math problems for the same reasons that I spent hours every week dancing or reading everything I could about horses -- it was intrinsically appealing and satisfying! I liked to tell my friends mathy riddles, draw mathy pictures, do modular origami, and talk math with my dad on our walks around the neighborhood.
    Young people who have that joy and amazement oozing out of their pours inspire us. Teaching young people, hanging out with my friends' kids, I feel like I can tap into their curiosity. And kids don't feel guilty about or try to justify their likes or dislikes. Our adult culture seems to have a vast spreadsheet of pre-approved purposes for why we should do something non-work-related.  When was the last time you bought a book because you wanted to "do your small part to help the economy"?  By always rationalizing the activities we do when we're not working, we negate the benefits of those activities while adding to our ambivalence about the work we're supposed to be doing. Don't feel like doing math? Not getting anywhere? Writers have writer's block, and they still think of themselves as writers. So, do something else you really enjoy for a while, and you'll still be a mathematician.
   Activities whose outcome is not directly tied into my economic well-being renew my sense of learning for learning's sake. I can immerse myself for a few hours in a physical and/or expressive activity  like swimming, dance, art, music, jogging, biking, or yoga. Appreciating the expressions of others through their art also fills this need to acquire "useless" information. Lastly, wandering around/exploring a city or the outdoors is"pointless" but awakens my senses and get me back into the mindset that I am on the lookout for beauty and not results.
   On this note, a friend told me recently: "I was driving home today, and I saw this beautiful sunset!  I thought to myself 'If only I had my camera!' But I couldn't enjoy it because I became so focused on getting home and getting my camera. When I finally got the camera and walked outside, the sunset was gone.  Why couldn't I have just enjoyed it while it was there?!" I think this illustrates that fact that beauty is transient, just like the inspiration you need to do mathematics, and it doesn't emerge because someone is pursuing it.
   So if I feel like biking around and taking in the scenery without a destination in mind, I just do it.
I swim back and forth in a lane while thinking about my breath and counting my strokes.  I move from one yoga poses to the next even if I fall over in between.  I walk out on the dance floor with a stranger and follow each movement as it arrives.  I turn my thoughts into lyrics as I walk home.
   Those of us who have the opportunity to put food on the table and feed our heads at the same are privileged.  We can retain a child-like curiosity even as we grow into new adult responsibilities.  Or we  bend our intellects to fit a "grown-up" world that is, in truth, incredibly immature and irrational.

Job Search!

So, I am searching for a job! Maybe you are too. Here are some tips from post-docs and professors at my school, UC Santa Barbara.

Before applying:
First Priority is doing good math! Study area you like, be open to new areas
i.e. Get papers written, go to conferences, give talks

General: make 100 applications (half post-docs, half teaching)
Apply even if they aren't necessarily hiring someone in your area/level
Apply even if no job is being offered (as long as you have some sort of contact there)
Read job ads carefully -- a tenure-track ad may also want post-doc
Look at list of EIMS, AMS (sign up on mail list), keep your eyes open for other opportunities not on Math-Jobs
Prioritize time, organize your materials well, be efficient about applying

Cover Letter:
Make a website with all of your application info so that you can email
professors at the institution to which you're applying

Use all of your human resources: other grad students who are also applying, your committee members

Make your cover letter focus on the specific institution

Top line of cover letter has you name, advisor, people you want to work with. Work hard on the ten places and apply everywhere even if you don't think you'll fit in. Have extra eyes -- You want to avoid "I want to work with Prof. X" who doesn't even work there -- to scan through.

Have two different cover letters (mention you'll be at Joint Meetings),

Recommendation Letters:
Who are you going to get to write letters? Advisor, committee, people from other institutions, teaching mentors
Letters of recommendation are the last thing read sometimes, but the letters need to support the image portrayed in the rest of the application (i.e. The person is an excellent researcher or excellent teacher or both and why)

Research/Research Statement:
Two research statements (non-expert level, expert level) supervising undergraduate research

How specific should this get: choose your own adventrue research statement ("for the expert:", "for the non-expert"). Convey enthusiasm! Think like a colloquium (general audience understands first fifteen minutes, last fifteen minutes throw in something that an officianado will understand
ignore above "colloquium advice" if you are aiming to work with a specific professor in which case you should email them separately and make the statement a little more technical.
Three or Four pages max

Teaching/Teaching Statement:
document your unique experiences (teaching), be proactive (have someone send an email referring you)
Remember: Michigan, Chicago, Texas all have Inquiry-based teaching centers

C.V.
Two different C.V.'s (long and short)
Look at peoples websites for examples of C.V.'s
Don't discount non-math interests

After you submit applications and are waiting to hear back:

1) Remind people that you applied without spamming them
a) My paper's been accepted
b) I got an offer

2) Be prepared for telephone interviews

Happy Hunting!!

D.I.Y. Disorienting Holiday Gift Ideas for the Mathematically Inclined

Above, we see a baby Felix Klein and the Klein bottle which was later named after him. Baby Felix's German family probably didn't give him lots of useless presents around the holidays, and he turned out okay (aka brilliant mathematician/physicist). All of this going out an buying extravagant gifts for the holidays just strikes me as silly, and the Holiday Sales are already starting, trying to entice all of us poor people to dish out dollars for useless junk. So I thought I'd start a list of math-themed gifts under $15 that are sure to please. Everyone knows that mathematicians love to be disoriented -- or at least they know that being disoriented is nothing to be ashamed of.

Make a Glass Klein Bottle for your favorite mathematician:
Don't remember what a Klein Bottle is? Take a cylinder and glue the ends together with opposite orientation. In other words, as you traverse clockwise around one end, glue the other end counterclockwise. Can't do it? That's because it can't be done in three dimensions without letting the cylinder intersect with itself. Acme Klein Bottles is a company that specializes in beautiful glass Klein Bottles that are, in general, pretty expensive. BUT, the good news is Acme Klein Bottles, there is a $10 option that may appeal to those conservationists/penny-pinchers among you.
The Jigsaw Puzzle comes with a free band-aid!

Make a Mobius Music Box for your loved one:

I ordered my very own DIY Music Box Kit, and made my own little music box! It was fun and you can do it too.
Take a gander at Think Geek's Kit.
AND Note that you can punch your own holes to compose an original backwards\upside down masterpiece.
By the way, August Mobius is the namesake of this strip, which was also discovered by a man whose last name was Listing. But which sounds cooler "Listing" or "Mobius"? Yeah, that's what I thought.


I have not tried this next one, but judging from my minimal knitting skillz, I may stick to the "Mobius Scarf", which I saw at Oiyi's Crafts Blog.




Make a Klein Bottle Hat to keep those precious brains warm:
Mathematician Sarah-Marie Belcastro generously provides instructions at her website for knitting these self-intersecting representations of the Klein Bottle. There is also a link to making hyperbolic baby pants.



Really, Real Projective Space deserves more recognition here. I mean, it's disorienting too!
In case you don't remember real projective space, it is what you get by taking a sphere and identifying (gluing together) antipodal points (i.e. the north and south pole). Since this space cannot be embedded (made accurately) in three dimensions (try it), we will make do with "Boy's Surface", which is an immersion (has self-intersection) just as our models of the Klein Bottle are. This immersion was part of Werner Boy's 1901 Thesis written under the Famous Mathematician Hilbert. The surface, which is pictured above, was discovered as a result of Hilbert's request for Boy to prove that no such immersion existed. So, even famous guys can be wrong!

Make Your Boy (or Girl) Boy's Surface:
Courtesy of Joe Field's website, you can use just good old fashioned paper, scissors, and tape to make Boy's Surface.

Lastly, I make the observation that all of these unorientable manifolds were first described by Germans -- coincidence?
Yeah, Probably. But a funny one!

Okay, I'd love to see some other disorienting and inexpensive gift ideas from my readership, which I imagine to be growing exponentially -- as in from 0 to 1 maybe :)