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.