GA2: Beauty and Geometry

I was just reading about the mathematician Tom Zhang and his fascination with "twin primes."

Please read this interview with this brilliant mathematician and consider at least one of the many things he's saying.  Choose something about his views of mathematics and write about it.

To get your creative juices flowing, a couple thoughts I had about the interview include (but are not restricted to):

1. The idea that mathematicians are born, not made.

2.  He views math as beautiful and interesting not at all for the application. He loves math for itself and for the way mathematics helps him use his mind.

3. I remembered something I read, written by Harold Jacobs in his Geometry text book from 1974:

Pythagoras was a Greek geometer who lived about 2500 years ago.  He wondered whether he could teach geometry even to a reluctant student.  After finding such a student, Pythagoras agreed to pay him an obel for each theorem he learned.  Because the student was very poor,  he worked diligently.  After a time, however, the student realized that he had become more interested in geometry than in the money he was accumulating.  In fact, he became so intrigued with his studies that he begged Pythagoras to go faster, now offering to pay him back an obel for each new theorem.  Eventually, Pythagoras got all of his money back. 

Now ok, a bunch of you admitted that you do some math in secret or in ways and times that you didn't think you were actually doing math. A bunch of you claimed that over use of technology contributes to math illiteracy.  Will any of you to admit that there's something in math --anywhere -- that you've found lovely, beautiful, cool, interesting, intriguing, puzzling, worth thinking about, or simply fun?  Oh, do share!

TPC: Who Uses Trig in His or Her Job and How is it Used?

Surveyors. That's the quick answer. Seems obvious and web searches will yield a plethora of sites that show trigonometry in surveying, but those surveyors have some deeper equations embedded into their programs.  Mollweide's Formula is one of them.  If you look at this equation, how many angles and how many sides of a given triangle are included in the formula?  Why, from a number theory standpoint, would this be a good thing for surveyors -- what would they use it for?  AND .....do you really think that the typical joe or jane surveyor would know this formula or even know of the existence of this formula?

Doctors.  Particularly, radiologists use trig as they aim gamma rays into the bodies of their patients to eliminate tumors without traditional scalpels. How do you think trig is used in this area? Orthopedists use angles in their work, just check out the abstract to this article:  http://www.ncbi.nlm.nih.gov/pubmed/7610093 or do a search -- there's lots available. Some images are not for those with a weak stomach.

Sailors. Just look for the triangles.                      

Rock Climbers and outdoor enthusiasts. Tyrolean Traverses  (Can you find the triangles, angles to measure....?)

Astronomers....

You get the idea.  In high school college-prep mathematics, trigonometry is often the first place students see direct applications of math in a variety of concrete areas.  For your blog, find an application,  a single application so you can develop it well, anything, and explain how trigonometry or the study of triangles applies to the field. Go ahead, ask parents, your friends' parents, anyone.  Do they use trigonometry? Where?  How?  You might find lots of people who say, "Nope, never used it."  Your mission is to find an application in the professional world.

Probability in the Pool: August 19, 2013

When my mom had a catastrophic stroke, I returned to my home town and found a pool in which to swim.  I swam mostly alone in a pool that was 50 meters by 25 meters.  Sometimes the lane lines were set to allow swimmers to swim "short course" (25 meters) or "long course" (50 meters).

On one of the short course days, I was alone in the pool.  There were 16 empty lanes.  I chose lane 8, one of the middle lanes.  I swam down-and-back (50 meters) in about 45 seconds.  This became a deliciously long swim as I moved back and forth alone in this pool with my own personal lifeguard.  Then, suddenly, I was jerked out of my fraction-calculating delerium (refer to previous post about Fractions in the Fast Lane) when waves overtook me.  A man dropped himself and his large belly covered in fur into the lane next to mine at precisely the moment I was at his end of the pool.

Ah, I thought.  A probability problem.   What is the probability that he would (a) choose the lane next to mine and at the same time (b) choose to enter the water during the roughly 7 seconds that I am vulnerable to the tsunami he created at the near end of the pool.  And is this probability small enough that I should think this individual inconsiderate?

On one of the long course days, I was again alone in the pool, now well spoiled and feeling like the Queen of Sheba in her own blue-glass lake (with lane-lines and a black line at the bottom.)  There were 9 lanes. I chose the middle lane.  I would swim 50 meters to the far end and 50 meters back, giving me new distances and fractions to consider.  I swam the same rate as in the short course setting in the pool.  Again, as I swam a deliciously long swim with new numbers flowing along side of me, another man, a more narrow man, entered the pool.  Now then, many of us in this world like to be individuals; we value our ability to make choices and to be unique.  This man decided to swim a uniquely different way in this pool.  He wanted to swim 25 meters, not 50 meters as the pool was set.  So he did, weaving his way across the pool, below the lane lines and intersecting my path perpendicularly.  My interest became piqued.  As I swam, I watched the red clocks surrounding the pool and observed that he swam breast-stroke quite regularly, finishing 50 meters (down and back, under the lane-lines) in about 100 seconds.

Ah, I thought. Another math problem. If we both start at the same place (let's say a corner of the pool, just to make this conceptually more straight-forward) and swim in paths perpendicular to each other, when (if at all) will we collide?  If we both leave one end of the pool at the same time and swim at our own constant paces, when (if at all) will we collide?  If a moment during our infinite-length swim is chosen randomly, what is the probability that we will be in a collision at that moment?  Consider that it would be about 7 seconds for me to be in his way; he'd be in my way for 1/9th of the way across the 25 meter pool.

TPC: Waves in the Pool

I swim at a constant rate.  Well, close enough; assume I swim at a constant rate, then we'll adjust that later in this blog.  That rate is roughly two meters for every second.  Well, ok, that's a little quick for this ol' lady, but those numbers work for us. So pretend.

Let's start by thinking of what a graph would look like as I cruise at my theoretical constant pace of two meters per second back and forth in a pool that is 25 meters long.  Let the x axis (independent variable) be time, measured in seconds, and let the y axis represent meters away from the wall, or side, of the pool.  Let's start our stopwatch at the exact time when I leave the wall and head to the other side of the pool.

How does the graph appear if I adjust my pace to 1 meter per second?  How about to 3 meters per second?  1/2 meter per second?

These kind of "waves" are not quite a sine wave. Wikipedia categorizes the waves as triangular:

http://en.wikipedia.org/wiki/Sawtooth_wave.

But there's more. Lots more.

How would the graph be different if I swam my 2 meters per second pace in a pool that was 50 meters long?

How would I need to swim differently to make the graph look like the other waves that the wiki pictures on the bottom right?  Which ones are not possible?  Spend some time with, of course, the sine wave and describe how I'd need to swim so that my distance from the wall would be a lovely, smooth sine wave.

So you've already seen that I do some interesting math while I swim. I can solve all the world's problems when I swim, you know. Now what happens if I'm swimming in the pool that's 50 meters by 25 meters.  I swim the long way at a constant 2 meters per second and another swimmer decides to swim the short way, perpendicular to my path and swimming a constant 1/2 meter per second. (No kidding, someone actually decided to do this last summer.)  I swim in lane one as does he (well, we don't but pretend we do) so we start at the same time and in the same corner. Because we are all thinking perfectly and precisely mathematically, we imagine triangle waves, but let's say we are so far advanced that both the other swimmer and I slow down and speed up appropriately so that the graphs of our distances from the wall with respect to time form perfect sine curves.  You've already worked out  how we adjust our speeds to make perfect sine curves, of course, because the previous paragraph asked you to do this.

So the bigger question is, assuming we can occupy the same spot simultaneously or that I can duck directly below the surface while maintaining my lovely sine curve pacing, how often do our paths intersect?

For your blog, either show graphs like I've described or take my idea a little further, maybe by answering the questions I pose, OR find your own application of a sine curve.  Where do YOU see it in the real world?

9%.

Fun to bike down but a workout to bike up, a 9% grade earns a failing grade in my gradebook. 

Outside of Otis, Massachusetts is a road with a very steep hill.  Put in neutral, our standard transmission car just cruised down the hill. On the way up, first gear was the way to go, so to speak. The rhomboid sign reported a 9% grade (“Test your brakes,” it warned). A biker was huffing her way up the slope while a second simply sailed down the hill.  What’s the 9% mean?  If 60% is passing and 90% is an “A,” what’s 9%? Doesn't seem like much; why the big deal on that hill outside Otis?  Folks seem to always aim for 100%, but that would be suicidal in an automobile or bicycle and certainly not preferable.  We, as humans, do our best to categorize (think: Kingdom, Phylum, Class, Order, Genus, Species or better yet square, rhombus, rectangle, parallelogram, trapezoid, quadrilateral); it seems we have categorized slopes (or grades) of hills as well. 

Wales has a road with a 25% slope; I-70 into Denver from the west has a cool 6% grade.  A handicap ramp has to be an inch vertically for every foot horizontally.  Are these ideas related? 

Your mission is to understand what these numbers mean and how engineers have come to categorize the grade of a road, ramp, or slope.  Nice word there, by the way, “Slope.” 

http://en.wikipedia.org/wiki/Grade_(slope). 

Yep, good ol’ Wikipedia actually has a description that works for us.  It may seem a little dense and might take some slower reading than, say, Ted Geisel’s stuff,  but it’s got all the ideas you need.  In the wiki, there are triangles, a protractor shape, a trigonometric function, and some other very familiar words.  Put the pieces together in your blog and you’re set for the week’s blog assignment. (Be sure you take out the irrelevant ideas for "grade" in my post -- this is meant to have nothing to do with the grade you get in class. That's a joke.)

More specifically, the assignment for both TPC and GA2:  the grade of the road has everything in the world to do with a trig function.  Which one? Why? Explain.  Use roads that you've seen or know about or find on line.  There's a couple different standards for handicap ramps (businesses vs private homes); find those if you'd like.  Go bananas on this one -- where else do you hear about grades?  What about the "angle of repose"?  What's that?  What about "railroad grades"?  Choose something that interests you; don't feel as though you need to cover absolutely everything, but DO cover the idea of what a "grade" is.   If you are one of those folks in GA2 fascinated by the number theory topic we touched on (Pythagorean Generators), you can choose to write on that instead of this whole idea.

GA2: Fractions in the Fast Lane: July 17, 2013, published August 19, 2013

"How did you happen to be so good with fractions," friends used to ask when I was in middle school.  Everybody knows everybody universally dislikes fractions.  For me, it was all about distance swimming.  I knew I could solve the world's problems during a long workout (though I'd forget the solutions to the world's most serious problems as I climbed out of the water); what I didn't know was how I was using swimming to solidify my working facility with fractions.  It was simple: as I swam 1,000 meters, I was constantly figuring out what fractions -- and what ratios were identical to the reduced fractions -- could represent how far I had swum and how much further I had to swim before I finished.  It started simply: if I swam 40 lengths in a 25 meter pool, then after 7 lengths, I was 7/40th done and had 33/40 to go.

Sometimes, however, I swam in the 20 meter YMCA pool and the numbers became different. I needed to focus and not just rely on memory.  I now had to swim 50 lengths to complete 1,000 meters.

My thinking soon became more complicated and required swifter calculations -- I moved to measuring what fraction of the swim I had completed for each stroke -- or even each partial stroke.

Bored with that, I began watching my teammates swimming in the neighboring lanes.  What were their ratios and how were their numbers different from mine?  At what points would we pass each other?

I long since moved away from my home town, stopped swimming, and became a math teacher. I forgot about fractions in the fast lane.

Then deep into middle age, I started swimming again.  And calculating fractions.  I kept this secret lest my lane-mates think me insane.

I don't always swim in pools.  There are lakes with cool fresh water, sunbeams that cut through the waves, and no visible bottoms. Plants grow through the water towards the source of the sunbeams, branching in infinitely smaller "Y" shapes at the same angles.  Bubbles surface and break into more and smaller bubbles from the depths; there are no numbers. Only fractals.  And chaos. And new things to think about.

Your blog: where do you use math in secret?  Or if you don't use math in secret, where might you start using math in secret or not in secret so you can increase your skills in math?

Integrity:June 2013

We live in a greater culture that does not appear to value integrity. We are witness to this lack of integrity when our generals make themselves vulnerable to blackmail and family crises when they have affairs; we witness it when accountants cook the books.  This lack of integrity pours into our school and our classrooms where students plagiarize their papers, lie to parents, teachers and administrators, and are often taught by the mainstream culture and by our responses that it’s sometimes better to lie.  One parent’s advice to his child who was a student last year at our school was the following: should you be busted for something, “Deny, deny, deny,” so you receive the best possible outcome.  The parent is a lawyer. Students this year took this attitude and, yes, they received the best possible outcome: no punishments by the school for transgressions.

The loss of integrity in the parent and student population begins to affect the integrity of our school when the lying is so pervasive that, this past spring, only one child out of 60 or 80 or 100 adults and students told the truth about a party that had alcohol.  The guilty and liars (the “and” term is used mathematically, as “intersection”) along with the innocent (think “union”) faced no concrete immediate consequences; indeed, it is very likely that some of the guilty liars earned major school awards at the end of the year; some of the guilty liars teased the guilty but honest student and called him, “Stupid,” (and worse) for telling the truth and facing the consequences. The lack of integrity bleeds to the administrators who deal harshly with the honest and cannot catch the guilty who lie – the administrators are put in a position where they appear to play favorites or where they are ridiculed behind their backs for not catching the guilty.   I don’t expect there’s an easy solution to this; my statement is not meant to be critical or to suggest that there’s something else that the administrators can do in response to a breach of integrity.

 This past week, while conversing with my two older children, one high-schooler said, “The school's response teaches that grades are more important than integrity.”  My other high-schooler chimed in, “Copying daily math homework is a near-daily task for most students.”   

As a teacher who believes we would ideally teach the whole child, I struggle to find ways to address integrity issues well in a mathematics classroom. I write about integrity in the “intro” page that I present the first day (what I write is included at the end of this report); I remind students of this statement prior to taking a test or at times when there’s a relevant story in the press.  I think many teachers do this. And they do it well. Clearly this is not enough.  But talk before the transgressions and reprimands after the transgressions are futile and not effective.      

In my first years teaching as a just post pubescent college graduate, I strongly believed that we needed to teach what I mistakenly called, “moral education.” This belief was brought about by my college statistics project where I had peers answer questions about cheating. I was appalled at the results (and do not remember them specifically now just that the numbers of cheaters were high.)  As a new teacher, I was attacked on all fronts by seasoned and brilliant and well-meaning colleagues who feared for censure and effects of religious conviction.  I did not have the words or framework to put my concerns into useful language or good ideas and could not adequately respond.

I do not want to stand idly by as I watch these transgressions occur.  However, I am one.  Of many.  I know that one person can have a profound impact (think of a single baby screaming on an airplane packed with 350 passengers), but it seems that we are like the Titanic traveling across the Atlantic: 1.  We are too arrogant to admit that we have weakness [I have colleagues that keep saying, “If it ain’t broke, don’t fix it.” Then, following up (choose one of the following):  “Just look at our test scores”… “I’m a great teacher”….”I’ve always done it this way and found success”…]  2.  We are too large and slow and encumbered by policies, standard practices, and bureaucracies to turn on a dime when needed.   3.  We will, however, sink if our integrity continues to be breached.   More importantly, I believe it is important to model and to teach, “doing the right thing.”  What’s the right thing?  Maintaining Honesty, Integrity, Compassion. Living and teaching the value and importance of integrity, hard work, and supporting each other.  I think there are basic tenants to which we all can agree.  Perhaps someone else can come up with better words for these things; I’m no wordsmith and failed miserably in a similar mission twenty-seven years ago.

So, presuming we want our kids to make “good choices” about their behaviors and approaches to academics, what do we do? 

Responding to breaches of integrity is not effective or always possible; cases in point are situations this past year.  If we call the school’s responses to transgressions, “downstream intervention,” we could think about “upstream intervention.”  Upstream intervention would be preventative, but would need to be larger and more meaningful than a talk and a blurb in black and white.

So what would I like to hear from a student?  It would be great to hear a student say with all earnestness to a peer who cheats, “Why would you do that?” as if they are asking why a peer would refuse to accept a gift.  A gift.  Now that’s how I want students to view what they learn in our classrooms. I don’t think that’s currently the case about what students learn in my classroom or about other classrooms.

If our assignments were meaningful, interesting and fun to do, then cheating might not happen. If students felt as if they could do the assignments, then cheating might not happen. If our problem sets were seen as instructive, purposeful and interesting, cheating might not happen. If our students had the time to feel as though they could do the job that they want to do, then cheating might not happen. Perhaps if students had a greater sense of community, a greater sense of emotional attachment to teachers, peers or others who are being deceived by the student to cheats, then the cheating might not happen.  Maybe I’m being idealistic by saying repeatedly, “Cheating might not happen.”  But even in the world of hardened criminals, isn’t there a code of conduct that says, you don’t mess with your mother or someone you respect or someone who has done something nice for you?

I believe we all need to re-think our assignments for students.  Are we giving kids so much work that they don’t have time to complete it all?  Are we asking kids to do assignments that are, quite frankly, over their heads and overwhelm them or that are meaningless to them for whatever reason or assignment that are too simplistic and boring?  Are we developing the kinds of relationships with students that foster mutual respect and a sense of community?  Is our emphasis on learning, understanding, and the joy of academic discourse or is our emphasis on grades?  Are we focusing so much about getting through an increasing amount of curriculum that we lose the ability to do tasks that are memorable and fun and still academic?  I have pockets full of ideas of projects kids at other schools have always thought were “cool,” “memorable,” or simply interesting.  It's deeply troubling to me that in my final ten or so year in education, I have never or rarely been able to pull out those projects from of my pockets because of time constraints, policies, curricular mandates, standardized tests and other bureaucratic ties.

I would like more freedom to teach some of these good ideas. I would like more time to be able to cover some extra topics or to allow students the opportunity of creative problem solving.  Whenever I insert an extra topic or project, the students love it, but I’m stressed and scrambling as I try to cover the other required material to complete the course, or to prepare them for a standardized test.  I find myself feeling that I give up teaching passion for math in favor of covering material for rigor.  Sure, I know we have students who thrive on rigor, but most of our students are not Bertrand Russell; they’d learn more problem solving and love more math if I could teach it differently.

I ask myself the question: What jobs and what kind of life am I preparing my current students to assume in 10 or more years?  Those jobs likely don't exist today (think: 1988's students were the ones who designed web pages, which didn't exist in 1988.)  Twenty-first Century Skills include problem solving, communication, cooperation, evaluation -- many higher order skills that don't enter the recipe for pummeling through current curricular topics. Twenty-first Century Skills include working with integrity (and being able to evaluate the integrity of media consumed), focusing intently beyond 220 characters, organizing information from a variety of sources, and recognizing the difference between opinion and fact, to name a few. 

Thomas Jefferson is known to have said,  "I hold it that a little rebellion now and then is a good thing," and indeed, "God forbid we should ever be twenty years without a rebellion." Perhaps it is time for a rebellion and a revolution in our current educational system.

The statement belos is from my “intro” sheet given to my math students each fall, during the last twelve of my teaching years, 2009-2021.  (This is written to be read aloud. There are 7 sentences;  7 students each read a sentence.)

Personal Integrity.

There is nothing more important in any person’s life than his or her personal integrity. 

Knowing that you “do the right thing,” and knowing that you live without regrets are invaluable to your successful life experience. 

Many things in your life you won’t have control over, but you will always have control over your personal integrity. 

Having and maintaining high moral standards is a wonderful hallmark for you to carry with you throughout your high school career and is far more important than any grade goal. 

When you graduate, the ending ceremonies and the accolades you receive are only meaningful if you have maintained the standards of integrity that your peers and faculty expect. 

Certainly, when you cheat or otherwise compromise your personal integrity, the school responds with swift and potentially severe consequences.  

The Math Department has prepared a statement on academic integrity.  You are expected to understand every part of this statement.  Should you have any questions about what is or is not cheating, please ask.