MY FAVORITE TAKEAWAY

We come to the end of the semester, and it seems like the time to share the pinnacle of the knowledge I will take away from this experience: THE TRIGONOMETRIC CIRCLE!!!  I have always been a visual learner, and to see the interconnectedness of all the principal trig functions as they relate to a unit circle was, well, like oxygen to a flame.  Suddenly it all made sense, and all I could think was, "Man, if I had been shown this in high school, I wouldn't be struggling so much with this now!"  And now I would like to share it with the world!

 

Okay, enough hyperbole.  Now let's explain how it works.  But first, let's start with the basics, the relationship of sine and cosine.  Take a look at this graphic:

 

This shows how sine and cosine cycle between 1 and -1 as we travel around the unit circle.  As the point on the perimeter travels towards the x-axis, sine approaches 0 and cosine approaches 1 or -1.  As the point approaches the y-axis, sine approaches 1 or -1 and cosine approaches 0.  It is from this basic relationship that all the other trig relationships are derived.

First, there's tangent, which is sine/cosine.  So, as sine approaches 1 or -1 and cosine approaches 0, tangent approaches positive or negative infinity, but it is undefined when cosine equals 0.  Conversely, as sine approaches 0 and cosine approaches 1 or -1, tangent approaches a value of 0. 

Second, there's cotangent, which is 1/tangent, or cosine/sine.  This means when sine is getting smaller and smaller, cotangent approaches positive or negative infinity, and it approaches 0 as cosine approaches 0.

Thirdly, there is cosecant, which is 1/sine.  We can see from the values of sine that cosecant can never be smaller than 1 or -1, and it stretches towards infinity as sine closes in on that zero value.

Fourthly, there is secant, which is 1/cosine.  Again, secant is never less than 1 or -1, and it, too, goes on to infinity as cosine approaches 0.

So, what do all of these six (!) functions look like on the unit circle?  See below:



Play around with the circle for a spell, especially in the first quadrant.  What do you notice?  As expected from our first graphic, sine and cosine oscillate between 1 and 0 and -1, depending on which axis we are approaching.  But it is the other four derived trig functions that are really interesting to observe here.  As the pink point moves closer and closer to the x-axis, sine approaches 0 and cosine approaches 1.  Take a look at the green tangent line; it approaches a length of zero as well, which is to be expected when we recall that tangent is sine/cosine.  Cotangent, aka tangent's inverse, heads towards infinite length, a likewise unsurprising result.  Secant, derived from cosine, heads towards its minimum value of 1, and cosecant, derived from sine, heads towards an infinite length.
Now, move the pink point to where the angle centered at the origin is 45 degrees, or as close as the graphic will allow (I have been unable to get closer than 45.04 degrees).  Notice the marvelous square symmetry of it all.  Sine is equal to cosine, secant is equal to cosecant, and tangent is equal to cotangent.  Again, all of this makes perfect sense when we recall how each function is derived, but now we can see that similarity in visual form. 
Finally, start moving the pink point towards (0,1).  Notice sine approaches 1, cosine approaches 0, tangent moves towards infinite length, cotangent towards zero, secant is likewise approaching infinity, and cosecant is heading towards a value of 1. 
The truly exciting aspect of this graphic for me is how simple it is to derive other trig identities by using a little Pythagorean algebra.  For example, sine and cosine are always two legs of an inscribed right triangle with a length of 1, so we can derive that sin²(a) + cos²(a) = 1.  We can also observe that tangent and the radius of the circle form the legs of a right triangle with secant as the hypotenuse, so we can now derive that sec²(a) = tan²(a) + 1.  The same process gives us csc²(a) = cot²(a) + 1.  Even if we forget the identities themselves, we can very easily figure them out if we remember visually exactly what is going on with the unit circle. 

So that's it.  My most fulfilling learning experience of the semester.  If and when I get around to teaching trigonometry, the unit circle seen here will definitely be my go-to learning device for helping students grasp the trig functions.  Hopefully I can save all my students the years of anguish and confusion I went through. 
Thanks for reading; it's been fun.  Cheers!

 

 

THE MATHEMATICS OF "FROZEN"

(Wait, there's math in Frozen?)

 

There are probably a great many children today whose first hearing of the word "fractal" was in the song "Let It Go" from the Disney film Frozen.  Come on, you know the song... sing with me:

          "My power flurries through the air into the ground.

          "My soul is spiraling in frozen fractals all around."

But just in case you forgot which part of the song this is, here is something to help you remember...

Ahhhh, those frozen fractals!

 

 Virtually everything you see in this shot is created using fractal geometry!

 

Okay, so exactly what is fractal geometry anyway?!?

I'm so glad you asked.  To some extent, a fractal is a repeating geometric pattern that is similar to itself at every scale, meaning, it looks the same as you zoom in and zoom out.  The word was first coined by Benoit Mandelbrot in the 1970's (which makes Elsa's use of the term pretty anachronistic, ja?), and he borrowed from the Latin word "fractus", which means "broken" or "fractured".  Beyond this, there is a lot of disagreement on how exactly to define a fractal.  Right now, it seems that a fractal simply must pass the eye test: "I don't know how to define it, but I know it when I see it."  To generate a fractal, one simply needs to take a basic geometric shape or pattern or rule or even a function, then repeat it over and over in an orderly fashion at increasingly micro and macro levels.  Still confused?  Here is a basic example to help explain....

 

The Koch Curve

Here is a simple fractal known as the Koch Curve:

Does this remind you of something?

This is what is known as a fourth power fractal.  What does that mean?  It means that a simple geometric shape is generated (first power) and then repeated three more times (fourth power), creating a geometric pattern of ever increasing complexity.  Still confused?  Let's look at the first power fractal of the curve:

 

The green line in this picture represents the basic shape used to generate the Koch curve, and since this is the first use of the shape, this is a first power fractal.  To generate the second power curve, simply repeat the pattern on each of the four green line segments, like this:

 



The lavender line is the second power fractal.  The picture above shows the second power fractal overlain on the first power one so you can see how the fractal scales down as it becomes more complicated.  We have gone from one large peak to four smaller ones.  The picture below shows the second power fractal by itself so we can see how with each repetition we are approaching the final shape created by the fourth power fractal:

Repeat this process again to get the third power fractal, like this:

Again, the third power fractal, shown by the purple line, overlays the second power fractal, and we have 16 even smaller peaks now.  Here is the third power fractal by itself:

 

One more repetition of the pattern gives us our final, fourth power fractal!  Put six of these fourth power fractals together, and you get something like this:

Eureka!  We have our snowflake!!

Looking at how much this very simple artificial snowflake begins to resemble the real thing, the utility of fractal geometry in a movie like Frozen becomes clear.  It is, in fact, no accident that this fractal resembles something from nature, for indeed fractals were first created to try and explain and explore the complicated designs of the natural world.  The fledgling computer animation industry of the late 1970's and early 1980's saw the usefulness of this type of geometry immediately.

 

Math versus Nature

The problem facing the first computer animators was finding a shortcut to create a simple digital landscape that emulates all the nooks and crannies and seeming randomness and complexity of a real landscape.  So, they turned to fractals.  Take a look at the following picture:

 

This is a fractal rendering to model the surface of a mountain, and one can easily see that as the power of the fractal increases, its approximation of a real mountain gets better and better.  The problem for early computer animators was that as the power of their fractals increased, it took longer for the computers to complete the calculations, known as renderings. In fact, rendering times increased exponentially as the powers of the fractals increased.  This was a severely limiting factor for the animators.  Some of the early results, while quite arresting at the time, seem positively quaint by today's supercomputer standards (as a side note, when Pixar was created in 1986, its primary purpose was to create animations to show off the computing power of Apple computers!).  Here is an early example from the 1985 film Young Sherlock Holmes:

This was the first example of a computer generated character in film.  The stained glass figure of a knight is able to come down from his window and walk around.  Because the character was made up of large, simple geometric shapes, the computers of the day were able to render the character in a reasonable amount of time.  In hindsight, he looks pretty dorky, but it was a start.  For another, perhaps better example, we turn to the 1982 film Star Trek 2: The Wrath of Khan.  Take a look at the following video:

Also Spock Zerathustra!  The spreading fire and the resulting mountain landscape were all rendered using fractals.  The geometry of the mountain ranges, while being a reasonable approximation for the real thing, is still pretty obvious.  However, as computing power has increased, so has the delicate complexity of CGI rendering, making it harder to tell truth from fiction.  Let's take a look at a few more examples:

 One of these ferns is the real thing, the other is a fractal.  Okay, so it isn't too difficult to tell which is which, but the fractal fern is a 20th power fractal and is more complex than anything in the Star Trek video.  Now we begin to see how more powerful computers allows for better approximations of nature's complexity. 

 

 Final thoughts

So, the next time you watch Frozen, keep a watchful eye for the fractals at play in creating the vaguely Norwegian world.  Chances are you won't notice most of them.  Some examples are easy to spot, like this:

The frost and the snowflake are created using fractal geometry.  How about this:

Elsa's ice palace is pretty obviously fractally generated.   And another:

Perhaps this one is not immediately obvious.  Fractals are also used to help simulate complex patterns of movement of large numbers of individual objects, like the snow particles in the picture above.  Fractals, fractals everywhere!!!

 

I will bet that from now on you will take more notice of the tremendous amounts of math that go into creating animated movies, but chances are most kids will not, and, at least until the movie is over, you might want to consider keeping it that way!!!  Cheers.

 

 

 

 

 



THE TROUBLE WITH RULES, or "WHAT DOES ZERO TO THE ZERO POWER EQUAL?"

So, what is your answer?  A number of you, I am sure, said "one", and I am sure some of you said "zero".  Maybe even a few of you said "undefined".  The truth is, there is no consensus to the answer on this.  The best answer one can give is, "well, it depends on your point of view", but more on that later.

First off, I wanted to do an informal, non-scientific poll to see what people think zero to the zero equals, so naturally I asked all my facebook friends to sound off.  The results went like this:

ONE                                                             ZERO                                                  UNDEFINED
   7                                                                    2                                                                1

These were the only sure answers.  There were some less certain responses.  What I found most telling, though, was the explanations for why people chose as they did.  Here are some of the choicest quotes......

• "It has to be 1 or you break math (italics are mine).  If it were not one you couldn't solve binomials or other exponential equations".
• "1-ish".
• "A very tough question!  Still calculating.  One maybe or maybe not.  I'll get back to you (ed.: she still hasn't gotten back to me...)".
• "Anything raised to the power of zero is one.  In order of operations, exponents come first."
• "So the general consensus with my math colleagues is that one is the answer, but you aren't wrong if you say 'undefined' (ed.: this is from a friend of mine who is a high school math teacher, and she polled her school's math department on my behalf.)".
• "I have to chime in with one".
• "Gosh, I thought it was zero".
• "I think it's zero (ed.: this was from the math teacher at Forest Hills Central High whose class I have been observing the last few months)".

Quite a bit of disagreement, even amongst those that agree, no?  Here's where I think the problem occurs.  In the study of exponents, we are given two different rules.  They are as follows:

1.     

 

   2.

So, which rule applies?  Which do we follow?  There is a third way to think of this problem:



 

 With this equation we can now take the limit as x approaches zero, which looks like this:

 



 

 

To truly understand what is going on here, let's look at a graph of all three functions:

In the graph above, the purple line at the bottom is, the green line at the top is, and the red line is the graph of .  The purple and green lines are destined never to meet as they doggedly head towards their expected values of zero and one respectively, but the red line shows us something interesting.  As x gets closer and closer to zero, the graph gets closer and closer to 1!  So, clearly it equals 1, right?

 

Not so fast.  Why does something to the zero power equal one anyway?  The standard explanation goes something like this:

 

We can also solve using another rule of exponents, the sum of powers rule, like so:

And here is where the trouble really starts; try replacing 3 with 0 and we get this:

 

 

We have long been told you cannot divide by zero, so now we are desperately stuck in a hopeless paradox brought about by a dastardly construction of contradictions! And I feel the real trouble is not the rules themselves, it is how they are presented to us.

 

THE REAL TROUBLE



All our lives, math has been presented to us in a very concrete fashion, with a seemingly overwhelming amount of rules to follow that, if implemented correctly, govern all mathematical problems that ever were, are, or will be, created by the likes of Leibnitz, Newton, Euler, Pythagoras, etc.  These great mathematicians are not to be questioned or doubted.  Our teachers teach us these rules as if they are absolute and unwavering, so when they fail us, we are left scratching our heads and (perhaps) questioning the meaning of life.  It turns out that math isn't so concrete, that sometimes the rules fail, and there are still vast stores of discoveries still to be made.  Students need to know that math is fluid, dynamic, expansive, and revealing more and more of its secrets to us even today. 

 

I truly feel the four most powerful words a teacher can say when asked a question are "what do you think".  This response opens up the possibility for personal discovery by the asker, not imposition by the asked.  So, if and when my students ever get around to asking me "what does zero to the zero equal", I am definitely going to respond with "what do you think?"  Because sometimes, as we have seen here, the lack of a definitive answer is really the worthier discovery.  Cheers!



Enrico Fermi, the face of a genius

 

Enrico Fermi and his problems

Enrico Fermi might have had problems, like most people, but that's not what this post is about.  I'm gonna talk about the type of problem that bears his name: (cue hero music) THE FERMI PROBLEM!  Fermi problems involve math, but they don't usually have a specific number solution, only a close estimate based on logic, deduction, math, physics, estimation, and even guessing.  These are the kind of problems that drive the average person to a "What?" and a shoulder shrug before they return to their pizza and beer and football game, but a curious sort might be driven to hours of distraction as he or she attempts to explore all the nuances that can lead one closer and closer to the right answer without ever achieving a FINAL answer!  Madness!!!

How did this quiet man of Italy come to have his name associated with these mathematical torture devices?  Well first of all, Enrico Fermi was a smart man.  I mean, like, really smart.  The kind of smart that makes a Mensa candidate look like the village idiot.  During his teen years he conducted experiments to test the density of the water supply of his hometown of Rome.  Just for fun!  He was so smart... (all together now, "how smart was he?")... he was so smart, that when he applied to university, the admissions panel admitted him not as a freshman, but as a doctoral candidate!  Yup.  Here, friends, was the Doogie Howser of physics and mathematics.  Enrico Fermi also had a nickname: (cue ominous music) "The Father of the Atomic Bomb".  Before the Manhattan Project became reality, it was Enrico Fermi that proved sustained nuclear fission was possible.  By nearly blowing up the city of Chicago! 

"The Italian Navigator has just landed in the New World"

Chicago Pile-1; from such humble beginnings...

 Enrico Fermi and his wife immigrated to America in 1938 to escape fascism in Mussolini's Italy.  Just a few years later, on December 2nd, 1942, Fermi and his team of scientists, in order to test his theories of sustainable nuclear fission, set up a rather large, orderly pile of wood and graphite and uranium, dubbed "Chicago Pile-1",  on a squash court underneath the football stadium of the University of Chicago.  The idea was to use large cadmium rods to absorb uranium neutrons to control the speed of the fission reaction.  However, if Fermi's calculations were wrong and the fission reaction raced ahead uncontrolled, there would have been a nuclear blast in downtown Chicago.  So, no pressure.  The test was a success, and the nuclear age was born.  To inform the White House of their success, a cable was sent bearing the following cryptic message: "The Italian Navigator has just landed in the New World".  Fermi's reward was being appointed one of the overlords of the Manhattan Project, which was tasked with creating an atomic bomb.

The Fermi Estimate

Enrico Fermi was well known for his ability to estimate answers to complex problems by using logic and deduction in lieu of hard and fast mathematical calculation.  The legend goes that he estimated the strength of the first atomic bomb, codenamed "Trinity", by dropping pieces of paper during the explosion and seeing how far they blew away from him.  His estimate was 10 kilotons, which was remarkably close to the accepted value of 20 kilotons, especially considering he came up with his estimate using no hard number calculations.  Below is a typical Fermi estimate attributed to Fermi himself and taken from Wikipedia:

"How many piano tuners are there in Chicago?" A typical solution to this problem involves multiplying a series of estimates that yield the correct answer if the estimates are correct. For example, we might make the following assumptions:

1. There are approximately 9,000,000 people living in Chicago.
2. On average, there are two persons in each household in Chicago.
3. Roughly one household in twenty has a piano that is tuned regularly.
4. Pianos that are tuned regularly are tuned on average about once per year.
5. It takes a piano tuner about two hours to tune a piano, including travel time.
6. Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year.

From these assumptions, we can compute that the number of piano tunings in a single year in Chicago is

(9,000,000 persons in Chicago) / (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 225,000 piano tunings per year in Chicago.

We can similarly calculate that the average piano tuner performs

(50 weeks/year)×(5 days/week)×(8 hours/day)/(2 hours to tune a piano) = 1000 piano tunings per year per piano tuner.

Dividing gives

(225,000 piano tunings per year in Chicago) / (1000 piano tunings per year per piano tuner) = 225 piano tuners in Chicago.

The actual number of piano tuners in Chicago is about 290.

Notice the number of different estimates involved in the above problem.  Assuming that each estimate is reasonably correct, the final answer will be a reasonable approximation to the actual value because any overestimating of one individual value will be cancelled out by underestimating on a different value.  Accuracy in a Fermi problem is usually measured in orders of magnitude, as some Fermi problems are extremely complex and have many components.  Take, for example, a formula called the Drake Equation, which is used to estimate the number of intelligent civilizations in the universe.  The equation looks like this:

where:

N = the number of civilizations in our galaxy with which radio-communication might be possible (i.e. which are on our current past light cone);

and

R_* = the average rate of star formation in our galaxy

f_p = the fraction of those stars that have planets

n_e = the average number of planets that can potentially support life per star that has planets

f_l = the fraction of planets that could support life that actually develop life at some point

f_i = the fraction of planets with life that actually go on to develop intelligent life (civilizations)

f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space

L = the length of time for which such civilizations release detectable signals into space

Clearly there are values in this equation that are unknown and will remain so for the foreseeable future, such as the number of planets with life or with intelligent civilizations.  We can only guess.  As guesses become more refined, the final answer, in theory, becomes more accurate.  Values for N in this equation range from 2 to 28x10^7, so clearly this is NOT a very good equation.  Plus, the lack of any contact with all of these supposed 280 MILLION civilizations has led to something called, get this, The Fermi Paradox!!

So, how is this idea of the Fermi problem useful?  Well, physicists and mathematicians and engineers will often use Fermi estimates before making actual calculations in order to not only provide context in which to do their actual calculations, but also to pre-test the validity of their final results.  How important is Fermi estimation?  Believe it or not, entire courses are taught on this at such prestigious places as MIT and the California Institute of Technology.  Often Fermi problems lead to more rigorous calculation, but the Fermi estimate at least gives one a starting point by figuring out parameters to refine.  This sort of logical problem solving has practical application in any number of fields of endeavor, such as engineering or city planning, architecture or chemistry.  Or just plain old mental torture frustration exercise.  So, try out this typical Fermi problem:

Archimedes of Syracuse was a philosopher, inventor, and scientist who lived and died in Sicily more than 2,000 years ago.  Assuming that the earth's atmosphere has been thoroughly mixed since then, how many molecules from his final breath are in your lungs right now?

Before you shrug, say "What?" and return to your pizza and beer, take a moment to think about not only the different values involved, but also what they might be.  Approximately.  When you have your Fermi estimate, or when you have simply given up, check the solution generally accepted below (to see the answer, highlight the dark text boxes below):

22.4 liters = 1 mole = 6x1023 molecules. Typical breath = 1 liter = 3x1022 molecules in

Archimedes’ last breath.

Volume of earth’s atmosphere = 8x1014 m2 (area of earth, see last question) * 30 km

(mean height of atmosphere) = 2.4x1018 m3 = 2.4x1021 liters. So 10 of Archimedes’ last

molecules are in each liter of the atmosphere.

Your lungs now hold roughly one liter. So there should be (on average) 10 of

Archimedes’ last molecules in your lungs now.

 Were you close?  Were you creeped out?  How many zeros did you have in your final answer?  Subtract your number of zeros from those of the final solution, and the absolute value of the resulting number shows you the order of magnitude you were in disagreement with the solution.  If you dare to try some more Fermi problems, click on the link below:

 

http://www.fermiquestions.com/play

or this one:

http://www.nytimes.com/interactive/2009/03/31/science/20090331-angier-quiz.html

Some sick, twisted person created a kind of Fermi estimate GAME!!!  You keep score by keeping a running total of your differences of order of magnitude.  Lowest score wins.  It's the new family game that will be sweeping the nation!

 

Final thoughts, or how I learned to stop worrying and love the bomb, or "If everybody is thinking alike, then somebody isn't thinking" -George S. Patton

When it comes to Fermi problems, for me personally it was love at first fight frustration *!@&#! noble attempt.  I deeply love the challenge of trying to come up with all the various components of a problem and estimate values for each one.  As an educator I put high value on demonstrating to math students that sometimes it is not all about coming up with the correct answer or regurgitating formulas and methods learned by rote. In the real world there are some problems that don't have set solutions or clearly mapped out paths to conclusion.  Sometimes we simply have to make our best guess.  A great deal of pressure is removed in such calculations, as it is not the answer but the process that has the greatest value.  You compare your answer with others, you re-evaluate, you live with the uncertainty of your final answer.  This, my friends, (cue cheesy music) is what makes the Fermi problem such a beautiful metaphor for life itself.

    



"If you create a machine to do the job of a man, you take something away from the man."

Don't get me wrong... I am not technology averse, nor do I entirely agree with the statement above.  It is, after all, only a quote from a mediocre STAR TREK movie.  A person's philosophy on life and existence should not be shaped and molded by a simple quote.  However, there is a fundamental danger in technology which gives this quote a small ring of truth.  Here's what I am getting at...

Technology comes into our lives a little bit at a time.  At first, it is novel, something to keep you ahead of the Joneses, then it becomes ubiquitous, then it becomes essential, then no one alive can remember a time without it.  Technology has allowed man (I use the term for convenience, not out of any misogynist tendencies) to do many wonderful things that he could never do otherwise.  Humans can't fly except through the use of technology, and the airplane has shrunk the world to only a day's travel from anywhere to anywhere else; Locomotives united east and west in the United States; the remote control fundamentally changed how we watch television, as did the VCR and the DVD player; computers, a thing of wonder relegated to classrooms and university labs merely 25 years ago, can now be worn on your wrist by anyone anywhere.  Can any of us imagine life without any of these technological advances?  Yet, for each one of these, there was a time when we did without them because they simply did not exist. 

When I was in high school, it seems like a foolish thing now, but calculators were hotly debated.  Were they a helpful tool to the math educator or an easy way out for the lazy math student?  There were specific rules that some of my math teachers had as to what qualified as an acceptable calculator problem.  This was usually when the math was long, ugly, complicated and prone to mistakes.  If you were caught walking into an SAT test with a calculator, including the wristwatches with the calculator built in, at the very least it would have been confiscated, and at worst you would have been asked to leave.  Yet now calculators are practically mandatory for an SAT test.  I have seen students pull out their calculators (including my own 11-year old daughter) to calculate things as simple as 35 divided by 5!  Or even easier, anything times ten!!  In greater and greater numbers our math students are relegating the basic skills to a machine, yet there is no great advancement in the complexity of the math being learned.  Students still graduate from high school having learned basic calculus, if they are the elite.  We have taken away part of the man, and replaced it with nothing greater.

The laptop and ipad are the contemporary equivalent of calculators of my generation.  We have certainly been privy to a parade of nifty things, such as Desmos and Geogebra and 101qs, that the computer allows us to utilize in teaching mathematics.  But are we as math educators on the slippery slope that allows the machine to do the job of a teacher?  One must never forget that behind every math game and math activity is the word "MATH".

What are we to do when the technology fails us?  Now, I am not referring to a doomsday apocalypse or global EMPs frying all electronics.  Learning disasters can be much simpler than that.  I have witnessed more basic problems, like network connectivity issues or forgotten passwords, that caused entire class periods of math instruction to be wasted not only because the teacher kept trying desperately to solve the issue, but because the teacher seemed unable (or perhaps just unwilling) to, for lack of a better term, teach math the old fashioned way.  This is eerily similar to the bright student who pulls out the calculator when they are perfectly capable of solving the problem themselves.  As technology becomes so integral to education that it becomes essential, a teacher must be vigilant to not become like his or her students and rely on technology exclusively to do all the educating for them.  Always the learning must show the mathematical truths that are at the foundation.  Otherwise we are only teaching how to play a game or use a program without critical thinking.  This is not the path to innovation and advancement; it is the path to indifference and apathy.

Computers in classrooms are here to stay.  Soon the day will arrive when no one alive will remember a time without computers and acers and ipads as an integral part of learning and teaching.  Let us all strive to make sure that they are a tool to open up doorways to rooms of knowledge and learning that we only dreamed of entering, that the next generation has a better grasp of math than we did, that they do not simply relegate math to the machines.  Because when that happens, we truly TRULY will not be able to do for ourselves, and something vital will have been taken away from us indeed!

Developing Students' Understanding of a Variable, by Ana C. Stephens.

This article deals with helping secondary students fully grasp the concept of a variable.  Three secondary classroom teachers were given a mouse thought exercise, where one has to come up with all the various ways that eight mice can be shared between two mouse cages connected by a tunnel.  Students developed charts, tables, even pictures to represent the ways, and they were encouraged to figure out a way to express this concept algebraically.  What struck me most was that so many students could not accept openly that two separate variables, say m and n, could share the same value.  This opens up my deeper concern for myself as a secondary math teacher: because so many math concepts have always come so easily to me, how much will I struggle in adapting concepts to make them more easily assimilated by the average or below average math student.  I recall in my elementary school days that we had math problems to do of the variety, __ + 5=8, and we had to figure out what went in the blank.  Then in middle school the blank was replaced with a letter, a variable, and this was an easy transition for me to make.  When this transition is a difficult one for a student in my classroom, I am hopeful that I will be both creative and understanding enough to allow the student to grasp the concept without completely being turned off by math.