Fractions AKA American Troublemakers

As I read Ms. Mariner's blog about fractions, and Americans not knowing the correct sizes of fractions, I was not surprised.  Although I cannot recall a specific time when I have encountered this misconception of fractions, I totally believe that it is a real issue.  Sometimes we Americans, admittedly myself as well, are just plain dumb, but who's to blame? Math teachers? Students?

The Common Core is quite frankly something that annoys me, a lot.  Ms. Mariner commented on how her and other independent school math teachers don't go through any sort of training.  I think that this is WAY better than them going through training.  If we wanted all of our math teachers to be exactly the same, we would be taught by robots, but robots would not be effective!!! By them not going through training, they are able to teach math in a way that makes sense to each individual student.  I think that's our country's education problem.  We want everyone to be the same! Well America, BUMMER DEAL.  We're not all the same, and our learning needs are not either, so suck it up and teach our students the way they need to be taught.

As far as the A&W/McDonald's issue mentioned in "Why Do Americans Stink at Math?" I think that it is a direct result of American's not understanding what they're learning.  Sure, they memorized that 4 is bigger than 3 when they learned how to count to 10 in kindergarten, but did they ever understand what fractions meant?  It was all memorization, so when they went to apply it in the real world, making purchasing decisions, they didn't get it.  Pure memorization of simple concepts allows absolutely no room for understanding and application of slightly more complex concepts.

This is just a little bit of evidence for why Common Core won't work.  We are not robots, our teachers are not robots, and we should not act as if we are.  We are individuals, with different skills, and different ways of discovering things.  We must cater to that if we truly want to understand and apply learned skills to our lives.

BIG Numbers

It is easy to be overwhelmed when we think of big numbers.  Not big numbers like 100, but REALLY big numbers. When I think about how to describe this overwhelming sensation, my mind goes to little kids.  It reminds me of when you're little, and you stretch your arms out as wide as you possibly can, and your left hand is as far as humanly possible.  You look up at your mom or dad and say "I love you THIIIS much!!"  and your parent looks back at you and says "I love you to the moon and back!"

...Do we ever think about just how much that is? I don't think we do, and I would argue that as we get older, we really don't understand huge distances any better than we did when we were 5, 6, or 7.

When a little kid stretches their arms as far as they can, that distance between their arms is their "forever" distance.  That is their standard for "the most in the whole wide world."  When we tell people that we love them to the moon and back, that is our "forever" distance.  Yes, we logically know that there exist distances larger than that, but we don't consciously comprehend that.  A little kid knows that there exists a distance larger than the space between their palms, but they don't consciously comprehend it.

This is all due to standards.  We created standards in our minds, and that is what we compare everything we discover after the standard is set to.  To test my theory, I asked a 7 year old girl, Maci, what the biggest number she knows is.  She said that 100,000 is the biggest number she knows.  100,000 is Maci's standard.  She cannot comprehend numbers bigger than that, because they are too big when compared to her standard.  When she thinks of small numbers, she compares them to her large standard of 100,000, and she is able to comprehend them.  As we get older, our standard gets larger, but when we hear big numbers we are still comparing them in the same way that 7 year old Maci does.
Everyone has different standards for their "forever" numbers, and that is why we individually comprehend different numbers in different ways.  An example of this is a runner.  Someone who runs 3 miles normally, cannot comprehend running 10.  This is because 3 is their standard.  Someone who runs 26 miles normally, can easily comprehend a 10 mile run.

Thus, it is all in what we are used to.  Standards are the key.

Transcendental Functions

I started by pulling up Webster's online dictionary and searching the word, "transcendental." I was directed to the definition of "transcendent". This is defined as "going beyond the limits of ordinary experience," or "far better or greater than what is usual."  Now, I don't think that reveals to me how the word is related to math, but I do think that I can think of it in such a way that it does.  For example, maybe we are studying transcendental functions because we want to take our minds to a place where they go beyond the limits of an ordinary mathematical experience, or because we want to study mathematics in a way that is far better or greater than what is usual.  These definitions and applications intrigued me.

Next, I found out what transcendentalism is.  It is a practice that I am not sure I am in agreement with, but that is interesting to ponder.  It is a practice of people who do not accept things as religious beliefs, but rather as a way of understanding life's relationships.  This also got my "math brain" thinking.  Maybe we are studying transcendental functions because we want to study the relationships between numbers and equations.  We want to discover how things work together in a mathematical context.

Finally, as I did research on transcendental functions themselves, I found my wonderings to be almost correct.  Transcendental functions have the name they do because they cannot be expressed simply.  They cannot be expressed using simple algebra.  They allow us, and force us, to go deeper into our mathematical thinking.

Sinusoidal Curves in Ski Racing

As I thought about this blog, and what I should write about, I just couldn't find much inspiration.  I began to think of ways that I could relate sine, cosine, or tangent to my life.  Since ski season is just getting started, I naturally thought of ski racing.  I believe that a sinusoidal curve can be used to model the turns that a ski racer makes around the gates on any given course.
Here's a picture to refer back to, in order to make what I say more clear:

As you can see, there are various different aspects of sine curves that are evident in a series of ski turns:  

Observation 1: The horizontal distance from red gate to a point collinear with the blue gate is the amplitude of the sine curve.
Observation 2: The vertical distance from the red gate a point collinear with the blue gate, is the wave length of the sine curve.
Observation 3: The distance from blue gate to blue gate, is the period of the sine curve.

I think that it is neat that I could apply this to my skiing.  While I was skiing this past weekend, I was thinking about these sine curves and what I was creating with my skis. Math really is everywhere!

The Give and Get Balance

Stefano Bertozzi's speech was very thought provoking.  Although I think that he has a generally good philosophy, I don't totally agree with everything he is saying.  He didn't seem to say that we should strive to be the best that we can be.  He said that it is okay to do bad stuff, as long as you do good stuff too.  Although I think that he probably is a generally "good person," and does good things for the world, I wasn't particularly fond of what he left out of his speech.  Although we are all human and very prone to sin, we must strive to be the very best we can be.  We don't need to dwell on all of the ways we've messed up (if we did that, we would most likely be very depressed), but that doesn't make them okay.  I enjoyed his speech because it made me think about my own opinions.  Although I don't agree with everything he said, I think he has some sort of good foundation for a good idea.  It is at least worth thinking about.

Ski Racing Parabolas

As an athlete, the concept of parabolas in sports caught my eye.  There are the obvious ones, such as the parabola of a soccer ball or football in mid-flight after being kicked, but I didn't want to just talk about something that everyone can think of and has already expanded on.  I do two sports competitively.  I play soccer, but I also ski race.  Since I already knew of parabolas that occur in soccer, I started thinking, "What parabolas occur in ski racing?"  It took some thinking, but after going through the process of a race, I think I am on to something.

Above is a diagram of a turn of a ski race.  Although there is some information you do not need that is given to you in the picture, it is a good way to illustrate what I am thinking about.  In a ski race, there are alternating colored gates that you have to turn around.  Something that my coach is always talking about is my line.  What line should I take in order to make it down in the fastest time?  The line is a series of parabolas that have origins at each gate.  I think it is very fascinating because each individual racer creates their own individual parabolas as they turn their skis around the gate.  I never thought about how math applied to ski racing, and I think it is very fascinating :-)

Why Only 3?

As I read Mrs. Mariner's blog and then  parts of Trevor's blog, I was in awe at a lot of the discoveries and ideas explored.  However, one question stood out to me and intrigued me the most.  That was, "Why can only 3 regular polygons tessellate?"  I am doing something that might be slightly different.  I haven't fully explored the question yet.  I am in the process of exploring it while I am writing this blog. 
To start off, I didn't want to just Google the question; That would be too easy and there would be little to no exploration involved.  First, I wanted to come up with my own ideas, and then find the answer.  The definition on mathisfun.com of tessellation is a pattern of shapes that fits together without any gaps between shapes.  That is simple, and makes sense.  So why can't that be done with more than 3 regular polygons?
Below is an image I found when I Googled, "Octagonal pattern."

This image kind of frustrates me.  In my head, I feel like regular octagons should fit together perfectly! I mean, why not? That's the question.  It seems to me as if the eight vertices of the octagons almost get in the way! The hexagon is the regular polygon that can tessellate with the largest possible number of vertices, six. 
Below is an example of that.

As I compared the two images of the octagonal pattern and hexagonal pattern, I realized it wasn't the vertices that are getting in the way.  It is in fact, the sides.  With the hexagons, you can make a straight line of hexagons, with two opposite sides touching other hexagons, and then you can fit the vertices of other hexagons adjacent to the sides that are meeting the other hexagons to form the line of hexagons.  Wow, that was a super complicated explanation of my thinking, and I don't know if it made any sense to you.  I don't exactly know how to explain it differently. 
Here is my attempt using the Paint program on my computer:

 Red= hexagons in a line
Yellow= two opposite sides of hexagons that are touching other sides of hexagons
Blue= hexagons that are on the sides of those in a line
Green= vertices that fit in to the line perfectly

When looking at the octagonal pattern, they don't have those vertices.  Rather, there are "extra" sides that get in the way, forming dead space in the shape of squares.  I don't really know why that is the case, but that is my observation!
For more explorations of tessellations (say that 3 times fast), you can go to:
http://www.mathsisfun.com/geometry/tessellation.html

I hope you enjoyed following my train of thought!