Thursday, April 15, 2010
Wednesday, February 3, 2010
Make Your Own Fractal!
Earlier in the year we learned about Fractals. For our last blog assignment I want us all to do something FUN. I want us all to make our own Fractals!
Go to the above website and create your own pattern. Turn it into a fractal and watch the magic happen!
Print out your original pattern, the final piece, and name it. Bring it into class and we will share our geometric "works of art."
Experimentation or Memorization?
When you think about Mathematics, which of these words comes to mind?
Experimentation?
or
Memorization?
When I was a child, my parents were always so impressed at how "advanced" I was in mathematics for such a young boy. To them, it was like I had some sort of a gift. In 7th grade, however, something changed. My math teacher and many of her colleagues were convinced that I just didn't "get it." They would administer tests; I would solve the problems I was given and I would usually (almost always) come up with the right answer, but when I got the test back... it was marked WRONG! "Why?" I would ask.
"Because you are not doing it the way it was taught." she would say "You are coming up with the right answer, but you are not following the steps that we taught you. We are not convinced that you know what you are doing."
In 8th grade, when all of my friends were advancing to their first Algebra class, I was stuck in a class known as "Intro to Pre-Algebra!" INTRO to PRE Algebra?! It was there that I learned to conform, and to do the equations as they were taught. I was taught that Mathematics is a series of Memorization - not Experimentation!
But is this true? How could I have always gotten the right answer even when I didn't memorize the steps? Was I just a good guesser? What do you think?
Is math Memorization or Experimentation? Reflect back on some of your experiences with math. Did your teachers promote Experimentation or Memorization?
As you ponder this question, take a look at the discoveries THESE children have made:
http://www.mathman.biz/html/discover.html
Experimentation?
or
Memorization?
When I was a child, my parents were always so impressed at how "advanced" I was in mathematics for such a young boy. To them, it was like I had some sort of a gift. In 7th grade, however, something changed. My math teacher and many of her colleagues were convinced that I just didn't "get it." They would administer tests; I would solve the problems I was given and I would usually (almost always) come up with the right answer, but when I got the test back... it was marked WRONG! "Why?" I would ask.
"Because you are not doing it the way it was taught." she would say "You are coming up with the right answer, but you are not following the steps that we taught you. We are not convinced that you know what you are doing."
In 8th grade, when all of my friends were advancing to their first Algebra class, I was stuck in a class known as "Intro to Pre-Algebra!" INTRO to PRE Algebra?! It was there that I learned to conform, and to do the equations as they were taught. I was taught that Mathematics is a series of Memorization - not Experimentation!
But is this true? How could I have always gotten the right answer even when I didn't memorize the steps? Was I just a good guesser? What do you think?
Is math Memorization or Experimentation? Reflect back on some of your experiences with math. Did your teachers promote Experimentation or Memorization?
As you ponder this question, take a look at the discoveries THESE children have made:
http://www.mathman.biz/html/discover.html
Careers in Mathematics OR Mathematics in Your Career
When it comes to people who study mathematics, there can be two types of people:
One type of person actually enjoys math (a person like me). This type of person studies mathematics because they find it intellectually stimulating. They believe in math and the answers it beholds. This type of person may want to go on to study math in college.
Another person, however (and for some strange reason) doesn't like math! This person studies math because they feel like they have to, not because they want to. They will do what they must to achieve a goal (whether that be a high school diploma, a college degree, any type of certification, etc.) - and if studying math is a means toward that goal, they will do what they must.
And some people are a little bit of both, but for the purposes of this blog assignment, I would like you to choose which one of these people sounds most like you. Do you truly enjoy math? Do you think you might be interested in learning about math to the depths of calculus and beyond? Or do you simply study math because you have to?
For those of you that would study math by choice, do some research. Find out what types of careers one might pursue with a degree in mathematics (Bachelor's Degree, Master's Degree, or Doctorate). Do any of these careers peek your interest? Comment back with a list of careers you have found and pick one that you think you would enjoy. Describe the amount of study that is required and what a person in that field actually does for a living.
For the rest of you. Start by choosing a career that you think you would enjoy. Essentially, start with an end and describe the means by which you would get there. (For example: If you want to be a Doctor, how would you begin? What types of degrees are required? And most importantly, to what level of mathematics would you have to study and how much mathematics would you use on the job.)
I will start by giving you MY outline.
Though I do enjoy mathematics, I did not go to college with a strong curiosity or interest in studying mathematics. So for the purposes of this assignment I will start with my "end."
Goal - Become a College Professor in the field of Educational Psychology.
As a college professor, I would like to teach some form of educational psychology.
To become a college professor in the field of Education, I must first have educational experience.
Choice of Educational Experience - High School Mathematics
Levels of education/experience required -
College Professor -
1. Bachelor's Degree in either Psychology or Education
2. Master's in Education
3. Doctorate in Educational Psychology
4. Teaching Experience (Choice - High School Mathematics)
Teaching Experience -
1. Post Baccalaureate in Teacher Education
2. Certification in Mathematics
Level of Mathematics Required -
1. Calculus I, Calculus II, Calculus III
2. Three upper level math courses (Linear Algebra, Differential Equations, & Foundations of Mathematics)
In chronological order in looks like this...
1. Bachelor's in Psych. - 4 years
2. Post Baccalaureate + Cert. in Math - 3 years
3. Teaching Experience - ? years
4. Master's in Education - 4 years (part-time)
5. Doctorate in Educational Psychology - 4 years(part-time)
On the job - As a college professor I will be required to do a lot of research in my field. A lot of research means a lot of STATISTICS, so as you can see not only will math be important as I teach it, but will continue to carry great significance throughout my career.
One type of person actually enjoys math (a person like me). This type of person studies mathematics because they find it intellectually stimulating. They believe in math and the answers it beholds. This type of person may want to go on to study math in college.
Another person, however (and for some strange reason) doesn't like math! This person studies math because they feel like they have to, not because they want to. They will do what they must to achieve a goal (whether that be a high school diploma, a college degree, any type of certification, etc.) - and if studying math is a means toward that goal, they will do what they must.
And some people are a little bit of both, but for the purposes of this blog assignment, I would like you to choose which one of these people sounds most like you. Do you truly enjoy math? Do you think you might be interested in learning about math to the depths of calculus and beyond? Or do you simply study math because you have to?
For those of you that would study math by choice, do some research. Find out what types of careers one might pursue with a degree in mathematics (Bachelor's Degree, Master's Degree, or Doctorate). Do any of these careers peek your interest? Comment back with a list of careers you have found and pick one that you think you would enjoy. Describe the amount of study that is required and what a person in that field actually does for a living.
For the rest of you. Start by choosing a career that you think you would enjoy. Essentially, start with an end and describe the means by which you would get there. (For example: If you want to be a Doctor, how would you begin? What types of degrees are required? And most importantly, to what level of mathematics would you have to study and how much mathematics would you use on the job.)
I will start by giving you MY outline.
Though I do enjoy mathematics, I did not go to college with a strong curiosity or interest in studying mathematics. So for the purposes of this assignment I will start with my "end."
Goal - Become a College Professor in the field of Educational Psychology.
As a college professor, I would like to teach some form of educational psychology.
To become a college professor in the field of Education, I must first have educational experience.
Choice of Educational Experience - High School Mathematics
Levels of education/experience required -
College Professor -
1. Bachelor's Degree in either Psychology or Education
2. Master's in Education
3. Doctorate in Educational Psychology
4. Teaching Experience (Choice - High School Mathematics)
Teaching Experience -
1. Post Baccalaureate in Teacher Education
2. Certification in Mathematics
Level of Mathematics Required -
1. Calculus I, Calculus II, Calculus III
2. Three upper level math courses (Linear Algebra, Differential Equations, & Foundations of Mathematics)
In chronological order in looks like this...
1. Bachelor's in Psych. - 4 years
2. Post Baccalaureate + Cert. in Math - 3 years
3. Teaching Experience - ? years
4. Master's in Education - 4 years (part-time)
5. Doctorate in Educational Psychology - 4 years(part-time)
On the job - As a college professor I will be required to do a lot of research in my field. A lot of research means a lot of STATISTICS, so as you can see not only will math be important as I teach it, but will continue to carry great significance throughout my career.
Monday, February 1, 2010
Mathematics in Nature
Has anyone ever heard of the Fibonacci Numbers? They are a sequence of numbers that follow a fairly simple pattern.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and so on.
Figure out the pattern yet?
If you haven't, here's the spoiler: each number in this sequence is the sum of the 2 previous numbers. (0+1=1, 1+1=2, 1+2=3, 3+2=5 etc.)
Sounds pretty simple right? Well what if I told that this pattern of numbers may be the most important sequence of numbers ever discovered? What if I told you that this sequence of numbers holds the key to understanding the mathematical processes of our natural world? What if I told you that the Fibonacci
Numbers appear in nature more frequently than any other pattern of numbers? Would you be curious?
The fact is, Fibonacci numbers ARE found in nature quite frequently. Perhaps what is most prevalent is the ratio of these numbers...
If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:
1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666..., 8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538...It seems that as we continue along this pattern, we seem to be narrowing down to one specific number. That number is known as "Phi" - but in numerical terms it is about 1.618033.
This ratio we are referring to is also known as the "Golden Ratio" and can be found almost everywhere in nature!
Try Googling "Golden Ratio" or "Fibonacci Numbers." Report back with some interesting photos and/or natural patterns you have found.
Extra credit - What is the relationship between this photo below and the Fibonacci Numbers? (E-mail me the answer to this question, or hand in a hard copy in class.)
Everyday Mathematics
Everyday, whether we realize it or not, we use some form of mathematics. For the purposes of this blog entry, I would like you to make a log. Choose one day to begin logging every instance in which you use mathematics. You might be surprised, if you really pay attention, to how much math you use without even knowing it. Here is an example of my log:
7:45 Am - Wishing to spend less than $10 on breakfast, I add the prices of my items before I purchased them this morning. 5 hour energy - 2.99, Soft Pretzel - .65, Breakfast sandwich - 1.99, Large Bottled Water - 1.35. Without actually adding the numbers, I rounded, and found I'd be spending about $7.
8:05 Am - Will I be late? I need to get to class by 8:30. On the parkway this morning I wondered if I would have enough time to get to class on time. I started at mile marker 25 and needed to get to mile marker 45 in 25 minutes. If I obeyed the speed limit at 65 mph, could I make it on time? 60 mph is the same as saying 1 mile for every minute. So assuming I go 60 mph the whole way, I would be there in 20 minutes. Considering I go about 75 on the parkway, I have plenty of time.
9:45 Am - I have 4 hours between classes with 3 assignments to do. Subtract an hour for food breaks and bathroom breaks and I should spend no more than 60 minutes on each assignment.
2:10 Pm - Calculus class. =D
6:00 Pm - Stop for gas on the way home. I drove 405 miles from my last fill up and (according to my receipt) burned 17.2 gallons of gasoline. My tank holds 18 gallons so I was .8 gallons away from running out of gas.
Also, I wanted to see if my car was running as efficiently as it should. I should be getting about 25 miles per gallon if everything is running correctly. So I checked. If 17.2 gallons gets me 405 miles then 405/17.2 = 23.25 mpg. Hmm... maybe I should check my tires.
I also added my mileage for the week. I wanted to see how much money I should fit into my budget for gas. I drive about 325 miles per week. At 23.25 mpg, I'd be looking at buying about 14 gallons of gas per week. At $2.50 per gallon, I should figure on spending $35 a week on gas.
11:00 Pm - For what time should I set my alarm? I have to be at class at 10:30 Am. Subtract 30 minutes for commute. Subtract 20 minutes for getting ready (getting dressed, brushing teeth, etc.) and 10 minutes for a shower. For a total of 1 hour. I should set my alarm for 9:30 Am.
7:45 Am - Wishing to spend less than $10 on breakfast, I add the prices of my items before I purchased them this morning. 5 hour energy - 2.99, Soft Pretzel - .65, Breakfast sandwich - 1.99, Large Bottled Water - 1.35. Without actually adding the numbers, I rounded, and found I'd be spending about $7.
8:05 Am - Will I be late? I need to get to class by 8:30. On the parkway this morning I wondered if I would have enough time to get to class on time. I started at mile marker 25 and needed to get to mile marker 45 in 25 minutes. If I obeyed the speed limit at 65 mph, could I make it on time? 60 mph is the same as saying 1 mile for every minute. So assuming I go 60 mph the whole way, I would be there in 20 minutes. Considering I go about 75 on the parkway, I have plenty of time.
9:45 Am - I have 4 hours between classes with 3 assignments to do. Subtract an hour for food breaks and bathroom breaks and I should spend no more than 60 minutes on each assignment.
2:10 Pm - Calculus class. =D
6:00 Pm - Stop for gas on the way home. I drove 405 miles from my last fill up and (according to my receipt) burned 17.2 gallons of gasoline. My tank holds 18 gallons so I was .8 gallons away from running out of gas.
Also, I wanted to see if my car was running as efficiently as it should. I should be getting about 25 miles per gallon if everything is running correctly. So I checked. If 17.2 gallons gets me 405 miles then 405/17.2 = 23.25 mpg. Hmm... maybe I should check my tires.
I also added my mileage for the week. I wanted to see how much money I should fit into my budget for gas. I drive about 325 miles per week. At 23.25 mpg, I'd be looking at buying about 14 gallons of gas per week. At $2.50 per gallon, I should figure on spending $35 a week on gas.
11:00 Pm - For what time should I set my alarm? I have to be at class at 10:30 Am. Subtract 30 minutes for commute. Subtract 20 minutes for getting ready (getting dressed, brushing teeth, etc.) and 10 minutes for a shower. For a total of 1 hour. I should set my alarm for 9:30 Am.
Thursday, January 28, 2010
Mathematical Beauty
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry." - Bertrand Russell
1/28/10 - http://en.wikipedia.org/wiki/Mathematical_beauty
The above image is an example of "Fractal Art." Fractals are based on a geometric principal of self-similarity. What this means is that essentially the final product (or the whole) is based on infinitely smaller and repeated patterns. Here is a less complex example of a fractal:
As you can see, what starts as a simple pattern soon becomes relatively more complex and detailed simply by repeating it's pattern upon itself. This concept can be applied in great complexity and detail as to create fractal art such as these:
1/28/10 - http://en.wikipedia.org/wiki/Mathematical_beauty
The above image is an example of "Fractal Art." Fractals are based on a geometric principal of self-similarity. What this means is that essentially the final product (or the whole) is based on infinitely smaller and repeated patterns. Here is a less complex example of a fractal:
As you can see, what starts as a simple pattern soon becomes relatively more complex and detailed simply by repeating it's pattern upon itself. This concept can be applied in great complexity and detail as to create fractal art such as these:
Mathematical Beauty can be defined in other ways as well. Some believe that the interconnection of theorums and infalliable truths of mathematics is beautiful. Much like an intricate puzzle, all of the pieces fit together so perfectly as to form a (yet unfinished) masterpiece of symbolic logic and an underground network of natural mechanics.
Please read about one of the following from http://en.wikipedia.org/wiki/Mathematical_beauty and comment:
1. Beauty in Method
2. Beauty in Results
3. Beauty in Experience
4. Beauty in Philosophy
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