Thursday, December 10, 2015

Why math?

Why math? That is the question many math majors get asked. I know that there are many different answers and many different reasons why I want to teach mathematics for the rest of my life. Math is a difficult topic for many individuals and it takes a dedicated student to work on the problems that they will be faced with. For me, I know I enjoy the challenge of finding the answer and how you can have many paths to the same answer. I think that the reason many math majors choose this path, is because we all have found satisfaction in being able to solve challenging problems.
My love for math began with multiplication. There was a game my teacher use to play called “Brain vs. Calculator.” The object of the game was to solve the simple one step problem that was given. If you are the brain you did it in your head and said the answer out loud. If you are the calculator you typed it in and then showed the answer on the screen. I loved playing and I always wanted to be the brain because I could do almost any multiplication or division problem in my head. I know that now in school, teachers are steered away from these activities, because most classrooms have calculators that students can always use. To me, this is a disadvantage to the students because they will have a hard time figuring out a multi-step problems. I’ve seen this firsthand when I tutor high school students in mathematics. A combination of many aspects have made me realize that I want to teach such aspects like: loving seeing students understanding, students having a disadvantage with technology and tutoring these student. This is part of the reason why I  am planning on becoming a high school math teacher. While teaching math in general is my goal I would prefer teaching ninth and tenth grade which is Algebra and Algebra 2 classes.
When I was in high school I had a teacher that made me go from understanding math to loving it. I ended with an understanding of math that I hadn’t known before. Being able to look at a problem and know what steps are needed is a major part of what I learned in high school. this knowledge of made not only helped in my education, but helped me gain more confidence in myself. In high school I was fighting a losing battle, because I started in the special education class in middle school only to be placed in advanced in high school. This made me want to work on my math skills even more. I know that it was because of one man that I was able to do this. He made learning math enjoyable and easy, which the goal when teaching. Hopefully once I have my own class, I can have a the experience of helping a student find their passion in school, like he did for me.
When I hit college I thought that math could not get more fun than the way it was in high school, but things change. I know one thing that has changed for me is the way I attack a math problem. If you had told me freshman year that I would like, or even love Geometry, I would have called you crazy. Then I took Euclidean Geometry and saw a totally different way of thinking and seeing the problems at hand. In my Euclidean Geometry class we worked with axioms and parallel lines. We also worked with shapes and graphs as they rotate. There are many different types of math that have made me see this in college, but Euclidean really made the connection to the visuals.  There are not always light happy topics in my math history. I was told in college that I was not smart enough and that I needed help to continue on with my goal of becoming a teacher. I hope that there are never experiences like this for any other students because it makes it hard to believe you can do the things you once knew you could. I loved math before my teacher and adviser told me I was not good at it. It is because of her that I found a passion to prove what I had known all along. I know now that I will be a teacher and I will have students that I can make a lasting impression on. My hope is that when my students look back at me, they remember the fun things and also some (if not all) the things that they learned. To make learning better for my student I want things to be active and fun, as well as interesting. Not every student will enjoy what I teach them, but at least they will understand the subject matter.   
Since being in college I have had classes varying between many different focuses, but I particularly enjoyed Math 229. In this class we made an exploratory activity to help show that not all math is the same boring topics, but it can be used to make things fun depending on the ways it is presented. There was one activity that was on proofs and how to teach your students about them without getting loaded down with the formal write up. It started as if you were a detective and you had to put together all the pieces of the story to prove how it works, why it works, and how it can be used. This is a ways of teaching that can be used in all levels that won’t only help introduce it, but it will help you write up a formal draft once you found all the missing pieces.When teachers are able to makes their students enjoy what they are learning then it will make it easier to learn. It is an interesting way of teaching when you bring an scenario that makes your student have fun while learning.  I love the idea that you can take a difficult topic and make it something fun that will teach at the same time. College has given me a real desire to teach what I have learned to my students.
In my own classroom I want to have more of a diverse teaching pattern that will show both visual and algebraic forms. I cannot expect my students to understand math the way I do, or even love it the way I do, but I do want to give them the best chance at learning it. To me there is nothing more important than helping students’ with their learning. There are many aspects of math ranging from Geometry and Algebra to Discrete and Applied Math. All of these need the same foundation that are multiplications, division, and pattern recognition. The foundation needs to be solid so that they can continue to learn. Some bad things about high school are that by the time students get there they already have their foundations built. However, it is not always a strong foundation which makes it hard to teach many topics. I hope that when others like me graduate they put an investment in the futures of their students as I plan on doing with my students.
I know that math is right for me because there’s always an answer and you will always have more to learn about it. You can never know all there is to know about a topic because the topics are always changing. Math has never been and will never be a consistent topic because mathematicians will always be finding new and useful discoveries. These discoveries can be applied to topics differently. Considering how math has changed while I have been in school, we went from needing everything memorized to having calculators with you at every step of the way. Also with me you can see that I went from having difficulties in school to being advanced and actually enjoying it. These to me are just some of the reasons why math will always be part of my future. I know that math takes more focus and patience than most students are willing to give it but I hope that I can help my students learn it just the same. I hope that I can show even a little bit of what I have learned, and I believe that I can help students understand new things and have great ideas for future advances in math.

Monday, November 23, 2015

Bernhard Riemann

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   Bernhard Riemann was the second of six kids, two boys and four girls, born to Friedrich and Charlotte. Their father was their teacher for the first few years of their lives will Bernhard turned ten when a local school teacher took over for their father. 
   in 1840 Bernhard went to Lyceum in Hannover where he was a good student but not exceptional in anyway, he did show interest in mathematics and was allowed to focus more directly on it at school. In 1846 Riemann enrolled in university of Gottingen and studied theology because his father had wanted him to but that didn't stop him from attending mathematics lectures. He did this till he could convince his father into letting him switch to mathematics. He was finally able to and he studied under Moritz Stern and Gauss. 
   This would have been an amazing place to study under Gauss but he was only teaching elementary courses and Riemann still only looked like a good student. The teacher to notice Riemann's potential was Stern which made him move to Berlin university in 1847 to study under Steiner, Jacobi,Dirichlet, and Eisenstein. 
   Moving back to his original University Riemann got his PhD under the supervision of Gauss.He then went on to write his Doctoral thesis which was looked at and accepted bu Gauss in 1851. 
   Gauss recommended Riemann to be appointed to become a lecture at his own home University. In these lectures Riemann have two parts:First he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call Riemannian space. In his second part of his lectures he posed deep questions about the relationship of geometry to the world we live in. 
   Once Gauss left Riemann fought for his chair at that University and wasn't given is but two years later he was appointed to professor and in the same year 1857 he had another punished paper. It wasn't till 1859 that Riemann was given the chair of mathematics at Gottingen.
   In 1858 Riemann was visited by Betti, Casorati, and Brischi and they discussed his idea of topology. Later he reported his greatest masterpiece "On the number of primes less then a given magnitude."
   To a more personal side of Riemann in 1862 he married Elise Koch and they had one daughter. In that same year he got tuberculosis. He then traveled to warmer climates to try and get better because in his family everyone died young. He passed away in 1866 while in Selasca, Italy.  After his death  his work published into a book focusing on geometric approach in math. 

Sunday, November 1, 2015

Euler: The Man!

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    From the beginning of Euler he was destined to be a genius, his father Paul Euler having studied theology at University of Basel and studied under Jacob Bernoulli. Not only did Paul learn from Jacob but he and Jacob's brother Johann both lived with Jacob while in school. Once finishing school Euler's father married and soon had Leonard Euler in the town of Basel.
    Euler's first teacher was his father and since he had studied under Jacob Bernoulli he was a well trained math teacher and was able to pass what he knew onto his son. Research shows that Euler wasn't sent to the greatest schools while growing up and self taught mathematics by reading text books. Euler was suppose to follow his father into the church but Johann Bernoulli found the great potential that Euler had and started private tutoring him. Even though his passion was in mathematics Euler followed his father's wishes and began studying theology. After realizing that there wasn't a future for him in the church he got his father's blessing to change his studies and focus on math.
    Euler by the age of 20 in 1726 had countless papers in print, short articles on isochronous curves in a resisting medium and was submitting papers into the Paris Academy Grand Prize and took second place behind Bouguer. After this achievement he was offered a post in St. Petersburg teaching in applications of mathematics and appointed to the mathematical-physical division. In the coming years he was appointed to the senior chair of mathematics and was able to start a family. In 173 he became sick and almost died, while being sick his eye sight started to diminish till he was completely blind in one eye and going blind in the other.
    In 1741 he and his family left St. Petersburg and went to Berlin where he worked at the Academy of Science. Euler spent the next 25 years in Berlin where he wrote 380 articles, books Calculus of variations, planetary orbits, artillery and ballistics, navigation, motion of the moon, and differential calculus. By the year of 1759 Euler assumed leadership of the Berlin Academy. He only stayed there for another 7 years when he decided to move back to St. Petersburg about this time Euler went completely blind, this however didn't stop him from doing his mathematics because he had a remarkable memory and was able to write articles with the help of his students. But in the year 1783 at nearly 77 years old Euler died.

    In his 77 years the mathematics that he had a part in grew and took off into what we know today. Here are just some of the topics he had a part in:

  • Geometry and Trigonometry 
    • considered sin and cos
  • Calculus
    • Differential calculus
    • Introduced 
      • beta 
      • gamma
      • integrating factors
  • Number Theory 
    • if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1
  • Lunar theory with Clairaut
    • Three body problem
    • wave theory of light
    • hydraulics 
    • music
    • acoustics
  • Eulers notations
    • f(x)
    •  i  for the square root on -1
    • π  for pi
    • ∑ for summation
    • finite differences Δand  Δ2y
  • The Basel problem
    • find a closed form for the sum of the infinity series ζ(2) = ∑ (1/n2)
  • And much much more.
   Looking back over Euler's life it is astonishing the things that he was able to accomplish and was able to do even close to the end when he was blind. I can say that I believe that math wouldn't be what we know it to be without Euler and what he gave future mathematics.


Monday, October 19, 2015

The Number Mysteries: A mathematical Odyssey Through Everyday Life

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    You want to be a teacher? You want to be able to answer that annoying question of "Where will I ever use this?" This book is an interesting read that will help you answer your students. It will help you see where these topics show up in the world and how they are important they are to what we know today.

   Not only did this book explain out the thinking behind the topics and cover in detail different examples but it gave online resources for you to continue your learning of the topics that you didn't understand. The five chapters in this book relate to:
  1. The Curious Incident of the Never-ending Primes
    1. Fibonacci number
    2. Golden ratio
  2. The Story of the Elusive Shape
    1. Finding length of coast line
    2. Snow flakes 
  3. The Secret of the Winning Streak
    1. Rock paper Scissors
    2. Lottery
  4. The Case of the Uncrakable Code
    1. Morris Code
    2. Semaphore
  5. The Quest to Predict the Future
    1. What hits the ground first a feather or a soccer ball
    2. Chaos vs Laminar Flow
    It is a very interesting they way that so many seeming unrelated topics are actually all interconnected is a very cool way to look at math and I think that it would be awesome to incorporate this into our everyday teaching method. If we get our students to be interested then we will see them understanding and actually enjoying what they are learning more.

   Marcus Du Sautoy the Author is an experienced mathematician and for many different topics one of the most known is his work with group theory and number theory. He put this book together and it shows that math is a very deep part of what our lives are and what we do on a daily basis. 

Book Notes: The Number Mysteries.

This book was broken into five chapters each covering one of the million dollar math problems. The topic of each:

  1. The Curious Incident of the Never-ending Primes 
  2. The Story of the Elusive Shape
  3. The secret of the Winning Streak 
  4. The  Case of the Uncrackable code
  5. The Quest to Predict the Future
Within each of these chapters there are activities that are online that connect you to concepts that are being discussed in the section. I found these as very good activities to understand what he was talking about.

In chapter one the million dollar question is called Riemann hypothesis.

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All the examples that I worked on in this chapter were related to how numbers build and will continue to add up in a never ending pattern. 

Chapter 2 
What is the shape of our universe? Seems like a question we get asked in science but this is actually related to math because we have to think about it in a logical way because we can only guess with science. 
This problem was solved but the mathematician  that found the answer didn't accept the million dollars he just likes doing the problem.

Chapter 3
Strategy to beat Rock Paper Scissors .....Where was this when i was growing up??? There are many things in this chapter that I never would have thought of and how it is explains them helped my understanding of things that I learned without having a real understanding of. To start the chapter it asks you to pick five numbers between 1-49 and then it ends with the Winning numbers....I failed but it was an interesting way to get interaction. The final problem is the Traveling sales men which is a type of question I remember getting asked in 341 class. 

Chapter 4
Code breaking....What kid doesn't have dreams about being a spy?
There are many types of codes that are based in math and even if you don't think its based on math i bet you would be interested in that there is some form of math hidden inside your codes. This really was interesting in how to read codes and write them. kinda makes me feel like a spy. 

Chapter 5
Thinking about the Cubs this year makes me wonder if Doc. Brown actually came to the future and found that they would win.....What is chaos? how would we predict chaos? I didn't understand this chapter as much as the others it was a stretch for me to understand where they were going and what they were doing.  

Sunday, October 11, 2015

Math VS. Science

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Is math a part of science, is science a part of math, or are they related at all?
   This is the question that brings us here today. I think there are many ways that this idea can be looked at and approached. Now I am going to show why I think math and science are related by explaining science and showing the math within and analyzing math and showing the science inside.

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    When we work in science we follow the steps above and we reach a point in the processes that we have to take what we have found and interpret what the data has given us. In many if not all cases this data has some numerical data weather it be height, temperature, etc. This isn't the only time math shows up in science experiments, anytime you need to use a formula to find missing values you are using math in science. There you are using mathematical formulas to find out aspects of science you couldn't find before. Not only that but you also can use the data to find patterns in science to predict how future data would proceed.
   If we now look for science in math we see that math is the process of working with numbers and logic so once we have our answers from given formulas we can then make sense of the data. This step of applying a meaning to data is doing science and then understanding the math that comes out of it.
  Next by looking at both of them we can observe some very interesting overlap:

  •     Tables
    • What is the best way to show numerical data in a table? What do both scientists and mathematicians use to present there data? The answer is tables because it gives all the desired data in one neat form. Tables are used in other areas of study where they are showing data but other areas of study that use tables are taking data that was derived from either mathematical data or scientific research 
  •     Communication 
    • The ideas are meant to be shared with the public and have to be understandable by the general public. This means that their ideas have to be stated in ways that everyday people can understand. 
  •     Proof
    • For the topics to be believed by many others you have to have evidence proving that it is true. This is true in both math and science in both areas you have to be able to prove you reasoning and show that there is always a way to reproduce it. 
   If were to play devils advocate and had to defend why I thought that math and science weren't related I would have to bring up the idea of how you can work on just pure math problems that have no connection to any story. This is like when you work on non-story problems when it's just working through a problem. Science is a different story though I don't believe that you can do pure science without having math show up at some point in the experiment. So my devils advocate would be that you can have just math or science with a hint of math, because you have can't have science without math.
   I know that Science vs. Math is a personal question and we all have our own views on it but this post gives my views on why I think that they are codependent of each other. As you can see, the evidence that both math and science are heavily related is strong and stands for itself in many areas therefore it should be agreed with.

Monday, September 28, 2015

Time to Net

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Nets are things that seem easy once you have the shape and just have to cut out and tape together but have you tried making your own, things get a little trickier. In class we made our own nets for a cube that had three different size pieces. The picture that follows is just one of many ways to make this shape.
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In my group we created three different nets that made the same end product. Which just illustrates that there are many different variations of how to draw net next for the same shape. I believe our professor wanted to see how many ways we could visualize the net because the one example was keep up at the front and we were to make the net with only looking at the shapes. This made the task harder and more of a challenge because we were using our imagination to make the shapes which didn't always work but in the end made what our nets all vary in ways.  
As our professor illustrated  its a good thing that in math there are many different ways to come to the final solution and nets are the same. So what are some strategies in finding nets? 
  • How I make them is starting with the the solid shape in front of you and trace that side that is on the paper. Following that you roll the shape across the paper and every time a face is flat on the paper trace it and only do ever face once. This will give you the shape as if you flattened it, then all that is needed is to add tab areas and you will have a working net. 
    • This can and will give many different net because you don't have to roll it the same way every time you start a net. An example of this is net number two below you can tell this because you can roll the shape up and makes the final cube. 
  • Some people just can just look at the 3D shape and see it flattened on a piece of paper. 
    • This will give the base as the center of your net and that makes it easier to build the shape.Most visualized nets will resemble net number one because you can see the sides are wings and the center makes up a four sided box.  
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These are the most common ways to make geometric nets without the use of technology. But since we are mathematician we like to think about how to optimize these nets and how can we find ways to make it easier, we are lazy and like using our technology. In my resent research I have found many web sites that will help and make shapes that you want nets for. Wolfram alpha has a nice set up for making 2D nets into 3D objects. But then there is always the math major's favorites of Geogbra, Sketch pad, and many other online resources. 
I think that this is a very interesting topic that isn't covered very well in schools. I think that being able to take 3D shapes and turn them into 2D shapes and vise verse are very important ideas to help with students learning and understanding of geometry. It gives students hands on knowledge of how to make and build what they are have been learning about. I know that even in college I enjoyed the challenge of how to build what I saw in front of me. I think this activity is a good active that is needed more in schools because classrooms focus more on the abstract instead of hands on learning once students get into higher level math classes which is sad because there are many activities that can be done in high school math classes.   

Thursday, September 10, 2015

Diophantus's Riddle

This is Diophantus's riddle it wants you to solve to find out how old Diophantus is when he dies. This riddle is somethings that many people don't want to work on because it has the awful fraction that everyone hates but as you work through it the fraction all start to work out and work together. Once you work through it you come to a whole final number. The following is my work on solving this problem.

If D equals Diophantus's age and S is the age of his son you will get the following expressions:

These numbers come from the riddle:
 is how long he spends as a boy
 is his life with a bread
 his life before marriage 
5 years married without child 
4 years after his son died till he died

Now doing the parenthesis first:
Find the common denominator witch is 84 we get 
(33/84) D+S+9
Replace S with (1/2)D and since we know that (42/84) is the same as (1/2) we can replace that as well
Move that D's to the same side,
Even though it isn't asked for we can now find the son's age by using 84 in our equation for S.

So taking what we have just gone through it's time to make our own and since we are all math nerds I will use "messier" fractions. I will being doing this in steps, stick with the steps. 

Step 1: Pick proportion.

   I found what their common denominator is.
Step 2: Variables and additional data.

Step 3:Write the problem
   You're going shopping with your friends and you find your favorite band is selling concert tickets for 1//8 your total bank balance. Continuing shopping you find the best outfit and shoes for the concert which cost 1/15 and 1/20 respectively of you balance. Once you get home you look and see that you have 18/30 of what you started with plus the $10 you have left from lunch. How much did your bank balance start at?

Step 4: Check!
1/8-concert tickets
S-how much left 
10-your change
Common denominator.
Plug in S.
Common denominator.
Multiply the reciprocal of (20/120) to both sides.
That is the final step and since we found a real answer that makes sense with the question asked we have a successful problem to make it more we could always ask:
   How much were the tickets?
   How much was the outfit?
   How much were the shoes?
This is how I worked through the Diophantus's riddle but there are many different ways to do it. Whats your's?
   This problem makes me wonder what else can be shown to students in better ways. This problem is a very good way of showing how fraction can work and correspond to problems that are strictly fraction heavy. I know that during high school I like many other hated fractions and this type of problem I think would help student make a connection to fraction. We all know that once students have a connection to what they are learning then they will not only learn it easier but have a better chance of remembering what they have learned.  

Wednesday, September 2, 2015

5 milestones of Math

What is math? This is a question that has been asked, you would think that being at the end of our math degrees we would have an answer to this but it is the most difficult question. So through this class and this blog I will be creating/thinking/learning what I believe the answer to this question is.
In the history of math there has been many major discoveries, but what are some of the main most important ones. I believe an important one would be discovery of infinity. This was a hard topic for many people to understand and even accept but is the most well known today. I think that without it there would be a completely different math world out there.
Throughout my college career I had to take Euclidean Geometry, which was unlike any other math class I have taken because it makes you think about things you can't automatically see or understand. The idea that there are more then just what we know as Geometry was a wide awakening to me. Which bring me to thinking that this is another major discovery in math of Euclidean.
We would have gotten no where in math without the basic operations of adding and subtraction. To me this was the very beginning of people learning and putting words with what they are doing. There are so many area of life that use these ideas and it to me would be the first and most important discovery.  
  Leading off of adding and subtracting I believe that multiplication and division would also be very important in our math history. Yes it is built of adding and subtracting but being able to use these idea really made math a more universal tool. 
Finally I think that in math when it comes to milestones that we need to know how to communicate what we know to others outside the math world. I believe that over the years we have become better and continue to get better at how it is communicated. I believe that milestone to be the opening up of the math word to the normal person.