What is Math?

I remember at the beginning of this semester I had a very difficult time defining math. I have been asked this question many times in my life, but I am never able to give an answer that I think fully reflects everything mathematics covers. However, this semester, I feel like I got a good, quick look at many aspects of mathematics and how it has developed over time.

At the beginning of the semester I said math is a logical way of explaining everything in the world and you can find math everywhere you go. I still think I would define math in a similar way. However, now my own definition that I created means even more. Throughout the semester I have seen example after example of math involved in ways I had not thought of before. For example, doodling and math, the Fibonacci sequence found in plants, and tessellations. I feel like these examples show the art and beauty of math, while most people only think of topics like calculus and algebra when considering the question “what is math?”

I think math is more integrated in our lives than many realize. Scientists use math while investigating, for example, using data or weighing objects. Physicists and Statisticians use math while solving equations. Art is full of mathematics patterns like symmetry, space filling doodling, and the Fibonacci sequence. On a day to day basis, everyone uses math to buy items, measure food, and tell time.  

I have come to realize that math can be complicated like abstract algebra and non-Euclidean geometry; however, it can also be simply explaining everything surrounding you. Math is found in everything. Either how it was made, how it is formed, or patterns within it.

Additionally, at the beginning of the class we were asked to list the five most important moments in the history of math. I was a little embarrassed that I could not come up with a good list of five moments or even a good list of mathematicians. None of my teachers or professors had ever focused on the history of math in any of my classes. The most I ever learned was maybe the name of the mathematician who discovered what we were learning. But I never was able to learn about their lives and all that they had proved and done for math. Now, after taking this class, I can name off who I think are the most significant mathematicians in my opinion: Euler, Euclid, Newton, and Fibonacci.

Additionally, I feel like many people think everything in mathematics has already been found or discovered. But, it was fun to see that mathematicians continue to discover and prove ideas today.

I think in order to appreciate math fully, you need to understand how large of a field mathematics truly is. It is important to know how long ago some math concepts were discovered and the process mathematicians went through to discover what we know today. 

History of Math- Women in Mathematics

          Many times women in math do not receive the credit they deserve. It seems the focus is on the males and all of their accomplishments. I wanted to take a minute to explore the contributions of five female mathematicians and their accomplishments in the field of math. We will look at Hypatia, Agnesi, Germain, Kovalevskaya, and Noether. 

1. Hypatia of Alexandria (370-415)

Hypatia lived in Egypt and was one of the first women to help with the development of mathematics. She got her interest in math from her father Theon who was a mathematician and a philosopher. She taught math and philosophy at a Platonist school in Egypt with a scientific emphasis. Through her teaching, she became a symbol of learning and science which was against the Christians beliefs. Eventually Christians killed her because they were threatened by her knowledge. We do not have any evidence of her own mathematics, however, she helped her father write Ptolemy's Almagest and a new version of Euclid's Elements. She also wrote commentaries on Diophantus's Arithmetica, Apollonius's Conics, and Ptolemy's astronomical works. 

2. Maria Agnesi (1718-1799)

Agnesi lived in Italy and her father hired talented tutors to help her with languages and philosophy. Her father liked to show off her intelligence in front of other people. At one point in her life, she wanted to become a nun, but her father would not allow it. She ended up staying with her father instead of pursuing her dreams to become a nun. She decided to study religious books and mathematics on her own. Ramiro Rampinelli was a monk who helped her better understand mathematics. She wrote a commentary on one of de L'Hopital's works. She wrote Instituzioni analitiche which gives illustrations of calculus results and theorems. Later, she dedicated her life to working with charities and the poor. 

3. Sophie Germain (1776-1831)

Germain was a French mathematician. She first became interested in Mathematics when she learned about Archimedes. To pursue her interest she studied Newton and Euler. Lagrange read some of her work and decided to become her mathematics counselor even when he found out she was a woman. She had many correspondences with well known mathematicians like Legendre and Gauss on their works. Many times she disguised her self as a man with the name M. LeBlanc so she wouldn't be overlooked. However, once they found out she was a woman, they praised her work even more than they originally had. She contributed to Fermant's Last Theorem, Germain's Theorem, and she tried to formulate a theory on elastic surfaces. She spent the rest of her life studying mathematics. 

4. Sofia Kovalevskaya (1850-1891)

   

Kovalevskaya was born in Russia. She became interested in mathematics when her uncle, Pyotr Vasilievich Krukovsky, spoke to her about it. She had wallpaper covering differential and integral analysis in her bedroom which introduced her to calculus. However, she had a tutor that first formally taught her about the subject. These lessons stopped when her father did not want her to continue her studies so she secretly studied physics and algebra on her own. She later unofficially attended a university where women were not allowed. She wrote papers about Partial differential equation, Abelian integrals, and Saturn's Rings. Her gender caused serious problems when she was pursuing a career at a university. The only job she could find as a women was for elementary education. Eventually she held a chair at a European university which was a huge accomplishment because only two other females had done this before. She continued studying math, writing articles, and even won a few prizes for her accomplishments in mathematics.

5. Emmy Noether (1882-1935)

Emmy Noether was born in Germany and her father was a mathematician. She first got her certification to be a teacher, but later realized she wanted to attend a university to unofficially study mathematics since women were not allowed. In 1904 she was allowed to officially study mathematics at a university. One topic she focused on was the theory of invariants for the forms of n variables. Noether did research on her own and started publishing her work which gave her a good reputation. She has a theorem named after herself for theoretical physics. However, she also spent much time working on invariant theory and ideal theory as well which helped produce abstract theory. In 1933, she came to the United States as a professor at Bryn Mawr College. 

        I think it is important for everyone to understand that men and women are equally capable of contributing to mathematics. I think when I become a middle school math teacher, I will teach my students about the history of mathematics. I may start my lessons introducing famous mathematicians including both males and females. I would like my students to see that math is not a subject that boys are better at. Hopefully showing famous female mathematicians will help the girls in my class see that they could be mathematicians too.   

    

Sources

http://www-groups.dcs.st-and.ac.uk/~history/BiogIndex.html

Communicating Math- Euler's Line and Euler's Circle

         This week in class we discussed Leonhard Euler and all he has contributed to mathematics. He lived from 1707-1783 and is known as the last mathematician to know all of mathematics. One of my favorite areas of math is geometry. When I heard about Euler's Line and Euler's Circle, I thought they were incredible discoveries. The Euler line is a straight line through the orthocenter, triangle centroid, circumcenter, de Longchamps point, the center of Euler's Circle, and other centers of a triangle.
         When I studied Euclidean Geometry, I was unaware and often confused by all of the different centers of a triangle. So first, I am going to explain how to find each of these centers, and then explain Euler's line that he discovered.
          First, we will look at the orthocenter. The orthocenter, point D, is where the three altitudes of the triangle intersect as shown below.

         Next, we have the triangle centroid, point G below. This center is where the three medians intersect. The median is found by finding the midpoint of each side of the triangle and connecting the midpoint to the opposite vertex. 

             The circumcenter, point H below, is the point where the perpendicular bisectors of each side intersect.

           Another center of circle, de Longchamps point, point K below, can be found. De Longchamps point is found by reflecting the orthocenter over the circumcenter as shown below.

        

          Finally, we will look at the center of Euler's circle which is also known as the nine-point circle. This circle is found by connecting nine points together. You connect: the midpoint of each side of the triangle, the foot of each altitude, and the midpoint of the three segments that connect the orthocenter to the vertices. The Euler circle is shown below with center at point S.

           Once Euler has found all of these centers of a triangle, he realized you could draw a straight then through them.

            The pink line above is the Euler line passing through the orthocenter(point D), the triangle centroid (point G), the circumcenter (point H), the de Longchamps point (point K), and the center of Euler's circle (point S). Since this image is quite complex with all the lines you need to find these centers, I simplified it below.

           The only triangle that does not contain Euler's line is an equilateral triangle. For equilateral triangles, all the centers of the triangle fall on exactly the same point. Thus, you cannot create a line connecting all these points when there is only one point.

           I think that realizing that Euler's line holds true in all triangles except equilateral triangles is quite incredible. When I first learned about it I was amazed. It made me wonder how many of these types of discoveries in geometry I don't know about yet or have not yet been discovered. 

          Also, if I ever become a geometry teacher, I would definitely introduce my class to Euler and his line and circle. I think these concepts would grasp students interest and make them more excited to learn about geometry.  

The Joy of X A Guided Tour of Math, from One to Infinity by Steven Strogatz

             I read the book The Joy of X A Guided Tour of Math, from One to Infinity by Steven Strogatz. All topics of math are addressed in this book in a way that anyone can understand. You do not need to be a math major or a mathematician to understand and enjoy this novel.

            The book is split up into six parts: Numbers, Algebra, Shapes, Change, Data, and Frontiers. Numbers addresses negative numbers, multiplying versus addition, division, and representing numbers. The second part about algebra discusses how algebra involves variables, imaginary numbers, word problems, and functions. Next, the book talks about shapes, geometry, proofs, and pi. After Shapes, the fourth part is about change which includes calculus, differential equations, and vectors. The fifth part is Data which talks about statistics, probability, and linear algebra. The last part is Frontiers which covers number theory, group theory, topology, and spherical geometry.

            Although this may sound like too many topics to address in one book, the author does a nice job. First, the book starts with easier topics and then slowly moves to more challenging mathematics. Throughout each chapter, Steven Strogatz does an excellent job of giving examples for every topic. These examples explain how these complicated math concepts apply to everyone’s day to day life. Additionally, he gives many diagrams and visuals to help you further understand what he is explaining. In part six, for example, he describes how group theory can be explained through how you flip your mattress to make sure it wears evenly by giving visuals and easy solutions.

            The Joy of X is a great read for many audiences. Teachers could find examples that would help them explain to their students how the math they are learning in class applies to their real lives. Math majors receive an overview of many of the topics they have learned throughout their studies and they may find an explanation they had never heard before to find further clarification. Anyone who dislikes math can read this and learn how math is all around them. I think everyone will find more joy in math when they read this book.

            Personally, right from the beginning, the book captured my attention with the following quote, “math always involves both invention and discovery: we invent the concepts but discover the consequences” (5). I don’t think I have ever thought about math like this before. Mathematicians defined what numbers mean and what addition means, but have no way of controlling the results of what they have defined.

            I highly recommend you read this book. It is an easy read and will help you see math in a whole new way. You will finally understand the importance of what your teacher was teaching you in high school, or you will find the beauty of math and how it is found all around you. 

Nature of Mathematics- Is math a science?

          

            At first glance at the question, “Is math a science?’, I would say no. All throughout school I loved math class and disliked most science classes. So from that I do not want to say math is a science. Additionally, in every school math class and science class are separate subjects you must learn. However, that does not answer the question; that is just my gut reaction.

            Thinking about it further, I have still concluded that math is not a science. Math is used in science, but science does not have to be used in math. Science involves investigating and studying the physical and natural world while math uses procedures and operations to explore possible worlds.

            Some instances that science uses math include when scientists have to quantify an object’s mass or length. Scientists sometimes use mathematical data charts to organize their information and find patterns. Additionally, chemists use algebra to figure out the proportions of compounds to use to make a reaction. However, weighing and measuring objects, organizing information, and algebra are not strictly used for scientific information. Science borrows these tools and ideas from math. Without these tools, scientists would not be able to discover many things they now know.

            Math on the other hand does not ever need to use science. Math does not need to use real world examples to make discoveries. It can use possible worlds and be very abstract.

            I am not trying to say that there are no relationships between math and science. Both are very structured in the way new information is found. Mathematicians use proofs and scientists use the scientific method. However, proofs and the scientific method are not the same, but both are systematic. For proofs, definitions and axioms are used to create a logical argument that shows a statement is true. For the scientific method, you ask a question, do research, create a hypothesis, experiment, draw a conclusion, and then report results. These differences help me to think that math is not a science.

            I think that mathematics follows more rules and is more logical. Once something in math is proven, it will always be true. Scientists use experiments which can give varying results. It is not as black and white for scientists to be certain if the results of their experiments are true one hundred percent of the time.   
            In conclusion, I do not think math is a science. Math and science have many connections but I do not believe these connections are enough to say math is a science. I think math is used as a tool for science, but science does not need to be used in math. Math is so much more than the ways it is used in science which makes be believe it is not a science.

           Even though I think math is not a science, it is important to hear others points of view of the question and challenge what I think. Additionally, it is possible my students one day will ask me this and I will be open to hear their opinions and share mine with them.

Citation

http://mathforum.org/library/drmath/view/52363.html

Doing Math- Nets

In class on Wednesday, we created three nets that when put together made a cube. The three shapes you make are a triangular prism, a triangular pyramid, and a square pyramid. We discussed that the triangular prism makes up ½ of the volume of the cube. The square pyramid makes up 1/3 of the cube. Finally, the triangular pyramid makes up 1/6 of the cube.

During class all four members of our group tried to make our own nets. Towards the end of class we realized it would have been smarter for us to all work together because each of us were unsure what each of the nets should look like. I was able to complete the triangular prism and the square pyramid, but did not have time to complete the triangular pyramid.

Geometry is one of my favorite areas of math because I love visualizing different shapes in my head and it is fun translating that to paper. After class, I really wanted to make a new set of these nets so I could complete the cube. Also, I wanted to take a closer look at the measurements on the nets to get a better understanding of how they work and if each part actually is the right percentage of the volume.

First, I am going to make the triangular prism. To do this I visualized slicing a cube diagonally down the middle since is makes up 1/2 of the cube. I decided the dimensions of the cube would be (3cm)(3cm)(3cm). Thus, two sides of my triangular prism would be 3X3 cm. The other two sides would be these 3x3 sides cut in half to make two triangles. To find the hypotenuse of the triangles I used the Pythagorean Theorem as shown in the image below. Finally, I used the hypotenuse of the triangle to find the length of the rectangle. 

    Next I decided to make the square pyramid which makes up 1/3 of the cube. I knew I should start with another 3X3 square which makes up another face of the cube. From there, I realized I again had to cut this 3x3 square in half again to make two more triangles. Finally, I knew there were two more triangles that made up the square pyramid. I knew that one side would be the hypotenuse I found from the previous triangle- 4.2cm. Then, I used the Pythagorean Theorem again to find the hypotenuse for the next two triangles as shown below. 

     The last net I made was for the triangular pyramid which makes up 1/6 of the cube. This was the most challenging net for me to make. It should have been the easiest because I had the other two already made and I just had to fill in the blank, but for some reason I kept messing up when I was drawing it. 

      I noticed that this net contained two of the triangles made from cutting the 3x3 square in half. Then I used the previous nets I made to find the dimensions for the other two triangles. 

       I was very excited when I was able to put the three nets together and found that they did in fact form a cube. 

         Once I had formed all my nets and made the cube, I wanted to do the math to make sure that the triangular prism was infact 1/2 the volume of the cube, and the square pyramid was 1/3 of the cube, and the triangular pyramid was 1/6 of the cube. My math is shown below. 

          It turns out each of them gave the volume they were supposed to give. I was happy I was able to make my own set of these nets and gain a deeper understanding of how the shapes do in fact have the correct volumes. 

         For me, this activity was important because it made me wonder how and if I could incorporate creating geometric models into my classroom one day. I think this activity may be a little harder than sixth or seventh graders could handle, but I think they would learn a lot from creating other simpler nets in class. This would be a fun example to show them and challenge them once they have mastered simpler nets.

        Now, I want to see if I can take another shape, besides a cube, and cut it up into three new shapes. As many times as I have made nets, I have never thought about making nets that create shapes that can be put together.  

Nature of Mathematics- Euclid's Proofs

         Greek mathematicians, for example Euclid, were the first to use the concept of proof. They figured out that in order to show that a statement was true they must use logical arguments. Before Euclid, mathematicians such as the Egyptians and Babylonions “proved” statements from repeated observations and inductive reasoning. However, the Greek mathematicians realized that they needed to have logical explanations to prove that statements were true. These proofs needed to clearly show why the statement was true in a way that would last over time. In order for Euclid to use this new system, he had to create definitions and axioms that would be accepted by everyone. These were written and shared with others in “The Elements”. Some of the definitions in this book include point, line, and surface. “The Elements” also contain Euclid’s five postulates which he assumed to be true. Euclid used these definitions, axioms, and postulates to prove statements. When the statements had been proven, the statement became an accepted theorem. Most of Euclid’s proofs involved geometry, but ever since Euclid, all types of mathematics use his logical argument concept and create accepted definitions to prove statements.


                        

                                                              

            Euclid’s idea of proofs changed mathematics. He found a way to prove statements in a consistent way that is still accepted today. Euclid created the foundation for how mathematicians would solve proofs. Mathematicians start with a statement they would like to prove. In order to do this, they use the definitions and axioms to create a logical argument showing the statement is true. Once they have done this they have found a new theorem. Euclid changed the way mathematicians proved statements from a system of trial and error to forming logical explanations.

            No one has been able to disprove what Euclid found. He was a founder for Euclidean Geometry. However, after him, Non-Euclidean Geometry was discovered when you change one of Euclid’s main axioms or assumptions about parallel lines.

Citations

http://nrich.maths.org/5996

http://www.storyofmathematics.com/greek.html

http://www.biographyonline.net/scientists/euclid.html