"The number of tenured jobs offered to women has fallen from 36% to 13%. Last year, only four of 32 tenured job openings were offered to women," stated Susan Goldenberg in her article posted in The Guardian. The article was about the president of Harvard University arguing that men out perform women in science and math because they are biologically different. Since the beginning of time, women have been discriminated against and told they are not as good as men in just about everything. Women in math especially is rare because people do not think women are as smart as men when it comes to mathematically thinking. However, there are many women throughout history that prove that theory wrong (https://www.theguardian.com/science/2005/jan/18/educationsgendergap.genderissues).

To start, one of the earliest woman mathematicians is Hypatia. She was the daughter of one of the last known members of the library of Alexandria, Theon. She studied mathematics and astronomy and worked with her dad on  translating the commentaries of mathematical works. She also created her own commentaries, and taught math from her home. Hypatia was also a philosopher and a follower of Neoplatonism, "a belief system in which everything emanates from the One." She also would give public lectures on Plato and Aristotle to many people who would gather to listen. Although she was a very popular woman figure, she became involved in a debate between the governor and the city's archbishop and ended up being killed by a mob of Christians (http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?page=5).

Another woman that made an influence in history was Sophie Germain. She started studying and reading at a young age, when the revolutions started to take off in Paris. The study of Archemeaies and his death is what started her interest in mathematics and geometry. Although she was not allowed to study at École Polytechnique because she was a girl, she found a way to get the lecture notes and would submit papers to a facility member, Joseph Lagrange, at the school under a fake name. Once he figured out she was not a man, Joseph became a mentor and Sophie started corresponding with other mathematicians. Her work was not as great as other men because of the lack of resources she had, yet she won an award from the French Academy of Sciences for her work on theory of elasticity, and a proof of Fermat's Last Theorem (http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?page=5).


Ada Lovelace was another influential woman in history. Ada grew up never knowing her father and her mother was very overprotective, but encouraged her study of math and science. When she grew up, she started corresponding with Charles Babbage, an inventor and a mathematician. He asked her to translate a memoir written in Italian that was analyzing what he called his Analytic Engine. The Analytic Engine was "a machine that would perform simple mathematical calculations and [could] be programmed with punchcards and is considered one of the first computers." Not only did Ada analyze the Analytic Engine, but she wrote her own notes and figured out how to calculate a sequence of Bernoulli Numbers which was known as the first computer program (http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?page=5).
 
Additionally, Sofia Kovalevskaya was from Russia where woman were not allowed to attend university. Therefore, she married a paleontologist, Vladimir Kovalevsky, and moved to Germany. When she arrived however, she was not able to attend lectures so instead she was privately tutored and finally received her doctorate after she wrote treaties on partial differential equations, abelian integrals and Saturn's rings. Kovalevsky ended up dying and Sofia became a lecturer in mathematics at the University of Stockholm. She also became the first woman in Germany to become a professor. She continued to make contributions in math and won the Prix Bordin in 1888 from the French Academy of Science, followed by an award the next year from the Swedish Academy of Sciences (http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?page=5). 

Finally, Emmy Noether was known as "the most significant creative mathematical genius thus far produced since the higher education of women began” by Albert Einstein. Emmy grew up in Germany, yet due to rules against woman in Universities, her mathematics education was delayed. She received her PhD for a dissertation on abstract algebra and got a position at the University for many years. People started to call her the "unofficial associate professor" at the University of Göttingen. However, she lost her job in 1933 because she was Jewish. Therefore, she moved to America and became a researcher and lecturer at Bryn Mawr College and at the Institute for Advanced Study in Princeton. This is where Emmy discovered some mathematical foundations for Einstein's general theory and made many contributions to algebra (http://www.smithsonianmag.com/science-nature/five-historic-female-mathematicians-you-should-know-100731927/?page=5).

Clearly women crucial to the development of mathematics and it is sad that they are being unrecognized still today. However, the number of women receiving doctorates in mathematics is rising according to the article "Has the Women-in-Mathematics Problem Been Solved?" The article states that over 50% of doctorates in the math department were given to women at Dartmouth College throughout the past eight years. It also talks about a woman named Carolyn Gordon, who is the president of the Association for Women in Mathematics. She creates atmospheres where women in math is normal so that everyone feels equal in the classroom. The reputation for women in mathematics has changed drastically in the past twenty five years and the most visible problems have been eliminated. The only problems now, which are harder to see and are being over looked are not easy to fix or straightforward. The article states that this is the reason that the issue has not been addressed. Things may have been improved from previous years, however since the problem has been severely fixed and the new problems are not staring you in the face, society does not know how to handle it, thus ignoring it all together (http://www.ams.org/notices/200407/comm-women.pdf). 

In my opinion, I had never noticed that women were being over looked in the math department until it was brought up in class. Of course I had known about women's rights in the past, yet I had no idea that women in the mathematics department was still an issue. However, looking over data the results shocked me to say the least. I feel very fortunate to be receiving the degree in mathematics that I am, and not being told that I cannot do something because I am a girl. 

In class we have been discussing the work of Euler, and all his contributions to mathematics. While I had heard of Euler, and learned about topics that he discovered, I never knew anything about him or how much he contributed until now. 

Leonhard Euler was not only a mathematician, but a physicist, astronomer, logician and an engineer whom made important strides in calculus, graph theory and number theory. He was born in Basel, Switzerland on April 15, 1707. Soon after, his family moved to a town called Riehen, where his father befriended the Bernoulli family, who ultimately had a large influence on Euler's work. At the age of only thirteen, Euler attended the University of Basel and graduated three years later with a masters in Philosophy. In 1727 Euler entered the Paris Academy Prize Problem where he had to find the best way to put a mast onto a ship. He ended up taking second place to Pierre Bouguer that year, however entered again and won twelve more times (https://en.wikipedia.org/wiki/Leonhard_Euler). 

Euler worked in almost all areas of mathematics including geometry, physics, algebra, calculus, trigonometry and number theory. His works ended up occupying 60-80 quarto volumes, which is more than any other mathematician. He is also the only mathematician in history to have not one, but two numbers named after him. The first number, e, which is widely used in calculus and approximately equal to 2.71828. The other is called "Euler-Mascheroni constant Y (gamma) and is approximately equal to .57721, however it is not known if gamma is rational or irrational (https://en.wikipedia.org/wiki/Leonhard_Euler).

Mathematical notation is something that Euler made popular through his many textbooks as well. He was the first to denote the concept of a function, and write it as f(x) to denote the function f to the argument x. He also introduced the Greek letter Σ for adding sums, the letter i to denote imaginary numbers and finally he popularized the Greek letter π which denotes "the ratio of a circle's circumference to its diameter." While many of Euler's proofs in calculus were not accepted, he made advancements in the development of the power series, and expressions of functions as the sum of infinitely many terms. He directly proved the power series for e, ​and the inverse tangent function which led him to solve the Basel problem in 1735. From here, he was able to use exponential functions and logarithms in analytic proofs and discover various logarithm functions using the power series. Euler was also able to define logarithms for negatives complex numbers, and the exponential function for complex numbers, which in turn he found related to trigonometric functions. He then came up with what is now known as Euler's Identity, aka "the most remarkable formula in mathematics" according to Richard P. Feynman. He called it this because of all the single notations of addition, multiplication, exponentiation, equality and important constants (https://en.wikipedia.org/wiki/Leonhard_Euler). 

Another idea that Euler developed was the idea of number theory. Although he did not come up with number theory himself, he based his ideas of off the work of Pierre de Fermat and disproved some of his conjectures. Euler proved "the sum of the reciprocals of the primes diverges" and from here saw the connection between the Riemann zeta function and prime numbers. He also "proved Newton's Identities, Fermat's little theorem and Fermat's theorem on sums of two squares." He made contributions to  Lagrange's four-square theorem and invented the toilet function. The toilet function is denoted as φ(n), and is "the number of positive integers less than or equal to the integer n that are coprime to n." From this function, he was able to generalize ideas from Fermat's little theorem and created what we now know as Euler's theorem. Finally, he further proved, from Euclid, the relationship between Mersenne primes and perfect numbers was a one to one function (https://en.wikipedia.org/wiki/Leonhard_Euler).

Graph theory was yet another mathematical concept that Euler helped develop throughout history. This began when he presented a problem known as the Seven Bridges of Königsberg and its solution. He realized that there is no way to cross all seven bridges only one time and end up in the same place one started, therefore there was no Eulerian circuit. He also discovered the formula V-E+F=2, relating to the number of vertices, faces, and edges there are in a convex polyhedron (https://en.wikipedia.org/wiki/Leonhard_Euler).

Along with all of the contributions mentioned above, he also contributed to applied mathematics, music, physics and astronomy. Euler was by far one of the greatest mathematicians of all time and without his work, mathematics and many other subjects would not be the same. I am happy to have learned so much about what he contributed to that has made my math experience so much better (https://en.wikipedia.org/wiki/Leonhard_Euler). 

Although many people believe that doing tessellations is not math, it most certainly is and it is a very important skill to teach all ages k-12. Since tessellations do not depend on numerical skills, they can be created by students with varying mathematical skill levels. Tessellations introduce problem solving and many different mathematical concepts (http://www2.gvsu.edu/oxfordj/teach.html).

Tessellations can be used as a counting activity for the younger students. The students could count how many different shapes are in the tessellation, they could also count the number of a certain shape are in a row or column, they could count the amount of times the pattern repeats, and they could determine how many tiles are colored the same. Tessellations could also be used in geometry lessons to make new shapes. The students could combine tiles to make new geometric figures, draw out the new figure they discovered and figure out what it is called and its properties. Finally, tessellations are also an awesome way to teach the concepts of rotation and reflection. The students could determine the original shape or object, and figure out if it was rotated or reflected to make the next shape in the tessellation (http://www2.gvsu.edu/oxfordj/teach.html).

Tessellations were first used by the Sumerians in about 4000BC for wall decorations formed by clay tiles. One of the first studies of tessellations was done by Johannes Kepler in 1619. He wrote a book called "Harmonices Mundi" about harmony and congruence in geometric and physical forms. Within the book he wrote about regular and semi-regular tessellations by exploring the structures of honeycomb and snowflakes. Finally, in 1891 Yevgraf Fyodorov, a Russian crystallographer, discovered that all the periodic tilings in a plane contain one of the seventeen groups of isometries. This became one of the reasons mathematicians started to study tessellations (https://en.wikipedia.org/wiki/Tessellation#History). 

In class, we looked at many different types of tessellations, some easier and some more complex. I did one that contained three shapes and three colors, that was not hard to figure out and replicate. However I also completed another one that took a little more thinking to figure out how to make the pieces fit together to make a tessellation. 

The first one I created is featured to the right. I started out with the hexagon and realized I could put a triangle on each edge. From here I noticed that a rhombus fit in-between two of the triangles. I then noticed that this figure just formed another hexagon, so from there I tried to create another larger hexagon with the same pattern as before. I also noticed that if I kept repeating the pattern, I would eventually keep creating larger and larger hexagons. Also, if I were to only have one of the smaller hexagons, and cut it in half, it is symmetrical, and the entire large hexagon is symmetrical as well. This tessellations could especially be used in a younger elementary classroom to explain how certain shapes can make other shapes.
 
The second tessellation, to the left, I created was harder to figure out the pattern and where I had to stop the tiling, in order to be able to repeat the pattern. I started with four green triangles in the middle, all meeting at their points. I then realized I could put an orange square on the "bottom" of the triangle. I also noticed that the white diamond fit perfectly in-between the four triangles, and from there two more triangles could be put in-between the white diamond and the squares. I noticed this was a 12 sided polygon (also known as a dodecagon). However repeating the dodecagon was not going to fill all the gaps. Therefore, on the top right corner of each dodecagon, I placed a square that touched corners with the top square, and filled in the edges of the square with green triangles. From here, the pattern could be repeated to create a tessellation. This tessellation could be used in a classroom to help teach the ratio, for example what the ratio is of each color the the others (white to orange to green). 
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All in all, not only are tessellations fun, but they are also very informative and a great lesson to teach all ages. It is a way to get the students thinking and problem solving, in a way where they do not think they are actually "doing" math. I will for sure incorporate different tessellation lessons throughout my teaching career!

While my blog post 0 focused on what our class thought the definition of math was, and what my views on math are, the book Mathematician's Lament has changed my opinion not only on the definition of math, but on the ways math should be taught. Many people believe that math is a set of formulas and algorithms used to solve problems. Our class decided that math was a language composed of numbers and letters used to communicate and problem solve. I discussed in my previous post that math was a reward. However, Paul Lockhart discusses that math is an art.

Math is argued to be something that one must wonder about, play with, and imagine to create anything he/she wants in the book Mathematician's Lament. The book uses an example about a triangle that fits into a rectangular box. It states that one can make this triangle as big as he/she pleases and the edges can be perfect because it is what he/she imagined. Although this person comes up with the triangle and rectangle, he/she cannot choose the amount of space the triangle takes up within the rectangle. Therefore, using math to try and figure it out. Paul Lockhart discusses how he used thinking and trial and error to decide to put a line down the middle of the triangle. He realized that there was the same amount of space inside the triangle as well as outside of the triangle, meaning that the triangle took up half of the rectangle. Because Lockhart had had experience with mathematics and has engaged himself with different ideas about problems, he was able to think of drawing the line to solve the problem. 

Students nowadays in math classes are being deprived from imagination and creative thinking, leaving no room for engagement with the subject. Teachers are told to have students memorize formulas and apply them to problems over and over. The mathematics education as is needs to be scrapped and rebuilt into something new, rather than just coming up with new texts books to clarify material. Instead of focusing on what math really is, teachers are more focused what material to teach first and what notations to use to meet criteria. Teachers are always trying to relate mathematics to everyday life, when instead it should not be, it should be a fantasy that relieves children of his/her everyday life. 

The way Mathematician's Lament discusses how math should be taught is by teaching its history first. It states that no other subject is taught without background information, where math throws out formulas and algorithms to memorize without any back story. It also discusses how it is okay to have formulas and algorithms, however being able to think creatively is a large part of math that should be brought back to the classroom. Students should struggle and be frustrated with a problem, this is when he/she comes up with new ideas that lead to other questions about the problem. If a student is really struggling, that is when a teacher should help, but not too much, only enough to make the student think about new ideas regarding the problem. Teachers should give students problems that are suitable to the students level of experience and give he/she time to come up with conjectures of his/her own. A good math classroom is one where healthy criticism is given, and teachers are flexible as well as opens to change in curiosity. Finally, math should be taught by solving puzzles, playing games and problem solving. Students should be put into a situation where deductive reasoning is necessary and creative thinking is involved. 

After finishing the book however, I realized why people may be skeptical about why Paul's ideas may be a little far off. While I do believe that the math department should reconsider the way they are teaching mathematics, there is still a curriculum that needs to be followed. I do believe that teachers could switch up the way they are teaching the curriculum by doing fun lesson plans, playing games, doing puzzles, and having math be more hands on while still covering topics needed. For example, the book discusses how geometry completely kills mathematics and makes students hate it. Instead of learning formulas to solve problems about shapes, teachers could provide 3D manipulative for the students to play around with, then provide the formulas for the students to discover about the shapes. There are many ways to keep the curriculum in tact, and still make learning math fun. 

All in all, math is not a language that can be learned, it is an adventure. Math is an art done for pleasure, it is not memorizing formulas to complete problems out of a book. Teachers are more concerned with following the curriculum and having the students memorizing algorithms to solve problems, that math is essentially being taken away. Students do not have room to critically think, instead he/she applies a formula to solve the problem and moves on to the next. Students should be able to think creatively and independently and not be "trained" to do something. The book Mathematician's Lament does an amazing job discussing what math really is, and how it should be taught in k-12 classrooms. 

Everyone in the math world has heard of the pythagorean theorem, however not everyone knows where it came from and who Pythagoras was. Although there is very little about his mathematical achievements, there are many biographies written about Pythagoras written by authors who believed he was a god-like figure, and most refer to him as the first "true" mathematician. Some people believe the information in these biographies are legends, while some think it is important information about the past. 

Pythagoras was influenced by both Thales and his acquaintance Anaximander. While Thales was old when Pythagoras visited him, and did not teach him much about mathematics, he interested Pythagoras in mathematics and astronomy and suggested he travel to Egypt to learn more. Pythagoras also attended Anaximander's lectures on Miletus, and learned that Anaximander was interested in geometry and cosmology which influenced Pythagoras' ideas about mathematics as well. 

Pythagoras established a school in Croton in 530 BCE which had strict (and odd) rules to all members including being a vegetarian, never urinating towards the sun, and never eating black fava beans. The members of his school were divided up into two groups called the "mathematikoi" and the "akousmatikoi." The mathematikoi were considered the learners, these members continued Pythagoras' mathematical and scientific work. The akousmatikoi were the learners, these members focused on the ritual and religious aspects of the teachings. 

The motto of Pythagoras' school was "All is number." He believed that each number had its own meaning, with ten being the "holiest." This was because ten could be derived from the sum of one, two, three and four, which was a large contribution to Pythagoras' intellectual achievements. From here, he figured out that a system of mathematics could be created from geometric elements corresponding with numbers, and integers and their ratios. These were all the parts needed to "establish an entire system of logic and truth."

From here, the Pythagorean Theorem, "for any right-angled triangle, the square of the lengths of the hypotenuse is equal to the sum of the square of the other two sides (a^2 + b^2 = c^2)." The most common triangle is one with the sides of length 3, 4 and 5 that can be drawn with squares as units on each side (drawn below). However, there are infinitely integers known as "Pythagorean triples" that can be used in the Pythagorean Theorem. 

Along with the Pythagorean Theorem, Pythagoras and his followers also realized that if you take the sum of all the angles in a triangle, the result will always be 180 degrees. He also came up with the idea of number theory, and many different properties of square numbers. Such that if you take a number n and square it, it equals the sum of the first n  odd numbers. Finally, they discovered the first pair of amicable numbers (220 and 284). 
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Although there are many mathematicians who have made an impact on mathematics today, Pythagoras and his contributions of right angled triangles seem to be the influential. Mathematicians have since expanded on Pythagoras' ideas and they are used everyday life. For example, the ideas that Pythagoras introduced can be used to answer numerous questions about a baseball diamond (how far does a catcher need to throw the ball to get from home to second base?). Also, it is used in for contractors to build a layout for houses when they do not know the lengths of a side of a room. Therefore, we should always be thinking about the ideas that Pythagoras introduced. 
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https://en.wikipedia.org/wiki/Pythagoras
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html
http://www.instructables.com/id/Pythagorean-Theorem/ (Picture cite).
​http://www.slideshare.net/992751/pythagorass-effect-on-our-world-today-presentation

In class on May 10th, our professor asked us the question "what is math?" Although we are all math majors and have taken almost all of the same math classes, not one of us answered the same. However, we all did describe math in different ways such as "math is a language used to communicate and problem solve," "math is a way to describe the world using numbers and logic," and "math is creative problem solving  with numbers, letters and symbols." To me, math is all of these things plus more. Math is something that you cannot just memorize like other subjects, there are many different ways to solve one problem. Also, whether you know it or not, you are using math every day of your life. People always ask me what my major is and when I respond with "math," the reaction never fails to be "ew! Why math?!" People have such a negative outlook on math because it can be a very difficult subject to succeed in. However, this is what makes it fun for me, being able to problem solve and figure out a hard question is so rewarding. Therefore, to me math is a reward. 

Unfortunately, I do not know a lot about the history of mathematics, yet I am excited to learn this semester. However, I do know a few names that have had remarkable effects in the history of math such as Euclid, Euler, Pythagorean, and Fibonacci. Each of these people contributed to math in different ways. For example Euclid gave us Euclidean geometry. Euler was important in the discovery of graph theory, Pythagorean gave us the Pythagorean theorem, and Fibonacci discovered the Fibonacci numbers. Although I do not know much, I know these are very important figures throughout the history of math. I am excited to learn more about the history and discover how math came to be. ​