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

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.

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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

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(

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).

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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).

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! ]]>

Math is argued to be something that one must wonder about, play with, and imagine to create anything he/she wants in the book

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

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

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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 (

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).

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.

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

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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.

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

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. ]]>