For this weeks post I wanted to do something of my own. I work at the YMCA for the summer day camp program as one of the teachers/counselors. This past week we did an activity at camp that I thought was awesome and wanted to share. I know the students were doing math without knowing it in this activity and I wish I would've had time to talk to them more about it. Any way here is what we did.

These students were so excited and proud of their structure, and I think they ended up being our winners. I was very proud! I talked to one camper after in this group and asked him how he came up with this idea. He told me "the bottom triangle thing was very strong. The marshmallow kept falling over at first when we got it so then we used the string to hold it up!"

This next group used a cube as their base. Keep in mind that we never told the students to create a shape or anything. This was all their own creativity!
They next used the remaining spaghetti to create a pyramid figure on top of the cube to hold up their marshmallow. 
I think they also ended up taping the string to the table and tying it to the top for more stability. This structure was a little wobbly, but it still held up the marshmallow in the end.
These girls worked great together and told me they "really like cubes." I'm not sure where their love for cubes came from, but I thought it was quite hilarious and cute.
I was very impressed by the construction of this figure as well. These campers are all-stars! 
​

Lastly, we have a structure that could not hold up the marshmallow at first, but in the end it survived. These students used only 3 spaghettis for the original base of their structure. They then taped the diagonal 2 spaghettis to the top of this base. When these students got their marshmallow they could not get it to stand up. But one of them brilliantly decided to use the string as a stabilizer on the bottom of the structure. Some how tying all these spaghettis together in a knotted mess provided enough structure for the marshmallow to stay. 
I thought this was awesome. These kids had to problem solve to find out how to fix their falling marshmallow and they were able to persevere and find a solution. 

I cannot tell you how fun it was as a math educator to watch this process of building a marshmallow tower. These students really blew my mind and put their brains together to come up with some awesome stuff!
There were 2 groups I wasn't able to picture because their structures didn't make it very far. The first had a problem with an angered student continuously knocking down the structure whenever it wasn't working out. The other group made a very very tall spaghetti tower that was just a single one sticking out of the ground taped to more as it builded up. As you might guess this tower didn't make it very far. 
I think it was exciting for the students to have free reign on building their spaghetti towers. Often at camp we like to provide a ton of structure throughout the day with activities and rules for safety and such. This activity gave the students freedom to use their creativity and explore their strengths. I think that children crave these opportunities and this is why they got so engaged in the activity. 
I think this activity could be extended by providing either more materials or less materials. The marshmallow could also be substituted with a heavier object to provide for a much more challenging task.
Anyway thanks for reading about the math we did at the YMCA. It was a ton of fun and I was glad to be a part of this activity at camp. 

There is a certain stigma about women and math that has always made me a little angry. Some people believe that women just can't be good at math or that math is a man's expertise. I experienced this stereotype when I was young. When someone would ask me my favorite subject, I always would say I love math. I was a natural at it and numbers made sense to me. I excelled at math growing up. One boy's response to math being my favorite subject was, "But you're a GIRL!" I was not sure how to respond to that, so I just walked away very confused. I will never forget this conversation, and I will never forget how it bothered me.

So to show that woman can indeed do math I decided to do a little research on successful women mathematicians. Turns out there are tons of girls that love math, and many that are pretty dang good at it. 

This here is Hedy Lamarr. This woman has many talents. She is a great actress, but she also was pretty math savvy. Womansinventors.com says "Austrian actress Hedy Lamarr (born Hedwig Eva Maria Kiesler) also became a pioneer in the field of wireless communications following her emigration to the United States. The international beauty icon, along with co-inventor George Anthiel, developed a "Secret Communications System" to help combat the Nazis in World War II. By manipulating radio frequencies at irregular intervals between transmission and reception, the invention formed an unbreakable code to prevent classified messages from being intercepted by enemy personnel." This incredible invention patented in 1941 eventually became the technical backbone for cellphones, fax machines, bluetooth technologies, and other communications devices. The funny thing is that she hardly got recognized for her amazing invention until 1997 when she received the Electronic Frontier Foundation (EFF) Pioneer Award, over 50 years later!

If we go back in time to the pioneers of math we find Hypatia, a Greek mathematician. This incredible woman became head of the Platonist school at Alexandria in about 400 AD. There she lectured on mathematics and philosophy. Unfortunately, many Christians saw Hypatia's work with the sciences as paganism and she was brutally murdered for it by Nitrian monks. Hypatia is recognized as one of the first women to publicly make an impact on the development of mathematics. 

http://www.women-inventors.com/Hedy-Lammar.asp
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hypatia.html

Another great mathematician is Emmy Noether. Noether is known for her contributions to abstract algebra and theoretical physics. She developed the theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws. Noether also faced certain persecutions being a Jewish woman trying to get a good education in Germany. Later, she went on to be a teacher and professor. This woman truly was incredible and I hope to study her more in the future. 

https://www.agnesscott.edu/lriddle/women/noether.htm

I could go on and on finding wonderful women that contributed to the study of mathematics, but I think I will stop here for now. I'm glad that it really isn't hard at all to prove that women are great at math! I appreciate the work that all mathematicians have done in the field, both men and women!
As for me and my math story I couldn't be more proud of my progress. If someone told me that I would be a math major when I was in middle school or even high school I probably would not have believed them, but my natural ability for math and my interest in numbers has gotten me a long way. I have one more semester of math classes (3 more) before I can complete that part of my degree. Ever sense starting college I have never felt looked down upon for my ability in math because I am a girl. I do often ask a lot of questions though, and I might be the annoying kid who doesn't get it at first sometimes. Nonetheless, I am glad I pursued a career in math education and I cannot wait to start teaching some children!

    I did not know what to expect when I picked up the book titled Love and Math. Of course, I was hoping for an odd mixture of math and romance because I'm a softy and I love a good chick flick. And oddly enough this book did tell a story about math and love. Not a love between two people, but a man's love for mathematics. 
    Overall, I thought this book had good insight for a future teacher of math. I had never read a book like this one before, and I think it is pretty unique. This book takes you through the story of Edward Frenkel's discovery of his love for math and explains his struggles with becoming a mathematician and antisemitism in the Soviet Union. I enjoyed reading through the narrative about Edward and his adventures through college and beyond, and the people that had an influence on his math journey. However, Frenkel tries to explain some pretty complex math in the book including symmetry groups, and other modern math ideas. Many of these explanations went way over my head, because I had not done math like this before. It was frustrating to feel like however many times I read over a section, I wasn't going to get the math being explained. I think the author's goal was to explain these ideas in ways that a person with no math background could understand, but even I, a math major, had a hard time with most of them. He did give it a good try though, and I don't think I could've done a better job. I think his explanations were most understandable when he used analogies and used regular objects to describe math related things. For example, he explained math as the Kingdom of Mathematics where different animals make up different parts of math like algebra or geometry and these animals are related in different ways. 
    Something that I appreciated about the book is that it opened my eyes to the hardship that Jew's went through in the Soviet Union. Frenkel was denied acceptance to a college simply because of his jewish heritage. It is crazy to me that people can be so prejudice against people different from them. I am glad that we have come a long way, but I am not ignorant and I know that discrimination still exists. It is important that I am aware of this and can strive for inclusion in every aspect of my own life.
    Another thing that I liked was that Frenkel really appreciated the work of teachers. There was one quote from chapter 11 that impacted me in this area. Frenkel says
"Now that I’ve had students of my own, I appreciate even more what Borya has done for me (and what Evgeny Evgenievich and Fuchs did for me earlier). It’s hard work being a teacher! I guess in many ways it’s like having children. You have to sacrifice a lot, not asking for anything in return. Of course, the rewards can also be tremendous. But how do you decide in which direction to point students, when to give them a helping hand and when to throw them in deep waters and let them learn to swim on their own? This is art. No one can teach you how to do this."
I really like how he explains being a teacher as hard, yet rewarding work. This is basically how I view teaching. I know that it will be hard work, and I have experienced a glimpse of that so far, but I also know that I can have a great impact on my students and I can see them grow and learn. I know that watching them learn will be extremely valuable and rewarding. 
    Reading this book also reminded me that my students will have a wide variety of strengths and interests when it comes to math. Edward was not interested in math at all until a mathematician introduced him to ideas of modern math like symmetry groups, but once he found the aspect of math that he loved he went to great lengths to learn it. He even scaled walls to get into math lectures at a college! It is important to remember that some students will love physics, some algebra, some number operations, and/or algebra. It is my job to help them discover and pursue the areas that they find interest.
    Lastly, Love and Math showed how important collaboration is when it comes to math. Many of the authors projects included help from other professors, scholars, and peers. Math isn't meant to do all on your own and I think that collaborating with others is a great way to be able to work through problems. Frenkel also talks about the value in making mistakes. He explains math as the "productive struggle" and "making mistakes in the right direction." This view of math makes it more enjoyable and less restricting.
    I would recommend this book to other teachers interested in math. The book definitely has good insight for teachers and hopefully other math people would have better luck understanding the math that Frenkel tries to explain. I appreciated his efforts, though. Maybe when I finish up my math classes I'll have a better chance of understanding some of the complex concepts in the book. For now I will stick to novels about real love stories ;). 

I did a little research on Omar Khayyam this weekend. I had never heard of this man before our class and I was very interested in his work with cubic equations that we talked about. Khayyam did some awesome work with cubic equations by using geometry in his book Treatise on Demonstration of Problems of Algebra (1070). He also wrote about a lot of other mathematical discoveries like the triangular array of binomial coefficients, known to us as Pascal's triangle. (Pascal's book on the triangle wasn't published until 1665. So, why in the world is it named after Pascal? Something to think about…) 
Here is a quote that Professor Golden commented :
​“When I want to understand what is happening today or try to decide what will happen tomorrow, I look back.”
― Omar Khayyám

Anyway, Khayyam was able to create a parabola and circle and find solutions of cubic equations using the intersection of the two. And here is whats really crazy: He did all this before we had graphing calculators and computers and all that!

Below is a neat animation from Geogebra that illustrates Khayyam's use of circles and parabolas. 
The animation is from https://tube.geogebra.org/m/rTV3y4Bb and gives the description:
"Khayyam showed that the solution could be found from the intersection point of a parabola with a semi-circle. By varying the scale of the parabola and the diameter of the circle you can solve the cubic equation for any positive value of b and c."

Using the red diamonds A and D the equation can be changed to find the solutions to different cubic equations.

https://tube.geogebra.org/m/rTV3y4Bb
​
Khayyam was clearly a very intelligent man. It totally blows my mind that he did all this graphing before any type of graphing technology was invented. 

Well the binomial theorem shows what happens when we multiply a binomial by itself n times. A binomial is a polynomial with two terms, or two terms added like 2x-6 or 7x+3x^2. 
If we want to see how the binomial theorem works we can try something like:

When we expand this equation we get:

Tada! We just used the binomial theorem, heres how:  The binomial theorem tells us that the coefficients for each term will end up following the corresponding row of pascals triangle. Here we use row 6 because we are expanding (a + b) to the 6th power. 

Furthermore we can see that the exponent for a counts down from 6 for each term and the exponent for b counts up from 0 to 6 for each term. Each terms exponents will sum to 6. ​

You can use this expansion rule for any positive integer n, or you can use the equation for the binomial theorem which is:

Now I don't know about you, but I certainly am pretty impressed by Omar Khayyam. Remember that he not only discovered the whole graphing deal with parabolas and circles and cubic equations, but he also had all this binomial theorem stuff worked out. Khayyam certainly knew what he was doing when it came to mathematics, and he doesn't seem to get too much credit. I mean I didn't hear anything about him until just a few weeks ago! More people should know about the great mathematician Omar Khayyam. 

One property was "Positive or negative numbers when divided by zero is a fraction the zero as a denominator." As we discussed this property as a class we immediately agreed that a number divided by zero is undefined and cannot simply be written as a fraction. In the modern math world we do not accept dividing by zero. 
This, however, led us to another questionable property. Brahmagupta said that "zero divided by zero is zero." At first I wanted to agree to this and it led to an interesting discussion. After talking about this as a group, I eventually came to the conclusion that zero divided by zero must be undefined because you can divide something by 0 as many times as you want and it will stay the same. 
After talking about these properties of zero and positive and negative numbers Professor Golden brought up the common question: why is a negative times a negative a positive number? This proved to be a very challenging question to answer, and although Brahmagupta wrote it down as a property, he did not explain why. Our class began racking our brains to explain this phenomenon. ​
The explanation that made the most sense to me used a number line. My colleagues pointed out that multiplication is like jumping along a number line. A negative changes the direction, so a positive times a negative will cause jumping in the negative direction. A negative times a negative will then change the direction again to cause jumping in the positive direction. This can be seen below with 4 times 2. Positive 4 times positive 2 is represented by the pencil in the positive direction. Positive 4 times a negative 2 is blue ink in the negative direction and negative 4 times negative 2 is the green ink in the positive direction. As you can see the negative signs are what change the direction on the number line. 

Brahmagupta clearly made mathematical advances that were important in his day, but I am glad that we have come a long way with understanding zero since then. Hooray for modern mathematics! 

I am glad that we talked about tessellations in class. I love both math and art so when the two subjects are put together I get really excited! I taught a lesson on tessellations in the fall at West Michigan Academy of the Arts and Academics (WMAAA) for Professor Golden's MTH 322 class. In the lesson we taught the students about different transformations that can be done on objects and how to create their own tessellation.
​ Below Is an example of the tessellation that one of the students successfully created. 

In class on Thursday we were given the opportunity to create our own tessellations. I originally started with pattern blocks and created an image with a hexagon in the middle and built out from the middle. It looked something like the picture below. Soon Professor Golden came by and reminded us that we should be making tessellations, and although our designs were neat, they were not a repeated pattern over a plane. A tessellation should look like a geometric shape that can be repeated consistently over a plane to create a pattern. 

So then I decided to take a new approach. I got some of the isometric cube paper and tried drawing out some shapes with that. I ended up with a neat pointed arrow shape that I noticed repeated through slides. I decided I wanted to use that shape and make my tessellation more interesting. So I took this original shape and rotated it to create a pinwheel shape. I noticed that the pinwheel would also tessellate across a plane with 3 rhombuses added to one side. I colored the shapes in and ended up with a pretty decoration. I really liked this little project and I am proud of my final tessellation. 

I definitely think that students can benefit from doing a project like this in class. Students can learn transformations like rotation, translation, and reflection. Students can also find out what polygons can tessellate and explore with different geometric shapes. I think that this is a great lesson for a geometry class, and can help reach the students that are interested in art.

In our first class we had an interesting conversation to answer the question "What is Math?" Surprisingly, in a classroom of students with many math classes under their belt this proved to be a difficult question. Math can be defined in many different ways and is a pretty broad concept. Despite these challenges we were able to come up with a few different responses. A few of them include "Math is a language 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/or symbols." Personally, I think there is truth to all of these simple definitions of math. Throughout my life I have found math as something in school that I could excel at. I have used math as a way to solve problems, analyze patterns, and play with numbers. I think math is interesting, challenging, and often fun. However, many people don't see math this way. Many people ask me why in the world I would choose Math as a major, or why I even find it interesting. I think many people have had poor past experiences with math and it is portrayed with a negative connotation. I am glad to be someone who enjoys math amongst many who don't and I think it is great to take math classes with other people who excel and enjoy math. I hope that in my classroom someday I can help my students see the practical important uses of math, and that they can start to appreciate math if they don't already. 

As far as math history I am sad to say that I really don't know much. I hope to learn more about the history of math in this class. One event that I am familiar with is the discovery of the Pythagorean Theorem by Pythagoras the Greek Mathematician. I think Pythagoras was credited with this because he was the first to write a proof about it, but it may have been discovered before his time, and/or by other people in different parts of the world. I also know that Albert Einstein was responsible for some important equations and ideas about math including E=MC^2. Like I said, I hope to learn more about math history in this class because I think it is important that I know more than I do now for the sake of my future teaching and future students.