Weekly 3: Nature of Mathematics

The Role of our Number System..in Education

Ever since the beginning of our time, numbers have been another lesson that is taught to us during our developing years.  Numbers are included in early development as a milestone of understanding.  First, we learn to verbalize numbers while slowly developing the meaning.  This order relates to how our brain develops.  The ability to speak will come before the ability to understand.  Where did this understanding come from?  There has to be teachers before there are learners. Where did this system originate and why has it become a norm for education?  This is the topic I want to explore.  I want to discover the purpose of the number system in education.  From a future teacher perspective, I learned the role of our number system through other teachers who insisted its importance.  What benefits does knowing the number system give to students?  Finally, what is important for me as a teacher to educate myself with?

The 'Number System' can included many different aspects.  In the most broad sense possible, it can be summed up in the ways that all people reason about numbers.  For each person, the definition will be different.  Not only does each person have a different definition, but regionally, culturally and historically, the definitions have changed.  Here in the United States, we commonly see numbers in our speech and learning as one, two, three....(1, 2, 3.....).  We have both a numerical representation as well as a visual representation.  So, in teaching, the introductory grades are just learning to incorporate this language into their vocabulary.  As their fine motor skills develop, they begin to write their visual representation.  Then, as students grow and develop, meaning starts to develop.  Students begin to incorporate numbers with meaning.  This knowledge is brought into everyday life.  The connection between learning and using in crucial.  Students must be able to make the connection between what they are learning and what type of benefit it provides them.  In education, is important for learning to be purposeful.  If your students cannot connect with the material presented, then it will not become an integrative part of their everyday life.  As a teacher, this learning needs to be meaningful and easily applicable.  So, making that connection for students is absolutely crucial.

However, our number system did not just appear.  It took many centuries from its development to become what it is today.  So, the historical aspect is not to be neglected.  Students need to understand the different steps it went through.  Knowing this aspect makes the content seem more deeply rooted.  Rich history can amplify a lesson for students.  We know that counting in its simplest form can start with tallying.  Students can think like the earliest counters by using this method.  Then from there, we can dig into how other cultures used their earliest counting methods.  This learning can create links to history and/or geography.  There are so many ways to integrate the learning of the number system into education.  The importance must be emphasized over and over so that students can deepen their comfort with these topics.  After all, counting, reasoning, and numbers are everywhere.  There is not an aspect of life that does not utilize our number system.  As teachers, we must educate our newest learners into number system geniuses.

All in all, teachers have already been educated about the importance of the number system.  We see its importance and the role it plays in society so we teach it to our students.  Then, students just see numbers.  I think that if we introduce the role of numbers in their most historical form we would see a deeper appreciation.  It should not take a college degree for us to see where numbers came from.  If we educate students in the proper order we might see a stronger response to the number system.  I understand that students cannot be so overwhelmed with the history of every aspect of math, but we can prioritize.  The number system is going to be a part of their lives forever.  Lets teach them right.

Weibull, Nikolai. "A Historical Survey of Number Systems." Math Chalmers. N.p., n.d. Web. 25 May 2014. <http://www.math.chalmers.se/Math/Grundutb/GU/MAN250/S04/Number_Systems.pdf>. 

Weekly 2: Doing Math

Working with Tessellations

Tessellations can be defined as a repeated pattern on a flat that uses geometric shapes with no spaces or gaps. When I think of ones that I see in everyday life, they are present on floors, wall tiling, and tapestry.  There are patterns present everywhere.  The more intricate the pattern, the more angles and shapes become a factor.  The way the shapes interact is absolutely fascinating.  Below I used GeoGebra to test these interactions.  For the most part, I used circles (midpoints) and lines to create grids/boxes. 

This was the starting shape to
the first tessellation.

When overlapped, this pattern is created.  

For the second pattern, I used the same size circles, but with different frequency and placement.
It's amazing how similar, yet different the two look. 

This is what one segment of the pattern looks like blown up.
 It appears to be composed of individual curved shapes
and not the product of circles and lines.  

Next, I put lines through every vertex in every direction.  The lines created a horizontal and vertical grid with their diagonals. The new shapes that were created make this pattern more dynamic and visually appealing.

Tessellations introduce and explore many parts of mathematics.  Students can work with these at many different levels.  An important lesson that students can learn is how shapes can interact.  In the examples above, we could see that overlapping circles can create ovals that make a floral pattern.  Then, moving to the next example, we can see that a slight shift in position can create a pattern completely unlike the rest.  Students can be creative and make discoveries on their own. They can answer questions like: Is this a linear pattern?  What happens if one more shape is added? How could lines be utilized?  What if only circles or squares were used?  Can you make different shapes from the beginning?  What types of ideas have you explored?

These patterns provide an inside look at the role of angles, lines, and circles.  When simple shapes are put together and overlapped in a repeating patterns, it appears to create a complex image.  This was the greatest mathematical lesson that I gained.  Not only can a simple object be made to appear complex, but knowing that complexity can always be made simpler. 

Weekly 1: History of Math

Euclid’s Fifth Postulate is one of the most historically discussed postulates in the history of mathematics.  Many of the most famous and recognized mathematicians have worked on the success of this postulate.  Some failed, some succeeded, and some work that aided in its discovery was not recognized until after the mathematician’s life ended.  The amount of effort and intelligence that went into this postulate is something to be recognized.

           At first, Euclid created five postulates.  The first four postulates describe the properties in exactly one single line.  It is not until the fifth postulate that parallel lines are introduced.  Mathematicians thought that the parallel (fifth) postulate could use all four previous postulates to help prove it.  The reason for so much discussion over the fifth postulate was because unlike the first four, there were many more discrepancies in the fifth. Mathematicians worked on the postulate until they found something to fall through in their work.

            The infamous Fifth Postulate is also known as the Parallel Postulate because it can draw conclusions about the characteristics of parallel lines.  This postulate was necessary to prove Proposition 29.  Euclid’s first 28 propositions could be proven with his first four.  Therefore, no further propositions, including parallel lines, could be proved without the understanding and correctness on the Fifth Postulate.  Interestingly, in Euclid’s original writings of the postulate, he assumes the creation of the parallel lines.  This did not sit well with other mathematicians, so the work began.

            There were many consequences that were created by the fifth postulate.  Many of these consequences looked to help conquer the ambiguity contained in the postulate.  They looked to observe the possibilities and behavior of the interaction between two lines.  It was not until the work of Saccheri and his approach to finding the answer.  Instead of trying to prove what Euclid had written, he wanted to prove that there existed a contradiction.  Although his work did not arrive at a concrete conclusion of contradiction, his method of using quadrilaterals was not completely a waste.  

            Following Saccheri were mathematicians looking for answers, but never found any.  The next big attempt came from Bernhard Riemann.  In Euclid’s Fifth Postulate, the length of the line is not defined and many people assumed that it meant infinite length.  Riemann found that an undefined length did not necessarily mean that the length was infinite.  This led to the discovery Elliptic Geometry.  Using a three dimensional plane, postulates 1, 2 and 5 were modified using circles.  In Elliptic Geometry, all the perpendiculars have equal length and the angles’ sum is determinable unlike plane geometry.  Riemann’s discoveries used a new and unique approach to geometry. 

            All of the failures and successes of mathematicians led to many discoveries.  Some work may have made accidental discoveries on the way, but without unique points of view, geometry would not be dynamic.  It took the work of Euclid, Riemann, Lobachevsky, Bolyai, and Gauss to challenge the work of others and make lasting discoveries.  Euclid laid most of the groundwork when it came to these discoveries.  If it was not for his work, he would have not paved the way for the other mathematicians.    Euclid’s Fifth Postulate created frustration for years, but without this frustration and determination, geometry would not have been redefined. 

The main message that I can take away from Euclid's work is that persistence is key.  He may not have been able to finish proving his work like he wanted to, but he paved the way for other mathematicians.  We need to understand that even mathematical tasks are done better by teams.  Math is not meant for one genius to try and conquer.  If you look at the other historical details of mathematics, you will see that multiple people have done their part to contribute to certain discoveries.  Yes, we know Euclid's name because of the work, successes, and troubles he had.  Experiencing difficulties only proves that he is human, which might have been a question I have had before.  

"Euclid as the father of geometry." Khan Academy. N.p., n.d. Web. 12 May 2014. <http://www.khanacademy.org/math/geometry/intro_euclid/v/euclid-as-the-father-of-geometry>.

"Parallel Postulate." - AoPSWiki. N.p., n.d. Web. 12 May 2014. http://www.artofproblemsolving.com/Wiki/index.php/Parallel_Postulate 

What Is Math?

What is Math?

This is probably the toughest word to give a definition to. How can you sum up a complex subject into a select few words? Math to me is not only a subject, but it can be an approach to life. Math is an undeniable part of our human experience. This is why it is necessary to know and understand. From the day we entered school, we have been presented with the term "math". For some of us, we may have a positive of negative opinion. Our opinions are developed based on our experiences or how it was presented. School is what should be held most responsible. Students begin to correlate negative feelings behind quizzes and tests and forget to see the real life applications math can hold. This is where my education major comes out. There is so much to learn and appreciate about the subject.  
When it comes to the history of mathematics, the part of math that I do understand is that it has come a long way from the beginning of time. Many people have come in and out of the content area. Some feelings unaccomplished and others making strides toward new discoveries. This is how our society experiences changes with discoveries. Every small piece of knowledge that is added to our world starts to form it into something new.

Probably one of the most influential milestones that I have studied in detail would be Euclid and his postulates. The work of one person has withstood the test of time and people after him have critiqued or used his work.

I am sorry to admit, that without any research, I cannot name any other milestones.