Weekly 7- Fractals (Nature of Mathematics)

What is a fractal?  Until this semester I have never been introduced to fractals.  They are an extremely unique and interesting concept.  A fractal is a pattern that “repeats itself at every stage.”  A fractal can easily be created by a person or you can find them several different places in nature, both of which will be shown in this blog post.

Although Benoît Mandelbrot was not the first mathematician to work on developing the concept of fractals, he was the one who was given credit for fractals.  Some of Mandelbrot's predecessors include: "Karl Weierstrass, Georg Cantor, Felix Hausdorff, Gaston Julia, Pierre Fatou, and Paul Lévy."  An advantage that Mandelbrot had over his predecessors was the use of technology, such as a computer.  These mathematicians started to discover fractals by looking at functions.  Gaston Julia found a function that was continuous, but had to draw the function by hand because he did not have computer technology.  Once Mandelbrot found another function which was continuous, he named a set after Julia, called the Julia Set. 

We will start by looking at a fractal:

In the fractal above you can see it starts with a larger circle with three smaller circles inside of the larger circle.  Then inside of each of the smaller three circles, those circles contain three more circles. and so on.  As you can see the same pattern occurs over and over again with the scaling getting smaller each time.

In order to better understand the concept, I created a fractal of my own:

Another neat observation I have made about fractals, is that when you combine more and more generations of the same pattern they end up making addition shapes that repeat as well.  An example of this in my fractal is that the Large purple rectangle for the first generation has two rectangles from the second generations of the same color attached to its side and then the second generation rectangle has two rectangles from the third generation of the same color attached to its side.  This pattern continues in both the yellow and purple.  If we would continue to add more generations the pattern would continue with each new generation as well.

Both of the images above were created by people or technology that people are using.  Next we will look at some of the fractals that occur in nature.  Some examples include; snowflakes, lightening, broccoli, trees, and shells, which can be seen below:

These images are only a few of the fractals you can see in nature.  Fractals are all over, next time you take a walk in a park, look around and see if you can find a fractal, I'm sure you will!  As you can see mathematics is all over the place in life.  Anywhere you go you can see something that relates to mathematics whether you know it or not.  Knowing that mathematics is all around you, helps when teaching math to students, because it will help to either create real world examples or to even show students that math exists everywhere and they will use it in their life.

Google Images  

http://en.wikipedia.org/wiki/Fractal

http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html

"Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement" Book Review

      The first book I read this semester was "Vision in Elementary Mathematics" which focused on different ways you can teach different mathematical concepts.  The second book I read this semester was "Accessible Mathematics:10 Instructional Shifts That Raise Student Achievement" by Steven Leinward.  The second book talked about different shift you can make in your classroom in order to raise student achievement.  Although I didn't get the chance to read the whole book, one of the chapter that caught my attention talked about taking every opportunity that you get to support the development of number sense with students.  It talked about ways you could work number sense in your classroom.  The book also talked about about using multiple representations while teaching students.  I already try to use many different techniques when I teach in order to connect with as many students as possible.  I liked this book, however I think it would be better to read it once I am in the classroom and can actually incorporate these shifts into my classroom.  It is a lot to take on at once so I would recommend trying to work on one shift at a time in your classroom.  I will use this book while I am teaching, but I will start by incorporating one shift and then each year add another shift to work on.  I highly recommend this book to teachers who are already in the classrooms teaching!

Historical Women in Mathematics

Historical Women in Mathematics

            In class we have been talking about how gender stereotypes still exist in mathematics.  Often times women are not recognized near as much as men in mathematics.  We will now talk about some of the different historical women in mathematics.

Agnesi: Maria was born into a wealthy and literate family, where she was the oldest of 21 children.  By the time Maria had reached her teens, she had mastered mathematics.  She would attend many of the gatherings at her family’s home, with many intellectuals of her time. In 1738 Maria published a collection of essays called Propositiones Philosophicae.  These essays were based on many of the conversation she had at her families gatherings.  However these essays were not her most famous work.  In Maria’s twenties she started working on Analytical Institutions.  Once her book was published is became one of the most complete work on finite and infinitesimal analysis.  “Maria Gaetana Agnesi is best known from the curve called the "Witch of Agnesi" (see illustration from her text Analytical Institutions). Agnesi wrote the equation of this curve in the form y = a*sqrt(a*x-x*x)/x because she considered the x-axis to be the vertical axis and the y-axis to be the horizontal axis [Kennedy].” (http://www.agnesscott.edu/lriddle/women/agnesi.htm)  Maria gave up her work in mathematics once her father died in 1752.

Noether: Noether was in the upper middle class growing up.  At the time women were not allowed to attend college, so instead Emmy went to finishing school.  In 1900 Noether wanted a university education in mathematics, so she audited classes at Erlangen.  She went to the University on Göttingen as an auditor, however a year later she went back to Erlangen when they started to enroll women.  In a matter of three years Noether got her Ph.D.  After  receiving her Ph.D, she worked on Erlangen for seven years without pay.  This is when she worked with Ernst Otto Fischer on theoretical algebra.  “In 1915 she joined the Mathematical Institute in Göttingen and started working with Klein and Hilbert on Einstein's general relativity theory. In 1918 she proved two theorems that were basic for both general relativity and elementary particle physics. One is still known as "Noether's Theorem."” (https://www.sdsc.edu/ScienceWomen/noether.html)  Noether also did a lot of work on ring theory, abstract algebra, and number theory among other things. 

Kovalevsky:  Sophia started her education in mathematics by studying her father’s calculus notes, which were used for wallpaper in her room.  At fourteen she was able to do trigonometry.  In 1870 Sophie studied under Karl Weierstrass at the University of Berlin for four years.  Under Weierstrass she produced three papers, one of which called, On the Theory of Partial Differential Equations, was published in Crelle’s journal.  After several years of trying to find a job, Sophie got an offer to lecture at the University of Stockholm, she was there for five years.  After that she had many accomplishments, including: a tenured position and becaming an editor for a mathematics journal.  However Kovalevsky’s greatest achievement was her paper, On the Rotation of a Solid Body about a Fixed Point which won the competition for the Prix Bordin. (http://www.agnesscott.edu/lriddle/women/kova.htm )

Weekly Five- "Vision in Elementary Mathematics"

“Vision in Elementary Mathematics”

            The book, “Vision in Elementary Mathematics”, is about different topics in mathematics that a teacher could use to teach their students in an elementary classroom.  It gives several ideas and techniques to teach students so that they can discover mathematics on their own.  There are a variety of topics covered in the following chapters: “Even and Odd”, “Divisibility”, “An Unorthodox Point of Entry”, “Tricks, Bags, and Machines”, “Words, Signs, and Pictures”, “Sudden Appearance of a Practical Result”, “A Miniature Problem in Design”, “Investigations”, “The Routines of Algebra I”, “The Routines of Algebra II”, “Graphs”, “Negative Numbers”, and “Fractions”.

            While reading this book I gained insight on how to get students to discover different techniques in mathematics rather than just being told.  This concept was applied throughout the entire book.  I strongly agree that it is important for students to discover concepts on their own rather than just being told a procedure and following it.  By discovering a concept on their own, the students tend to remember the idea the problem was trying to convey.  A quote from the chapter on divisibility states, “It is necessary for children to understand what is involved in a calculation.  If they understand what is happening they can devise methods for themselves and can test by their own thinking correctness of their work.”

            There are several methods of teaching discussed in this book.  I believe most of the methods in this book are aimed at students who are visual learners.  One of the examples that stood out to me was in the chapter “Investigations”.  The book was discussing how many students believe (x+y)² is the same as x²+y².  However, that is not the case.  Let’s say x=3 and y=4.  In this case (x+y)² is 7² which equals 49 and x²+y² is 3²+4² which equals 25, and 49 ≠ 25.  The way the book illustrated this was by creating a 7x7 table with one square representing 3² and one representing 4² and showing that there are still blank areas that would need to be completed in order to fill the square as shown below.

           Overall, I would strongly recommend this book to college students who are going to school to be elementary teachers or an existing elementary teacher who is struggling to connect ideas with students.  It shows many techniques that visual learners will be able to relate to and understand.  It also does a great job on how to explain these concepts to your students in different ways.  I found this book to be very interesting and of such value that I couldn't put it down.  The value came from the various techniques and ideas on how to get the students to discover math versus just being told how to do something.

Communicating Math- Bungee Jump

Today we did an experiment in class where each group had different action figures or objects.  We were going to attach our object to rubber bands and create a bungee cord out of them.  Our group had a hippo toy which we were to use.  For this experiment we were given our object along with a meter stick and ten rubber bands to start.  With these ten rubber bands we were allowed to test out the lengths we believed the rubber bands would stretch when attached to our hippo.  Our data can be seen below:

As you can see we tried 1, 2, 3, 4, 5, and then 7 rubber bands.  Once we did the first five trials, we took the average distance between each trial and noticed the average was 22.625 cm.  Since we originally measured the height of the drop to be 199 in which we converted to 505.46 cm.  We then took 505.46 cm divided by 22.625 cm and found that we would need a little over 22 rubber bands.  Then to make sure our prediction was accurate we knew three times seven was 21 rubber bands so we decided to test seven rubber bands where we found the distance of our object to drop 165.5 cm.  This number was very close to 158.375 cm if we would have multiplied 7 rubber bands by the average distance of each rubber band which, was 22.625.  This indicated to us that no matter how many rubber bands we added to our hippo, each rubber band would stretch approximately the same distance.  Therefore, we multiplied 165.5 cm by  three to find that 21 rubber bands would allow the object to drop 469.5 cm.  The object of this experiment was to get the object as close to the ground as we could without hitting the ground.  Since we had found the distance each rubber band would stretch was constant, and our average per rubber band was 22.625 cm we knew we could add one more rubber band to the 21 rubber bands giving us 22 rubber bands and our object would reach 491.5 cm.  This number(491.5) was roughly 14 cm from the ground.  We knew if we were to add an additional rubber band the object would hit the ground.  We then went to test our object by dropping it and seeing how far from the ground it was.  We were told that our object was roughly 17 cm from the ground when it was dropped with the bungee.  Therefore our prediction was very close to the actual result.  I believe our method worked because we tested several different instances and came up with the same result that on average a rubber band would stretch 22.625cm.  Since we had found a constant we were then able to find an equation to help us find the total number of rubber bands. When we divided the total number of centimeters the jump was (505.46 cm) by the average distance each rubber band stretched (22.625 cm) we found an answer of 22.34 rubber bands would be needed in order for it to reach the ground.  Since the goal was to get as close to the ground as possible without actually touching the ground, we knew that 22 rubber bands was the closest we could get to the ground, when only using whole rubber bands.

Overall, I found this experiment to be helpful in demonstrating a different method of teaching mathematics.  I find that many elementary students are not interested in math and therefore teachers are left trying to find ways to capture students attention in order to get them interested.  I believe students would find this a nice way to learn about estimating as well as creating a equation in order to find the number of rubber bands needed.  By only giving us 10 rubber bands to test ideas with, he made it so that we could not find the result by tying 22 rubber bands together.  He made us dig deeper into our knowledge and use many different math techniques, such as multiplication, division, converting measurements, creating and testing hypotheses.  I will use this method of teaching in my future classroom to get my students up and engaged in their learning.  My favorite way of teaching is to have students explore and find results on their own by personal experience rather than providing step by step instructions for them to follow.  I believe when students create or come up with a strategy on their own they are more likely to remember the strategy in the future.

Doing Math- Nets for Geometric Models

     In a previous class I took we had discussed how to create nets of different geometric shapes.  Then in class we also discussed making nets in order to create a square.  Although there are many different ways to create a net for each shape these are the nets I came up with.  I created nets for the following shapes (which can be seen below): square pyramid, triangular pyramid, triangular prism, right cylinder, hexagonal prism, and a cone.

Doing Math- Geometric Tessellation

Below is a geometric tessellation I created: 

     In order to create my tessellation, I started by drawing the squares that are colored yellow and green.  Once those were drawn I thought about what I could add next and I noticed there were little squares which are colored orange and purple in between the yellow and green squares.  I felt like the squares themselves were not enough and decided to draw diagonal lines through the squares in order to create triangles within the squares, keeping in mind that all shapes had to be polygons. I then drew horizontal and vertical lines splitting the paper into 20 squares.  From doing this I created a pink and blue square outline around my original yellow and green squares.  Once the outline was drawn I decided to color the tessellation in a pattern as well because when you look at a geometric tessellation not only the shape pattern continues but also the color pattern.

     I could see myself introducing tessellation in my classroom once I become a teacher in order to have my students explore how different geometric shapes can connect and create a different polygon.  For example how four triangle can create a square which can be seen in my tessellation below or how six triangles can create a hexagon.  Another observation they could make it that hexagons can connect to one another and not have any gaps.  I think creating a tessellation could help students understand shape properties as well as having them work on problem solving and puzzle skills, if they are given certain shapes and asked to make as many different combination as possible where there are no gaps.  I think in elementary or middle schools I would prefer to have students work with manipulatives of the different shapes, so they are not limited to the shapes they can easily draw by hand.  Overall I think using tessellations can help further students learning in several different content areas.

History of Math - Why is Euclid Important?

About 300 B.C., Euclid wrote The Elements, a book which contained his five postulates on which he based all of his theorems. This became one of the most famous books ever written and this system of mathematics came to be known as Euclid’s Geometry.  Euclid stated five postulates on which he based all his theorems.  It was obvious from the start that Euclid’s fifth postulate was different than the other four. 

Euclid’s fifth postulate, also known as the parallel postulate, states:  “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” (O’Connor and Robertson 1).  From the beginning, Euclid was not satisfied with the fifth postulate.  He tried to avoid using the fifth postulate and was able to in his first 28 propositions of The Elements.  There were several main flaws in Euclid’s development of the parallel postulate:  failure to recognize the need for undefined terms (in other words, he defined every term), he relied on diagrams to guide the logic in the construction of his proofs, and he assumed that straight lines were infinite but it is nearly impossible to check for the consistency of the fifth axiom.

In the centuries since Euclid’s fifth postulate, János was one of the first to doubt the truth that the parallel postulate was a valid theorem.  Bolyai’s real breakthrough was the belief that a new geometry was possible.  While there were numerous mathematicians who developed the new geometry, János Boylai’s contributions were significant in changing the view of geometry.  Because János ignored his father’s advice to avoid work on Euclid’s parallel postulate, he provided a new view to geometry: a geometry in which the parallel postulate did not hold true was possible.  Boylai’s appendix setup a series of mathematical proposals whose implications would blossom into the new field of non-Euclidean geometry.  Boylai’s findings later led to Beltrami’s proof that if Euclidean geometry was true, then non-Euclidean geometry was also true. 

Euclid’s geometry and his five postulates gave mathematicians a way to study the world around them, but provided some limitations.  János Boylai’s contributions changed the way mathematicians viewed their world and opened them up to the possibilities of other geometries besides Euclid’s.   Baylai provided the means of studying what lies beyond our world.  

Citations

Heiede, Torkil. "The History of Non-Euclidean Geometry." : 201-11. Print.

O'Connor, J J., and E F. Robertson. "Non-Euclidean Geometry." MacTutor History of Mathematics. N.p., Feb. 1996. Web. 14 Nov. 2013. <http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html>.

What is Math?

     What is Math?  Math can be interpreted many different ways.  For me math is about discovering new ideas and processes in which help you solve a problem.  It is also about being able to use logic with math such as thinking about real world examples and if measurements make sense.  There are many methods to solving problems which I believe students should discover on their own.  Some people think math is procedural while others believe it is about discovery.  I think math is both a procedure and also a discovery.      Since I want to be a teacher math is the process of students creating a strategy that works for them.  I could  teach a procedure to my students, however the chances of them actually retaining knowledge from that lesson will be slim to none, whereas if they discover a method of their own they will be able to recall it more easily in the future.
     As far as discoveries go, I have not done a lot of research.  However they main discovery I have learned about it Euclid's Postulates.  As well as what separates Euclidean and Non-Euclidean geometry.  Another big idea in math that I have studied is the Pythagorean Theorem.  This discovery was introduced to me in middle school and was continually brought up throughout my education.  Next, is the Quadratic Equation, this has also been brought up several times throughout my education starting in high school.  There are several methods that I have used, however these are the most common and the ones I have actually spent time discovering and learning myself.