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.