Book 2 Skimpression: Vision in Elementary Mathematics

The book Vision in Elementary Mathematics by W.W. Sawyer provides many insightful and useful teaching strategies for elementary and middle school mathematics.  The writing is easy to understand which means all teachers (no matter their area of expertise) and most parents would be able to read this book.  The book breaks down many math topics into digestible small segments and suggests drawing and visualizing numbers and stories as the best way to understand each component.  In Chapter 4, Tricks, Bags, and Machines, Sawyer indicates that we should use exploratory teaching methods that allow students to:

·        be successful

·        discover and reason

·        be intrigued and fascinated

Although it seems like the ideas presented in this book are new and novel approaches for understanding math, this book was written in 1964.  In addition, these lesson attributes remind me of something I recently learned in one of my math courses.  In Teaching Middle School Algebra, one of the most significant ideas I learned was that a rich and meaningful mathematical task should:

·        encourage student reflection

·        allow students to choose their methods and tools

·        be mathematically interesting and challenging

·        provide long-term insight and strategies (residue)

·        indirectly teach concepts and procedures

The ideas from Vision in Elementary Mathematics seem to align perfectly with the “new” concepts I just learned about.  If these teaching strategies and ideas have been around for at least 50 years, it makes me wonder why the concepts did not catch on until now.  Why didn’t my teachers use these common sense strategies when I was growing up and when my own children were in elementary school?  I feel that if I would have been taught math (and all subjects, for that matter) in an exploratory way rather than by rote memorization, all of the higher level subjects would have made more sense and would have been easier to master.  I feel strongly that true learning occurs when we make meaning of the material presented, not by memorizing facts.

This book is a great reference tool for elementary and middle school teachers, as well as parents.  I purchased this book to keep as resource in my teaching library.  I give Vision in Elementary Mathematics five stars!

History of Mathematics: Milestones in the Evolution of Infinity

I created a Glogster page about the milestones in the development of infinity.

Milestones in the Evolution of Infinity

There were several key players in the evolution of infinity, however, many other mathematicians have also contributed to infinity's development.  Infinity has been debated for thousands of years - is it real?  - does it even exist?  This Glog touches upon the main characters in infinity's ongoing saga.

Communicating Math - The Joy of X by Steven Strogatz

Book Review:  The Joy of X by Steven Strogatz

In this whimsical book about math, The Joy of X, Steven Strogatz creatively explains the many concepts of mathematics.  Strogatz uses entertaining analogies and stories to help clarify topics ranging from the mysterious e to the dreaded imaginary numbers. 

Readers begin their journey at the basic level of math – numbers – their origination and history into arithmetic.  Strogatz poses the question “Did humanity discover or invent numbers?” He proceeds to compare numbers to atoms, a theme we see emerge throughout the book.  Visual learners will appreciate the many illustrations and visual aids used to help reveal patterns for just about every concept including operations, imaginary numbers, algebra, story problems, quadratics, proofs, sine waves, and group theory.  Strogatz often connects math to art and science, a topic of debate within the world of mathematics.  I especially enjoyed the chapter about word problems and how to break down the often misleading implications that (sometimes deliberately) confuse the reader.   We wind our way through other topics such as functions, exponents, and logarithms,  where Strogatz explains that a logarithm is the “staple remover” for an exponent.  He delves into integral calculus and derivatives, including a chapter devoted to limits.  In this section, we read about how to continuously divide a circle, lining up its wedges into a rectangle, in order to show the connection between the area formula of a circle and the area formula of a rectangle.  Strogatz goes on to make sense of the confusing topics of conditional probability, linear algebra, and finally finishes the book at the idea of infinity. 

Throughout the book, we find examples of the practical applications of  math, often seeing connections to astronomy, engineering, and the medical field. Sprinkled into the chapters, readers will find references to and recommendations for some of Strogatz’s favorite related books, such as Nathan Carter’s Visual Group Theory.  Strogatz also references a terrific series of YouTube math videos by Vi Hart that creatively and artistically explore topics such as topology.  At the end of the book, the author includes a note section which digs further into some of the ideas he touches upon in each chapter.   

Although Strogatz lost me in a couple of chapters - I was not sure of the points he was trying to make in the chapters about statistics and infinity -  I appreciated his humorous and engaging writing style.  I think this would be a perfect book for college freshman; it would help them make sense of the connections and mysteries of mathematical concepts, possibly making math less intimidating. I also think this book could be a valuable teaching tool in middle school and high school (reading a chapter at a time) when introducing new math topics to students.  The Joy of X is a great and enjoyable read, sure to entertain and clarify concepts for people with a wide range of mathematical abilities.

History of Math

How is Pythagoras (500 BC) linked to Galois (1800’s) or Wiles(1900’s)?  I recently watched a terrific documentary about Sir Andrew Wiles.  The video follows Wiles’ quest to prove Fermat’s Last Theorem, which states that there is no whole number solution to x^n+y^n=z^n greater than 2 (a cousin to the famous Pythagorean Theorem).   Fermat claims that he did not have room to write the full proof in the margin of his copy of the Arithmetica.  Over the centuries no one, not even brilliant mathematicians like Gauss, Kummer, Euler, or Germain, has been able to prove what Fermat claimed as true. Up until 1993, even with the advantage of computer assisted algorithms, thousands of people had tried to prove it but no one had succeeded.

Wiles’ dream began when he was only ten years old, after reading a math book in his local library and understanding that Fermat’s 300 year old theorem could be proven.  He worked tirelessly for many years -  researching and contemplating the works of many mathematicians, bridging seemingly unconnected areas of math, and handwriting all of his thoughts on paper and chalkboard.  Using ideas, theories, and proofs from mathematicians including Taniyama, Galois, Iwasawa, and Flach, in order to formulate his own conjectures and theories, Wiles finally revealed his proof of Fermat’s Last Theorem in 1993.  However, several months later, there was a problem discovered with the last part of the proof.  Wiles went back and started picking his proof apart to see where he went wrong, only to find that the dots could be connected using a path he had previously abandoned. So in the end, Wiles really did uncover a proof to Fermat’s Last Theorem.  One of Wiles’ great accomplishments on this journey was the webbing together of so many areas, ideas, and branches of math.  I made the word cloud (below) which contains some of the theories and mathematicians whose ideas Wiles used to uncover his proof to Fermat’s Last Theorem.

Quote from Documentary –

ANDREW WILES: “Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it's all illuminated. You can see exactly where you were. At the beginning of September, I was sitting here at this desk, when suddenly, totally unexpectedly, I had this incredible revelation. It was the most—the most important moment of my working life.”

Modular Elliptic Curves and Fermat's Last Theorem

Nature of Mathematics - Math as a Language

Think about it.  Math as a language.  Some say, "math is like a foreign language" - Greek, in fact;  they "don't get it" - don't understand it.  Setting aside the fact that mathematics actually has many great Greek contributors (and Islamic, Indian, Italian, to name a few more), if you look at the components of a language, math really is a language.

All languages have:

• symbols (alphabet)
• nouns, verbs, and descriptors (words)
• rules, methods, and practices that are related to the use of these symbols (grammar)
• sentences
• people who use that system of symbols in order to communicate

So does mathematics -

• numerals and variables are the symbols
• the way the numerals and variables are used give us nouns (numerals, constants, and representations that are fixed), verbs ((in)equalities, operations), and pronouns (variables),
• conventions of usage
• variables a,b,c... usually indicate constants
• variables i,j,k... usually indicate counting numbers
• variables x,y,z ...usually indicate unknown variables to be obtained
• sentences read left to right
• order of operations

• sentences are in the form of equations and formulas (2x + 4 < 20)
• people around the world use this system of numbers to communicate about real world issues both informally and formally -  counting, measurement, money, data, statistics, probability, reasoning, geometry, calculus, physics, engineering

Some say mathematics is not a language in itself because

• Math is primarily a written form of communication (really?)
• Math is an art (it is created by it's own rules, it is beautiful)
• Math is a science (it is experimental, it explains realities according to set rules)

I would argue that mathematics is a sweet combination of all of these.  I think math is a language, with it's own set of symbols, conventions, and means of communicating; it is an art that is beautifully created from it's own set of rules; it is a science that is experimental in nature, explaining realities of the real world.  Just because math is an art or a science, doesn't mean is is not a language.  I believe that mathematics IS a language.

Doing Math

Nets for Geometric Models

Inspired by the book Adventures Among the Toroids by B.M. Stewart, I searched for a polyhedron net with "holes."  What I found was a 3 dimensional flexagon, which is hinged together at common sides.  As opposed to typical flexagons, this one is not a static figure, rather it is a kaleidocycle that can move and transform into different shapes.  The type of kaleidocycle I made is a type of hexaflexagon, which displays six triangular faces at the same time and can be transformed to display a different set of six triangles.  I am posting pictures of some of the steps I took in creating my kaleidocycle.


Here is the link for the instructions which I used to make my 3D hexaflexagon.:
3D Hexaflexagon

My hexaflexagon:

This kaleidocycle inverts to display a different pattern on the other side.  This was a really interesting and fun activity that I hope to bring to my students someday.  I could incorporate lessons about vertices, edges, faces, polygons, perimeter, area, and volume.
 
As I was exploring on the Internet, I came across this great math lesson for exploring area, perimeter, and volume, using polyhedra and flexagons.  This lesson aligns with 5th grade Common Core State Standards. 
Understanding Polygons and Polyhedrons Using Flexagons

Here is a real world example of a polyhedron with a hole. It is a little distorted, but it is a lamp.  This is different than a flexagon since its shape is static in nature.  It does not change but it was created with a hole in the middle, staying true to the properties of the polygon.

Nature, History, Explanation, and Doing of Math

Al-Khwarizmi made this week's post really easy (although time consuming) due to the rich contributions of his work.  This brilliant mathematician allows me to touch upon a little of everything for this course -

• Explaining math (the origin of our number system)
• The Nature of Mathematics (linking geometry and algebra)
• The History of Mathematics (the role al-Khwarizmi played at the House of Wisdom)
• Doing Math (creating a tiling inspired by the Islamic tilings)

This is so cool! I found this image as I was researching our most recent mathematician, al-Khwarizmi.  While the image is what caught my eye, as I dug deeper, I found out that al-Khwarizmi actually invented our numeral system by determining the number of angles formed by line segments. This is one of the most basic representations of how algebra and geometry are linked.  Rather than being English numbers, these are actually considered to be Hindu-Arabic numbers. 

Al-Khwarizmi is known as the Father of Algebra.  At the end of the forward for his book, the Compendious Book on Calculation by Completion and Balancing, al-Khwarizmi gives credit to God for encouraging him to persevere through difficult times to ultimately write such a concise and useful work of mathematics.  The word "algebra" is a Latin derivative the Arabic word Al-jabr, which came from his Compendious Book.  Attached is a link to a short and easy to follow description of what algebra means: The Origin of the Word Algebra  .

We can find further evidence of Al-Khwarizmi linking algebra and geometry through his use of geometric shapes (squares and units) to create algebraic expressions, as we saw when using algebra tiles in class on Thursday.  Here is an example of a problem I did using the virtual algebra link  Virtual Algebra tiles provided in class:


As one of the original contributors to the the House of Wisdom (a library, translation establishment, and school in Baghdad, Iraq established in the 700's), al-Khwarizmi is known as one of the most famous mathematicians at the House of Wisdom.

Inspired by the Islamic Tilings, I tried to create a tessellation with the letter K.  Seeing that could be rather challenging, I decided to think of letters or numbers that might fit easily together.  The number 4 came to mind.  Since my daughter is involved in 4-H, I thought I might be able to use that combination of numbers and letters for a tiling.  My original attempt at this tiling failed because the stem of the 4 was not long enough to allow for the tiling as I was picturing it.  When I extended the stem of the 4 - "Eureka!" (oh, wrong mathematician - Archimedes said that!), it worked!  I was able to tile in the H along side of the 4, separated by a hyphen (light blue square).   Here is the tiling I created:

After exploring al-Khwarizmi on my own, he became very interesting and inspirational to me.  I saw the depth of his workings in so many facets of math:  history, nature, communicating, and doing math.  Prior to being introduced to this brilliant man in my math capstone class, I had not realized that Islamic mathematicians had such a profound impact on math as we know it today.  Al-Khwarizmi sparked my interest in math history, a subject I have usually steered clear of.  I thought, if this guy is so interesting, and his math ideas are so engaging, then there must be other mathematicians who are also intriguing and fascinating.   Al-Khwarizmi is the reason I decided to do my semester project about the great mathematicians throughout history.  Math history anyone?