About Me

Hello.  My name is Kali Orenstein.  I am a 4th year student here at Grand Valley State University.  I am attending GVSU as a Mathematics and Education major and Elementary Certification minor.  Outside of school, while I am living here, I work as a nanny and an at home daycare teacher.  When I go home to a suburb outside of Chicago, I work at a different daycare.  When I am not working or doing something school related, I enjoy trying to find a good book, spending quality time with my family, or hanging out with my friends.

I have used technology in previous classes to write/read blog post and made a weebly about myself that is up and running.  I plan to use technology in my classroom, depending on the school district I am in, by allowing access to computers or Ipads, and using a doc camera to show my work or my computer for videos or examples.

My sisters are the most important thing to me.  I don’t know where I would be with out them.  (Left to right: Me, Alissa, Emily).

Knapp Forest Elementary is the school I am currently Teacher Assisting at and I love it!

Kanpp Forest Elementary is part of Forest Hills Public Schools..

Communicating Math: MTH 495

Optional: has your work this semester changed or refined your idea of what mathematics is and what mathematicians do?

 

I do not think that my work has changed my idea of what mathematics is.  However, I do believe that the class and my work has refined my ideas of math.

I think through a lot of discussions within the class, I have gotten the chance to hear about so many new ideas and points of view.  I now see many parts of  mathematics differently and am more open to new ideas behind them.

For example, “Is math a science?”  was a big topic of discussion recently in the class.  Although I am still at a firm “no” with the question, I loved getting the opportunity to not only express my thoughts correspondingly with a group of my classmates, but also here the opinions of more than half of my class who disagreed with me and my group.

Another aspect that I liked was learning about where math came from.  We learn the formulas and problems our whole lives but we never think about where they originated or who thought to prove what.  It was interesting to take a different spin on a math class an really dig in deep to the material at hand.  One of my favorite parts of this was exploring who “invented”/discovered  calculus.  I never knew that this was a debated and still don’t know who I believe did this to their future generations(joke) but do feel more knowledgeable on where it came from and how it came about.

Knowing that mathematicians did much more than just proving one theorem after another gives me much more of an interest in the subject than I ever imagined I would gain from taking the course.  I am glad I was a part of the class as a whole and will continue looking for more and more ways this subject sparks my interest.

Communicating Mathematics: Amicable Numbers

When I first started researching about amicable numbers, I was confused on what they were.  Not only that but I felt like I should know what they are because one of the the first websites I opened about them stated that if your are familiar with number theory you should know what amicable numbers are.

After reading more into it and exploring other websites, the explanation of what they were finally clicked. After that I remembered learning about these numbers in class, however I do not recall learning about them prior to my senior year capstone class in college.  Therefore, I would not find it surprising that others do not recall what they are or how they work.

I found through research that there are 42 pairs of amicable numbers below 10^6 with 9 new ones found.  Finding these pairs require a lot of time and precision.  I would not say it is “hard” work to find them once you get the hang of the pattern, but it is most definitely time consuming.

How I understood these numbers and the best way for me to explain this concept is through looking at an example as to how they work.

The simplest form of amicable numbers is the pair (220,284).  To start off we will like the factors of both of those numbers.

220

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

284

1, 2, 4, 71, 142

Now that we see the factors of both numbers, our next step is to add up all of the number in 220 and then again in 284.  When added together, we see the following:

220

1+2+4+5+10+11+20+22+44+55+110=284

284

1+2+4+71+142=220

Since the factors of 220 add up to 284 and the factors of 284 add up to 220, this is therefore an example of a pair of amicable numbers.

A list of more, found from my research is found at my given websites:

Link 1

Link 2

Why am I telling you about these numbers? Well, I was just interested at seeing how many pairs were out there.  Clearly there are more than six, in fact, as said earlier, there are more than 42 pairs which is a lot more than people originally thought.  The first pair, displayed above was said to be originally founded by Pythagoras.  Although there is no yet use for these numbers other than to compare with one another within the pair, when discovered, these pairs did more than just that for the people of that time.

During the time of the discovery, these numbers were said to be magical.  They were used by astrologers to prepare horoscopes.  They were also known to create special ties between two individuals. (I believe that is because of the special tie the numbers had in the pair.)

Back on how many pairs there are.  Even though the research I have found has said that there are around 42-51 pairs of amicable numbers, I have continued my research to find that there are 390 pairs of these numbers.  The people involved in this discovery included the following:

  • Fermat
  • Arab al-Banna
  • Descartes
  • Euler
    • Who found a huge amount
  • Escott
    • offering the other pairs adding up to 390.

So now my research is up to 390 pairs, but I believe there to be much more.  In fact, with infinite being the highest form of a “number” we have, I think that there must also be an infinite number of amicable pairs.  I have no clue how I would solve or prove this, but I am sure it has been or needs to be done.

I liked doing this research.  I feel much more knowledgeable about the subject and found the whole thing very interesting.  I was intrigued by the subject just because of how many unknowns there still are within this subject matter.  I am curious to know if more research is being done and when/if anything more will be discovered.

Doing Math: Tessellations

As I am enrolled in my Capstone class, I am currently working on my capstone project meant to reflect my learning on at least one area of mathematics.

During the semester, we talked a bit about tessellations.  We got the opportunity to make and color our own during and outside of class.  Although this was very time consuming, I thoroughly enjoyed the chance to explore the different ways symmetry can be applied through mathematical pictures.

Tessellations are symmetric designs featuring animals, people, shapes, etc. which can fit together like a puzzle.  Escher, known as the father of tessellations, is my inspiration for my project.

Because of my interest in these types of tessellations, and the fact that I am going to school to become an elementary school teacher, I have decided that I would like to make my very own coloring book for ages 6+.  In these tessellations, there will be a key that will start off matching colors with numbers then work its way up to matching colors with shapes .

To go about doing this, I will make a set of tessellations and from there order them in what I think is simplest to most difficult, so the readers level of thinking will go up the further in the book they get.

An example of the finished look that I will expect to see is attached below, made by me:

tessellation 1

After looking over this tessellation I think that this would be one of the last tessellations in the book.  This was not only very time consuming to make, but would also be difficult for the reader to understand and color in.

While making this tessellation, I had to think about how I wanted it to look in the end and what forms of symmetry I wanted to display.  I had never made a tessellation prior to this one so I started my thinking with a blank slate.  I really just started with drawing out one of the K’s.  From there I decided that I wanted to move horizontally reflecting over a line each time.  However, when I moved vertically, it was simply just a translation.  Therefore, moving diagonally gave me a glide reflection.  After positioning all of the K’s where I wanted them, I saw an opportunity to make the tessellation more complex and full of lots of colors.  My thinking was that every time there was a closed up shape, there was a new color.  When that closed up shape was repeated, it was the same color.  I closed up as many shapes as I thought looked good and from there colored everything in giving me my final look.

What will a student get out of coloring?  I believe this is a great idea for young learners because it gives them the chance to see what symmetry, translations, reflections, rotations, and more are.  When they go about coloring each page, my hopes are that they get tougher for them and they begin to see a pattern of what a tessellation is.  Since my book will begin with them filling in shapes with numbers, they will just get a visual of what a tessellation is.  From there, the next section is filling in shapes with matching shape colors.  This will allow the student to see what tessellations are made of, different geometric shapes in my case.  And the next section is DIY, allowing them to explore what they have learned with the help of my and their imagination.

Summary:

My coloring book will be for those from the ages of 6+.  Towards the end of the book, it may be too difficult for 6 year olds.  In the start of the book, it will be too trivial for, let’s say, 7th graders.  Although learning about translations and geometry comes around middle school, this book will give younger learners the chance to explore and spark their interest at an early age.  The book contains three sections, 1. Matching numbers with colors 2. Matching shapes with colors 3. DIY.  This allows the student to grow their thinking as the book progresses.

If I had more time to expand this book even further, there are a couple things that I would do.  First, I would make the book lengthier, adding multiple pages to each section.  From there I would give the reader/student an answer key in the back.  I would try to do this by making a copy of each page, coloring it fully in by myself, scanning the outcome of each to the computer, shrinking them, and put more than one answer on each page in the back.  This reminds me of a  crossword puzzle book, where there is a place to fall back on when stuck or checking your outcome.

 

 

 

 

Communicating Math: The Relationship Between Mathematics and Physics

Before writing this blog post, I looked around to find a good article that portrayed my thoughts on the relationship between math and physics.  I think I found one that supports my ideas.

In this blog post, I will be going into depth about an article that I found by Benjamin Plybon called “The Relation between Mathematics and Physics.”  After reading the article, I made connections, found disagreements, and learned many new things.

The first thing that really caught my attention in the article was when the author began to describe where the differences in the two subjects take place.  He argues that physics and mathematics build off of one another and you can’t have one without the other.  Mathematics is used as a symbolic language to express physical problems. The disconnect happens when we start thinking about the way we think.  Physicists think like physicists.  Mathematicians think like mathematicians.

In mathematics, everything must be rigorously defined, examined, explained, and proved in a logical order.  Because of this, many students can become increasingly frustrated and timid when discussing mathematical concepts with a physicists mindset around people who have a mathematical mindset.

To go into more depth on this topic, I have made a connection to my personal life.  My major is Mathematics and Educations with an Elementary Certification minor.  With that being said I have taken quite a few education courses at this point in my education.  In many of my classes we have had in depth group discussions about how important it is to have an open environment where no one is scared to answer a question “wrong.”  This is simply because any answer that someone may think is wrong is really just a great opportunity for more discussion that can lead the class to a more complex and clearer understanding of the mathematical concept.

In the article, Bejamin mentions that students who think like physicists may become scared to answer a question wrong in a math class because of any surrounding math thinkers who typically thrive on finding and fixing mistakes.  This is not a good thing.  I think that we need to understand this issue an

love and math–book review

Love and Math

For the first half of our semester, I have been reading “Love and Math” by Edward Fenkel.  When I first began reading, I had no clue what to expect.  The summary of the book gave me a little insight but I had never read a math book that is not a text book.  I felt as though the preface was long and drawn out so immediately regretted my choice in book. I was very wrong.

After reading the first few pages, I realized this book wasn’t just explaining mathematics to someone but it was more so a narrative of the author’s life.  I found this very interesting.  Reading is not my strongest suit but when it is more of a story than an article it makes it much easier for me to understand and relate to.  I loved this aspect of it.

I learned just how hard of a life Edward had growing up and trying to get into the best schooling he could. I also discovered how hard he had to work to overcome his struggles to achieve greatness in his academic life.  After reading all of the accomplishments that Edward has made, I was surprised that I did not recognize his name when I first discovered his book.

There were a few things that did bug me about this book.  In it, the author explains how much he used to dislike math and how bad he was at it, however, after reading in between the lines that clearly was not true at all.  He skipped grades, loved reading mathematical articles, and used his time to keep exploring mathematical ideas and concepts.  Another thing that I was not particularly fond of was how long some of his point dragged on for.  There were multiple times while I was reading that I had to skip forward a couple paragraphs because I was positive I understood the point he was trying to make.  There were also times when the math was way over my head so I couldn’t read into too much.

I did not feel bad when approaching these difficult portions of the book because in his preface he explained if I, as the reader, come to a portion I am not understanding, just skip over it and maybe the next time I read the book I will understand it much more.

I recommend this book to anyone who has an interest in mathematics.  I was able to connect to Edward in more than one way and put myself in his shoes when there wasn’t a connection.  It was a fast easy read but very interesting and makes me want to learn more about the author and his mentors.

Nature of Mathematics: Number Systems

One of the oldest branches of mathematics is number systems.  When we analyze when and where exactly the number system came from we end up working our way all the way back to the development of mathematics in general.  Some say that the first discovery of a placement of any number is the highest achievement of ancient elementary arithmetic and the foundation of mathematics in general.  Relating this to my capstone class now, where we do a lot of mathematics that people would use in ancient times, I cannot imagine trying to think about any form of mathematics without having a scale to relate it back to.  It makes me curious as to what mathematics actually took place prior to this discovery of number placement.  The discovery of this positional principle was made during the Babylonian mathematics around 2000 BC.  After the discovery of the positional principle of a number representation, came the decimal number system at around the 8th century.

A few mathematicians that we know of today were some of the original supporters of this “unusual” numerical organization.  A couple of examples of these people is the famous French mathematicians Laplas and Leonardo Pisano, better known as, Fibonacci.  After doing some research, I came to a quote of his that I really liked where he said,

“The nine Hindi numerals are the following: 9, 8, 7, 6, 5, 4, 3, 2, 1. Using these numerals and the numeral of 0 called “zephirum” in Arabian, one may write some number”.

This quote allows me to put myself into other peoples’ shoes and realize what a drastic change, and upgrade this must have been.  To simplify what I mean, if you look at the number 5,555,555, even though this one number consists of all of the same number, 5, each one of these 5’s means something different to the whole number.  The reason why is the placement of it.  This is a very complex and confusing problem to someone who has been doing mathematics in a completely different way before discovering this new way.

A great link that I found had a interesting visual on how to talk about the number system and why we use it.  I liked getting the chance to explore other’s thoughts and ideas on this matter.  The link is written below:

https://prezi.com/-75t7sg6pd4_/essentials-of-numeracy/?utm_campaign=share&utm_medium=copy

 

The “So what?” for my students?  That’s simple.  The reason we must know about the number system is to have something to relate to.  For every math problem we do, there is a way to show it through this system.  The number line is a good example of that.  Whether dealing with measurement, fractions, addition, multiplication, etc.  Showing it on the number line provides a clear precise way of what we are doing when be asked to solve a problem.

 

History of Math

What has surprised you about Greek mathematics that you wish you’d known before?

Greek mathematics is something that I have not truly explored until now.  I have researched and read articles about it and have been surprised about multiple things.  Some examples are of the following:

  • How many known mathematicians were involved in greed mathematics
  • Where proofs came from
  • How schooling worked.

I had no idea that the idea of proofs originated in Greek Mathematics.  I knew that Euclid did many proof to show how certain aspects of our everyday math exists but I didn’t know that a formal proof was not around before Greek mathematics.

One fact that I learned through the research done is that there were specific school just for learning about these newly learned mathematics.

While studying mathematics throughout my educational career, I have come to recognize quite a few names that are famously known in the math world.  Now that I got the chance to do some background research on Greek mathematics, I have come to find out that most of these names that I recognize when talked about are known from there impact they had in Greek mathematics.  A few names come to mind when I say this and that is Pythagoras, Aristarchus, Thales, and Euclid.

Pythagoras was one of the first Greek mathematic thinkers.  He spread his thoughts and ideas with a group of his followers who then continued to teach them to others.  These Pythagoreans were known to put things in order.  They said that math contained the rules of the world around us.  He is best known for proving that the Pythagorean Theorem is true.

Aristarchus was the first person to suggest that the earth revolved around the sun and not the other way around like most people thought. He also figured out the size of the moon and discovered that the stars are far away and the sun is much bigger than the moon or the earth.

Thales was the first person to predict an eclipse of the sun successfully.  He also figured out a way to measure one of the Egyptian Pyramids through measuring shadows based off of his own height.  He also proved many mathematical facts such as a diameter bisects a circle, the bottom angles of an isosceles triangle have the same angle measurement, when two straight lines cut into each other, and if two triangles have two angles and a side in common, the triangles are identical.

Euclid is a very interesting character.  He has no known photos of himself, and some people don’t even know if he existed.  Euclid’s goal was to prove things by using reason and logic.  He also taught how triangles and circles work, along with irrational numbers and three-dimensional geometry.

One paragraph that I found very interesting during my reading was the following:

“Thales may have been Anaximander’s teacher, and Anaximander was Pythagoras’ teacher. Some ancient writers say that Pythagoras, when he was young, actually visited Thales, and that Thales advised Pythagoras to go study in Egypt. Thales died in 543 BC, only a few years after his city was conquered by the Persians.”

What is Math?

What is math?  Math to me is using logical and analytical thinking to derive solutions to the problems we see from all directions.  In almost all of my math classes I have been asked this exact question.  In each one I hear a wide variety of answers from each student.  Math is used every day.  Whether you are solving a mathematical equation or just figuring out how long you need before leaving for your next class. Every time I am asked to answer this question I try my hardest to remember my previous answer since it is such a difficult question, and yet I never can.  I think the reason behind that is because there is no “right” answer to this question.

I myself have learned math in the steps of any other person in my shoes:

  • Numbers and Counting
  • Addition/subtraction
  • Multiplication/division
  • Algebra
  • Geometry/Trigonometry
  • Calculous
  • Euclidean Geometry
  • Etc.

The reason I bring up my history of math is simply because I feel as though the milestones of math throughout history were brought up in the same manner.  The only thing I might switch around is Euclidean Geometry because that was brought up whenever Euclid had discovered it.

Teaching Portfolio

My Additional Lesson Plan comes from a website I found online after searching for lessons for statistics and probability.  I like this lesson quite a bit because it calls for the use of a computer.  I think students would like this and become more engaged in the lesson with this interactive piece added.  After reviewing some of the main points of probability and statistics, the students apply a real life situation to it and then follow up with the chance to make a game board.  When they are on the computer they get the chance to explore “brainpop”  which has to do with chance and variability and with influence them while they are creating their own game.  I would say this is a 3 with alignment to the GAISE Report because although this is a very enriching and educational activity for the students I did not recognize any familiar words or phrases with the GAISE Report’s.