Tuesday, June 28, 2016

How Testing and Advance Sports Statistics Relate

In my one of my education classes in the winter semester of 2016 we had a debate in class. The debate was about whether standardized testing was important or not. I argued for the side that believed that standardized testing was important and should be valued.

One of my main arguments lead me back to my favorite things in the world: SPORTS. You may be wondering, "what do sports have to do with testing?" Hopefully by the end of this post you will understand the connection.

In sports we have a lot of stats that we look at. There are normal stats that many of us may know like rebounds in basketball, rushing yards in football, or home runs in baseball. Then there are advanced stats, sometimes called "analytics." Many of these stats require complex calculations like Player Efficiency Rating (PER) in basketball or Wins Above Replacement (WAR) in baseball. Advanced stats like these take everything a player does and boils them down to a number. When scouting players, you can rank them by there advanced statistics. It must be easy to decide who the beast players are then right? Not at all. There are many other factors to look at when scouting a player like their personality, their work ethic, are they good under pressure, are they exceptionally good or bad at one aspect of the sport. All of these play an important role in evaluating a player. The stats are just a tool to help you determine the best player. They are not the be all end all for evaluation. Lets look at an example. Michael Jordan, regarded as arguably the greatest basketball player of all time, had a career PER of 27.91 which is the best all time. Magic Johnson is sometimes argued to be the greatest player of all time but most people put him second behind Jordan. Magic had a career PER of 24.11 which is 14th all time. No one in the right mind would say that Magic Johnson is the 14th greatest player of all time. There was so much more to Magic than the stats show.


All of this relates back to standardized testing. We can look at testing like advanced statistics. Testing boils each student down to a number. But when we evaluate students we shouldn't be looking just at their test scores. There are other important factors that go into evaluating students like their GPA, work ethic, problem solving skills, critical thinking skills, and what subjects they are exceptionally good and bad at. I can use an example from my life. I got a 24 on my ACT while my friend Ben got a 29. When you see that you may assume he is a better student then me. That's not necessarily the case (even though we argue about who is smarter all the time). In high school my GPA was 3.6 while his was 2.3. What does this show you? Clearly I was a much harder worker in school than he was even though he got a higher score on his ACT. Testing doesn't mean everything. There was more that I brought to the table as a student than my ACT score showed.


The ACT or any test like it does not define a student. It does however give colleges, school boards, and parents a tool to track how well students are doing. You can make a generalization about a student based on their test score but you have to learn more about a student that just their test score to get a real evaluation of them. The same goes for an athlete. A statistic does not define the athlete. It can be a tool to help you evaluate an athlete but you need to learn more about the athlete if you are going to get a good evaluation of them. Testing is not the be all end all. It is an important tool for evaluating students.

Wednesday, June 22, 2016

Communicating Math - Communicating in the Classroom

This summer I am working in the Camp Fire West 4C Believe to Become program in Grand Rapids, MI. I am one of the assistant teachers. Everyday the students, who are all in 6th, 7th, or 8th grade, come into my classroom and I lead a math lesson for about 40 to 50 minutes.

It has only been a week and it has already been one of the more challenging weeks of my entire life. My biggest problem with these students is communicating with them. Sometimes they have trouble communicating their work to me and other times it seems that I have trouble communicating to them. Many of these students are in the Grand Rapids Public School system which is not always known for being one of the best school systems. Many of the students I have can't perform some simple tasks that they should have already known by this time in their school career.

The video I have linked is sometimes what I feel like when I am talking to them.


Today we played a game that involved solving simple calculations using fractions. Here were some examples of student work,

(3/4) + (4/8) = 7/11   There is no other work shown. It is clear that this student does not understand the concept of a common denominator.

(5/7) x (3/11) = 15/something. This student understood that when we multiply fractions, we multiply straight across. However they didn't know that 7 x 11 = 77.

(1/3) + (1/3) x (3/4) = (2/3) x (3/4) = 6/12. This student knows how to add and multiply fractions but was unaware of the order of operations. They did addition before multiplication which is not correct.


The fact that these students were unable to perform these simple calculations made me want to rip my hair out. It made want to go find their math teachers and smack them upside the head. There is no way that these student should not already know this material. 

Now it's not all bad. There were plenty of students that were able to play the game and succeed. They could do these simple problems and I could tell they were getting a little bored so I began to give more challenging problems such as below.

(2/3)^2 x (3/4) = (4/9) x (3/4) = (12/36)

It made feel much better to see that there was a chunk of students who understood the ideas. There were also some that informed that they understood the ideas better once I was able to show them how I thought about a problem. 

Overall this has been a good experience so far. I still have 5 more weeks with these students. Hopefully by the end of the summer, the students will understand these concepts much better. I am looking forward to continuing to work with them. This will be great experience as I will be student teaching in the Grand Rapids Public Schools this fall.

Sunday, June 12, 2016

Review of "e: The story of a number"

As a math major I found this book to be relatively interesting. The number "e" is an important part of mathematics. I was very curious about how this book was going to be written. How can you write 200 or so pages about a single number? Not even a single concept. A single number known as "e".

As the book progressed it made sense. They started by talking about the discovery of logarithmic functions. This lead to them trying to find the "natural log" (which we now know has a base of e). IN chapter 3, the number e is discovered by using compound interest which uses the formula (1+1/n)^n. In this formula n stand for the amount of time. As n goes to infinity, the formula approaches 2.7182.... which we now know as "e". When mathematicians realized this phenomena, they discovered limits, which then allowed them to discover calculus. It then goes on to discuss what calculus and the number "e" allowed us to discover after that.

As a mathematician is it is amazes me that one number could be written into one book. While I did find the book interesting, there were many parts that were very dry. There were a lot of calculations derivations that were very confusing to me. I enjoyed the history and the discoveries, but the pure math was way over my head.

If you are extremely interested in math, I would recommend this book as it talks about some of the most important features and topics of mathematics. If you have curiosity about how e, logarithmic function, and calculus came about, I would recommend this book. If you don't have a background in math at all, then you would hate this book. It probably would make little to no sense you. If you do have a background in math but approach math by saying "just give me the formula" then this book is not for you. You would not care about how any of these concepts cam to be.

In the end I would give this book a 3.5 out of 5. It was interesting to learn about the concepts in more detail and to learn how they were discovered. The only problem was that the pure math derivations were way too complicated for me.

Sunday, June 5, 2016

History of Math - Fibonacci

Leonardo Pisano is one of he greatest and most important mathematicians in mathematical history. That name might confuse you since thew title of this post is Fibonacci. You would expect that this post is about Fibonacci. This post is about Fibonacci! His real name is Leonardo of Pisa. The name Fibonacci is short for "Filis Bonacci" which means " son of Bonnaccio" (his father's name was Guglielmo Bonaccio). Fibonacci is pictured below.

Fibonacci had a huge impact on math and changed the course of math history. During the early 1200s, Europeans were still using Roman Numerals. It was Fibonnaci that started to convince Europe that they should use the Arabic symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. He used his book "Liber abbaci (meaning Book of the Abacus or Book of Calculating) to persuade the Europeans to use this system. In the book he explains how to do the 4 basic math operations (add, subtract, multiply, and divide) using his new system. Many of us today would find it comical that professional mathematicians had to learn these basic operations but we have to remember that this was a completely new system. It was like learning another language. You are not going to pick it up with a lot of hard work and studying. 

This new system allowed us to perform these basic operations much easier than using Roman Numerals. It also saved us alot of space because numbers like 1994 are writen as MDCCCCLXXXXIIII in Roman Numerals. If you want to see how difficult it was using Roman Numerals in Math follow this link http://turner.faculty.swau.edu/mathematics/materialslibrary/roman/

When we hear the name Fibonacci, the first thing that comes to mind is the Fibonacci numbers. That would make sense as it is mostly what he is known for (not as many people know he helped move us away from Roman Numerals). In his book "Liber Abbaci" put in the "Rabbit problem" and the solution below.

How Many Pairs of Rabbits Are Created by One Pair in One Year
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.

Here is the solution according to Fibonacci: 

Because the above written pair in the first month bore, you will double it; there will be two pairs in one month.
One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs;
of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month;
there will be 144 pairs in this [the tenth] month;
to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.
To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above written pair in the mentioned place at the end of the one year.You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above written sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.


From this we get the Fibonacci sequence of 1, 1, 2, 3, 5, 8, 13......etc. While Fibonacci studied the Arabic system of numbers, he also studied the arithmetic application of the new numbers from some other local countries. So it is quite possible that he did not invent the problem or the series that is named after him. He was simply the man who showed it to the rest of the world and that is probably why it is named after him.

The concept of the Fibonacci sequence gives us this idea of recursive. For something to be recursive, it means that it has to be based on something previously done. We can see that the Fibonacci numbers are recursive. It takes the 2 previous numbers, adds them together, and that gives you the next number.

As we can see Fibonacci was quite important. He allowed us to move away from Roman Numerals which saved us a lot of time. Imagine some of the crazy calculations we would have had to make if we were still using Roman Numerals. Mathematical proofs would be exponentially longer if they were written using Roman Numerals. Fibonacci was able to show the world the Fibonacci Sequence that also introduced us to the idea of something being recursive. He has had and will continue to have a lasting impact on the world of mathematics

Wednesday, May 25, 2016

Nature of Mathematics - What is a number?

If I asked you "What is a number?", you would probably laugh at me. You would say "that's an easy one." That's what me and many of my classmates thought when we were posed this question in our capstone math course. When we actually thought about the question, we realized we didn't really have a good answer. We all felt like this guy:


As we shared with the class, we all had similar but different definitions of what a number was. When I googled "What is a number?" the definition I got was:

Number: an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification. 

When I looked at this definition it made me wonder about things like infinity, zero, fractions, complex numbers, rational, and irrational numbers. Are they really numbers? 

To me there are only 2 categories that I listed which are not numbers. Those two are complex numbers and infinity. They words in the definition for me are SHOWING ORDER.

Infinity has no order. The idea of the word order in math is that if you take a number, add another number, then that new number is bigger than the original. An example would be if I have the number 1 and I add 0.2, I will get 1.2 which is larger than 1. For infinity that is not the case. Infinity + any number = infinity. Therefore infinity has no order and it not a number.

Complex numbers also have no order. Lets look at two complex numbers like 3i + 7 and 10i + 3. How do we know which one is larger? Do we assume the one that has a larger real part is larger? Do we assume the one with the larger imaginary part is larger? Do we add both imaginary and real parts together and that determines which is larger? WHO KNOWS!! Since we cannot order complex numbers I do not believe they are numbers.

As I was writing this post I became skeptical of irrational numbers being numbers. My initial thought was that irrational numbers are numbers because they can be put in an order. The problem is that irrational numbers go on forever without repeating therefore they can't be represented as a fraction which makes them nearly impossible to compare.  If the first 200 decimals of 2 irrational numbers are the same, but we don't know the 201st decimal, how do we know which is bigger? This question has left me puzzled and feeling confused like Homer. I really can't say definitively whether I believe irrational numbers fit my definition. If I had to choose I would say they are numbers because they can still be ordered when looking at a set number of decimal places.


What is a number? It is not an easy question to answer. We all have our own definitions and beliefs to what a number is. I believe that showing order is the most important part to being a number. That is why I believe that complex numbers and infinity are not actually numbers. Irrational numbers have me puzzled. It is possible for them to show order if we knew every decimal place but they can't be represented in the real world. Hopefully one day we will have the perfect definition for what a number is. I'm not sure if that day will ever come.

Wednesday, May 18, 2016

Doing Math: The Riddle of Diaphantus

In class we discussed a riddle and I wanted to discuss the way I thought about and solved the riddle.

The riddle: 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'

Mathematician Diophantus

The only foreseeable strategy I could see was to use algebra. I started off by assuming that "x" was going to equal the length of Diophantus' life. Then I broke the riddle down:

boyhood was 1/6 of life = (1/6)x
grew whiskers was 1/12 of life = (1/12)x
marriage was 1/7 of life = (1/7)x
after 5 years he had a son = 5
child reached half of father's age = (1/2)x
he died 4 years later = 4

Then I added all of these values together and set them equal to x which is the length of Dionphantus' life. Thus we get:

(1/6)x + (1/7)x + (1/12)x +(1/2)x + 5 + 4 = x
(25/28)x + 9 = x
9 = (3/28)x
x = 84

So I believe that Diophantus lived to be 84 years old.

A question posed was: can you solve this problem without using algebra. After discussing with my classmates we came to the conclusion that no it would not be possible. The only way to possibly get an answer would be to assume a length of childhood and extrapolate form there. But even then you will not be able to get a correct answer when using algebra.

An example would be to assume that boyhood was 12 years. Based on that you can assume that life was 72 years long. But now lets add up the values.

boyhood was 1/6 of life = 12
grew whiskers was 1/12 of life = 6
marriage was 1/7 of life = 10.28
after 5 years he had a son = 5
child reached half of father's age = 36
he died 4 years later = 4

This all adds up to 73.28 which is not equal to 72. Therefore the only way to solve this riddle is to

Wednesday, May 11, 2016

What is Math?

On the first day of my capstone math class at GVSU, our professor asked us "What is Math?" Initially I didn't know how to answer it. There are so many different aspects of math that there is no real way to give a clear cut definition. 

But I will make an attempt. To me "Mathematics is using formulas, equations, and processes to solve and analyze numerical problems".

That is my best attempt to capture all of mathematics. I know that many people have their own definitions, even my classmates had much different definitions. But we all had the same general idea which was to try and capture all of mathematics in one sentence.

With a subject as complex as math, I think we can all agree that there is no one clear cut answer to the question "What is Math?"

Mathematics has a great and wonderful history. Unfortunately I don't know a whole lot about the History of Math. In many of my classes we have learned about mathematicians and the work that they did. Those like Euclid who gave us Euclidean Geometry or Pythagoras who gave us the Pythagorean Theorem. Other names such as Archimedes, Euler, Fibonacci and Newton are some of the names that I can recall at this moment. I look forward to this semester and refreshing my knowledge about these great mathematicians. I also look forward to potentially learning new material that I may not have known about these mathematicians.