Book review

I read the book Vision in Elementary Mathematics by W.W Sawyer. I loved this book and felt that every teacher should read it who teaches mathematics to elementary or middle school students. This book was written awhile ago, but a lot of this information was very helpful.

The book started out by talking about odd and even numbers. There was a great explation and great pictures to go along with it. The most common example that could be used with elementary students is the one with the dots. For whatever number you have you create that many dots. Next you would create circles around the pairs of dots. If there are no more dots left you have an even number and if you have a dot remaining you have an odd number.

The author also talks about how in so many math classes you talk about conversions and the student actually has to do the math and figure out those conversions. The author says that most students don't remember what those different units area, for example, foot, inch, furlong, etc., the student only remembers how to convert those units. The author states that it would be better if those students could do those conversions but just have some basic ideas on what length those units actually are. 

One of the problems they had in this book was dealing with a problem that invloved a man and two sons. They talked for many pages about this problem and how they could go about this. This was a very interesting problem to me because there were so many different ways you could look at the problem. 

This book was a great read. I did feel that I had to read this book really fast in order to get through it for this class. Therefore, I will be going back and re-reading it while taking my time. That way I can learn even more! 

Who Is Better? Men or Women? (History of Math)

The answers in class to the question about if men or women are better in mathematics was interesting. I thought for sure many people would say men because that is the stereotype. When thinking about mathematics classes, I think about how many guys are in the classroom and how many women are in the class. Usually there are more men in the classroom and that shows you have more men go into this field or have mathematics as a major.

I knew that the typical stereotype is that men are better than women at mathematics, but after reading several articles this is something that has been proved wrong. A researcher and his colleagues at a University of Missouri wanted to attempt why there are more men than women in top level mathematic fields. The researchers found many studies that claimed "men are better at math"- believed to undermine women's math performance- had major methodological flaws, utilized improper statistical techniques, and many studies had no scientific evidence of this stereotype.

This stereotype was first published in 1999 in the Journal of Experimental Social Psychology. This is due to the stereotype that women are worse than men in math skills, females develop a poor self-image in this area, which leads to mathematics underachievement.

After reading several articles dealing with men and women and test scores, I have realized that this stereotype is not always true. Many studies stated that women scored better on test when guys weren't in the room. All the women test together and all the men test together. This seems to help the women and they score better than if they were taking the test with men.

There also was a study that stated in high school math classes the teachers weren't happy because many women weren't taking the higher math classes and only guys were. "Women perform as much as 12 percent better on math problems when tested in a setting without men, according to a study of Brown University undergraduates led by a graduate student of psychology".

I feel like this would be a great topic to come back to. There is so much information and studies that deal with this stereotype. It would be really cool to look at all the higher level math classes and look at how many men and women are in each one. We could also look at the grades that the men and women get in these classes. I would love to spend more time on this topic!

http://munews.missouri.edu/news-releases/2012/0118-%E2%80%9Cwomen-worse-at-math-than-men%E2%80%9D-explanation-scientifically-incorrect-mu-researchers-say/

https://www.brown.edu/Administration/News_Bureau/2000-01/00-023.html

Book Review

I was going to write a small summary of every chapter, but then I realized that would be a lot of information since there were so many chapters in this book. There were some chapters that were really small and I really didn't learn that much in that chapter. Therefore, I decided that I would break this review into the different parts of the book.

Part 1 - Numbers (1-6)
Part 2- Relationships (7-11)
Part 3- Shapes (12-16)
Part 4- Change (17-21)
Part 5- Data (22-24)
Part 6- Frontiers (25-30)

Part 1- Numbers 
In these beginning chapters they talked about negative and positive numbers and how difficult those numbers can be for students that are just learning about them. Textbooks, online sites, etc do so many thing possible to have the negative numbers standing out, printing the numbers in red and putting parenthesis around them so they stand out. Us, as mathematics have come up with all sorts of funny little mental strategies to side step the dreaded negative sign. We will try everything to get those negative numbers gone from our equation. The author states that maybe why we are so confused about negative numbers is because a negative times a negative equals a positive.
     3*1= (1) (1) (1)
     3*(-1)= (-1) (-1) (-1)
   
-1*3=-3
-1*2=-2
-1*1=-1
-1*0=0
-1*-1=?
We add 1 to the number before it, therefore the next number would be 1.

The Commutative Property- Is this law so obvious?
Look at the array of 3*7 and 7*3.  The number of dots are always the same, just the array is turned.

The last chapter of this part talked about Roman Numerals.
MDCCCVII - MDCCCLXXIV
            1807 - 1874
The Roman Numerals are hard to read and cumbersome.
What are some good ways to represent numbers?
 * Tallies, Roman numerals (this was a challenge)
 * Hindu-Arabic system - all numbers can be expressed with the same 10 digits, just placed in the right spot.
 * Place Value- Addition and Subtraction

In these chapters I really did learn a lot. Looking at the different types of numbers that many people used before our time was interesting. Some of these chapters were things that I already learned in other elementary math classes, but it was good to refresh my memory .

Part 2- Relationships
These chapters are the ones that I seemed to learn the most and enjoy the most. There were simple concepts in this chapter, that were put into prescriptive. Some of the diagrams and maps really seemed to help.

On page 47, they have a square that is 50x by 50x. We are always obsessed with solving for x. We do whatever is possible to move x to the other side or get x by itself.

Taking the square root of a negative. This is something that many people get confused about and this book does a great job on explaining that and showing pictures.

In chapter 9 they take apart the 50x by 50x square to show you how to come up with the answer and how the quadratic formula comes about. (This is shown in detail in my notes)

Part 3- Shapes
In the first part they start talking about the Pythagorean Theorem. There are many different types of proofs in this chapter. One of the proofs that shows the squares on the side of the side lengths is one that I actually proved in MATH 341, so I was very familiar with that one.

"The square on the hypotenuse is the sum of the squares on the other two sides"

The graph of the time of the sunset and the time of the sunrise was a very interesting graph. The person said that it looked like "opposing waves". That is true because that graph takes into affect the days growing longer and shorter with the changing of the season.


Part 4- Change 
The problem that interested me the most was the one about the hike through a snow covered field. This was interesting because it took into consideration the different paths that the hiker could take. The hiker didn't want to walk in the snow for that much time, so what would be the shortest way for him to walk without walking in the snow for that long? This is the question that the student would be trying to figure out. In chapter 17, the pictures really show a great representation of this. This could be a problem that the students could work out together and figure out which one would best suit the hiker.

Reading about vectors took me back to MATH 345 since we talked about vectors a ton in that class. Looking at the relationship between two different concepts or objects. Looking at how vectors can be added and subtracted was a different concept to me. This something that we have talked about, but it would be nice to look deeper into this concept. In chapter 21 it takes about the concept of Tennis. That is a good way to think about vectors. (Running velocity, ball sails wide, and aim down the line).

Part 5- Data
This is part of the book that I seemed to know a lot about already. I have taken many courses at Grand Valley that has talked about data, collecting data, analyzing data, and making conjectures about data. One problem that I found to be very interesting in these chapters was the question about breast cancer in chapter 23. This is a question that I read and was like, " How can we figure that out?" But after looking at the problem for a little bit I figured out what needed to be done. The author asked 24 doctors to answer this question to the best of their ability, and most of them didn't get this right. What is wrong? Why did so many well educated people get this wrong? This is something that is a hard concept for people to think about. Looking at the probability of different things and figuring out what part needs to go first and what part needs to come second, last, etc.

Part 6- Frontiers
I learned what a twin prime was in this section. A twin prime is a pair of prime numbers close to each other, almost neighbors, but between them is an even number, keeping them from touching. For example, 11-13, 17-19, 41-43, etc.

The graphs on page 205 are very interesting. The graphs show the number of primes on one graph and the number of odd numbers on the other graph. There is a lot to be said about these two graph and a lost of comparing that could be done.

I THINK THIS WAS MY FAVORITE PART OF THE BOOK. When they talked about mattress math in detail. When I was first reading this chapter I really didn't know where they were going to go with it. But after reading a little bit farther I was literally laughing out loud to myself. This is a great way for students to visually see the different flips and rotations. This is a visual activity that could be acted out and made into an art project. The students could physically make the beds and cut to show the vertical or horizontal flip or rotation.

Overall, this was a great book that had a lot of information in it. It was a little bit on the longer side so I have to read through it really fast to get finished with it. It would have been nice to take my time and go into detail about every chapter!

Important Role of Our Number System- (Nature of Mathematics)

When thinking about our number system a ton of different ideas come to mind. In our number system there are so many different concepts and rules that come to mind. In order to follow along with our number system you would have to know this rules in order to come up with an answer. So, with our number system , what are the different types of numbers that come to mind?

Natural Numbers- (also known as "counting numbers")- Infinitely many natural numbers- 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 12, 13 etc.

Whole Numbers- The natural numbers together with 0

Integers- The set of real numbers consisting of the natural numbers, their additive inverses, and zero is included.

Rational Numbers- These are the numbers that can we written as a ratio between two integers, written as a fraction.

Irrational Numbers- These are the numbers that can't be written as a fraction or ratio. In the decimal form, it will never repeat. You can get close to an integer but you won't be able to get that integer right now.

Real numbers- This is the set of all numbers containing all of the rational numbers and all of the irrational numbers. I usually remember this as being...all the numbers that are on the number line.

Complex Numbers- These are the numbers that use i, imaginary unit.

While looking up information about our number system, I came across this website that I couldn't stop looking at.
Fun Website!

On this website there are many different lessons for middle grade math. In the lessons students use interactive models and pattern blocks. Below are a list of some of the lessons.

Addition and Subtraction of Rational Numbers (Part 1 and Part 2)
Multiplication and Division of Rational Numbers
Repeating Decimals and Fractions
What is Root 2?
Comparing Pi's and Roots

These are just some of the lessons that the students could look at to help them understand our number system.

Amicable Numbers- Communicating

What are Amicable Numbers? (Communication Math)

When I first saw this question I really didn't know what it meant. This is not something that I have heard of, or at least remembered about. Therefore, I had to do some digging and figure out what this means. I found a couple different definitions for this.

Amicable numbers are a pair of numbers such that the sum of their proper divisors (not including itself) equals the other number.
     After reading this definition I thought I knew what they were talking about, but as I started to try and come up with some numbers I started having a hard time. Therefore, I had to research some examples before I could start to come up with my own!

(220, 284)- These numbers are said to be amicable numbers.

First we will take 220 and come up with all the proper divisors
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
If we add these numbers together 1+2+4+5+10+11+20+22+44+55+110=284

Now, we can come up with all the proper divisors for 284
1, 2, 4, 71, 142
If we add these numbers together 1+2+4+71+142=220

Therefore, these numbers are amicable numbers, or amicable pairs.

After looking at this example it really seemed to help. But, I couldn't help but wonder how difficult this would be to just pick two numbers. It would take me forever to pick two numbers and see if they were an amicable pair. So, I decided that I would look for another pair and see if there was a pattern.

We will take 2620 and come up with the proper divisors.
1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310
If we add these together 1+2+4+5+10+20+131+262+524+655+1310=2924

Now, we take 2924 and come up with the divisors
1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462
We add these together 1+2+4+17+34+43+68+86+172+731+1462=2620

Therefore, (2620, 2924) are an amicable pair.

I figured out that even after having the numbers of the amicable pair, it was difficult to find all the divisors. That actually took me a long time to do. I had to think about all the rules you learned in high school dealing with the different test.

This was a lot of fun to look up and learn. I had never heard of this before so it was interesting to learn!

Geometric Tessellation (Doing Math)

Doing Math

In class we worked on the Tessellations on paper and with pattern blocks. I started the above Tessellation in class, but I finished it over the weekend. I added a couple more patterns to the Tessellation and added color. The color really seemed to help. It makes the Tessellation look better and it's easier to see the pieces that are reflected and rotated. 

While creating this pattern I had to start a couple of times. I knew that I needed something that that would repeat, but the repeating needed to be something simple in order to go back and create the colors. This is something that took some time because I kept making it too hard and it was very hard to follow. When it is that hard to follow its very hard to figure out which color you need to use and the shapes keep going together. After thinking about my strategy, I realized that I needed to create simple shapes that don't overlap each other. When the shapes started to overlap it became difficult for an on looker to see the patterns and the tessellation. 

I didn't spend an hour on the above piece so I decided to create another Tessellation...but this time I was going to create a geometric Tessellation. I did this using an app for my iPad called Mandalar. 

This app was very easy to use and would be great to use with students in the classroom. I think this app could really help the students understand shapes and Tessellations. The only problem with this app is that it was really hard to fill in the shapes along the outside of the picture above, that is why you don't see the blue and yellow around the outside.

I have made tessellations from when I was a student in elementary class, but coming back to these really after going through the College of Education, I have a whole different meaning. While creating these tessellations I was thinking about how the students would enjoy this and what they would learn from it. It would be a great way to involve technology into the classroom and have the students work with this app. While creating these tessellation, like stated above, you have to think about what shapes you are going to put next and where the shape would make sense. I feel this would be a great idea for the students to learn and a great problem for the students to work through. 

This is an app that I will save and play along with and try to create even more Tessellations! 

Who is Euclid?

I spend my hour and a half reading articles upon articles about Euclid. There were so many concepts and ideas placed in these articles that I couldn't figure out what was true and what were rumors. Below is the information that I gathered about Euclid. Most of the facts and ideas seem to be true since I found them on more than one site, but since there are so many rumors out there, some of these might be rumors.

Who is Euclid?
"Father of Geometry"
Not too much is known about Euclid's early life. He was born around 330 B.C. in Alexandria. Euclid's life has been confused with the life of another Euclid, therefore it makes it difficult to believe any of the trustworthy information about this mathematician.

Euclid's Career
He is known as the "Father of Geometry" for a reason. He discovered the gave and really gave it its form. Euclid spent most of his time at the Alexandria library where he did most of his studies and thinking. He started his studies focusing on geometry. " He began developing his theorems and collated it into a colossal treatise called "The Elements"." "The Elements" sold more copies then the Bible and was used numerous times by mathematicians and publishers. There was no end to Euclid's geometry, and he continued to develop many different theories and theorems based on numbers, basic arithmetic, etc. "The Elements" was the first kind of geometry that people of the modern era developed.

Axioms
"Euclid stated that axioms were statements that were just believed to be true, but he realized that by blindly following statements. there would be no point in devising mathematical theories and formulae." He realized that axioms needed to be proved by proofs. Next, he started to develop logical evidences that would stand against his axioms.

Euclid has a lot of other work as well. He has a wide range of other works actually that are still used and referred to this day. These other works were positions backed with solid proofs.

Death and Legacy
Mankind doesn't know when or why Euclid died. I is said that he might have passed away around 260 B.C.His books, theorems, axioms, and proofs are still used to this day. He left behind a legacy that would be used in mathematics for a numerous amount of years.

http://www.britannica.com/EBchecked/topic/194880/Euclid
http://www.thefamouspeople.com/profiles
http://en.wikipedia.org/wiki/Euclid