In an 8th grade mathematics classroom students are between the ages of 13 and 14. In an 8th grade mathematics class students will have a lesson on the volume of a cone, sphere, and cylinder.

Most likely the class that I will be instructing in the future will be similar to the school I am in now. Kentwood Public Schools are very diverse. When taking in all of the classes I participate in and students in the hallways, I would say that there are approximately

70% African American

5% Asian

25% White

On average there are either twice as many females in a class or twice as many boys in one class. So, for the hypothetical classroom I’ll say there are two males for every female. Majority of student’s attitudes towards school in general is that they do not want to be there. Especially in the inner city schools where a lot of students have a lot of other issues in life and school is not their main worry. When talking specifically about math, most students think that it is a waste of time and it is impractical. Students mainly find math to be too difficult and not fun at all.

Most boys around this age are interested in some type of sport, mainly football, baseball and soccer. A lot of kids play video games and have certain television shows they watch every week such as the walking dead. A lot of girls are interested in sports such as volleyball and soccer. This is also the age where girls are becoming interested in their appearance and fashion. Both girls and boys are hitting the stage where they want to be accepted so they are very impressionable.

]]>Most likely the class that I will be instructing in the future will be similar to the school I am in now. Kentwood Public Schools are very diverse. When taking in all of the classes I participate in and students in the hallways, I would say that there are approximately

70% African American

5% Asian

25% White

On average there are either twice as many females in a class or twice as many boys in one class. So, for the hypothetical classroom I’ll say there are two males for every female. Majority of student’s attitudes towards school in general is that they do not want to be there. Especially in the inner city schools where a lot of students have a lot of other issues in life and school is not their main worry. When talking specifically about math, most students think that it is a waste of time and it is impractical. Students mainly find math to be too difficult and not fun at all.

Most boys around this age are interested in some type of sport, mainly football, baseball and soccer. A lot of kids play video games and have certain television shows they watch every week such as the walking dead. A lot of girls are interested in sports such as volleyball and soccer. This is also the age where girls are becoming interested in their appearance and fashion. Both girls and boys are hitting the stage where they want to be accepted so they are very impressionable.

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There was a study done in Japan, Germany and the United States, where teachers were video taped and researchers studied the videos to find out what the differences were among the teaching styles. The conclusion was that Germany has a "developing advanced procedures" teaching style, Japan has a "structured problem solving" teaching style, and the United States has a "learning terms and practicing procedures" teaching style.

The teaching style I have observed thus far in my class (mind you I have only been in class for a total of 5 days) is the American Style of teaching. What I mean by this is that the teacher goes over a simple process and expects the students to copy down the process. She then does some examples and goes over definitions and assigns the students homework problems that mirror the practice problems she had already done with the class. The main difference I have noticed is the type of questioning she asks the students and the type of questions in the book. The book asks more conceptual questions such as is this example true or false and why. Putting the why in there is asking students to be able to explain, which will demonstrate the students complete understanding. Also, during the initial note taking and examples, the teacher tends to ask a lot of conceptual questions such as "What is the difference between (-4)^10 and -4^10?" and "compare and contrast an integer and a whole number." It is this type of questioning that puts a little more of the "developing advanced procedures" technique, because the students will be able to carry their conceptual knowledge to more advanced questions. Another thing is the teacher will bring up more challenging examples or concepts that are not in the book, such as "Can we write (2)(2)(2)(a)(a) in exponential form?" This requires students to fully understand the concept of base, exponent and how the grouping works. Students came up with a variety of answers, but in majority of the classes students were able to come up with (2^3)(a^2). I say it is more of an American style teaching because although she introduces a more complicated problem, she still goes mainly from the book, and does not start off the class with any complicated problems to allow students to come up with their own questions. ]]>

With out the work of Galileo who knows where the development of our country would be. The understanding and mathematics behind gravity is essential to so many things that our world uses daily, such as planes, guns, and many other things that have become such a huge part of life. With out this understanding our world would be completely different. |

Nature Of Mathematics Click Here

Nature of Mathematics Click Here

]]>Now I know that sounds impossible, but the way I look at it is that math today is not just what we do, but how we do it. So yes, mathematicians discovered a lot of things about the world and used those discoveries to make Theorems, but as far as the way we communicate mathematics such as "6+2=8" is invented. I say this because it can change at anytime, just like the Roman Numeral system changed. Our symbols and language of mathematics is invented, but their meanings are discovered. Before this class, I would have said that mathematics is invented, not realizing that all of the mathematical foundation that I have been using since I was in kindergarten is based off of my surroundings. I think that this is a very important aspect to think about, and being that I am a future educator I would love to incorporate this type of critical thinking in my classroom.

I think that students these days view math as the enemy because to them it is a list of rules to constantly follow and if you mess up a little at any point the whole problem is wrong. I want students to realize how important mathematics is and how much more there is to it other than steps and procedures. In my class we did plenty of activities such as bungee jumping, magic squares, tessellations etc. that all dealt with using the things around us to learn off of. All of the activities were very beneficial in learning and they definitely made me ask myself more questions, which is what I want my students to do. From doing all of these activities and gaining the background knowledge of these mathematicians I was able to form an answer to the questions "is math invented or discovered" with a reasoning! I would love to incorporate this question in my classroom so students are able to say what they think and have a good reasoning. The best part about this question is that there is no right or wrong answer, as long as the person has a good reasoning. Critical thinking and reasoning is very important for students to learn and I believe that using activities and questions that need reasoning is a good place to start. ]]>

Another great woman mathematician is Hypatia. She is one of the first women to study and teach math, astronomy and philosophy. It is believed that she was born between 350 and 370 in Alexandria, Rome. The story is that a mob of Christian zealots led by Peter the Lector accosted a woman’s carriage and dragged her from it and into a church, where they stripped her and beat her to death with roofing tiles. They then tore her body apart and burned it. Her father was a well known astronomer and mathematician as well. It was known that Hypatia had a difficult life growing up due to the culture she grew up in. Hypatia big mathematical achievement was the design of an astrolabe, a kind of portable astronomical calculator that was used until the nineteenth century.

Now, if you are like me I know you are thinking so what, this all happened so long ago it doesn't affect me. But it does. Even in today's society women are strongly discouraged to achieve highly in mathematics, and even if a woman does make an achievement they are not recognized on the same level as the males. This happens in the classroom, media, even family. There have been studies that show in our subconscious, even women teachers do not give female students the same amount of time or encouragement as they do to male students. We need to be aware of the difficulties of our previous female mathematicians and notice how important their accomplishments in order to keep moving forward to improve our society.

(Read more: http://www.smithsonianmag.com/womens-history/hypatia-ancient-alexandrias-great-female-scholar-10942888/#OrAyij9ujIfgRul2.99)

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By: Jennifer Lezell

Rating 1-5: 3.5

Mathematician’s Lament is a very different book in some ways and a very similar book in others. One way it is different is the style of communication. The style of writing in this book goes from the thoughts of the author, Paul Lockhart, to dialogue between two people, Simplicio and Salviati, which contained the answers to the questions I was thinking while reading this book. This book is also different because it contains many different thoughts about mathematics that I have never thought about before, such as “7” is just a symbol, not a number but what the number

I have had many different readings on the education system and one thing is for sure, no one is happy with the education system. Paul Lockhart explains what is wrong with the education system, from teaching style to curriculum. He throws good, real life problems, in the book to make it more interesting. He explains not only what would make the education system better but why it would make it better, which is a very common type of reading among the education readings.

Paul Lockhart did a good job communicating his thoughts and reasoning. He is very persuasive and easy to understand in this book. He makes the arguments valid by bringing problems into the text to explain his thinking. There are plenty of different ideas in this book about mathematics that will help keep the reader’s attention. When reading this book I felt passion, which is something not very common among books about mathematics. The style of writing made this book fun to read. There were so many different, off the wall, ways that he explained mathematics that I just had to keep reading to see where he was going with these crazy references and stories. Also, Paul Lockhart has such a different perspective on the way that mathematics should be taught.

Although this book brought up many good pointers and ideas, there are a few things that could have been done better. I felt as though this book, along with many other books, is too unrealistic. It even states that the types of changes that are suggested are impossible to do in the schools. After that, I felt as though I had wasted my time reading that because I can’t change it. Also, I feel as though this book was written by a mathematician only, not an educator. Paul Lockhart is correct on his idea of what an educator is, but the author is not realistic with how the educator should teach. The suggestions are very unrealistic when there is one teacher and thirty kids in a classroom with 45 minutes a day. I would say for the most part that his advice on what to change in the schools is very impractical. Many ideas had been repeated at least twice. Between discussing the classrooms and the curriculum, the same ideas were brought up about what is wrong and why it is wrong. Stating that mathematics is an art, after about the fifth time, became a boring idea to me because I knew what would be said next. There should have been real life examples or times when the author used this teaching method on students and some type of data or evidence to support the claims. There was no evidence or real life examples to show that the argument on how to fix the education system would work at all.

Contrary to Paul Lockhart, no matter what class students are in, vocabulary is an important concept and it helps when communicating to others. The idea that we do not need the definitions in the math classroom that we teach and use today would cause tremendous communication problems in the math world. Imagine if no one came up with the number system that we have today. The common understanding of the symbols of the number system makes it a lot easier to communicate with others what we are doing and saying. I also believe it helps advance the math world because without this common language of mathematics it would have made it very difficult for different mathematicians to communicate their ideas.

A mathematician’s Lament is an intriguing book with many of ideas, but I believe that reading a more practical book that can give ideas that are applicable would be more beneficial for the education world. This book is more beneficial for the gain of knowledge in the mathematics world than the education world.

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Now, when majority of people think about math they think that there is no evidence or reasoning. Most people just think that there are just facts in math because someone a long time ago thought it would be a good idea. This is thought is completely wrong. Math is a science. Mathematicians have used many different experiments and collected a lot of data in order to make conclusions that we now-a-days take for granted and understand to just be common facts. For example, pi. Many people do not know why pi is 3.14159265359... but they know that when they plug it into a calculator using a formula they get the answer they need, such as the volume of a sphere. The crazy thing is that a mathematician, Archimedes, started with similar rectangles and found the ratios of the perimeter to the longest side to always be the same. He then thought about an octagon, decagon, and shapes with so many sides that it gets closer and closer to the shape of a circle. As he kept finding the ratio of perimeter to the longest side, he noticed the ratio was approaching 3.14159... what we know today as pi. When we think about the "perimeter" of a circle we think circumference, and the longest side is the diameter.

There are many other mathematicians who have done many and many of different experiments to determine theorems in mathematics. Another reason I find mathematics to be a science is because in the end there is one answer, like in science there is one answer. For example, when a science teacher asks the names of the planets there are exactly 9 planets he/she would expect the student to say, in no certain order "Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto." In mathematics, when we ask what is the square root of 36 we expect to hear positive or negative 6. Yes, there are many different types of questions in both subjects that may allow for room to reach a conclusion in different ways, but in the end, the conclusion should be the same.

In the end, although math and science are different in some way, I believe the main concepts of both are aligned well making math a science. There have been many historical mathematicians and scientists who have spent their whole lives devoted to finding evidence, proof, and data to reach conclusions and it is those conclusions that allow us to use science, including math, in our everyday lives.

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4*0= 3*0= 2*0= 1*0= 0*0=

4*1=4 3*1=3 2*1=2 1*1=1 0*1=

4*2=8 3*2=6 2*2=4 1*2=2 0*2=

4*3=12 3*3=9 2*3=6 1*3=3 0*3=

4*4=16 3*4=12 2*4=8 1*4=4 0*4=

Notice, I left all of the answers for the multiples of zero blank. That is, because I want you to look at the pattern among the answers and tell me what you think the next answer should be off the patterns shown, not off of memorization. What patterns do you see? Now, lets look at what the rest of the chart would look like based on this pattern.

3*-3=-9 2*-3=-6 1*-3=-3 0*-3=0 -1*-3=3 -2*-3=6 -3*-3=9

3*-2=-6 2*-2=-4 1*-2=-2 0*-2=0 -1*-2=2 -2*-2=4 -3*-2=6

3*-1=-3 2*-1=-2 1*-1=-1 0*-1=0 -1*-1=1 -2*-1=2 -3*-1=3

3*0=0 2*0= 1*0= 0*0= 0 -1*0=0 -2*0 =0 -3*0=0

3*1=3 2*1=2 1*1=1 0*1=0 -1*1=-1 -2*1=-2 -3*1=-3

3*2=6 2*2=4 1*2=2 0*2= 0 -1*2=-2 -2*2=-4 -3*2=-6

3*3=9 2*3=6 1*3=3 0*3=0 -1*3= -3 -2*3=-6 -3*3=-9

What I want you to imagine is trying to figure out from all positive values what the next number is going to be. This is how the historical mathematicians viewed multiplying. They were able to notice that pattern among the rows and columns allowing them to place the negative or leave it out. They noticed, that the column starting from the bottom was decreasing by three, so once it got to zero, it would be zero, and when it got to 3*-1 it would have to be -3 because 0-3=-3 and so on. This pattern also emerges when looking at the rows going from left to right. Notice, the numbers are decreasing by 3 making 0*3=0 since 3-3=0. Now, observe the whole rows and columns of zero, this is due to the fact that 0-0=0. Therefore, the previous number subtracted the multiple is zero. I hope this chart helps make sense of the pattern among numbers as much as it has helped me!

Through patterns and consistencies, mathematicians have been able to make great discoveries, and although negatives have just been understood as a fact, we should try to make some sense of it. It is patterns and consistencies like these and many others that have allowed us to use math in so many different ways to evolve the world to what it is today. Although this is not how mathematicians found these properties, it helps us explain them. ]]>