To help with my exploration of this question, I asked some friends and acquaintances for their answers. Here are some of the answers people gave me:
"Math helps to make the world make sense in formulas."
"Math is the study of calculation and the language of science."
"Math is hard."
"Math is using derived equations and functions to solve numeric problems."
"Math is working with numbers."
"Math is the abstract study of quantity, using letters, numbers and structures to solve equations."
To me, math is necessary. The world and everything in it has to do with math. I do not have a great background in math but I do know that it is all around me. Even the computer I am typing on right now is creating binary equations to allow me to do everything I need to do for my homework.
In order to further explore this question, I did some research. I found this article (http://www.livescience.com/38936-mathematics.html) by Elaine J. Hom, who says : "Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building blocks for everything in our daily lives, including mobile devices, architecture, art, money, engineering, and even sports." Hom seems to have considered this question a fair bit and I learned a lot from reading her article. Although I had known that math has been around since the beginning of time, I never considered the fact that society demands mathematics - and the more complex the society, the more complex the math needs to be. I also had not considered how old math is. You tend to think of people in ancient times as not intellectually advanced because of their lack of world knowledge and technology. But algebra, that branch of mathematics that had me tearing out my hair in high school, was invented in the 9th century. The NINTH century! So when the Vikings invaded Ireland and Scotland, people were doing algebra. That alone amazes me. The type of number manipulations and equations that those ancient people where able to do, 12 centuries ago, is the same thing that people are learning today in high schools. Hom has all sorts of information in this short, very readable article that I had never known about math until now.
To expand upon the subject in question, I then asked "what does it mean to do mathematics?" Some people gave me these answers:
"It means to work with numbers to solve an equation."
"It means writing down something on an exam to try and make it look like you know what you're doing."
"It means using equations and formula to get from a question to an answer."
To expand upon the subject in question, I then asked "what does it mean to do mathematics?" Some people gave me these answers:
"It means to work with numbers to solve an equation."
"It means writing down something on an exam to try and make it look like you know what you're doing."
"It means using equations and formula to get from a question to an answer."
"It means measuring, counting and calculating just about everything in your everyday life."
Then I decided to see what Google had to say about this. Keith Devlin of Stanford University (http://www.maa.org/external_archive/devlin/devlin_04_05.html) writes on his blog "'Doing math' involves all kinds of mental capacities: numerical reasoning, quantitative reasoning, linguistic reasoning, symbolic reasoning, spatial reasoning, logical reasoning, diagrammatic reasoning, reasoning about causality, the ability to handle abstractions, and maybe some others I have overlooked. And for success, all those need to be topped off with a dose of raw creativity and a desire - for some of us an inner need - to pursue the subject and do well at it." This was the first time I ever considered math as being an outlet for creativity. I suppose if you had a mathematician's level of knowledge, you could look at a problem and get creative, coming up with lots and lots of different ways to answer the one question. Then Dr. Stordy read us a book in class called The Math Curse and I saw very clearly, in the form of a children's book, another way to get creative with math. The book used math in unconventional ways and talked about all the different ways people can use math in the run of a day but generally don't, or do and don't even realize they're doing it, such as when they are buying something or deciding who gets how many cupcakes.
The third thing I asked my subjects was "If you are thinking mathematically, what exactly are you doing?" I received these answers:
"You're calculating in your head."
"You are doing equations for little things without even realizing it."
"You think in mathematical terms."
"You come up with the most logical answers to questions by having done the math in your head."
"You can solve problems really well and really quickly because your mind is good at it and used to it from all the time you've spent doing math."
I actually think there might be something to that last answer, although at first glance it doesn't seem to make a whole lot of sense. I think thinking mathematically really means that your mind works in such a way that it does small equations and calculations without it being obvious and you can understand why things are the way they are without having to write everything down. For example, a child without a mathematical mind might see a tall skinny glass of water, and a short fat empty glass, and decide that there is no way that the water is going to fit into the short glass because the tall glass is 'so much bigger'. A child with a mathematical mind will probably have no issue knowing that it's going to fit just fine because the volume of water is the same, just the shape of the container is changing. To think mathematically means that we probably wouldn't even think twice about this because it just makes sense to us, mathematically.
Thinking mathematically, on one extreme, I think is like thinking in another language. This would apply to all those mathematicians out there - the Sheldon Cooper's of the world. He is a physicist but also a mathematician and some of his sentences are so convoluted, you have to really think about what he's saying to decipher it. I understand he is a fictional character, but I actually know some people that are near his level of genius and that is how they speak. They inadvertently make everyone around them feel inferior because their language is so complex. Keith Devlin of Stanford University is seen here in a video in which he is introducing his online course that he began offering in 2012. Ignoring the information about the course, the way he explains and defines mathematical thinking is very straightforward and sums up just about everything I've read on the topic - thinking mathematically is about thinking outside the box. http://www.youtube.com/watch?v=YFs06zgBfMI
Stay tuned for thoughts from our class discussion!!
This is a link to an e-book called Thinking Mathematically which I found online, after reading an article about how amazing this book is. Here is the link if anyone is interested:
http://f3.tiera.ru/2/M_Mathematics/MPop_Popular-level/Mason%20J.,%20Burton%20L.,%20Stacey%20K.%20Thinking%20Mathematically%20(2ed.,%20AW,%202010)(ISBN%209780273728917)(O)(265s)_MPop_.pdf
This is a link to another e-book called Learning to Think Mathematically. This is a textbook written by professors of the graduate education program in California University. It does a fantastic job of helping readers to understand mathematical thinking, what it is, how it came to be, and how to use it.
http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf
Then I decided to see what Google had to say about this. Keith Devlin of Stanford University (http://www.maa.org/external_archive/devlin/devlin_04_05.html) writes on his blog "'Doing math' involves all kinds of mental capacities: numerical reasoning, quantitative reasoning, linguistic reasoning, symbolic reasoning, spatial reasoning, logical reasoning, diagrammatic reasoning, reasoning about causality, the ability to handle abstractions, and maybe some others I have overlooked. And for success, all those need to be topped off with a dose of raw creativity and a desire - for some of us an inner need - to pursue the subject and do well at it." This was the first time I ever considered math as being an outlet for creativity. I suppose if you had a mathematician's level of knowledge, you could look at a problem and get creative, coming up with lots and lots of different ways to answer the one question. Then Dr. Stordy read us a book in class called The Math Curse and I saw very clearly, in the form of a children's book, another way to get creative with math. The book used math in unconventional ways and talked about all the different ways people can use math in the run of a day but generally don't, or do and don't even realize they're doing it, such as when they are buying something or deciding who gets how many cupcakes.
The third thing I asked my subjects was "If you are thinking mathematically, what exactly are you doing?" I received these answers:
"You're calculating in your head."
"You are doing equations for little things without even realizing it."
"You think in mathematical terms."
"You come up with the most logical answers to questions by having done the math in your head."
"You can solve problems really well and really quickly because your mind is good at it and used to it from all the time you've spent doing math."
I actually think there might be something to that last answer, although at first glance it doesn't seem to make a whole lot of sense. I think thinking mathematically really means that your mind works in such a way that it does small equations and calculations without it being obvious and you can understand why things are the way they are without having to write everything down. For example, a child without a mathematical mind might see a tall skinny glass of water, and a short fat empty glass, and decide that there is no way that the water is going to fit into the short glass because the tall glass is 'so much bigger'. A child with a mathematical mind will probably have no issue knowing that it's going to fit just fine because the volume of water is the same, just the shape of the container is changing. To think mathematically means that we probably wouldn't even think twice about this because it just makes sense to us, mathematically.
Thinking mathematically, on one extreme, I think is like thinking in another language. This would apply to all those mathematicians out there - the Sheldon Cooper's of the world. He is a physicist but also a mathematician and some of his sentences are so convoluted, you have to really think about what he's saying to decipher it. I understand he is a fictional character, but I actually know some people that are near his level of genius and that is how they speak. They inadvertently make everyone around them feel inferior because their language is so complex. Keith Devlin of Stanford University is seen here in a video in which he is introducing his online course that he began offering in 2012. Ignoring the information about the course, the way he explains and defines mathematical thinking is very straightforward and sums up just about everything I've read on the topic - thinking mathematically is about thinking outside the box. http://www.youtube.com/watch?v=YFs06zgBfMI
Stay tuned for thoughts from our class discussion!!
This is a link to an e-book called Thinking Mathematically which I found online, after reading an article about how amazing this book is. Here is the link if anyone is interested:
http://f3.tiera.ru/2/M_Mathematics/MPop_Popular-level/Mason%20J.,%20Burton%20L.,%20Stacey%20K.%20Thinking%20Mathematically%20(2ed.,%20AW,%202010)(ISBN%209780273728917)(O)(265s)_MPop_.pdf
This is a link to another e-book called Learning to Think Mathematically. This is a textbook written by professors of the graduate education program in California University. It does a fantastic job of helping readers to understand mathematical thinking, what it is, how it came to be, and how to use it.
http://hplengr.engr.wisc.edu/Math_Schoenfeld.pdf
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