Thursday, April 15, 2010
Wednesday, February 3, 2010
Make Your Own Fractal!
Earlier in the year we learned about Fractals. For our last blog assignment I want us all to do something FUN. I want us all to make our own Fractals!
Go to the above website and create your own pattern. Turn it into a fractal and watch the magic happen!
Print out your original pattern, the final piece, and name it. Bring it into class and we will share our geometric "works of art."
Experimentation or Memorization?
When you think about Mathematics, which of these words comes to mind?
Experimentation?
or
Memorization?
When I was a child, my parents were always so impressed at how "advanced" I was in mathematics for such a young boy. To them, it was like I had some sort of a gift. In 7th grade, however, something changed. My math teacher and many of her colleagues were convinced that I just didn't "get it." They would administer tests; I would solve the problems I was given and I would usually (almost always) come up with the right answer, but when I got the test back... it was marked WRONG! "Why?" I would ask.
"Because you are not doing it the way it was taught." she would say "You are coming up with the right answer, but you are not following the steps that we taught you. We are not convinced that you know what you are doing."
In 8th grade, when all of my friends were advancing to their first Algebra class, I was stuck in a class known as "Intro to Pre-Algebra!" INTRO to PRE Algebra?! It was there that I learned to conform, and to do the equations as they were taught. I was taught that Mathematics is a series of Memorization - not Experimentation!
But is this true? How could I have always gotten the right answer even when I didn't memorize the steps? Was I just a good guesser? What do you think?
Is math Memorization or Experimentation? Reflect back on some of your experiences with math. Did your teachers promote Experimentation or Memorization?
As you ponder this question, take a look at the discoveries THESE children have made:
http://www.mathman.biz/html/discover.html
Experimentation?
or
Memorization?
When I was a child, my parents were always so impressed at how "advanced" I was in mathematics for such a young boy. To them, it was like I had some sort of a gift. In 7th grade, however, something changed. My math teacher and many of her colleagues were convinced that I just didn't "get it." They would administer tests; I would solve the problems I was given and I would usually (almost always) come up with the right answer, but when I got the test back... it was marked WRONG! "Why?" I would ask.
"Because you are not doing it the way it was taught." she would say "You are coming up with the right answer, but you are not following the steps that we taught you. We are not convinced that you know what you are doing."
In 8th grade, when all of my friends were advancing to their first Algebra class, I was stuck in a class known as "Intro to Pre-Algebra!" INTRO to PRE Algebra?! It was there that I learned to conform, and to do the equations as they were taught. I was taught that Mathematics is a series of Memorization - not Experimentation!
But is this true? How could I have always gotten the right answer even when I didn't memorize the steps? Was I just a good guesser? What do you think?
Is math Memorization or Experimentation? Reflect back on some of your experiences with math. Did your teachers promote Experimentation or Memorization?
As you ponder this question, take a look at the discoveries THESE children have made:
http://www.mathman.biz/html/discover.html
Careers in Mathematics OR Mathematics in Your Career
When it comes to people who study mathematics, there can be two types of people:
One type of person actually enjoys math (a person like me). This type of person studies mathematics because they find it intellectually stimulating. They believe in math and the answers it beholds. This type of person may want to go on to study math in college.
Another person, however (and for some strange reason) doesn't like math! This person studies math because they feel like they have to, not because they want to. They will do what they must to achieve a goal (whether that be a high school diploma, a college degree, any type of certification, etc.) - and if studying math is a means toward that goal, they will do what they must.
And some people are a little bit of both, but for the purposes of this blog assignment, I would like you to choose which one of these people sounds most like you. Do you truly enjoy math? Do you think you might be interested in learning about math to the depths of calculus and beyond? Or do you simply study math because you have to?
For those of you that would study math by choice, do some research. Find out what types of careers one might pursue with a degree in mathematics (Bachelor's Degree, Master's Degree, or Doctorate). Do any of these careers peek your interest? Comment back with a list of careers you have found and pick one that you think you would enjoy. Describe the amount of study that is required and what a person in that field actually does for a living.
For the rest of you. Start by choosing a career that you think you would enjoy. Essentially, start with an end and describe the means by which you would get there. (For example: If you want to be a Doctor, how would you begin? What types of degrees are required? And most importantly, to what level of mathematics would you have to study and how much mathematics would you use on the job.)
I will start by giving you MY outline.
Though I do enjoy mathematics, I did not go to college with a strong curiosity or interest in studying mathematics. So for the purposes of this assignment I will start with my "end."
Goal - Become a College Professor in the field of Educational Psychology.
As a college professor, I would like to teach some form of educational psychology.
To become a college professor in the field of Education, I must first have educational experience.
Choice of Educational Experience - High School Mathematics
Levels of education/experience required -
College Professor -
1. Bachelor's Degree in either Psychology or Education
2. Master's in Education
3. Doctorate in Educational Psychology
4. Teaching Experience (Choice - High School Mathematics)
Teaching Experience -
1. Post Baccalaureate in Teacher Education
2. Certification in Mathematics
Level of Mathematics Required -
1. Calculus I, Calculus II, Calculus III
2. Three upper level math courses (Linear Algebra, Differential Equations, & Foundations of Mathematics)
In chronological order in looks like this...
1. Bachelor's in Psych. - 4 years
2. Post Baccalaureate + Cert. in Math - 3 years
3. Teaching Experience - ? years
4. Master's in Education - 4 years (part-time)
5. Doctorate in Educational Psychology - 4 years(part-time)
On the job - As a college professor I will be required to do a lot of research in my field. A lot of research means a lot of STATISTICS, so as you can see not only will math be important as I teach it, but will continue to carry great significance throughout my career.
One type of person actually enjoys math (a person like me). This type of person studies mathematics because they find it intellectually stimulating. They believe in math and the answers it beholds. This type of person may want to go on to study math in college.
Another person, however (and for some strange reason) doesn't like math! This person studies math because they feel like they have to, not because they want to. They will do what they must to achieve a goal (whether that be a high school diploma, a college degree, any type of certification, etc.) - and if studying math is a means toward that goal, they will do what they must.
And some people are a little bit of both, but for the purposes of this blog assignment, I would like you to choose which one of these people sounds most like you. Do you truly enjoy math? Do you think you might be interested in learning about math to the depths of calculus and beyond? Or do you simply study math because you have to?
For those of you that would study math by choice, do some research. Find out what types of careers one might pursue with a degree in mathematics (Bachelor's Degree, Master's Degree, or Doctorate). Do any of these careers peek your interest? Comment back with a list of careers you have found and pick one that you think you would enjoy. Describe the amount of study that is required and what a person in that field actually does for a living.
For the rest of you. Start by choosing a career that you think you would enjoy. Essentially, start with an end and describe the means by which you would get there. (For example: If you want to be a Doctor, how would you begin? What types of degrees are required? And most importantly, to what level of mathematics would you have to study and how much mathematics would you use on the job.)
I will start by giving you MY outline.
Though I do enjoy mathematics, I did not go to college with a strong curiosity or interest in studying mathematics. So for the purposes of this assignment I will start with my "end."
Goal - Become a College Professor in the field of Educational Psychology.
As a college professor, I would like to teach some form of educational psychology.
To become a college professor in the field of Education, I must first have educational experience.
Choice of Educational Experience - High School Mathematics
Levels of education/experience required -
College Professor -
1. Bachelor's Degree in either Psychology or Education
2. Master's in Education
3. Doctorate in Educational Psychology
4. Teaching Experience (Choice - High School Mathematics)
Teaching Experience -
1. Post Baccalaureate in Teacher Education
2. Certification in Mathematics
Level of Mathematics Required -
1. Calculus I, Calculus II, Calculus III
2. Three upper level math courses (Linear Algebra, Differential Equations, & Foundations of Mathematics)
In chronological order in looks like this...
1. Bachelor's in Psych. - 4 years
2. Post Baccalaureate + Cert. in Math - 3 years
3. Teaching Experience - ? years
4. Master's in Education - 4 years (part-time)
5. Doctorate in Educational Psychology - 4 years(part-time)
On the job - As a college professor I will be required to do a lot of research in my field. A lot of research means a lot of STATISTICS, so as you can see not only will math be important as I teach it, but will continue to carry great significance throughout my career.
Monday, February 1, 2010
Mathematics in Nature
Has anyone ever heard of the Fibonacci Numbers? They are a sequence of numbers that follow a fairly simple pattern.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and so on.
Figure out the pattern yet?
If you haven't, here's the spoiler: each number in this sequence is the sum of the 2 previous numbers. (0+1=1, 1+1=2, 1+2=3, 3+2=5 etc.)
Sounds pretty simple right? Well what if I told that this pattern of numbers may be the most important sequence of numbers ever discovered? What if I told you that this sequence of numbers holds the key to understanding the mathematical processes of our natural world? What if I told you that the Fibonacci
Numbers appear in nature more frequently than any other pattern of numbers? Would you be curious?
The fact is, Fibonacci numbers ARE found in nature quite frequently. Perhaps what is most prevalent is the ratio of these numbers...
If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:
1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666..., 8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538...It seems that as we continue along this pattern, we seem to be narrowing down to one specific number. That number is known as "Phi" - but in numerical terms it is about 1.618033.
This ratio we are referring to is also known as the "Golden Ratio" and can be found almost everywhere in nature!
Try Googling "Golden Ratio" or "Fibonacci Numbers." Report back with some interesting photos and/or natural patterns you have found.
Extra credit - What is the relationship between this photo below and the Fibonacci Numbers? (E-mail me the answer to this question, or hand in a hard copy in class.)
Everyday Mathematics
Everyday, whether we realize it or not, we use some form of mathematics. For the purposes of this blog entry, I would like you to make a log. Choose one day to begin logging every instance in which you use mathematics. You might be surprised, if you really pay attention, to how much math you use without even knowing it. Here is an example of my log:
7:45 Am - Wishing to spend less than $10 on breakfast, I add the prices of my items before I purchased them this morning. 5 hour energy - 2.99, Soft Pretzel - .65, Breakfast sandwich - 1.99, Large Bottled Water - 1.35. Without actually adding the numbers, I rounded, and found I'd be spending about $7.
8:05 Am - Will I be late? I need to get to class by 8:30. On the parkway this morning I wondered if I would have enough time to get to class on time. I started at mile marker 25 and needed to get to mile marker 45 in 25 minutes. If I obeyed the speed limit at 65 mph, could I make it on time? 60 mph is the same as saying 1 mile for every minute. So assuming I go 60 mph the whole way, I would be there in 20 minutes. Considering I go about 75 on the parkway, I have plenty of time.
9:45 Am - I have 4 hours between classes with 3 assignments to do. Subtract an hour for food breaks and bathroom breaks and I should spend no more than 60 minutes on each assignment.
2:10 Pm - Calculus class. =D
6:00 Pm - Stop for gas on the way home. I drove 405 miles from my last fill up and (according to my receipt) burned 17.2 gallons of gasoline. My tank holds 18 gallons so I was .8 gallons away from running out of gas.
Also, I wanted to see if my car was running as efficiently as it should. I should be getting about 25 miles per gallon if everything is running correctly. So I checked. If 17.2 gallons gets me 405 miles then 405/17.2 = 23.25 mpg. Hmm... maybe I should check my tires.
I also added my mileage for the week. I wanted to see how much money I should fit into my budget for gas. I drive about 325 miles per week. At 23.25 mpg, I'd be looking at buying about 14 gallons of gas per week. At $2.50 per gallon, I should figure on spending $35 a week on gas.
11:00 Pm - For what time should I set my alarm? I have to be at class at 10:30 Am. Subtract 30 minutes for commute. Subtract 20 minutes for getting ready (getting dressed, brushing teeth, etc.) and 10 minutes for a shower. For a total of 1 hour. I should set my alarm for 9:30 Am.
7:45 Am - Wishing to spend less than $10 on breakfast, I add the prices of my items before I purchased them this morning. 5 hour energy - 2.99, Soft Pretzel - .65, Breakfast sandwich - 1.99, Large Bottled Water - 1.35. Without actually adding the numbers, I rounded, and found I'd be spending about $7.
8:05 Am - Will I be late? I need to get to class by 8:30. On the parkway this morning I wondered if I would have enough time to get to class on time. I started at mile marker 25 and needed to get to mile marker 45 in 25 minutes. If I obeyed the speed limit at 65 mph, could I make it on time? 60 mph is the same as saying 1 mile for every minute. So assuming I go 60 mph the whole way, I would be there in 20 minutes. Considering I go about 75 on the parkway, I have plenty of time.
9:45 Am - I have 4 hours between classes with 3 assignments to do. Subtract an hour for food breaks and bathroom breaks and I should spend no more than 60 minutes on each assignment.
2:10 Pm - Calculus class. =D
6:00 Pm - Stop for gas on the way home. I drove 405 miles from my last fill up and (according to my receipt) burned 17.2 gallons of gasoline. My tank holds 18 gallons so I was .8 gallons away from running out of gas.
Also, I wanted to see if my car was running as efficiently as it should. I should be getting about 25 miles per gallon if everything is running correctly. So I checked. If 17.2 gallons gets me 405 miles then 405/17.2 = 23.25 mpg. Hmm... maybe I should check my tires.
I also added my mileage for the week. I wanted to see how much money I should fit into my budget for gas. I drive about 325 miles per week. At 23.25 mpg, I'd be looking at buying about 14 gallons of gas per week. At $2.50 per gallon, I should figure on spending $35 a week on gas.
11:00 Pm - For what time should I set my alarm? I have to be at class at 10:30 Am. Subtract 30 minutes for commute. Subtract 20 minutes for getting ready (getting dressed, brushing teeth, etc.) and 10 minutes for a shower. For a total of 1 hour. I should set my alarm for 9:30 Am.
Thursday, January 28, 2010
Mathematical Beauty
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry." - Bertrand Russell
1/28/10 - http://en.wikipedia.org/wiki/Mathematical_beauty
The above image is an example of "Fractal Art." Fractals are based on a geometric principal of self-similarity. What this means is that essentially the final product (or the whole) is based on infinitely smaller and repeated patterns. Here is a less complex example of a fractal:
As you can see, what starts as a simple pattern soon becomes relatively more complex and detailed simply by repeating it's pattern upon itself. This concept can be applied in great complexity and detail as to create fractal art such as these:
1/28/10 - http://en.wikipedia.org/wiki/Mathematical_beauty
The above image is an example of "Fractal Art." Fractals are based on a geometric principal of self-similarity. What this means is that essentially the final product (or the whole) is based on infinitely smaller and repeated patterns. Here is a less complex example of a fractal:
As you can see, what starts as a simple pattern soon becomes relatively more complex and detailed simply by repeating it's pattern upon itself. This concept can be applied in great complexity and detail as to create fractal art such as these:
Mathematical Beauty can be defined in other ways as well. Some believe that the interconnection of theorums and infalliable truths of mathematics is beautiful. Much like an intricate puzzle, all of the pieces fit together so perfectly as to form a (yet unfinished) masterpiece of symbolic logic and an underground network of natural mechanics.
Please read about one of the following from http://en.wikipedia.org/wiki/Mathematical_beauty and comment:
1. Beauty in Method
2. Beauty in Results
3. Beauty in Experience
4. Beauty in Philosophy
Math: A Universal Language?
It has been said by many that mathematics is the only true "universal language." Is this true?
"Very few people, if any, are literate in all the world's tongues -- English, Chinese, Malay, Tamil, Hindi, Bengali, and so on. But virtually all of us possess the ability to be literate in the shared language of mathematics. It is this shared language of numbers that connects us with people across continents and through time."
1/28/09 - http://sps.nus.edu.sg/~huyihuyi/maths.html
One of the reasons for believing that mathematics is a universal language is the fact that it is a universal truth. No matter what state, country, planet, solar system, or galaxy you belong to 2+2 will always equal 4, Pythagoreas' theorum will always hold true, and Pi will always exist.
Because of this, its reasonable to believe that life on other planets have discovered these same truths (or invented them depending what you believe), and thus communication with intelligent lifeforms on other planets is feasable. However, opposing views do present themselves.
"People often say that mathematics is a universal language. There's an important truth there, but we should always keep in mind that the phrase is a play on an ambiguity. Obviously, the notations in which we write mathematics are not universal, and certainly the terms attached to the notations, and in terms of which we explain them, are not universal... ...The play on words involved in saying that mathematics is a universal language lies in this: mathematics is universal, and mathematics is a language; but it is not universal as a language, nor is it a language insofar as it is universal. The underlying principles, the things discussed, are universal; quantities and structures of various kinds and the logic, so to speak, of how they can relate to each other. But we human beings do not have immediate intellectual access to these things, so we build up to an intellectual understanding of them by efforts of the imagination -- cognitive processes leading to expression in talking, writing, and drawing."
1/28/10 - http://branemrys.blogspot.com/2009/04/on-mathematics-as-universal-language.html
Lastly, consider this scene from the movie "Contact" and formulate an opinion.
The sources I have above are, of course, limited compared to the amount of discussion that exists on this topic. So as with all blog assignments, extra credit will be assigned to those who cite their own sources.
"Very few people, if any, are literate in all the world's tongues -- English, Chinese, Malay, Tamil, Hindi, Bengali, and so on. But virtually all of us possess the ability to be literate in the shared language of mathematics. It is this shared language of numbers that connects us with people across continents and through time."
1/28/09 - http://sps.nus.edu.sg/~huyihuyi/maths.html
One of the reasons for believing that mathematics is a universal language is the fact that it is a universal truth. No matter what state, country, planet, solar system, or galaxy you belong to 2+2 will always equal 4, Pythagoreas' theorum will always hold true, and Pi will always exist.
Because of this, its reasonable to believe that life on other planets have discovered these same truths (or invented them depending what you believe), and thus communication with intelligent lifeforms on other planets is feasable. However, opposing views do present themselves.
"People often say that mathematics is a universal language. There's an important truth there, but we should always keep in mind that the phrase is a play on an ambiguity. Obviously, the notations in which we write mathematics are not universal, and certainly the terms attached to the notations, and in terms of which we explain them, are not universal... ...The play on words involved in saying that mathematics is a universal language lies in this: mathematics is universal, and mathematics is a language; but it is not universal as a language, nor is it a language insofar as it is universal. The underlying principles, the things discussed, are universal; quantities and structures of various kinds and the logic, so to speak, of how they can relate to each other. But we human beings do not have immediate intellectual access to these things, so we build up to an intellectual understanding of them by efforts of the imagination -- cognitive processes leading to expression in talking, writing, and drawing."
1/28/10 - http://branemrys.blogspot.com/2009/04/on-mathematics-as-universal-language.html
Lastly, consider this scene from the movie "Contact" and formulate an opinion.
The sources I have above are, of course, limited compared to the amount of discussion that exists on this topic. So as with all blog assignments, extra credit will be assigned to those who cite their own sources.
Wednesday, January 27, 2010
Math: A Discovery or an Invention?
Below are some excerpts from various websites/blogs/articles that take part in the long lasting debate of whether mathematics is a discovery or an invention. Please read them and reply. Take a position in this great debate and defend it as best you can. There is no right or wrong answer. (For extra credit, cite a source you have found on your own.)
"When Newton saw an apple drop from an apple tree he had an epiphany. That epiphany was the concept of gravity. Even though no one before Newton had ever thought of gravity, it had always existed and had always made apples drop from trees. Newton "discovered" and put a name to the concept of an object being pulled towards the earth; he did not "invent" gravity. Just as gravity wasn't "invented", math wasn't "invented". It was "discovered.""
1/27/10 - http://www.megaessays.com/viewpaper/59728.html
"Although the Neanderthal did not have the knowledge of numbers, he still had a basic understanding of math when he established that more deer would be better than less deer. This sense of math was his ability to distinguish quantities. For example, a Neanderthal knew that the more deer he killed, the better. As long as there is nature and natural events, there is math. Before math was discovered, it existed as relationships in nature. Now that we analyze math and invent symbols and numbers to express it more efficiently, it is still the same math that the Neanderthal experienced, just in a different form."
1/27/10 - http://www.megaessays.com/viewpaper/59728.html
"...how do we know these mathematical truths? Where do we get them from? These mathematical entities, according to the Platonist School of Thought, are “out there” for us to discover, and they exist independently of the human mind. They are abstract and non-spatiotemporal. It parallels Plato’s belief in a “World of Ideas”; that unchanging ultimate reality that the everyday world tries to imitate (but imperfectly). These mathematical entities, such as triangles and fractals, are inherent in nature and cannot be ‘invented’. Our only way of knowing their existence is to try very hard to discover them... ...Pythagoras’s Theorem may be seen as an invention, but even the great Mathematician himself acknowledged that it was a discovery of his."
1/27/10 - http://www.answerbag.com/q_view/420161
Discovered
"When Newton saw an apple drop from an apple tree he had an epiphany. That epiphany was the concept of gravity. Even though no one before Newton had ever thought of gravity, it had always existed and had always made apples drop from trees. Newton "discovered" and put a name to the concept of an object being pulled towards the earth; he did not "invent" gravity. Just as gravity wasn't "invented", math wasn't "invented". It was "discovered.""
1/27/10 - http://www.megaessays.com/viewpaper/59728.html
"Although the Neanderthal did not have the knowledge of numbers, he still had a basic understanding of math when he established that more deer would be better than less deer. This sense of math was his ability to distinguish quantities. For example, a Neanderthal knew that the more deer he killed, the better. As long as there is nature and natural events, there is math. Before math was discovered, it existed as relationships in nature. Now that we analyze math and invent symbols and numbers to express it more efficiently, it is still the same math that the Neanderthal experienced, just in a different form."
1/27/10 - http://www.megaessays.com/viewpaper/59728.html
"...how do we know these mathematical truths? Where do we get them from? These mathematical entities, according to the Platonist School of Thought, are “out there” for us to discover, and they exist independently of the human mind. They are abstract and non-spatiotemporal. It parallels Plato’s belief in a “World of Ideas”; that unchanging ultimate reality that the everyday world tries to imitate (but imperfectly). These mathematical entities, such as triangles and fractals, are inherent in nature and cannot be ‘invented’. Our only way of knowing their existence is to try very hard to discover them... ...Pythagoras’s Theorem may be seen as an invention, but even the great Mathematician himself acknowledged that it was a discovery of his."
1/27/10 - http://www.answerbag.com/q_view/420161
Invented
"To some extent this is a matter of definition and semantics. My opinion: Mathematics, as a formal system, is invented in the sense that one starts with a certain set of elements (usually numbers, but not necessarily so) and defines a set of rules regarding various relationships between, and operations of, those elements. And you can invent any rules of the game you desire."
"Mathematics is an invention according to the Formalists and Intuitionists, who believed that mathematics is an invention of the mind because (1) there is no place in this world for mathematical concepts such as negative and complex numbers, thus they must be a construction of the human mind; and (2) they want to fully explain the absolute certainty of mathematics, and since mathematics is an invention of the human mind, then its certainty is inevitable. Mathematics does not inform us anything about the world that we live it; they were constructed for purely practical purposes."
1/27/10 - http://www.answerbag.com/q_view/420161
"We created math. Math is just something we use to subsitute our world [with] digits or X and Y's. We created math, not discovered it. Discovering something is finding something that already exists, which is not the case [here]. Numbers are inside our minds, we created them."
Friday, January 22, 2010
Mathematical Anxiety
Before we get into the philosophies of mathematics, I would like first to discuss with you your opinions of math and "Mathematics Anxiety." There is no question that many students dread their math classes and have a high level anxiety toward the very idea of solving an algrebraic equation. Why is this? Below is a section of Wikipedia's analysis of the causes of "Mathematics Anxiety." Please read it and respond in reference to the questions that follow.
Students often develop mathematical anxiety in schools, often as a result of learning from teachers who are themselves anxious about their mathematical abilities in certain areas. Typical examples of areas where mathematics teachers are often incompetent or semi-competent include fractions, (long) division, algebra, geometry "with proofs", calculus, topology. In many c
ountries, would-be math teachers are required only to obtain passing grades of 51% in mathematics exams, so that a math student who has failed to understand 49% of the math syllabus throughout his or her education can, and often does, become a math teacher.Math is usually taught as a right and wrong subject and getting the right answer is paramount. Unlike most subjects, there is almost always a right answer. Additionally, it is often taught as if there is a right way to solve the problem and any other approaches are wrong, even if they get the right answer. When learning, understanding the concepts should be paramount. With a right/wrong approach to teaching math, students are encouraged not to try, not to experiment, not to find algorithms that work for them, and not to take risks.“Teachers benefit children most when they encourage them to share their thinking process and justify their answers out loud or in writing as they perform math operations. … With less of an emphasis on right or wrong and more of an emphasis on process, teachers can help alleviate students' anxiety about math.” [12]
While teaching of many subjects has progressed from rote memorization to the current Constructivist approach, math is still frequently taught with a rote learning behaviorist approach. That is,
1. a problem set is introduced
2. a solution technique is introduced
3. practice problems are repeated until mastery is achieved
Constructivist theory says the learning and knowledge is the student’s creation, yet rote learning and a right/wrong approach to teaching math ensures that it is external to the student.
Teachers who actually understand what they are teaching tend to encourage questions from the students. Those teachers who do not understand much about their subject, on the other hand, impose fear on the students to prevent them asking questions which might expose the teacher's ignorance.
It has long been well established that anyone (other than a tiny minority who have serious learning disabilities) can learn any area of mathematics, given a desire to learn, a coherent presentation of the information, and adequate practice. Nevertheless, many educational administrators continue to profess the belief that anything more complex than simple arithmetic is too difficult for most people.
In spite of the unfortunate design of the modern school system, a remarkably high percentage of schoolchildren continue to find mathematics interesting, relaxing, easy, and enjoyable.
______________________________
Feel free to react openly to this selection... but also consider the following questions:
1. What is your opinion of mathematics in the classroom? Interesting? Boring? Stressful?
2. What are some other reasons you believe students fear math?
3. Do you think YOU are capable of ANY level of mathematics? If not, why?
4. Name your favorite math teacher and why he/she was your favorite.
Thursday, January 21, 2010
Introduction
Dear Students,
The purpose of this blog is to possibly (and hopefully) open your eyes to new ways of viewing mathematics. So many times I hear students complaining that math is boring or math is just a bunch of useless memorization. This, though you are all entitled to your opinion of course, is not entirely true. Mathematics can, at times, be very interesting; interesting in a way that public schools do not often teach. This blog is my attempt at showing all of you that math is about more than just numbers and equations, and hopefully throughout the course of the year, you may learn to appreciate mathematics in ways that could not be acheived inside the classroom.
Some questions we may discuss will be considered as follows:
1. Is mathematics a science that has been discovered or invented?
2. How does psychology/preconception/experience affect one's perception and ability towards mathematics?
3. What is mathematical beauty?
4. What is the relationship between mathematics and epistemology?
5. Career choices deeply rooted in mathematics.
6. Everyday mathematics.
7. Explaining/Exploring nature through mathematics.
These, among other topics, will be discussed throughout the year. Do not be intimidated by the depth of some of these questions. I will be here to guide you along, and your grade is only dependent on your participation and contribution. Most of these topics are opinion based (theoretical/philosophical) so NEVER be afraid of being wrong! Please, express freely and openly. You will be notified in class when the next blog is posted. Have a great day!
Subscribe to:
Posts (Atom)