MATHEMATICAL THINKING AND SCIENTIFIC WORK
Mathematics is divided into two categories. They are applied mathematics and pure mathematics. Applied mathematics is about mathematics and the application in our daily life. While pure mathematics is related to prove or find new theorems in Mathematics.
Thinking about Mathematics, we have an object here that is our idea. It is an abstract object that can not be touched and seen with our eyes. It is just can be found in our mind because we have ourselves idea. And in fact, our ideas can be different than others. While in our daily life, we have some concrete object. There are two ways to get some ideas from concrete object:
1. Idealization
It means that we should assume that something is perfect. In fact, there is no pointed thing in this world because everything consists of molecules and atoms which has rounded shape. So, even we say that needle is pointed, in fact it does not.
But to think in Mathematics, we have to assume something is perfect. To know whether an angel of a triangle is cute or obtuse, we have to assume that cute is really pointed so that we could count the size of an angel.
2. Abstraction
It means that we have to learn some theorems. I wonder when learning History of Mathematics in my last semester because long time ago, in the age that we usually call it as an ancient age. But in fact, there were so many mathematicians that found so many theorems with its proof. Even, they were smarter than mathematicians today. Look around! If we are asked to proof a theorem, we know that it is very difficult. While, the mathematicians long time ago could find many theorems and prove it. What ashamed!
In Mathematical thinking, there are some aspects that could affect it:
a. Consistence
In doing any mathematics problems, we have to be consistence. It will always accord to the beginning rules. If there is a mistake, it will not be used again forever. It teaches us that we have to be the one who always consistence. In doing our principles of our life or our mission or agreeing some promises.
b. Logic
It is included daily logic and formal logic. In our daily life, there will be always some relationships between mathematics operations and our life’s problem. We can use our logic to get some conclusions. In Yogyakarta State University, there is a lesson “Logika dan Himpunan Matematika” for Mathematics Education which can make us easier to get some conclusions. And it gives us a ‘training’ to use our logic in solving many problems in our life.
3. Thesis and antithesis
In a Physics rule, if there is an action, there will be a reaction. If there is a thesis, there will always be a antithesis.
The further action after Mathematical thinking is scientific work. We can make scientific work based on:
1. Characteristic
Scientific work has its own characteristic. Its characteristic can show the quality of the scientific work.
2. Impersonal
What means by impersonal is not related to any problems that we have (our feeling or our emotion) in doing scientific work. If we make some questions for a test, for example, we must not include our feeling in the statement there. Because it will be disturb the scientific work itself.
3. Standard
To have a good quality, scientific work must have a standard. It is not only about the quantity of scientific work that we did but also the quality is more important.
Rabu, 13 Mei 2009
Rabu, 15 April 2009
Examples of solving mathematics' problems.
Examples of solving mathematics' problems.
1. We will prove that the square root of two is irrational number by showing the contradiction of this problem. So, we will suppose that the square root of two is rational, that is the square root of two equal a over b where a and b are relatively prime integers. It means that they have no common positive integral factor other than unity. The operation of the square root of two equal a over b becomes a equal b times the square root of two if we multiply each sides with b. Then, if we quadratic each sides we will get a square equal two times b square. A square is twice b square, while first we have assumed that a and b have to be relatively prime. So, it is impossible to find some numbers that relatively prime while the square number of one of them is twice than other square. So, it is proved that the square root of two is irrational number.
2. We will show how to indicate that the sum angles of triangles is equal to one hundred and eighty degree by drawing a triangle first. Then, we cut the triangle by scissors. Named each angles of the triangle by A, B, and C. So, we have angle A, angle B, and angle C now. What we have to do then is that cut one by one the angles of the triangle. So that we get three different angles. The next step is integrating those three different angles by making them closer each other in a line like in a jigsaw game. We get a straight line which is composed from angle A, angle B, and angle C. We know that a straight line has one hundred and eighty degree, which is the sum of those three angles of the triangle. So, it is proved that the sum angles of triangles is equal to one hundred and eighty degree.
3. We will find where phi from is. There are some steps to get there. The first is that we have to prepare wire forty-four centimeters in length and make a circle with radius of seven. Second, we have known that the length of is circumferences of the circle. And when we divide it by seven times two, it will generate twenty two seventh. The third steps is redoing the first and second steps with in one hundred centimeters long, one hundred and fifty four centimeters, and one hundred and forty four centimeters and then make each of those a circle. Then, divide each examples by each radius times two (for example, divide one hundred by the radius times two). The result will be close to twenty-two seventh or three point fourteen. This kind of number, which called as phi.
4. We will find out the area of region bounded by the graphs of y equal x square and y equal x plus two. If we want to see it clearly, we may sketch the graph. From which we should make x and y-axis, then we will sketch y equal x square by drawing a curve through zero point zero coordinate. Then, we sketch the graph of y equal x plus two, which is through the intersection point between y equal x square and y equal x plus two. Then, we will find the intersection point between y equal x square and y equal x plus two by substitution. So, we substitute y equal x square into y equal x plus two. It generates negative x square plus x plus two equal zero. So, we get the x is two or negative one. And the intersection point are two point four coordinate and negative one point one coordinate. The next step to find the area is integrated the area that bounded by y equal x square and y equal x plus two. The operation will be integral of negative x square plus x plus two about x from x equal negative one to x equal two. We solve the integral which equal negative one third times x cube plus half of x square plus two times x from x equal negative one to x equal two. The result is thirty-nine sixth. So, the area is equal thirty nine sixth. The solution is thirty-nine sixth.
5. We will find out the determine of the intersection point between the circle x square plus y square equal twenty-five and y equal x plus one. It is better for us to sketch the circle and line y equal x plus one. First, we may have to find the intersection point between the circle and the line. So, we substitute y equal x plus one into the circle x square plus y square equal twenty five. If we quadratic each sides, we will get x plus one in bracket square equal twenty-five minus x square. Then x plus one in bracket square equal x square plus two times x plus one equal twenty minus x square. Then the operation will be two times x square plus two times x minus twenty four equal zero. Then, if we divide each sides by two, we will get x square plus x minus twelve equal zero. Then we get x equal negative four or x equal three. So, the intersection point are three point four coordinate and negative four point negative three coordinate. So, the solution are three point four coordinate and negative four point negative three coordinate.
Median
The definition of median (of a triangle) is a line that made from a vertex of a triangle so that it divides the edge in front of its vertex into two same points.
How to draw a median?
First, make a triangle by three different point named as A, B, and C. The second step, from points B and C, make a bow, which have some radius so that the bow will intersect on two points. Then, give a name as D and E. The third step is connecting point D and E so that it cross a point in line BC and named as F. The fourth step is connecting point A and F, then it will be make a median.
If we make three medians in a triangle, it will be crossed in a point. Moreover, the point can be given a name as centroid or center of gravity. And the special character of a median if it is crossed in a point, the ratio will always two and one. Two is for longer line.
The length of median can be found by using Stewart’s postulate. If you want to know how to find it, it will be posted in different title.
Examples:
There is an ABC triangle.CD is the median. And the length of AB is fourteen centimeters. While BC has ten centimeters long and AC is six centimeters in length. How long is CD?
The way to get the length of CD is counting by a formula like this. CD square equal to open bracket half times BC square close bracket plus open bracket half times AC square close bracket minus open bracket a quarter times AB square close bracket.
TAUTOLOGY
Tautology is plural statements, which always true for each substitution of any truth value in its single statements.
It can be separated in two groups. They are conditional tautology and biconditional tautology.
There are so some ways to decide the truth value of plural statements. They are making the table of truth values and arithmetical procedure. Plural statements can be translated into arithmetical procedure. First, negative a can be changed into one plus a. second, a and b can be changed into a plus b plus a plus open bracket a times b close bracket. Third, a or b can be changed into a times b. fourth, if a then b can be changed into arithmetical procedure as open bracket one plus a close bracket times b. Fifth, a is if and only if b can be changed into a plus b.
Example of solving problem:
1. Show that if p then in bracket p or w close bracket or r is tautology.
2. Find that open bracket if p then q close bracket if and only if open bracket if q then p close bracket is tautology or not.
The answer of number one will be like this. If p is False, than the implication will be True because the antecedent is False. While if p is True so, the implication will be True too because the consequent is True for any truth-value q and r. so, it is showed that that if p then in bracket p or w close bracket or r is tautology.
Then, we will find the answer of number two. If p is True and q is False so the implication of p to q is False while if q then p is True, the biimplication above will be False. So, open bracket if p then q close bracket if and only if open bracket if q then p close bracket is not tautology.
The standard deviation will be posted in different title. Thanks.
1. We will prove that the square root of two is irrational number by showing the contradiction of this problem. So, we will suppose that the square root of two is rational, that is the square root of two equal a over b where a and b are relatively prime integers. It means that they have no common positive integral factor other than unity. The operation of the square root of two equal a over b becomes a equal b times the square root of two if we multiply each sides with b. Then, if we quadratic each sides we will get a square equal two times b square. A square is twice b square, while first we have assumed that a and b have to be relatively prime. So, it is impossible to find some numbers that relatively prime while the square number of one of them is twice than other square. So, it is proved that the square root of two is irrational number.
2. We will show how to indicate that the sum angles of triangles is equal to one hundred and eighty degree by drawing a triangle first. Then, we cut the triangle by scissors. Named each angles of the triangle by A, B, and C. So, we have angle A, angle B, and angle C now. What we have to do then is that cut one by one the angles of the triangle. So that we get three different angles. The next step is integrating those three different angles by making them closer each other in a line like in a jigsaw game. We get a straight line which is composed from angle A, angle B, and angle C. We know that a straight line has one hundred and eighty degree, which is the sum of those three angles of the triangle. So, it is proved that the sum angles of triangles is equal to one hundred and eighty degree.
3. We will find where phi from is. There are some steps to get there. The first is that we have to prepare wire forty-four centimeters in length and make a circle with radius of seven. Second, we have known that the length of is circumferences of the circle. And when we divide it by seven times two, it will generate twenty two seventh. The third steps is redoing the first and second steps with in one hundred centimeters long, one hundred and fifty four centimeters, and one hundred and forty four centimeters and then make each of those a circle. Then, divide each examples by each radius times two (for example, divide one hundred by the radius times two). The result will be close to twenty-two seventh or three point fourteen. This kind of number, which called as phi.
4. We will find out the area of region bounded by the graphs of y equal x square and y equal x plus two. If we want to see it clearly, we may sketch the graph. From which we should make x and y-axis, then we will sketch y equal x square by drawing a curve through zero point zero coordinate. Then, we sketch the graph of y equal x plus two, which is through the intersection point between y equal x square and y equal x plus two. Then, we will find the intersection point between y equal x square and y equal x plus two by substitution. So, we substitute y equal x square into y equal x plus two. It generates negative x square plus x plus two equal zero. So, we get the x is two or negative one. And the intersection point are two point four coordinate and negative one point one coordinate. The next step to find the area is integrated the area that bounded by y equal x square and y equal x plus two. The operation will be integral of negative x square plus x plus two about x from x equal negative one to x equal two. We solve the integral which equal negative one third times x cube plus half of x square plus two times x from x equal negative one to x equal two. The result is thirty-nine sixth. So, the area is equal thirty nine sixth. The solution is thirty-nine sixth.
5. We will find out the determine of the intersection point between the circle x square plus y square equal twenty-five and y equal x plus one. It is better for us to sketch the circle and line y equal x plus one. First, we may have to find the intersection point between the circle and the line. So, we substitute y equal x plus one into the circle x square plus y square equal twenty five. If we quadratic each sides, we will get x plus one in bracket square equal twenty-five minus x square. Then x plus one in bracket square equal x square plus two times x plus one equal twenty minus x square. Then the operation will be two times x square plus two times x minus twenty four equal zero. Then, if we divide each sides by two, we will get x square plus x minus twelve equal zero. Then we get x equal negative four or x equal three. So, the intersection point are three point four coordinate and negative four point negative three coordinate. So, the solution are three point four coordinate and negative four point negative three coordinate.
Median
The definition of median (of a triangle) is a line that made from a vertex of a triangle so that it divides the edge in front of its vertex into two same points.
How to draw a median?
First, make a triangle by three different point named as A, B, and C. The second step, from points B and C, make a bow, which have some radius so that the bow will intersect on two points. Then, give a name as D and E. The third step is connecting point D and E so that it cross a point in line BC and named as F. The fourth step is connecting point A and F, then it will be make a median.
If we make three medians in a triangle, it will be crossed in a point. Moreover, the point can be given a name as centroid or center of gravity. And the special character of a median if it is crossed in a point, the ratio will always two and one. Two is for longer line.
The length of median can be found by using Stewart’s postulate. If you want to know how to find it, it will be posted in different title.
Examples:
There is an ABC triangle.CD is the median. And the length of AB is fourteen centimeters. While BC has ten centimeters long and AC is six centimeters in length. How long is CD?
The way to get the length of CD is counting by a formula like this. CD square equal to open bracket half times BC square close bracket plus open bracket half times AC square close bracket minus open bracket a quarter times AB square close bracket.
TAUTOLOGY
Tautology is plural statements, which always true for each substitution of any truth value in its single statements.
It can be separated in two groups. They are conditional tautology and biconditional tautology.
There are so some ways to decide the truth value of plural statements. They are making the table of truth values and arithmetical procedure. Plural statements can be translated into arithmetical procedure. First, negative a can be changed into one plus a. second, a and b can be changed into a plus b plus a plus open bracket a times b close bracket. Third, a or b can be changed into a times b. fourth, if a then b can be changed into arithmetical procedure as open bracket one plus a close bracket times b. Fifth, a is if and only if b can be changed into a plus b.
Example of solving problem:
1. Show that if p then in bracket p or w close bracket or r is tautology.
2. Find that open bracket if p then q close bracket if and only if open bracket if q then p close bracket is tautology or not.
The answer of number one will be like this. If p is False, than the implication will be True because the antecedent is False. While if p is True so, the implication will be True too because the consequent is True for any truth-value q and r. so, it is showed that that if p then in bracket p or w close bracket or r is tautology.
Then, we will find the answer of number two. If p is True and q is False so the implication of p to q is False while if q then p is True, the biimplication above will be False. So, open bracket if p then q close bracket if and only if open bracket if q then p close bracket is not tautology.
The standard deviation will be posted in different title. Thanks.
Solving Indefinite Integration Equation
Solving Indefinite Integration Equation
A. We will find y, from the known statement of dy over dx equal four times x square. The first way to get y is that we should multiply each sides with dx. So that, we get dx times dy over dx equal four times x square times dx. Moreover, it is became dy equal four times x square times dx. Then, to get y, we should integrate each sides. So that, indefinite integral of dy equal indefinite integral of four times x square times dx. In addition, the result is y equal four third times x cube plus constant. Therefore, the solving problem is y equal four third times x cube plus constant.
B. The next videos explain about adding problems. Here are some of the examples.
1. We will find x where x minus five equals three. The first step is adding each sides with a certain number that can cause x equal some number. Here, we can add five to each side. Then, the operation will be x minus five plus five equal three plus five. So that, we get x equal eight. The solution is eight.
2. We will find a where seven equals four times “a” minus one. First, we should add a number (one) to each side. The operation will be seven plus one equal four times a minus one plus one. Then, we get eight equal four times “a”. The next step is multiply each side so that we can get some number equal “a”. Here, we can multiply each other with a number which is the opposite of coefficient of “a”. Then we get eight times a quarter equal a quarter times four times a. In addition, the result is two equal a. So, the solution is two.
3. We will find x where two third times x equal eight. We should multiply each other with the opposite of two third which is half of three. (Why? The reason can be read in problem number two). Then the operation will be half of three times two third times x equal eight times half of three. It became x equal twelve. Therefore, the solution of this problem is twelve
4. We will find x where five minus two times x equal three times x plus one. The first step is eliminating one of the variables in one side. For example we may eliminate x in the right side, so that we have to subtract three times x in each sides. We get five minus five times x equal one. The second step is subtracting them with five. So that we get five minus five minus five times x equal one minus five. It gives negative five times x equal negative four. The third step is multiply them with negative one fifth. It leads negative one fifth times negative five times x equal negative one fifth times negative four. And the result is x equal four fifth. So, the solution is four fifth.
5. We will find m where three minus five times open bracket two m minus five close brackets equal negative two. The first step is open the brackets by multiply five with the number in brackets. So we get three minus ten times m plus twenty five equal negative two. Then the operation becomes negative ten times m plus twenty-eight equal negative two. Then the next step is adding each side with twenty-eight. It becomes negative ten times m equal negative thirty. Then the last step is dividing them by negative ten. Therefore, we get m equal three. So, the solution is three.
C. The next interesting video is about logarithm. We will proof one of the character of logarithm.
Logarithm base x of A plus logarithm base x of B equal logarithm base x of A times B. Then, suppose that logarithm base x of A is l and it is equivalent with x raise to the l equal A. Next, suppose that logarithm base x of B as m and we get x raise to the m equal B. Then we get logarithm base x of A over B equal n. It becomes x raise to n equal A over B which is equal x raise to l over x raise to m equal x raise to open bracket l minus m close bracket. While n is l minus m. So that is proved that logarithm base x of A over B is logarithm base x of A minus logarithm base x of B.
A. We will find y, from the known statement of dy over dx equal four times x square. The first way to get y is that we should multiply each sides with dx. So that, we get dx times dy over dx equal four times x square times dx. Moreover, it is became dy equal four times x square times dx. Then, to get y, we should integrate each sides. So that, indefinite integral of dy equal indefinite integral of four times x square times dx. In addition, the result is y equal four third times x cube plus constant. Therefore, the solving problem is y equal four third times x cube plus constant.
B. The next videos explain about adding problems. Here are some of the examples.
1. We will find x where x minus five equals three. The first step is adding each sides with a certain number that can cause x equal some number. Here, we can add five to each side. Then, the operation will be x minus five plus five equal three plus five. So that, we get x equal eight. The solution is eight.
2. We will find a where seven equals four times “a” minus one. First, we should add a number (one) to each side. The operation will be seven plus one equal four times a minus one plus one. Then, we get eight equal four times “a”. The next step is multiply each side so that we can get some number equal “a”. Here, we can multiply each other with a number which is the opposite of coefficient of “a”. Then we get eight times a quarter equal a quarter times four times a. In addition, the result is two equal a. So, the solution is two.
3. We will find x where two third times x equal eight. We should multiply each other with the opposite of two third which is half of three. (Why? The reason can be read in problem number two). Then the operation will be half of three times two third times x equal eight times half of three. It became x equal twelve. Therefore, the solution of this problem is twelve
4. We will find x where five minus two times x equal three times x plus one. The first step is eliminating one of the variables in one side. For example we may eliminate x in the right side, so that we have to subtract three times x in each sides. We get five minus five times x equal one. The second step is subtracting them with five. So that we get five minus five minus five times x equal one minus five. It gives negative five times x equal negative four. The third step is multiply them with negative one fifth. It leads negative one fifth times negative five times x equal negative one fifth times negative four. And the result is x equal four fifth. So, the solution is four fifth.
5. We will find m where three minus five times open bracket two m minus five close brackets equal negative two. The first step is open the brackets by multiply five with the number in brackets. So we get three minus ten times m plus twenty five equal negative two. Then the operation becomes negative ten times m plus twenty-eight equal negative two. Then the next step is adding each side with twenty-eight. It becomes negative ten times m equal negative thirty. Then the last step is dividing them by negative ten. Therefore, we get m equal three. So, the solution is three.
C. The next interesting video is about logarithm. We will proof one of the character of logarithm.
Logarithm base x of A plus logarithm base x of B equal logarithm base x of A times B. Then, suppose that logarithm base x of A is l and it is equivalent with x raise to the l equal A. Next, suppose that logarithm base x of B as m and we get x raise to the m equal B. Then we get logarithm base x of A over B equal n. It becomes x raise to n equal A over B which is equal x raise to l over x raise to m equal x raise to open bracket l minus m close bracket. While n is l minus m. So that is proved that logarithm base x of A over B is logarithm base x of A minus logarithm base x of B.
Rabu, 25 Maret 2009
Some Great Vidoes
Last Thursday I got many experiences again from the videos that played by my English Lecturer, Mr. Marsigit. He played six videos that very interesting. But, I think either my ears or the speakers are not good enough. So that, there were just a little information that I can review from that day. But it doesn’t matter because from that little information I caught, I could get a lot knowledge. Here are some of the reviews.The first video was titled as Dead Poets Society. I do not really know the meaning of the title and the relation between the video itself. However, there, a lecturer said to their students who were waiting to look for something more than an hour. Then, the lecturer said that they had to look at things in different way. He asked them to stand up in front of the class one by one. Then he said that if they had to know something they had to try. They had to try to find their own voices. Some sentences from the lecturer gave me a new paradigm. In my life, I usually look at things in the same way how people around look at them. There is no different idea to look in other ways. Therefore, when some problems come, I just solve it by some solving problems that have been there before. However, after I listened from the video I am aware that what I have done is just followed others and there is no creativity there. Therefore, I would like to change my mind, to start new things, and do better by looking at things in different way. I think those sentences said by the lecturer in this video were very great.The second one showed a young boy that became a motivator in a stage. He was about ten years old and I think it is very unpredictable when a child in that year can speak up in front of many people. He gave many spirit words, but the most important word that I could catch is that he can be anything, do anything, and dream anything because he always believes himself. In addition, the third was about a song by two boys which one of the lyrics is a question. Moreover, the question is what you know about Math. They said about some subjects in Math like trigonometry, integration, etc. It showed to me that there are many things in Math that I have not known yet. Even, I have been familiar with them. However, in fact there are so many things that I should learn more. The fourth, fifth and sixth videos will be showed in another posting.
Rabu, 18 Maret 2009
My Reflection in Learning Mathematics
My Reflection in Learning Mathematics
Last Thursday, 12th of March, I had a surprised English test from Mr. Marsigit. That was really difficult for me to remember some familiar words in Mathematics but they were missing from my mind. It was really funny to be thought, because I usually use those Indonesian words in my class, Mathematics Education but what a shame am I when I didn’t know the English words.
I know that I would never be a perfect person, but I am aware in that day, that there were a lot of things that should be known. And from Mr. Marsigit, I can study a lot and learn how to improve my knowledge. For example through posting some blogs to him, I can improve my knowledge and know how to use some references.
Well, talk about last test in Thursday, I got an important big experience about how important an honesty is! Not just in some test, but truly in our daily life. It was very important to have and to take everyday.
Learning Mathematics also always need that honesty. Because Mathematics is honesty itself. 5 + 3 never equals as 10, but it always equals 8. That’s why, it is important to know what we will have in learning Mathematics. It’s not just some pure theorems which always related to some formulas or numbers, but it has some application to our daily life.
Pure Mathematics is always started by our assumption and then It is followed by some definition of a certain concept and there will be some axioms. But, when we talk about pure Mathematics, it is an analyze, not an experiment. So that, its truth depends on our logic. That’s why we need logic when we analyze Mathematics.
When I listened about what is explained by Mr.Marsigit which has not been familiar to my ears, I am aware that there is just a little that I have known about Mathematics.
Even though, when Mr. Marsigit talked about one of his follower from California who wants to do research in Indonesia related to Mathematics. That was very great for her because she was in age to me. And when we are compared, there will be a lot of differences in achievement between her and me.
Look around to the world, I can’t talk many more. Because there I should be aware that a lot of things I have not been known especially in Mathematics. Even, in elementary school or Senior High School I may be proud of ourselves that sometimes I got a good mark, but in fact it gives no wonder because there are many more things which should be learn. It is proved that I am nothing.
And the important thing is a changing. Nothing is better without any better changing. Therefore, I should move!
Last Thursday, 12th of March, I had a surprised English test from Mr. Marsigit. That was really difficult for me to remember some familiar words in Mathematics but they were missing from my mind. It was really funny to be thought, because I usually use those Indonesian words in my class, Mathematics Education but what a shame am I when I didn’t know the English words.
I know that I would never be a perfect person, but I am aware in that day, that there were a lot of things that should be known. And from Mr. Marsigit, I can study a lot and learn how to improve my knowledge. For example through posting some blogs to him, I can improve my knowledge and know how to use some references.
Well, talk about last test in Thursday, I got an important big experience about how important an honesty is! Not just in some test, but truly in our daily life. It was very important to have and to take everyday.
Learning Mathematics also always need that honesty. Because Mathematics is honesty itself. 5 + 3 never equals as 10, but it always equals 8. That’s why, it is important to know what we will have in learning Mathematics. It’s not just some pure theorems which always related to some formulas or numbers, but it has some application to our daily life.
Pure Mathematics is always started by our assumption and then It is followed by some definition of a certain concept and there will be some axioms. But, when we talk about pure Mathematics, it is an analyze, not an experiment. So that, its truth depends on our logic. That’s why we need logic when we analyze Mathematics.
When I listened about what is explained by Mr.Marsigit which has not been familiar to my ears, I am aware that there is just a little that I have known about Mathematics.
Even though, when Mr. Marsigit talked about one of his follower from California who wants to do research in Indonesia related to Mathematics. That was very great for her because she was in age to me. And when we are compared, there will be a lot of differences in achievement between her and me.
Look around to the world, I can’t talk many more. Because there I should be aware that a lot of things I have not been known especially in Mathematics. Even, in elementary school or Senior High School I may be proud of ourselves that sometimes I got a good mark, but in fact it gives no wonder because there are many more things which should be learn. It is proved that I am nothing.
And the important thing is a changing. Nothing is better without any better changing. Therefore, I should move!
Rabu, 11 Maret 2009
What is The Meaning Of Mathematics
What is The Meaning of Mathematics?
Everyone knows about numeric. Even when they can't read. Why? Because, be aware or not, numbers which is became a big part of Mathematics has been closer to us, even to people in some years ago. They knew how to sell and how to pay in their life, even they knew about accounting. Though, that was the old one and not like the modern one like we have today.
Mathematics, become so popular because when we entered our first grade of Elementary School or our Kindergarten, we had been familiar with numeric.
But, before we run out with Mathematics, do you know what is the meaning of Mathematics?
From www.knowledgerush.com, Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. The word "mathematics" comes from the Greek (máthema) which means "science, knowledge, or learning".
It will be so different when we talk about learning Mathematics in University and in School. Because learning Math in University means that we have to learn the pure and the applied Math, while learning Math in School means that we are just learning Pure Mathematics which is according to Proffessor P.J. Giblin, The Head of Department of Mathematical Sciences from University of Liverpool, means as mathematics which underlies all applications. We just learn some theorems and we don't have to know what's the proof.
When we talk about what will we learn, we may talk about the objects. While the objects of Mathematics are abstraction and idealization. Abstraction means a characteristic when something out of physical thinking. It can't be touched but it has a value. That's the important thing that we have to know. While according to www.changingminds.org, idealization is the over-estimation of the desirable qualities and underestimation of the limitations of a desired thing.
Mr. Marsigit gave an example of idealization when he taught in last Thursday, that there is nothing perfect in this world. Even, a needle which is popular as a pointed thing in this world is not truly pointed. Why? Because it has so many molecules and it consists a lot of atoms which have a circle shape. So, it's not really really pointed.
While the characteristic of Mathematics are logic and consisten. Mathematics is not just about numbers and calculation. But it gives us a way to think critically about some probelms in our life. Mr. Sukirman, a lecturer of Yogyakarta State University ever gave a question, "Why do we learn Logika Himpunan?". He said that we didn't need that theorems or some other proofs of the Logika Himpunan itself, but we need the knowledge as a bridge to make our mind think critically so that we can apply to our reality.
That' sthe fact. Mathematics is not just theorems. Because the main power of Mathematics is to make our critical thinking.
Everyone knows about numeric. Even when they can't read. Why? Because, be aware or not, numbers which is became a big part of Mathematics has been closer to us, even to people in some years ago. They knew how to sell and how to pay in their life, even they knew about accounting. Though, that was the old one and not like the modern one like we have today.
Mathematics, become so popular because when we entered our first grade of Elementary School or our Kindergarten, we had been familiar with numeric.
But, before we run out with Mathematics, do you know what is the meaning of Mathematics?
From www.knowledgerush.com, Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. The word "mathematics" comes from the Greek (máthema) which means "science, knowledge, or learning".
It will be so different when we talk about learning Mathematics in University and in School. Because learning Math in University means that we have to learn the pure and the applied Math, while learning Math in School means that we are just learning Pure Mathematics which is according to Proffessor P.J. Giblin, The Head of Department of Mathematical Sciences from University of Liverpool, means as mathematics which underlies all applications. We just learn some theorems and we don't have to know what's the proof.
When we talk about what will we learn, we may talk about the objects. While the objects of Mathematics are abstraction and idealization. Abstraction means a characteristic when something out of physical thinking. It can't be touched but it has a value. That's the important thing that we have to know. While according to www.changingminds.org, idealization is the over-estimation of the desirable qualities and underestimation of the limitations of a desired thing.
Mr. Marsigit gave an example of idealization when he taught in last Thursday, that there is nothing perfect in this world. Even, a needle which is popular as a pointed thing in this world is not truly pointed. Why? Because it has so many molecules and it consists a lot of atoms which have a circle shape. So, it's not really really pointed.
While the characteristic of Mathematics are logic and consisten. Mathematics is not just about numbers and calculation. But it gives us a way to think critically about some probelms in our life. Mr. Sukirman, a lecturer of Yogyakarta State University ever gave a question, "Why do we learn Logika Himpunan?". He said that we didn't need that theorems or some other proofs of the Logika Himpunan itself, but we need the knowledge as a bridge to make our mind think critically so that we can apply to our reality.
That' sthe fact. Mathematics is not just theorems. Because the main power of Mathematics is to make our critical thinking.
Senin, 02 Maret 2009
An Introduction To Bahasa Inggris II
Some people say that English is fun and easy. Look around! A lot of children in Australia or England or in other countries are very good in speaking English.(Of course?!) But some other people say that English is difficult and annoying! The way to spell or the way to understand what is the meaning of some sentences make them say that English is difficult.
But well, English is so easy when we have a strong willing to study it. Dr. Marsigit, one of an English lecturer in Yogyakarta State University said that we have to search by ourselves to learn English. He will give the 15% and the 85% is ours.
A man who have visited London for the first time in 1975, said that there were some ways of precondition to work due to communication in English, especially in the class of Bahasa Inggris II. They are:
1. Motivation
Someone who have strong motivation will work hard to do something. For example, in learning, maybe someone study because their parents ask him or her so that in the future they hope that she or he will have better life. In another condition, maybe someone study because of someone he or she loved.
But the highest motivation in doing everything is for Allah. We may do learning for our parents, to get some good marks, or others but the main point that we should have is that we do learning for Allah. To say thanks for everything given by Him, and to understand more that Allah is The Greatest one through objects we are learning about.
2. Attitude or behavior
A boxer will do a lot of exercises and eat some vitamins to get strong body. He will exercise regularly so that he win in some competitions. That's a boxer. What about a student, especially a university student?
University students have to behave as a university student, not like students of elementary school who always wait for their teacher to get some homework. University students have to search by their selves. Not just listen and write in their books when the lecturer is teaching, but they have to find some references to get the complete one. There are a lot of facilitates like library and internet to use.
3. Understanding
It's just the same with reading a French newspaper but we don't know what's the meaning when we don't understand about what we are learning about. We always listen and write what the lecturers say, but if we don't understand at all, we have nothing. Because what is the most important to do in our learning is understanding. Than we can do some other way like exercise or do practices when we have understood what we are learned about.
4. Skill
We live in globalization era and the technology is so wonderful. If we can not use the modern technology, we will be 'an outback human'. There is no different with people who can not read any letters because in the past they don't studied . Skill is very important for us who want to survive in the globalization era. But it does not mean that we have to follow what the most people do, because we have ourselves religion and we have ourselves God. Not only hard skill that we must have but also soft skill. Through some training or learn from books or join a seminar we can increase our skill.
5. Experience
Experience is the best teacher. That sounds very familiar in our ears. Because from elementary school we have had that sentences in the wall in our classroom. Through learning from many subjects, we are wished to have a lot of experiences. Not always go to abroad or other places that we have not been visited before, but from the subjects we are learning, we can get a lot of experiences. Join some organizations or competitions will increase our knowledge too so that it can be our new experiences.
6. Communication
That will be very funny when someone does not need any speaking to others. We live as a social creature and we need communication, of course. The most important between a lecturer and the students is communication too. Because without any communication, we won't understand about what others mean. Through Bahasa Inggris II, we are all wished to communicate well especially in English. Not just know about the grammar, or vocabularies, but know how to speak. And the important thing that we should know is what knowledge we are learning for.
Either life or English is a choice. It depends on our selves from what side we want to see. But, whatever English that we have to learn in Bahasa Inggris II is from and for ourselves, not the others even the lecturer.
But well, English is so easy when we have a strong willing to study it. Dr. Marsigit, one of an English lecturer in Yogyakarta State University said that we have to search by ourselves to learn English. He will give the 15% and the 85% is ours.
A man who have visited London for the first time in 1975, said that there were some ways of precondition to work due to communication in English, especially in the class of Bahasa Inggris II. They are:
1. Motivation
Someone who have strong motivation will work hard to do something. For example, in learning, maybe someone study because their parents ask him or her so that in the future they hope that she or he will have better life. In another condition, maybe someone study because of someone he or she loved.
But the highest motivation in doing everything is for Allah. We may do learning for our parents, to get some good marks, or others but the main point that we should have is that we do learning for Allah. To say thanks for everything given by Him, and to understand more that Allah is The Greatest one through objects we are learning about.
2. Attitude or behavior
A boxer will do a lot of exercises and eat some vitamins to get strong body. He will exercise regularly so that he win in some competitions. That's a boxer. What about a student, especially a university student?
University students have to behave as a university student, not like students of elementary school who always wait for their teacher to get some homework. University students have to search by their selves. Not just listen and write in their books when the lecturer is teaching, but they have to find some references to get the complete one. There are a lot of facilitates like library and internet to use.
3. Understanding
It's just the same with reading a French newspaper but we don't know what's the meaning when we don't understand about what we are learning about. We always listen and write what the lecturers say, but if we don't understand at all, we have nothing. Because what is the most important to do in our learning is understanding. Than we can do some other way like exercise or do practices when we have understood what we are learned about.
4. Skill
We live in globalization era and the technology is so wonderful. If we can not use the modern technology, we will be 'an outback human'. There is no different with people who can not read any letters because in the past they don't studied . Skill is very important for us who want to survive in the globalization era. But it does not mean that we have to follow what the most people do, because we have ourselves religion and we have ourselves God. Not only hard skill that we must have but also soft skill. Through some training or learn from books or join a seminar we can increase our skill.
5. Experience
Experience is the best teacher. That sounds very familiar in our ears. Because from elementary school we have had that sentences in the wall in our classroom. Through learning from many subjects, we are wished to have a lot of experiences. Not always go to abroad or other places that we have not been visited before, but from the subjects we are learning, we can get a lot of experiences. Join some organizations or competitions will increase our knowledge too so that it can be our new experiences.
6. Communication
That will be very funny when someone does not need any speaking to others. We live as a social creature and we need communication, of course. The most important between a lecturer and the students is communication too. Because without any communication, we won't understand about what others mean. Through Bahasa Inggris II, we are all wished to communicate well especially in English. Not just know about the grammar, or vocabularies, but know how to speak. And the important thing that we should know is what knowledge we are learning for.
Either life or English is a choice. It depends on our selves from what side we want to see. But, whatever English that we have to learn in Bahasa Inggris II is from and for ourselves, not the others even the lecturer.
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