Reflection #2

The Different Between Traditional Method of Teaching and Innovative Approach of Teaching Learning Mathematics

On Monday, February 25^th, 2013, Mr. Marsigit tells us about traditional method and innovative approach of teaching leaning mathematics. Firstly, he draws the illustration of traditional way to teaching Mathematics like in below.

Traditional method is teacher just transfer knowledge to the student or teacher centered. In traditional method, the teachers behave as if theyare only one resources of learning Mathematics and play dominant in all activity and initiative. The students is like empty barrel then they just receive from something that the teachers deliver, give and transform the knowledge to the students, so it is also called deliver method transfer of knowledge method or transfer of knowledge approach. From this situation, they able two character the teacherbehavior, the student behave and the situation in the class. Traditional method is very easy to do but it has many impacts for the students. The students will not be happy but they will feel bored.

Innovative learning is very different with traditional method. In this method, students are more active and the teacher just as facilitator. The teachers facilitate the students need. Students play dominant in all of activity and initiative. The students also can construct their knowledge so not only use formula from the book but they can find the knowledge by themselves. In below, there is an illustration of innovative approach.

In Indonesia, many teachers still use traditional method. Whereas, the innovative learning is important so implement. Easy to speak but it is too hard to implement because it not yet our culture, in primary, junior, still traditional method, in international. It is so bad for education in Indonesia.

The good teacher is the teacher that can facilitate. In order to facilitate, the teachers need tool, technology, or equipment, facilitation and students worksheet. Technology is very important to facilitate, and for primary and junior, the more important is student’s worksheet.Student’s worksheet can be online study or IT or web blog anddeveloping web blog. This is the part to develop material to teach. In here, the students need to construct their own knowledge, language, and mathematics. If you have timeline, so in the end your timeline is you can achieve in TOEFL as the formal criteria for English.

To achieve it, we must have need motivation, fell happy, positive thinking, enthusiasm and trust, because of it have influence to the way getting knowledge the speed, the outcome, and the process. The big problem in Indonesia, now, is how and what kind that I can implement, develop this kind method of teaching learning mathematics. We need to understanding for example the nature of mathematics, school mathematics, student learn mathematics, all aspect of teaching learning process. We must also know that there are two kinds of mathematics, that is, formal Mathematics and school mathematics. Formal Mathematics contains axiomatic, definition, theorem, and etc. These kinds of school are for senior high school and students in university. However, school Mathematics contain the nature of mathematics. School Mathematics is for junior high school and elementary school. The problemsof teaching learning process of Mathematics at primary schools are mostly coming from theteacher not from the student, because theydo not understand the nature of mathematics.

According to Ebutt and Straker (1995), teachers do not use axiomatic Mathematics definition, but they must use Mathematics as school mathematics. With this definition, the student will like mathematics. The following is the nature of School Mathematics by Ebutt and Straker :

1.    Mathematics is a search of pattern or relationship

In this point, the student should be motivated to determine their pattern or relationship between in material to the other material. Therefore, the students have a freedom to determine their own pattern or relationship with their method. In the last, the student will be more understanding with the teacher’s explanation.

2.    Mathematics is problem solving activity

What is the goal learn mathematics? The goal of learning Mathematics is to find the solution of each problem. As we know that Mathematics happen in daily life situation, inevitably and in real life situation, pencil and paper do not always come in handy. Therefore, learning must be meaningful, so number should be make useful and sense. However, remember when the students solve the problem, they have another way to solve it. Therefore,these activities are useful to develop the student’s thinking.

3.    Mathematics is investigation activity

What do you think about Mathematics as investigating activity? It means that to solve the problem of mathematics, we must investigate it. When investigating it, the students have many ways to solve the problems. They use their pattern or relationship that it depends of their understand about the material. So this activity will develop the student’s creativity, so many ways that they can use. These activity can make the student think that Mathematics is so easy and not rigid or limited with one ways, but Mathematics can be solve with many ways.

4.    Mathematics is learning of communications

Mathematics as learning of communications works to:

a.         Encourage students to make examples of the nature of mathematics

b.        Encourage students to explain the nature of mathematics

c.         Encourage students to justify the need for mathematics

d.        Encourage students to discuss mathematical problem

e.         Encourage students to read and write mathematics

According to the nature of school Mathematics in above, teacher must know about the nature of student learn mathematics. So what is it?

1.      Mathematics students learn effectively is by  good motivation and perception

How can the student get a good motivation and perception? It depends of the teaching method. The teacher must inspire, challenge, and stimulate the students by the teaching method. However, we must know the psychology of our student because students have different character so the teacher must different way to be motivated each student.

2.      Students learn Mathematics individually

Students examine their own mathematics. He studied according to his understanding. Sometimes he is abstracting events in everyday life into the concept of mathematics. Every student has different ways of learning mathematics. It can be seeing directly when the students solve with a case of mathematics. It will be looked different way of solving the problems between students with the other students. This is not blame for Mathematics is essentially investigative activity.

3.      Students learn Mathematics in collaboration with the other

Learning Mathematics with the other can make us more expected to understand mathematics. In-group, the students can exchange their ideas and if they have some problem, they can solve it together.

4.      Students learn Mathematics contextually that bound by space and time

Students will be better understand mathematics if teachers pass them contextually. Young student learn about something concrete and daily living that close to them. With the experience, they develop their ability to understand abstract concept, axiom, prove, reason logically, manipulate symbol and generalize. Teacher can give the example that close to daily live so the student know the application of Mathematics in daily live not only just calculate the formula.

Reflection #3

On Monday, March 4^th, 2013, Mr. Marsigit asks us to make some questions. Based on the question, Mr. Marsigit tries to understand our question and answer in English.

Alisa             : How we as the student must study English in mathematics?

Mr. Marsigit  : Based on my theories and experiences in my explanation in LSM that there is a need for us to learn everything around us, there exist in there environment giving by the god. So, I perceive something should be naturally and learn naturally. The problem is how the curriculum not very hard to learn by students but how the curriculum more unite not mathematics formal. Therefore, the best way to translate, to be translated and to reflect, to be reflected. That’s way, Iamfacilitating you, in order you able to translate and to reflect your experiences for everything, although it is just small experience. In the Greek terminology, translate and called hermeneutics. Then for a long time, I try to we can define in Indonesia term. I found maybe the nearest meaning that is silahturahim. The best way to learn English is to make silahturahim. Please you, silahturahim to everything that consist of English, silahturahim to television broadcasting, dictionary, electric dictionary, internet, encyclopedia, native speaker, tourists that fluency speak in English, English book, to Mr. Marsigit, Marsigit blog. Not only in trying to learn English, but also everything including physics, language, mathematics, English, everything please do develop silahturahim. If you learn to try to uncoversociology, it is also silahturahim. If you learn to try to understand about the society phenomena, you do silahturahim especially in moslemday, like idulfitri, iduladha and etc.

Dian              : How to implement character education

Mr. Marsigit  : To implement character education in the class, we should understand about the effect of teach and learn mathematics. First, we should understand innovative teach and learn mathematics.to understand the innovative teach and learn mathematics, we should understand the nature of mathematics. You can read my blog, in there; you can find various definition of mathematics. We can produce more than 23 definition ofmathematics, definition mathematics based on Plato, Aristoteles, philosopher, mathematics, servant, and on you. Theoretically, the number of mathematics is certain of the number of people that think about mathematics. But, main week, we can divide two category, first, formal mathematics or axiomatic mathematics, mathematics or pure mathematics, the second is school mathematics, realistic mathematics or concrete mathematics. As an innovative teacher, we should understand that there are manymathematics, not only one. On Saturday, Prof. Widodo tells about high mathematics, oh mathematics is very beautiful, but for elementary student that in valley and will climb into the top of mountain, oh mathematics is like hell.

Faqih             : Why you ask the teacher to write?

Mr. Marsigit  : Reflection is the best method. Reflect is the highest order of thinking because reflectionis very complete, start from record, rewrite, and think, the reflection must belong to you. So if you sincere, you will not feel if your English improve.

Wildan          : How to make meaningful learning?

Mr. Marsigit  : To translate and to be translated. You must translate in large meaning. The main key is hermeneutics or silahturahmi.

Dian              : How to teach mathematics effectively?

Mr. Marsigit  : Ido not like this of question. I prefer to make question how the students learn mathematics effectively. I donot believe in mathematicsinstruction, but I do believe in student learning mathematics. Therefore,I try to develop teaching and learning of mathematics. For me,mathematicsinstruction just likes a myth not real. The real one is how to facilitate the student on order they to learn. To facilitate including give a chance, time, reference, and etc.

Marissa         : What is the main idea of the nature of school mathematics?

Mr. Marsigit  : For younger student, knowledge is activity so dimension of mathematics define at activity. Therefore, the definition for the younger is school mathematics. AccordingEbbut and Straker, the nature of school mathematics is first, activity search pattern and relationship, activity to investigate, activity to problem solving, and activity for communication. The best mathematicians aremathematician that can adapt in the situation. So if we talking with young student, mathematicsis activity.

Shella            : How to change the thinking of student that mathematics is difficult?

Mr. Marsigit  : To change it, give them facility, the change of thinking will be running. This is my spirit.

I think that’s all of my reflection for last Monday. I’m sorry for my mistakes. Thank you. 

Reflection #4

English_What you know about math?

First, we saw the video with the title how you know about math? This is the lyrics of the song.

Part 1            : What you know about math?

                     What you know about math?

                     What you know about math?

Part 2            : Hey, don’t you know I represent Math League

                     When I add shorty subtract

                     Freshmen backpack where I’m holding all my work at

Back to Part 1

I know all about math

Answers 44

It’s real easy cuz of sig figs

You got 45 you rounded high

Your answers too big

Back to Part 1

I know all about math

TI-84

Solar Edition you know I’m shining dawn

Extra memory, I look back to do my natural log

You know we multiply

While memorizing pi

Take limits to the sky

be sure to simplify

Graphing utility that’s trigonometry

One hundred I’m math b don’t you cheat off me

Distance is rate times time

the sine graph ain’t a line exponential decline

but your score can’t beat mine

We’re memorizing grades for our mathly states

against the mathly greats not getting many dates

I got to find a mate but girls just play hate

and always make me wait (can’t even integrate)

Back to Part 2

Back to Part 1

I know all about math (heh)

English_Degree

Degree is measured by one full counterclockwise rotation of terminal side of angle back to its starting point measures (360⁰) three hundred sixty degrees. Three hundred sixty degrees (360⁰) angle is one move minute hand in counter clockwise way around clock back to original position to make a circle. Then, three hundred sixty degrees degree (360⁰) is equal to a circle then 1 degree is equal one over three hundred sixty degrees(360⁰) of full revolution and 11 degrees is equal to eleven over three hundred sixty degrees of full revolution. Ninetydegrees (90⁰) is equal to one fourth of full revolution and it is called right angle. If we move the terminal side of the angle in counterclockwise direction in the additional ninety degrees (90⁰), the angle measure is one hundreds and eighty (180⁰). It is called straight angle.

Beside degree, the other unit that we can used to measure angle is radian. The radian uses radius of the circle to figure out measurement of an angle. Radians and degrees have a measurement relationship. As we know that 360⁰is single counterclockwise revolution and one full revolution is equal to 2π. Then, we know that 360⁰ is equal to 2π radians. To convert radians to degrees or vice versa, we must simplify the equation. First.360⁰ is equals 2π radians then we simplify it until we get that 1 degree is equal to π/180 radians and one radian is equal to 180⁰/π. For example, we want to convert 120⁰ into radians in to degrees. If we have 1 degree is equal to π/180 radians, remember for algebra, to do it we must multiply both sides of equation by 120, so any time you alter one side of equation you have to do it to the other side too. Therefore, 120⁰ is equal to 2 π/3 radians. The other example is if we want to convert 11π/12 radians to degree. Same with above, we have 1 radians is equal to 180⁰/π then remember we can doing something to one side without doing something on the other side, so multiply it with 11π/12 radians. Thus we get that 11π/12 is equal to 165⁰

English_Limit by Inspections

To determine limit by inspection, we must know two conditions. First, this matter only can apply if x goes to positive or negative infinity and second is if limit involves a polynomial is divided by a polynomial. For example, , this problem fits the 2 condition because it is polynomial that divided by a polynomial and x approaches infinity. The key to determining limits by inspection is in looking at powers of x in the numerator and the denominator.

This is the shortcut to determine limit by inspection. First shortcut rule, if the highest power of x is greater in numerator then the limit is positive or negative infinity. For example, , you can see that x approach to infinity. Then, the highest power of x in numerator is 3is greater than the highest power of x in denominator is 2, so the limit of this expression can be positive or negative infinity. Since all the numbers are positive and x going to positive infinity, the limit must be positive infinity. If you can’t tell if the answer is positive or negative infinity, you can substitute a large number of x and see you end up with a positive or negative number. Whatever sign you get is the sign of infinity for the limit. Second shortcut rule, if the highest power of x is in the denominator then the limit is zero. For example, , since the highest power of x in numerator is 2 is less than the highest power of x in denominator is 3, so the limit of this expression can be positive or negative infinity. Thus the limit of the inspection is zero. Now, the last shortcut rule is little bit trickier. The trick is used iwhen the hghest power of x in numerator is same as the highest power of x in denominator. If this is a case, so  is equal to the quotient of the coefficients of the two highest powers. Remember that the coefficient is the number that goes with a variable, for example 2 is the coefficient of . This is the example of the last shortcut, . You can see that the highest power in denominator and numerator is 3. According to this rule, let’s mean that lim is equal to coefficient of ’s over each other.  The coefficientof ’s in the numerator is 4 and the coefficient of ’s in the denominator is 3 so the limit is equal to 4/3.

English_The Golden X

The first equation look like this is ax=b, where x is variable, and a, b is constant, because of a and b isn’t number so we don’t need to solve it. Let’s take a equation when the value of a is 4 and b is 12, that means we got an equation 4x = 12. Remember we try to figure out the value of x, therefore we can get variable x on side by itself. Thus, we must simplify each side by 4, then in the left side 4 divide by 4 is equal 1, so they cancel out and 12 divide 4 is equal 3. So the answer is 3, we always can check it by substitute the answer to the original question, then we get 4(3) is equal 12. Thus, we get the right value of x. why we divide it with 4? It is because the term 4x means that 4 and x are multiplied together. To get the value of x, we must do the opposite of multiplication that is division. The other example is , it is means that 7 times x. to get x, we can divide both side by 7, so we get x=9. We can check it to original equation that is 7(9) is equal to 63, so 9 is the right value of x. the other equation form is , where a, b, and c are constant. For example, . In this case, we need to move 3 first because 3 in the left side so we need to subtract both side with 3. Therefore, we get , then we can divide it by 5, so we get x = 3.

English_Integers

Integers are the whole number and their negatives. Whole numbers are not fractions or decimals. If we take number 1, then we keep to add 1 to it, all of the number that we get is whole number. Therefore, 2.000.000, 134, and 5 are whole numbers. Back to the definition of integer, integer can be positive, negative, or zero. The easy ways to visualize integer is number line. Number line is vertical or horizontal line that is marked at event intervals or units similar with thermometer. It is the number line works. When moving to right or up of the number line, the numbers become greater. When moving to left or down of the number line, the numbers become smaller. Any number above or to right of zero is positive or greater than zero but any number below or to left of zero is negative or less than zero. The number is negative if it has a minus sign in front like -5. The number line will be useful to world to we get the operation of negative number. The end of the number line goes on forever because there is no end or the numbers go on infinity. Integers are made from digits. Digits are simply the numbers 0 to 9. Every digit goes in a certain place. The first number in whole number is unit place. The tens place is just next on left. One place over from the ten place is hundreds place. To the left of the hundreds place, thousands, ten thousands, hundred thousands, millions, ten millions, and soon. Just like we saw the integer in the number line, the digit go on forever because the number go on to infinity. Let’s take a look at the number 1492. For this number, 2 is the unitplace, moving to the left is 9 in ten place and the 4 is hundreds place. Finally, the 1 is the thousand place.

English_MultiplyingEksponent

The first rule is multipying exponents, we can use it when the exponents are same for both base number that we are multiplied, then multiply the bases and keep same exponent. The algebra form is a n-th power times b to n-th power is equal to a times b to the n-th power.For example, 3⁵x4⁵, we just multiply the base that 3.4 that is 12, and keep same exponent, so the result is 12⁵. If we see something raise to the second power, it is called square. When we see something raise with the third power, it is called cubed.

The second rule is just like the first rule, only this time it is only divide instead of multiply. We use the second rule when dividing two numbers with same exponents. The algebra form islikea n-th power is divided by b to n-th power is equal to a over b to the n-th power. For example,  , we just  divide the base numbers, six is divided by 2 is equal to 3 and use the common exponent so the result is 3³.

The third rule is little different with the other. If we have base number raised to a power and completely exponential term raised to another power, then we can multiply the two exponents. The algebra form is like a to the n-th power to the m-th power is equal to a to the n-th times m-th power. For example, , we can multiply the exponent that is 3 times 2 is equal to 6, and keep base the same, so the result is 2⁶.

The fourth rule is multiplying integers with different exponents and the same base number with add the exponents and use the common base. The algebra form is like a to the n-th power times a to the m-thpower is equal to a to the (n+m)-th power. For example, 2³ x 2⁵, first keep the base 2 and then add the exponents 3+5 =8, so the result is 2³ x 2⁵=2⁸.

The last rule is same in the fourth rule, the difference is in dividing and subtracting. The rule is dividing integers with different exponents and the same base number with subtract the exponents and use the common base. The algebra form is like a to the n-th power is divided by  a to the m-thpower is equal to a to the (n-m)-th power. For example, , first keep the base 2 and then add the exponents 5subtract3 is equal 2, so the result is  is equal 4².

English_Function Terminology

The basic of building algebra is function. Function is an algebraic statement that provides a link between 2 or more variables. It is used to find the value of 1 variable if you know the values of the others. For example, , if you know x, you can find y. Let the ,  so y is equal to 10. This situation occur if one variable apperas by itself on one side of the equation. For example, , y is the function of x, because y appears by itself. It is important that y is underdone by the other symbol like  is just y. Whenever you compute the right side, you will find the value of y. So a function is a codependent relationship between x’s and y’s without get x, you can’t get y.

The official meaning of function is very specific kind of relation in which each element of one set is paired with one and only one, element of the second set. The relation is any numerical expression relating one number, or set of numbers, to another. There are two kind of relations, that is, equations and inequalities. Relation can be simple thing by equation  and the inequality when the specific numbers are being related to each other. Algebra explore relation between the nonspecific number or variable like x and y that represent the whole set of possible numbers. Variable relations is the expressions that contain variables like . The kind that can be used to determine just one value for one of the variables that exactly like , when you substitute a value for x, you can calculate just 1 value for y. An equation with one variable by itself on 1 side and we say that one variable is a function of whatever variables appear on the other side.

Function of x or f(x) or f of x is equal to y. For example, , so y is express the function of x because  so . is the standard form to express a function.