Sumber utama: Perkuliahan Filsafat Pendidikan Matematika oleh Dr. Marsigit, MA pada 21 April 2011

Dari kehidupan sehari-hari mulai sejak dahulu kala, peradaban manusia hingga sekarang adalah suatu fenomena dalam filsafat. Peradaban-peradaban manusia, ada Mesopotamia, Babilonia, Mesir dan terutama Yunani mulai berpikir tentang matematika. Orang-orang Yunani mulanya berpikir tentang matematika dengan cara abstraksi dan idealisasi untuk menghilangkan atau membebaskan matematika dari ruang dan waktu. Abstraksi dan idealisasi dilakukan untuk memperoleh bukti.

Dari alam transenden pikiran manusia disebut sebagai noumena, segala sesuatu dianggap menjadi dua, yaitu segala sesuatu itu bersifat tetap (oleh Permenides) atau segala sesuatu itu bersifat berubah (oleh Heraclitos). Segala yang ada di pikiran manusia bersifat tetap sedangkan yang ada di kehidupan sehari-hari atau fenomena bersifat berubah.

Suatu sistem, struktur, dan bangunan memiliki fundamen. Fundamen ini bergantung pada darimana ia memulai, jika dimulai dari sesuatu yang jelas maka disebut fundamentalism sedangkan jika ia dimulai dari yang tidak jelas maka disebut intuitionism. Dilihat darimana matematika dimulai maka matematika bisa berarti tunggal, dual, multi, atau plural. Matematika yang dual maka bisa berarti matematika itu absolut atau relatif. Oleh karena itu, manusia perlu berpikir secara ekstensif dan intensif sehingga secara filsafat dapat menterjemahkan atau menemukan maknanya.

Dalam filsafat matematika terdiri dari tiga jalur, yaitu
1. Ontologi Matematika
2. Epistemologi Matematika
3. Aksiologi Matematika.

Suatu ketika berkembang pemikiran matematika berkembang dari sebuah awal. Matematika ini adalah matematika fundamentalis, formalis, dan aksiomatis dengan sifat-sifatnya rigor atau apodiktif, konsisten, tunggal, pasti, koheren, identitas, dan absolut. Tokoh matematika yang seperti ini adalah Hilbert. Akan tetapi, matematika ini adalah dalam ranah pikiran manusia yang terbebas dari dimensi ruang dan waktu, dengan kata lain matematika ini disebut pure-mathematics.

Di sisi lain, dalam kehidupan sehari-hari terutama di sekolah, matematika yang dibutuhkan adalah matematika yang terikat oleh ruang dan waktu sehingga matematika sekolah ini bersifat kontradiktif, relative, plural, dan korespondensif. Oleh karena itu, akan sangat bahaya bila pure-mathematics diterapkan dalam sekolah. Jika pure-mathematics benar-benar diterapkan dalam sekolah yang terjadi adalah sebuah kehancuran. Matematika dianggap sulit oleh siswa.

Abstraksi dalam bahasa Inggris adalah asitraction. Kata ini berasal dari bahasa Latin abstractio (dari abstrabere yang artinya “menarik dari"). Kata abstractio dapat disejajarkan dengan kata Yunani aphairesis. Secara harfiah abstraksi berarti memisahkan suatu bagian dari suatu keseluruhan.
Segala sesuatu yang ada di dunia ini dapat diabstraksikan. Manusia melakukan abstraksi dari bumi menjadi sebuah bentuk titik ‘.’ . Titik tersebut bias berada di dalam pikiran atau di luar pikiran kita. Dari abstraksi tersebut dikembalikan lagi ke bumi, gunanya adalah untuk menterjemahkan bumi. Karena bumi bergerak dalam ruang dan waktu maka titik sebagai abstraksi tersebut terdapat fakta dan potensi. Jika titik tersebut dapat disadari maka titik tersebut akan bermakna.
Bagaimana titik yang sebagai abstraksi ini menjadi suatu daya untuk menterjemahkan dunia? Dari sebuah titik kita dapat membuat apa saja, bisa garis, bidang datar, lingkaran, kurva, bentuk kubus, bola, dan sebagainya. Jika dari titik bisa dijadikan sebuah garis maka titik tersebut adalah potensi dan garis tersebut adalah faktanya. Jika titik tersebut dibuat ke dalam spiral dan dunia sebagai isinya maka dunia tersebut bergerak dalam ruang dan waktu. Jika kita menterjemahkan dunia dengan mengunakan analogi dan itu ada di dalam pikiran kita maka itu hanyalah setengahnya dari dunia sehingga setengahnya lagi yang diperlukan adalah fakta, pengalaman, atau realita.
Dalam sebuah konsep, abstraksi dari kehidupan nyata dapat dibuat sebagai kurva normal. Dalam kurva normal kita tahu bahwa terdapat standar deviasi atau besarnya penyimpangan dan tanda keputusan atau batas toleransi. Bagian normalnya ada bagian tengah yang paling cembung. Dalam kehidupan nyatanya, yang melebihi batas toleransi adalah sebuah masalah dan bagian normalnya adalah kehidupan yang bahagia. Sesuatu yang menjadi masalah perlu “ruwatan”. Ruwatan disini adalah sebagai suatu penjelasan. Dalam ilmu filsafat hal yang paling penting adalah penjelasan. Penjelasan menghindari sesuatu yang disebut mitos, karena mitos adalah musuh terbesar dari filsafat. Maka penjelasan itu lah yang membuat mitos menjadi logos. Jika tidak mampu memberikan penjelasan maka itu disebut dogma atau otoritarian.
Immanuel Kant menyatakan bahwa dalam pikiran manusia terdapat empat kategori, yaitu kualitatif, kuantitatif, kategori , dan relasi. Sedangkan dalam otak kita ada dua cara berpikir yaitu naik dan turun. Naik jika dihubungkan dengan logika dan apriori, sedangkan turun jika dihubungkan dengan sintetik dan pengalaman.

Student’s mathematical thinking has three aspects, they are mathematical attitudes, mathematical methods, and mathematical contents.
I. Mathematical Attitudes
1. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
(1) Attempting to have questions
(2) Attempting to maintain a problem consciousness
(3) Attempting to discover mathematical problems in phenomena
2. Attempting to take logical actions
(1) Attempting to take actions that match the objectives
(2) Attempting to establish a perspective
(3) Attempting to think based on the data that can be used, previously learned items, and assumptions
3. Attempting to express matters clearly and succinctly
(1) Attempting to record and communicate problems and results clearly and succinctly
(2) Attempting to sort and organize objects when expressing them
4. Attempting to seek better things
(1) Attempting to raise thinking from the concrete level to the abstract level
(2) Attempting to evaluate thinking both objectively and subjectively, and to refine thinking
(3) Attempting to economize thought and effort
II. Mathematical Thinking Related to Mathematical Methods
1. Inductive thinking
2. Analogical thinking
3. Deductive thinking
4. Integrative thinking (including expansive thinking)
5. Developmental thinking
6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)
7. Thinking that simplifies
8. Thinking that generalizes
8. Thinking that specializes
9. Thinking that symbolize
10. Thinking that express with numbers, quantifies, and figures
III. Mathematical Thinking Related to Mathematical Contents
1. Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)
2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)
3. Attempting to think based on the fundamental principles of expressions (Idea of expression)
4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)
5. Attempting to formalize operation methods (Idea of algorithm)
6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)
7. Focusing on basic rules and properties (Idea of fundamental properties)
8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)
9. Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)
(Mathematical Thinking and How to Teach It?, http://pbmmatmarsigit.blogspot.com/)

Topic: Linear Equation System with Two Variables
Aim : Knowing about student’s learning linear equation system.
I asked to student at third grade junior high school about linear equation systems. I make an example with the variable is books and pencils. If I bought three pencils and two books, the cost is Rp12.000 and if I bought two pencils and three books, the cost is Rp13.000. How the cost of each book and pencil?
She can answer that book’s cost is Rp3.000 and pencil’s cost is Rp2.000. To solve that problem, she makes idealization that pencils are same and books, too. She makes assuming that each book has a same cost and each pencil has a same cost, because if each cost is different she cannot solve that problem.
I asked her, how is the color? Are you use that to solve? She said nope because the important thing is the cost.
She makes abstraction by changed pencil with variable x and by changed book with variable y. It makes her easy to solve the problem. She can easily solve the problem with variable x and y. It makes her use elimination or substation easily.
Conclusion:
She use mathematical thinking to solve the problems, because she makes an idealization and abstraction. Idealization and abstraction including a mathematical attitudes, mathematical methods, and mathematical contents.

A category is a pure concept of the understanding. The understanding is defined as the faculty of the mind which deals with concepts. Immanuel Kant believe that human mind can regulate experience with space and time, but there is a category before an experience.

Kant arranges the forms of judgment in a table of judgments which he uses to guide the derivation of the table of categories. He creates a list of categories by first enumerating the forms of possible objective judgment which are endowed with their objectivity by virtue of their inherent apriory concepts.

Quantity of Judgments, there are universal, particular, and singular.

Quality, there are affirmative, negative, infinite.

Relation, there are categorical, hypothetical, disjunctive

Modality, there are problematic, assertoric, appodeictic

Then, Kant differentiated twelve pure concepts of understanding four classes of three, they are:

Categories of quantity, there are unity, plurality, and totality.

Categories of quality, there are reality, negation, and limitation.

Categories of relation, there are inherence and subsistence (substance and accident), causality and dependence (cause and effect), and community (reciprocity between agent and patient)

Categories of modality, there are possibility-impossibility, existence-nonexistence, and necessity-contingency.

These category is a native conception of understanding and a pure concept of understanding. Then Kant said that thought without the content with perception supply are empty. Kant said that representations must have some common ground if they are to be the source of possible knowledge, this ground of all experience is the self-consciousness of the experiencing subject. So, categories feature is an important thing for the experience.

Each category has a schema. Schemata are needed to link the pure category to sensed phenomenal appearances because the categoties are heterogeneous with sense intuition.

Sumber:

http://en.wikipedia.org/wiki/Category_(Kant)

http://en.wikipedia.org/wiki/Critique_of_Pure_Lesson

http://id.wikipedia.org/wiki/Immanuel_Kant

To make a scientific work, we have to know about scientific characteristics.
First characteristic is impersonal. It means that the scientific work might not be contained with our personal. Scientific woks should be universal and global.
Second, have a standard or criteria. It can be report, proposal, etc.
Third, have ethical code and no plagiarism.

We use mathematical thinking to make a scientific work. We can use mathematical thinking of we know about mathematics. What is mathematical object??
Mathematical object just lye in our mind. Mathematical object is coming from a concrete object and abstract object. Abstract object cannot be manipulate.
There are two way to get mathematical object:
1. Idealization --> with assumption that the object is absolutely perfect.
2. Abstraction --> just to learn a certain characteristic.

And the important thing in mathematical thinking is about consistent and logic. Consistent is suitable with the agreement. If there is one is not consistent, then all is not consistent. Logic is coming from daily life.

PART 1

1. Explain how to prove that the square root of two is irrational number!

First, suppose that square root of two is rational number, that is square root of two is equal to a over b, a and b is relatively prime. We can say that a is equal to square root of two times b. So, a square is equal to two times b square.

a square and a are an even number because a square is twice an integer number.

Let say that a is equal to two times c. Then, we get four times c square is equal to two times b square. It is same with two c square is equal to b square. We see that b square is an even. So, b is an even too.

This is impossible to say that a and b is relatively prime. So, square root of two is not a rational number. Square root of two is irrational number.

2. Explain to show that sum angle of triangle is equal to one hundred and eighty degree!

First, we should make a triangle. Then, cut every angle. Arrange the angle, so the points coincide. We see that the angles of triangles make a straight angle. We just learn that the straight angle is equal to one hundred and eighty degree. So, sum angle of triangle is equal to one hundred and eighty degree.

3. Explain how you are able to get phi!

First, we can make many circles with different diameters. Then, we have to measure the circumference in every circle. Now, we have diameters and circumferences. Then, we divide every circumference by its diameters. So, we get the value is approach to three point one four. So we can find phi.

4. Explain how you are able to find out the area of region bounded by the graph of y = x² and y = x + 2

First, we must find the intersect point of y equals x square and y equals x plus two. We can substitute y. It become x square equals x plus two. We can subtract both sides by negative x minus two and we get x square minus x minus two is equal to nought. And we get x roots is two or negative one. Two and negative one is the intersect point in x.

The graph of y equals x plus two is above graph y equals x square. So, to find out the area, we can use integral with lower boundary x equals negative one to upper boundary x equals two, of x plus two minus x square in brackets dx.

So we get the area is equal to x square over two, plus two x, minus x cube over three, with lower boundary x equals negative one to upper boundary x equals two. We substitute the value of x. Then, we get two square over two, plus two times two, minus two cube over three, minus open bracket negative one square, plus negative one times two, minus negative one cube over three close bracket. So, the area of region is twenty seven over six.



5. Explain how to determine the intersection points between the circle x² + y² = 20 and y = x + 1!

First, we substitute x plus two in circle's variable y. So, we get x square plus open bracket x plus one close bracket square is equal to twenty. Then we get x square plus x square plus two x plus one is equal to twenty. Now, we add both sides by negative twenty. We get two x square, plus two x minus nineteen.
Now, we can get value of x. x is equal to a half plus a half of square root of thirty nine and x is equals to negative a half minus a half of square root of thirty nine. To get the value of y, we can substitute value of x in the equation of y equals x plus one.

1^st video : Dead Poets Society

This video tells us about look at something in a different way. We shouldn't look something in a common way because we can't be creative. We must look our own way.

2^nd video : Believe

In this video, there is a boy. That boy said on the stage. First, he asked audiences, "Do you believe in me?" And the audiences answered,"Yes!!". Then, the little boy replied, "Because I believe in me." This video tell us to believe in ourselves. Because we can do anything, say anything, think anything, and become anything if we can believe in ourselves.

3^rd video : What You Know about Math

This video's contents is a song about mathematics. This song is funny because the singers (two men with rapper style) give us a question that is what you know about math. Then, the singer tell us little about mathematics. I just can say that this song is very interest.

4^th video : Solving Differential Equation

Let dy over dx is equal to four times x square. To find variable y, first, we can multiply both sides by dx. So, we get dy is equal to four times x square times dx. Now, we must integrate both sides. We get integral dy is equal to integral four times x square times dx. Now, we can solve it. We get that y is equal to three fourth times x cube plus c. c is constant.

5^th video : Solving Linear Equation with One Variable

1. Let x – 5 = 3, Find the value of x!

To get the value of x, we can add both sides with five. It will be x minus five plus five is equal to three plus five. So we get x is equal to eight.

2. Let 7 = 4a – 1, Find the value of a!

We can add both sides with one. It will be seven plus one is equal to four a minus one plus one. We get eight is equal to four a. Then, we divided both sides by four. So, we get two is equal to a.

3. Let 2/3 x = 8, Find the value of x!

To get the value of x, we can multiply both sides by three second. It becomes three second times two third x is equal to eight times three second. So, we get the value of x is eight times three second, that is twelve.

4. Let 5 – 2x = 3x + 1, Find the value of x!

First, we can subtract both sides by three x. It becomes five minus two x minus three x is equal to three x plus one minus three x. We get five minus five x is equal to one. Then, we can subtract both sides by five. That equation becomes negative five x is equal to negative four. Now, we can divide both sides by negative five. So, we get x is equal to four fifth.

5. Let 3 – 5(2m – 5) = -2, find the value of m!

First, we should multiply negative five by two m minus five. The equation becomes three minus ten m plus twenty five is equal to negative two. We get twenty eight minus ten m is equal to negative two. Then, we can add both sides by negative twenty eight. We get negative ten m is equal to negative thirty. Last, we can divide both sides by negative ten. So we get m is equal to three.

6. Let ½ x + ¼ = 1/3 x + 5/4, find the value of x!

First, we should subtract both sides by a quarter. It becomes a half x is equal to one third x plus one. Then, we can subtract both sides by one third x. We get one sixth x is equal to one. Last, we can multiply both sides by six. So we get x is equal to six.

7. Let 0,35 x - 0,2 = 0,15 x + 0,1, find the value of x!

First, we can add both sides by negative nought point one. We get nought point three five x minus nought point three is equal to nought point one five x. Then, we add both sides by negative nought point one five x and we get the equation is nought point two x minus nought point three is equal to nought. Then, we add both sides by nought point three. We get nought point two x is equal to nought point three. Last, we can divide both sides by nought point two. And we get x is equals to one point five

6^th video : Proof Log Base x of A is Equal to Log Base x Minus Log Base x of B

Let say that logarithm base x of A is equal to B (log_x A = B). We can say that it is same to x to the B is equal to A (x^B =A). If we multiply log base x of A with C, we get C times log base x of A is equal to B times C (C log_x A = BC).

Go back to x to the B equal to A. If we rise x to the B to the power of C, we get x to the B to C power is equal to A to the C. We can say that x to the BC is equal to A to the C (x^BC = A^C).

Then, write x to the BC is equal to A to the C in logarithm expression. So, it becomes logarithm base x of A to the C is equal to BC (log_x A^C = BC).

Now, we can see that C times logarithm base x of A is equal to logarithm base x of A to the C (C log_x A = log_x A^C).

We know that C times logarithm base x of A is equal to logarithm of A to the C and we just learn that logarithm base x of A plus logarithm base x of B is equal to logarithm base x of A times B (log_x A + log_x B = log_x AB).

What happens if we change the addition with a subtraction?

Let say that:

1. Log base x of A is equal to l, it says that x to the l is equal A.

2. Log base x of B is equal to m, it says that x to the m is equal B

3. Log base x of A over B is equal to n, it says that x to the n is equal to A over B.

Now, we change A over B with x to the l over x to the m. We can write that x to the l over x to the m is equal to x to the l times x to the negative m or that also equal to x to the l minus m. We can see that x to the n is equal to x to the l minus m. So, n is equal to l minus m.

Then, log base x of A over B is equal to l minus m. That also equal to log base x of A minus log base x of B (because log base x of A is equal to l and log base x of B is equal to m). So, log base x of A over B is equal to log base x of A minus log base x of B.