Developing a Professional Vision of Classroom Events

Posted by husnulkhuluq | Mathematics | Tuesday 31 May 2011 09:50

Authored by: Miriam Gamoran Sherin (Northwestern University)

Professional vision is the ability of someone to see and to interpret a condition based on their background understanding that can distinguish them with the others in their professional aspect.
For the classroom event, there are at least two groups to develop the professional vision, they are: researchers, and teachers. Therefore, this journal will focus on two perspectives, of professional vision toward the classroom event, that are based on the personal experience of the Sherin as a researcher and David Louis as a teacher. Hoping to be able identifying the components of professional vision of both teacher and researcher, let’s check out of the following stories:

1. How I Learned to Interpret Classroom Practice
(Personal experience of Miriam Gamoran Sherin)
The story began when Sherin joined “Video Portfolio Project” which aims to design a performance assessment of mathematics teaching in form of portfolio video. After months of discussion and testing, they conclude that there are 4 aspects to concern, are: management, pedagogy, mathematical thinking, and climate.

For that, Sherin began to analyze her experience in teaching through videos analyzing by identifying the callouts and highlight some to focus attention.

Next, Sherin decided to use observation and videotape of mathematics classrooms to study the way teachers conduct the lesson. For this case, there are 2 steps he did: 1) identifying new criteria (focus on the way teacher teaches mathematics), 2) doing on-line assessment during live classroom observation. Through this process, Sherin developed professional vision of classroom events. So, her actual professional vision was not only to notice the events to focus attention but also to interpret what the teacher and the students said and to determine what this implied about their understanding.

Another thing she did was holding a regular meeting, with the teachers she was observing, to watch and discuss the excerpts of videotapes from the teachers’ classroom. She then explored the relationship between teachers’ interpretation of video and their classroom instruction. That’s why she met David Louis.

2. Changing Perceptions of Classroom Practice
(Story/ Personal experience of David Louis)

David is a mathematics teacher who also worked to design and implement units of curricula that he thought supported the Community of Learners principles. He collaborated with Sherin and Mendez to observe the efforts of mathematics teachers to implement the pedagogy reform Community of Learners.

In this research, David was identified initially focus on pedagogy and alternate pedagogy strategy he might have used. The problem he faced most was how to conduct an effective classroom discussion. Initially, David’s professional vision was the same as that of the other teachers.

A Shift in Professional Vision
Basically, there is a significant different between Sharen (researcher, which focus on interpreting mathematical ideas arose in class and David (teacher, which focus on what action to do in the given situation) after watching the videos of classroom activities. Therefore, Sharen encouraged David to analyze “what did happen”, not “what might have happened”. David did what Sharen suggested for him, and finally he became more comfortable to analyze what had occurred in the classroom through the excerpt of the videotape, o find the meaning of students’ comments, and to understand the mathematics had been discussed. However, it doesn’t mean that the researcher’s professional vision is better than that of the teachers, but it help in making the teachers’ standard focus on pedagogy quite analytical.

Impact on David’s Teaching
A shift in David’s professional vision, analyzing video, will of course have an impact on his teaching method. Now, David began to look easily at mathematical ideas arose in class during the lesson.
When asked how his experience in club influenced his teaching, David shared two things, are: 1) in the video club, he began to focus in a detailed manner on the ideas and comments that students raised, 2) through club video, he had developed different technique for reflecting on his teaching.

Issues in the Analysis of Professional Vision
Based on the experiences of those two people (Sherin and David), we can draw 3 factors contributing toward the development of professional vision of classroom events, they are: 1) our role in the classroom, 2) the medium through which we observe a class, and 3) the strategies we use to interpret the practice.

“Quadratic Function Card”

Posted by husnulkhuluq | Mathematics | Friday 15 April 2011 14:30

The post is made as the completeness of my Microteaching instructional packages

Objective: Increasing students’ ability in identifying graph of quadratic function based on its algebraic properties
Checking the students’ understanding toward the lesson they learned

Sample Card

Play Guidance
1. Teacher provides some cards on his table.
2. Teacher begins pin a card on the board.
3. Teacher calls a students’ number.
4. Every student (from each group) with that number comes in order (based on group’s order) to the teacher’s table and try to find appropriate card containing graph satisfying the condition.
5. The group whose delegation answers wrong will directly be eliminated from the game.
6. The group surviving until the card is over will be the winner.


“Teaching Aid”

Posted by husnulkhuluq | Mathematics | Thursday 3 February 2011 08:45


A. Introduction
Many people think that mathematics is very unpleasant, very difficult, and meaningless. They just imagine that mathematics is identically with complicated numbers, many long term equations which are hard to memorize and to understand. Several students moreover think that mathematics such a burden and scary lessons. There are some opinions implanted in students’ mind before learning math. As the consequence, most of the students achieve bad for this lesson. Those opinions are:

1. Mathematics’ formula is to memorize
This makes the students are lazy to learn and finally they got nothing. But actually mathematics is not a memorizing lesson. It is about concept understanding. Memorizing math formula will be meaningless moreover if they didn’t understand the concept.

2. Mathematics is an abstract knowledge and has no relation with real life
This term is definitely wrong. The fact shows that math is a very realistic knowledge. It is such an analogue of the real life. It is proved by many inventions and research of mathematics in such sectors as economy, technology, demography, social, and many others where math has a very significant role for it.

3. Mathematics is boring and rigid
This opinion isn’t true. Although mathematics problem generally has only one solution it allows many ways to reach that solution. It means that mathematics isn’t rigid.
(Mallarangan, 2009).

To prevent those opinions, we have to change our mathematics learning teaching process to be more applicative and real. The topic or the way we teach should consider the students’ necessary. We have to implant and ensure that the students love mathematics. One way is to habit them finding the concept through game or easy-learning-condition. For this case, I will try to give some illustrations about using Lego in teaching number patterns and concept of area.

B. Introduction to Lego
Lego is a line of construction toys with a colorful interlocking plastic bricks and an accompanying array of gears, mini figures, and various other parts. Lego bricks can be assembled and connected in many ways to construct some objects. (Christiansen, 2007).
This game is very popular toward the children. Not only kindergarten students but also elementary, junior, until senior high school. Playing with Lego can help us to increase creativeness, thinking on patterns, and many others.
Here are the examples of Lego:

Generally, lego consist of two elements, are:
1. Grand Base
Grand BaseThe grand base is the basic larger interlocking bricks on which the bricks are assembled. This is to ensure that the lego is well arranged.
2. Interlocking bricks

Interlocking brick is separated bricks that we are going to assemble.

C. Lego as a Teaching Aid
A teaching aid is a tool used by teachers, facilitators, or tutors to:
- help learners improve reading and other skills,
- illustrate or reinforce a skill, fact, or idea, and
- relieve anxiety, fears, or boredom, since many teaching aids are like games.
(SIL, 1999).
Based on those three requirements, I think lego can also be used, as a teaching aid, in some certain subjects include mathematics. Playing lego can improve the students’ skill in building or constructing some forms and analyzing pattern skill. Besides that, it can also help the students to reinforce and illustrate the idea in their mind by constructing some on the grand base. The most important also is by using lego the students will not be anxious, afraid, and bored in learning especially mathematics.
This media is made of terraced papers and colored papers. He used this media to teach number patterns. After observing the media, I guess that using lego will be more interesting to the students.
Here, I’m going to show you two concepts that can be taught using lego. However there are many other concepts that are available to be taught using lego too.

1. Using Lego in Teaching Number Pattern
According to the recent curriculum applied in Indonesia, number pattern is learned in the 6th and the 7th grade. Basically, number pattern emphasizes on students ability to analyze some certain ordered numbers. The task is to determine the term in a certain position. The ordered numbers is actually can be represented by the number of lego brick assembled to construct a certain pattern. Therefore, the students can be easier to find the pattern.

There are some steps that the teachers should do in teaching this lesson by using lego, are:
a. Teacher tells students the learning objectives

b. Teacher takes the grand base of lego

c. Teacher shows the students how to play lego and show them the 1st term

d. Students and teacher construct the 1st until the 3rd term together

e. Teacher ask the students to continue playing lego and identifying the number pattern

f. Students present the pattern they get from playing lego

g. Teacher checks students work by identifying the number pattern in front of the class and asks the students to make conclusion.

2. Using Lego in Teaching Concept of Area
Based on curriculum, the area of a plane figure is firstly learned in 5th and 6th grade of elementary school. I found that most students nowadays did not understand the concept of area eventhough they can finish the questions on that term. Most of students only memorize the formula of finding area of a plane figure but they didn’t understand the concept. So with, if the students are given non-basic plane figure, it will be difficult for them to find the area. For that, we can use lego as the solution to teach the concept of area to the students either directly on indirectly.

There are some steps that the teachers should do in teaching this lesson by using lego, are:
a. Teacher tells students the learning objectives

b. Teacher tells the students the concept of area

c. Teacher takes the grand base of lego and mark a certain area in grand-base that will be measured

d. Students are asked to close the area by interlocking the brick

e. Teacher tells the students the marked area by counting the brick gears.

f. Teacher gives another marked area and asks the students to find the area themselves.

g. Teacher checks the students’ work and help the students to make conclusion.


Christiansen, Ole Klirk. 2007. Lego. Accessed on December 30th, 2010 through website
Mallarangan. 2009. Meruntuhkan Mitos Matematika yang Menakutkan Menjadi Menyenangkan. Accessed on January 4th, 2011 through website
SIL International. 1999. What is a Teaching Aid? Accessed on December 30th, 2010.
Unit Laboratorium Matematika PPPPTK Matematika. Unknown. Daftar Alat Peraga Matematika. Accessed on December 30th, 2010 through website

“Queuing Theory”

Posted by husnulkhuluq | Mathematics | Thursday 16 December 2010 07:36

Queuing Theory is first declared by Agner Kraup Erlang in 1909 with the paper published entitle “Queuing Theory”. The idea comes from his observation toward the limited capacity problem of service telephone to service consumer demand in a certain time. There are some examples with some unique pattern problems like traffic, bank service on the month or every week. The unique pattern we mean here is the lately service which is caused by the service demand rate which is more than the capability of facility to serve. However, the unique of the pattern shows that there are many processes to recognize together with their assumptions.

A. Basic Concept of Model
Basic Purpose
Queuing models aims to minimize a direct cost to supply the service and individual cost for they are waiting for the service. The different between the number of demands and the capability of service facility causes two consequences, are queuing and empty capacity. A long queue implies waited-line that causes an opportunity cost to the costumer. Another issue, too many service facilities affect meaningless facilities. That’s why they both should be minimized.

System and Parameters
There are four dominant factors toward the system approach such as, System Boundaries, Input, Process, and Output. System boundary gives us a limitation of the area observing in queue. Input implies the people need to get a service from a supplier facility. The process means the service activity itself while output states the customer who is completely served.

There are two variables that influence waiting line shape that is arrival rate (λ) and service rate (μ). If arrival rate is greater than the service rate then there is waiting line. Therefore, we should assume λ > μ to guarantee that process do not stop because of a very high demand.
Next, we will show the simulation of queuing basic system.

Arrival Rate (λ) and Poisson Process
Let’s think about the observation of A. K. Erlang in Copenhagen Telephone, the pattern of costumers demand on continuous of time can be divided into several fixed intervals. In this case, the demand of the costumer is distributed randomly in each fixed intervals. It is known as Poisson Process.

From the example, there are 10 costumers came in 06.00 – 10.00. But, it is different from what happened in the other intervals. This is an example of phenomena observed by A.K. Erlang and followed by Poisson process which is often occur in any queuing cases. In this case, the assumption as follows:
1. Arrival costumer is randomly
2. Arrival costumer in each time interval didn’t influence each other.

From the example, the time interval is divided into four fixed intervals. If I is the amount of time interval, then

Where, Ii is the i-th interval

This case shows I1 = 1 interval with 6 arrivals; I2 = 1 interval with 1 arrival; I3 = 1 interval with 0 arrival; and I4 = 1 interval with 3 arrivals. Hence I = 4. If N symbolizes the number of costumers coming during the intervals, and there is Ki customers in interval Ii, hence the number of costumer during I is:

In this case, N = 6+1+0+3 = 10.
So the costumers come randomly to every similar interval. If every interval divided into n subintervals then with the same process and assumption, the time interval can be stated as Poisson distribution. Therefore, the arrival rate of the costumers in each fixed interval can be estimated as

Service Rate (μ)
Service rate is the average time needed to serve a customer. If the capacity of service facility can serve 4 costumers per hours, then the service rate is µ = 4/hours, and the service time (time needed to serve each costumer) is 1/µ = 15 minutes. Therefore, we know that the service time satisfies negative exponential distributions while the service rate satisfies Poisson distribution.

The Length of Waited-Line
In fact, the length of waited line bounded by the space capacity to wait before getting. But, finite waited line capacity need special development model. That’s why in common model discussion; it’s assumed infinite.

Queuing Rule
Waited line or queuing is formed when the service of facility is busy servicing the costumer, so the other costumers have to wait. If the service is finished, then the first customer in queuing line is will be first served. The rule is clear, first come first served.

Equilibrium Concept in System
Queuing model use equilibrium concept or balancing the number of costumers in system as a ground to develop the model. If there is N costumers in a system then a costumer is getting out of system after being served hence the number of costumer in the system become N-1, then a costumer is coming into the system hence the number of costumer become N.
If, λ = arrival rate
µ = service rate
Pn = probability of n costumer in system
Pn-1 = probability of n-1 costumer in system

THEN the concept of equilibrium can be written as OR

If λ as arrival rate and µ as service rate with λ > µ as assumption, then busy rate of the system denoted by ρ is

“Queuing Theory – Model Configuration”

Posted by husnulkhuluq | Mathematics | Wednesday 8 December 2010 16:23

As we can see in reality, queuing happened in some public services has various forms. It might be caused by the number of services they are going to have or the number of people is in the queuing. For this case, we can analyze four items, are:
 Length of System (Ls)
 Time Spent in System (Ts)
 Length of Queue (Lq)
 Time Waiting in the Queue (Tq)

1. Length of System (Ls)
The length of system or Ls is the number of customers is in system, including they are still in queuing or in serving. Ls is influenced by two parameters are arrival rate (λ) and service rate (μ). Another thing to concern, the busier the system is, the lower the probability of empty system, and vice versa. That’s why; it is logically to define Ls as the comparison of the business rate (λ/μ) with the probability of empty service.

2. Time Spent in System (Ts)
As the definition of Ls, we can guess to define the time of system or Ts as the total time during the customer entering the queue until totally serviced. Ts will of course is be influenced by Ls and arrival rate or λ.
Therefore, Ls = λ . Ts
Let’s compare it with the previous formula we got,

3. Time Waiting in the Queue (Tq)
Time waiting in the queue or Tq is defined as the total time during the costumer is in queue (entering system until having service). So, the difference from Ts itself is only the service time (1/μ).

Let’s substitute the formula of Ts into that equation,

4. Length of Queue (Lq)
The length of queue or Lq is influenced by arrival rate (λ) and the time waiting in the queue (Tq). Thus,

Here, we have also four configuration model of queuing that we always meet in our daily life, such as:
 Single Channel Single Phase
 Multi Channel Single Phase
 Single Channel Multi Phase
 Multi Channel Multi Phase

In this case, we define channel as the number of servicer in every activity to serve such as the number of cashier available in the supermarket, the number of ATM machine in a place, and many others. Phase itself is defined as the number of services that the customers should have to complete servicing.
1. Single Channel Single Phase
This type is the simplest model configuration of queuing. It only consists of one row queuing and a kind of service to have. In daily life, we can see it in a single ATM machine or in one-locket ticketing service. We can also draw it as the following:

2. Multi Channel Single Phase
In daily life, this model can be seen in some cashiers in supermarket. Where there are many cashiers in that place but we should only queue to one cashier to finish paying.
The main concept of this configuration is;
No queuing if n≤k or the number of costumer is at most equals to the number of facilities serving.
There is a queue if n>k or the number of costumer to serve is greater than the number of facilities serving.

3. Single Channel Multi Phase
Single channel multi phase describes another queue model configuration which involves more than one kind processes and only one server in each serving process. We can see this queuing model in police office when we want to make a driving license (SIM). Where there are some services to have such as healthy testing service, photo service, etc.

4. Multi Channel Multi Phase
Multi channel multi phase is the most complicated queuing model. It involves many activities and many services too in each activity we have. This model can be seen in SNM-PTN registration where there are many activities we face such as taking form, paying service, giving back the form, etc and there are also many servers in each activity we should complete.

“Introduction to Philosophy of Mathematics”

Posted by husnulkhuluq | Mathematics | Wednesday 3 November 2010 15:51


Galileo Galilei (1564 – 1642) said that:
“Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics”

1. Introduction to Philosophy of Mathematics
Etymologically, the word “philosophy” means the study of what the thing means to exist, what good and evil are, what knowledge is, or how people should live. It implies that to talk about the philosophy of mathematics is to talk about what the mathematics is, where it comes from, and how its characteristics are.
Another definition about philosophy stated by John Dewey that philosophy is a kind of thinking that consists in turning a subject over in the mind and giving it a serious and consecutive consideration. In relating with his statement, we can conclude that the philosophy of mathematics is a mean to think about mathematics over and intensively. Not only studying how to count or solving differentiation or integration problems but also studying the essential of it.
That’s why the study about the philosophy of mathematics addresses to the questions below:
a. What is the basis for mathematical knowledge?
b. What is the nature of mathematical truth?
c. What characterizes the truths of mathematics?
d. What is the justification for their assertion?
e. Why are the truths of mathematics necessary truths?

2. The Point of Philosophy of Mathematics
Based on 5 questions addressed to philosophy of mathematics, the study of philosophy of mathematics is concerned on 7 aspects, are:
a. Mathematical Epistemology
Mathematical epistemology means a reflective thinking on mathematics, such as the nature of mathematics, the history, assumption, foundations, and the truth of mathematics itself. So this part focuses on answering the question what mathematics is, how its properties are, and what it talks about.

b. Mathematical Ontology
Mathematical ontology makes the substance or the content of mathematics itself as problem to investigate. It studies every topic discussed in mathematics whether it is real or not.

c. Mathematical Methodology
Mathematical methodology studies the characteristics of the methods applied in mathematics. Mathematics nowadays uses axiomatic method which is involves such problems as the selection, independence, and simplification of primitive terms and axioms, the formalization of definitions and proofs, the consistency and completeness of the constructed theory, and the final interpretation.
d. Logical Structure of Mathematics
This part discusses about the language used in mathematics, its structure, and so on in connection with human logic. The study of the logical structure of mathematics will show us that mathematics is very rigor of logical laws, precision, and conclusion without denying the real condition. It implies one purpose of mathematics that is to construct our mind.

e. Ethic Implication of Mathematics
This case concerns on the existence of Mathematics in human daily life. In other word, we can say that this part guides us to know how important mathematics is in our life. In science philosophy, this term is known as the end of knowledge is to answer the question “what does science, now that we have achieved it, really amount to?”

f. Esthetic Aspect of Mathematics
Good mathematics must satisfy one of the three criteria, are; immediate usefulness on science, potential usefulness, or beauty. The beauty of mathematics is caused by the originality of the idea in mathematics, simplicity of theorems, creativity of thinking, or other properties within mathematics.

g. The Role of Mathematics in Human Civilization History
It is about mathematical history era by era which brings so many impacts in human activity.

3. Introduction to Absolutist View of Philosophy of Mathematics
In the beginning of the study about philosophy of mathematics, the study is dominated by an absolutist, which views it as a body of infallible and objective truth, far removed from the affairs and values of humanity. It is supported by an approach which is most adopted in epistemology that assumes the knowledge in any field is represented by a set of propositions, together with a set of procedures for verifying them, or providing a warrant for their assertion.
In this case, mathematical knowledge consists of a set of propositions together with their proofs. Since mathematical proofs are based on reason alone, without recourse to empirical data, mathematical knowledge is understood to be the most certain of all knowledge.
So that the absolutists conclude that the truth of mathematics is exact and unquestionable any more.

4. Assumption
Based on the characteristics of philosophy of mathematics, it is obviously seen that the study has a very important role toward the mathematics advance. It has to be a very strong and systematic foundation of mathematical knowledge. That is mathematical truth.
This assumption became a basic of foundationism which bounds up the absolutist view about mathematical knowledge.

Ernest, Paul. 1991. The Philosophy of Mathematics Education. London: The Farmel Press.
Gie, The Liang. 1999. Filsafat Matematika. Yogyakarta: Pusat Belajar Ilmu Berguna.
Manalaksak, Dwin Gideon. 2004. Tinjauan atas ‘Fungsi’ Berdasarkan Filsafat Matematika. Jurnal Filsafat, Desember 2004, Jilid 38, Nomor 3.
Marsigit. 2008. Matematika Ditinjau dari Berbagai Sudut Pandang.

Resensi buku “The Philosophy of Mathematics Education”, karya Paul Ernest

Posted by husnulkhuluq | Mathematics | Friday 22 October 2010 13:34

Para absolutis teguh pendiriannya dalam memandang secara objektif kenetralan matematika, walaupun matematika yang dipromosikan itu sendiri secara implisit mengandung nilai-nilai. Abstrak adalah suatu nilai terhadap konkrit, formal suatu nilai terhadap informal, objektif terhadap subjektif, pembenaran terhadap penemuan, rasionalitas terhadap intuisi, penalaran terhadap emosi, hal-hal umum terhadap hal-hal khusus, teori terhadap praktik, kerja dengan fikiran terhadap kerja dengan tangan, dan seterusnya. Setelah mendaftar macam-macam nilai di atas maka pertanyaannya adalah, bagaimana matematisi berpendapat bahwa matematika adalah netral dan bebas nilai ? Jawaban dari kaum absolutis adalah bahwa niai yang mereka maksud adalah nilai yang melekat pada diri mereka yang berupa kultur, jadi bukan nilai yang melekat secara implisist pada matematika. Diakui bahwa isi dan metode matematika, karena hakekatnya, membuat matematika menjadi abstrak, umum, formal, obyektif, rasional, dan teoritis. Ini adalah hakekat ilmu pengetahuan dan matematika. Tidak ada yang salah bagi yang kongkrit, informal, subyektif, khusus, atau penemuan; mereka hanya tidak termasuk dalam sains, dan tentunya tidak termasuk di dalam matematika (Popper, 1979 dalam Ernest, 1991: 132).
Yang ingin ditandaskan di sini adalah bahwa pandangan kaum absolutis, secara sadar maupun tak sadar, telah merasuk ke dalam matematika melalui definisi-definisi. Dengan perkataan lain, kaum absolutis berpendapat bahwa segala sesuatu yang sesuai dengan nilai-nilai di atas dapat diterima dan yang tidak sesuai tidak dapat diterima. Pernyataan-pernyataan matematika dan bukti-buktinya, yang merupakan hasil dari matematika formal, dipandang dapat melegitimasikan matematika. Sementara, penemuan-penemuan matematika, hasil kerja para matematisi dan proses yang bersifat informal dipandang tidak demikian. Dengan pendekatan ini kaum absolutis membangun matematika yang dianggapnya sebagai netral dan bebas nilai. Dengan pendekatan ini mereka menetapkan kriteria apa yang dapat diterima dan tidak diterima. Hal-hal yang terikat dengan implikasi sosial dan nilai-nilai yang menyertainya, secara eksplisit, dihilangkannya. Tetapi dalam kenyataannya, nilai-nilai yang terkandung dalam hal-hal tersebut di atas, membuat masalah-masalah yang tidak dapat dipecahkan. Hal ini disebabkan karena mendasarkan pada hal-hal yang bersifat formal saja hanya dapat menjangkau pada pembahasan bagian luar dari matematika itu sendiri.
Jika mereka berkehendak menerima kritik yang ada, sebetulnya pandangan mereka tentang matematika yang netral dan bebas nilai juga merupakan suatu nilai yang melekat pada diri mereka dan sulit untuk dilihatnya. Dengan demikian akan muncul pertanyaan berikutnya, siapa yang tertarik dengan pendapatnya ? Inggris dan negara-negara Barat pada umumnya, diperintah oleh kaum laki-laki berkulit putih dari kelas atas. Keadaan demikian mempengaruhi struktur sosial para matematisi di kampus-kampus suatu Universitas, yang kebanyakan didominasi oleh mereka. Nilai-nilai mereka secara sadar dan tak sadar terjabarkan dalam pengembangan matematika sebagai bagian dari usaha dominasi sosial. Oleh karena itu agak janggal kiranya bahwa matematika bersifat netral dan bebas nilai, sementara matematika telah menjadi alat suatu kelompok sosial. Mereka mengunggulkan pria di atas wanita, kulit putih di atas kulit hitam, masyarakat strata menengah di atas strata bawah, untuk kriteria keberhasilan penguasaan pencapaian akademik matematikanya.
Suatu kritik mengatakan, untuk suatu kelompok tertentu, misalnya kelompok kulit putih dari strata atas, mungkin dapat dianggap matematika sebagai netral dan bebas nilai. Namun kritik demikian menghadapi beberapa masalah. Pertama, terdapat premis bahwa matematika bersifat netral. Kedua, terdapat pandangan yang tersembunyi bahwa pengajaran matematika juga dianggap netral. Di muka telah ditunjukkan bahwa setiap pembelajaran adalah terikat dengan nilai-nilai. Ketiga, ada anggapan bahwa keterlibatan berbagai kelompok masyarakat beserta nilainya dalam matematika adalah konsekuensi logisnya. Dan yang terakhir, sejarah menunjukkan bahwa matematika pernah merupakan alat suatu kelompok masyarakat tertentu. Kaum ‘social constructivits’ memandang bahwa matematika merupakan karya cipta manusia melalui kurun waktu tertentu. Semua perbedaan pengetahuan yang dihasilkan merupakan kreativitas manusia yang saling terkait dengan hakekat dan sejarahnya. Akibatnya, matematika dipandang sebagai suatu ilmu pengetahuan yang terikat dengan budaya dan nilai penciptanya dalam konteks budayanya.Sejarah matematika adalah sejarah pembentukannya, tidak hanya yang berhubungan dengan pengungkapan kebenaran, tetapi meliputi permasalahan yang muncul, pengertian, pernyataan, bukti dan teori yang dicipta, yang terkomunikasikan dan mengalami reformulasi oleh individu-individu atau suatu kelompok dengan berbagai kepentingannya. Pandangan demikian memberi konsekuensi bahwa sejarah matematika perlu direvisi.
Kaum absolutis berpendapat bahwa suatu penemuan belumlah merupakan matematika dan matematika modern merupakan hasil yang tak terhindarkan. Ini perlu pembetulan. Bagi kaum ‘social constructivist’ matematika modern bukanlah suatu hasil yang tak terhindarkan, melainkan merupakan evolusi hasil budaya manusia. Joseph (1987) menunjukkan betapa banyaknya tradisi dan penelitian pengembangan matematika berangkat dari pusat peradaban dan kebudayaan manusia. Sejarah matematika perlu menunjuk matematika, filsafat, keadaan sosial dan politik yang bagaimana yang telah mendorong atau menghambat perkembangan matematika. Sebagai contoh, Henry (1971) dalam Ernest (1991: 34) mengakui bahwa calculus dicipta pada masa Descartes, tetapi dia tidak suka menyebutkannya karena ketidaksetujuannya terhadap pendekatan infinitas. Restivo (1985:40), MacKenzie (1981: 53) dan Richards (1980, 1989) dalam Ernest (1991 : 203) menunjukkan betapa kuatnya hubungan antara matematika dengan keadaan sosial; sejarah sosial matematika lebih tergantung kepada kedudukan sosial dan kepentingan pelaku dari pada kepada obyektivitas dan kriteria rasionalitasnya. Kaum ‘social constructivist’ berangkat dari premis bahwa semua pengetahuan merupakan karya cipta. Kelompok ini juga memandang bahwa semua pengetahuan mempunyai landasan yang sama yaitu ‘kesepakatan’. Baik dalam hal asal-usul maupun pembenaran landasannya, pengetahuan manusia mempunyai landasan yang merupakan kesatuan, dan oleh karena itu semua bidang ilmu pengetahuan manusia saling terikat satu dengan yang lain. Akibatnya, sesuai dengan pandangan kaum ‘social constructivist’, matematika tidak dapat dikembangkan jika tanpa terkait dengan pengetahuan lain, dan yang secara bersama-sama mempunyai akarnya, yang dengan sendirinya tidak terbebaskan dari nilai-nilai dari bidang pengetahuan yang diakuinya, karena masing-masing terhubung olehnya.
Karena matematika terkait dengan semua pengetahuan dari diri manusia, maka jelaslah bahwa matematika tidaklah bersifat netral dan bebas nilai. Dengan demikian matematika memerlukan landasan sosial bagi perkembangannya (Davis dan Hers, 1988: 70 dalam Ernest 1991 : 277-279). Shirley (1986: 34) menjelaskan bahwa matematika dapat digolongkan menjadi formal dan informal, terapan dan murni. Berdasarkan pembagian ini, kita dapat membagi kegiatan matematika menjadi 4 (empat) macam, di mana masing-masing mempunyai ciri yang berbeda-beda:
a. matematika formal-murni, termasuk matematika yang dikembangkan pada Universitas dan matematika yang diajarkan di sekolah;
b. matematika formal-terapan, yaitu yang dikembangkan dalam pendidikan maupun di luar, seperti seorang ahli statistik yang bekerja di industri.
c. matematika informal-murni, yaitu matematika yang dikembangkan di luar institusi kependidikan; mungkin melekat pada budaya matematika murni.
d. matematika informal-terapan, yaitu matematika yang digunakan dalam segala kehidupan sehari-hari, termasuk kerajinan, kerja kantor dan perdagangan.
Dowling dalam Ernest (1991: 93), berdasar rekomendasi dari Foucault dan Bernstein, mengembangkan berbagai macam konteks kegiatan matematika. Dia membagi satu dimensi model menjadi 4 (empat) macam yaitu : Production (kreativitas), Recontextualization (pandangan guru dan dasar-dasar kependidikan), Reproduction (kegiatan di kelas) dan Operationalization (penggunaan matematika). Dimensi kedua dari pengembangannya memuat 4 (empat) macam yaoitu: Academic (pada pendidikan tinggi), School (konteks sekolah), Work (kerja) dan Popular (konsumen dan masyarakat).
Dengan memasukkan berbagai macam konteks matematika, berarti kita telah mengakui tesis D’Ambrosio (1985: 25) dalam ‘ethnomathematics’ nya. Tesis tersebut menyatakan bahwa matematika terkait dengan aspek budaya; secara khusus disebutkan bahwa kegiatan-kegiatan seperti hitung-menghitung, mengukur, mendesain, bermain, berbelanja, dst. Merupakan akar dari pengembangan matematika. Dowling dalam Ernest (1991: 120) mengakui bahwa pandangan demikian memang agak kabur; kecuali jika didukung oleh pembenaran tradisi matematika.
Ernest, P., 1991, The Philosophy of Mathematics Education, London : The Falmer Press.

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