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Frustrating or Challenging Assessment?

One day I came to my class and found it really tiring for me for just concentrating on the subject. I was not there (my body’s there, but my mind’s not). I was thinking on my assignment that will be collected on the next day. I thoughtĀ  of it day by day and it drove me crazy.

I remember that as a teacher one day, I’m expected to give a challenging task for my students. Then, is this a kind of challenging assignment? Or, the opposite, this is a “frustrating” assignment? I ask myself, then how could I differ those two kind of assessment? how could I give a challenging assignment as my assessment without making my student frustrated?

Such an attractive question, isn’t it?

I try to look through my sources. I think there is one thing I should consider when I make challenging assessment, OBJECTIVE. My assessment should measure what student have learned. By doing this, students won’t be really frustrated since they know that what will be assessed is something they already learn in class (imagine if they learn A then you assess B, your student will fight youšŸ˜” ).

One other thing is motivation. What comes to your mind when I said this word? Maybe you imagine a teacher who said, “You can do it, kids” to his/her students. Yes, dialogue is one way to give your students assessment. But there are still so much thing related to it.

You can motivate your student by letting them know that you can help them when they have any difficulty. Beside it helps you to do formative assessment, it also helps students to be more motivated in accomplishing their assessmentšŸ™‚ . They realized that you support them.

I think that is my answer upon my question(s). Assessment will be no more frustrating when students know this assessment is useful for them to accomplish their objective and support is available for them.

(I find this post as reminder from two years ago. I re-post it from my class blog atĀ http://petamath.wordpress.com/2011/01/10/frustrating-or-challenging-assessment/Ā )
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Gallery

Some pictures of my sketches or drawing are posted on these pages. Before I posted these here, I posted them to my Instagram, that is why you can see some visual edits.

I drew this while learning to sketch a tree from a book.

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Di Dalam Kelas dan di Luar Sana

Tulisan saya ini berawal dari saat saya memperhatikan aktivitas sehari-hari murid-murid saya. Mereka belajar di kelas dari sekitar pukul 7 hingga pukul 3 sore. Setelah itu mereka lanjut mengikuti kegiatan ekstrakurikuler dari pukul 4 hingga sekiltar pukul 5. Setelahnya, mereka akan mencari apakah ada PR yang harus dikerjakan atau materi yang harus dipelajari karena ulangan. Melihat mereka, saya teringat pada kegiatan saya selama di SMA. Dulu, saat saya duduk di bangku kelas X SMA, saya belajar di dalam kelas dari sekitar pukul 7 hingga pukul 1 siang. Kegiatan ekstrakurikuler saya tidaklah padat, hanya seminggu sekali. Bagi saya, PR, Ulangan, dan ujian adalah segalanya. Kegiatan yang nantinya tidak tertera di rapor selalu bisa saya nomor-duakan. Saya menumpahkan konsentrasi saya pada apa yang terjadi di dalam kelas.

Suatu hari, saya dipanggil oleh guru Bahasa Inggris. Beliau mengatakan bahwa saya diminta mengikuti pembinaan untuk lomba debat bahasa Inggris. Saya ragu, bahasa Inggris saya biasa saja dan saya jarang ikut lomba di kala SMP. Ibu guru saya ini meyakinkan saya bisa karena saya pernah menang lomba telling story saat SMP. Jujur, saat itu, saya takut menolak karena guru yang saya hormati ini yang mengajak. Sebenarnya saya takut, nilai rapor saya akan turun kalau terlalu berkecimpung di ekstrakurikuler. Waktu saya untuk mengerjakan PR dan belajar akan berkurang. Namun, apa boleh buat, saya menjalaninya.Ā Saya mengikuti pembinaan dengan intensif bahkan kadang hingga larut malam. Saya sering kelelahan sampai rumah dan melihat buku pelajaran saja sudah tak sanggup. Belajar? Ah, besok saja kalau ada ulangan, saya sibuk.

Selain pembinaan ini, saya juga mengikuti organisasi dakwah sekolah. Saya sering menjadi panitia hari besar agama dan berurusan dengan berbagai macam karakter teman-teman saya. Beberapa sering membuat jengkel, namun banyak yang menjadi teladan bagi saya. Sama ceritanya, saya pulang dari sekolah dalam keadaan lelah.

Lama kelamaan, saya kewalahan. Saya sering lupa kalau ada PR atau ulangan. Saya mulai merasa pengaruh “tidak baik” dari kegiatan-kegiatan yang saya ikuti. Sekali lagi, nilai rapor bagi saya adalah segalanya. Saya mulai mangkirĀ dari kegiatan-kegiatan di luar kelas. Terkadang saya izin dengan alasan acara keluarga atau pura-pura tidak tahu ada kegiatan hari itu. Ya, saya berhenti dari apa yang di luar sana dan fokus pada apa yang di dalam kelas.

Namun, dalam periode mangkirĀ itu, saya merasakan perubahan.Ā TimelineĀ kegiatan sehari-hari saya yang biasanya saya atur dalam sebuah buku kecil, terlihat kosong. Setelah pulang sekolah, saya melihat rentang waktu kosong hingga pukul 5 sore, saat saya harus mengerjakan PR atau belajar. Biasanya, dalam 4 jam itu saya bisa melatih kemampuan bahasa Inggris saya, atau diskusi dengan teman-teman di organisasi. Tidak, tidak, toh saya bisa isi dengan belajar materi pelajaran. Saya pun mencobanya. Sepulang sekolah, saya membaca materi pelajaran atau kalau jenuh, membaca novel atau komik. Seminggu saya mampu bertahan dengan rutinitas ini. Selebihnya, saya jenuh sekali.

Saya pun kembali aktif di ekstrakurikuler. Saya tuliskan kembali kegiatan-kegiatan saya di buku kecil, berusaha mengaturnya. Semua PR, tugas, atau jadwal ulangan juga saya tulis, agar saya tidak lupa. Walau sempat kewalahan juga, namun saya banyak belajar mengatur waktu secara efektif. Saya mulai menyadari, saya harus investasikan waktu saya baik untuk hal-hal di dalam kelas maupun di luar sana. Saya belajar materi pelajaran, mengerjakan PR dan tugas untuk menambah ilmu pengetahuan. Saya ikut organisasi dan ekstrakurikuler untuk menambah skill, bertukar pendapat, bercanda-tawa dengan teman-teman. Dengan begini saya tidak bosan.

Saat saya kuliah, saya baru sadar bahwa apa yang saya Ā lakukan di SMA membantu saya untukĀ surviveĀ di kampus. Kemampuan mengatur waktu, berkomunikasi, dan ilmu-ilmu pengetahuan yang saya pelajari di SMA membantu saya menjalani kegiatan-kegiatan di kampus. Memang, saya masih perlu banyak belajar tentang itu semua, namun paling tidak saya punya bekal.

Bagaimana jika saat saya bekerja? Atau berkecimpung di masyarakat? Hingga saat ini saya percaya bahwa selain apa-apa yang saya pelajari di dalam kelas, yang saya pelajari di luar kelas tidak kalah penting. Selain sebagai variasi kegiatan sehari-hari agar tidak jenuh. juga sebagaiĀ bekal untuk jenjang hidup berikutnyašŸ™‚

 

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Fostering Mathematical Literacy in Elementary School

In elementary school, students start to learn mathematics as a discourse. They start to know what are addition, subtraction, division, and multiplication. Eventually, they begin to be aware of word problem. From a simple case of daily life, they are asked to solve it by transform it into mathematical symbol and operation.

For example, Anna is given two problems:

8 ā€“ 2 = ā€¦

Bobby has 7 balls. After giving it to Sally, he only has 3 balls. How many balls Bobby gives to Sally?

For the first problem, Anna easily solves it. However, for the second problem, Anna first should understand the problem and after that, transform this word problem with daily discourse into mathematical discourse. She may know that Bobby actually gives 4 balls to Sally but she cannot write the process of it as 7 minus 3 equals 4.Here, Anna is having difficulty in transforming daily discourse into mathematical discourse. She finds it difficult to interpret a case happened in real life and use mathematics to solve it. If this case happened, teacher will somehow get confused on assessing studentā€™s understanding whether the student do not understand the meaning of the problem, or they understand the meaning but cannot transform it into mathematical discourse to solve it.

One of the solutions to overcome this problem is by fostering mathematical literacy. Mathematical literacy is the ability to think numerically to interpret and analyze real life case in order to solve it and not merely doing mathematical operation (Gouthro & Griffore, 2004). Based on the Annaā€™s case above, mathematical literacy is not merely the ability to solve the first problem but also real life problem like the second problem.

The main aspect of improving mathematical literacy is the thought that mathematics is not a collection of procedures and formulas but a tool that is useful to solve daily problems. It is also a language and communication system used for exploring knowledge (Whitin et al, 1990).

Strategy to improve studentsā€™ mathematical literacy is by making open-ended problems (Whitin et al, 1990). Different from closed-ended problem, open-ended problems allow students to try various method to find the answer. Along the process, teacher allows multiple representations as his/her appreciation on studentsā€™ unique background of knowledge. Learning mathematics does not concern more on procedural and fixed algorithm given by teacher anymore but more on guiding students to find their own meaning and knowledge of the lesson (Dā€™Ambrosio, Johnson, & Hobbs, 1995, in Gouthro & Griffore, 2004). Students can try their way and reflect whether that way is effective and fitted to them. They will also independently learn some registers of mathematics and get adjusted to mathematics discourse. By doing this, students will have sound understanding because they learn in a flexible atmosphere.

Multiple representations

Examples of multiple representation

Fostering mathematical literacy is very useful for not only students in learning but also teacher in assessing studentsā€™ understanding. From studentsā€™ side, by having good mathematical literacy, students will be directed to more abstract discourse easily (NCTM, 2009). For example, if students can understand that based on the Annaā€™s case above 4 comes from 7 minus 3, it will be easier for them to solve the following case:

Bobby has x balls. After giving it to Sally, he only has y balls. How many balls Bobby gives to Sally?

By slowly moving from concrete example to abstract problem, students with good mathematical literacy will know that the solution for this abstract case is x-y.

Nowadays, the gap between students and mathematics exists because students do not think that mathematics has connection to their real world. Moreover, mathematics discourse is relatively different with daily discourse. Since fostering mathematical literacy should be related to real life situation, students will value mathematics more. They will think that mathematics is useful and worth to be learnt (Gouthro & Griffore, 2004). It increases the interest of students to learn Mathematics more.

From teachersā€™ side, fostering mathematical literacy will help them to assess studentsā€™ understanding (Whitin et al, 1990). The current trend of assessing studentsā€™ steps on problem is by grading each step they have. For example there is a problem:

Mom has three children. Each child is given four candies by mom. How many candies Mom give to all of her children?

Teacher will expect students to write the solution as simple as 3 x 4 = 12. However, the problem is that the information got from this answer is too little to make sure studentsā€™ understanding. Since in improving mathematical literacy teacher should allow multiple representations, because each student has their own style in transforming daily discourse into mathematical discourse, the variety of ways in solving problems will also increase. Students may want to draw a picture of the three children and four candies on each child, and simply write down 4 + 4 + 4 = 12. Responding to this answer, teacher can introduce multiplication as repeated addition. Here, students will be adjusted to multiplication as his/her new register and internalize it. Teacher will obtain more information to assess studentsā€™ understanding.

Fostering mathematical literacy to help students in understanding mathematical discourse is indeed not an easy job. It comes as a respond of the difficulties had by students and their assumption to mathematics subject. However, if teachers conduct the strategy, studentsā€™ difficulties in understanding discourse may be overcome and their opinion on Mathematics will be improved as they get adjusted to mathematical discourse through flexible, open-ended atmosphere. Teachers will also get the benefits in assessing students.

Further readings:

Gouthro, M, and Griffore, J. 2004. Leading Math Success: Mathematical Literacy. Ontario: Ontario Education, retrieved on November 16, 2011 from http://oame.on.ca/lmstips/files/lms/LeadingMathSuccess.pdf

NCTM. 2009. Guiding Principles for Mathematics Curriculum and Assessment. USA: NCTM

Whitin, D. J, Mills, H, and Oā€™Keefe, T. 1990. Living and Learning Mathematics: Series and Strategies for Supporting Mathematical Literacy. Portsmouth: Heinemann.

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Solving Problems of Tangent Line to Circle by Using Geogebra

Technology is undeniably helpful for todayā€™s education. The presence of technology is already adapted by teachers as a beneficial tool to improve studentsā€™ understanding. In mathematics itself, technology is often used to find concept and solving problems. Some topics in Mathematics are rather difficult to be visualized especially for students with low abstract way of thinking. Most used technology for this type of case is Computer Algebra System (CAS) which is called Geogebra. Geogebra is a software created to help teachers to teach topics related to Algebra and Geometry. Its visualization of function, figure, and geometric object is somehow useful for students to depict what exactly matter.

In this paper, writer will narrow her discussion on topic of tangent line to circle. This topic is considered to be a suitable one since it consists of combination between geometric and algebraic interpretation. However, the topic will not be delivered thoroughly conceptual, but by using problem solving approach. Collaborated with Geogebra, problem solving will be discussed in this paper as promising method in teaching students the concept of tangent line to a circle.

Students nowadays tend to think that mathematics problem is only about procedures and rules (Cai, n.d). This happens because the approach given to them make them unflexible to show what they think. Students somehow have many different ways to answer one same question. Iit will be different if teacher teach by using problem solving approach. This approach allows students to have unconventional way of solving problems since the problem indeed does not have exact algorithm or routine process (Schoenfeld, 1992, in Tripathy, n.d). Problem solving approach sometimes requires students to apply a very simple algorithm into a very broad and complicated application (Rusczyk, 2005).

There are many advantages given by problem solving approach. Problem solving approach can improve studentsā€™ skill on reasoning, communication, and connection making compared to traditional approach (Cai, n.d). Teaching through problem solving also encourages students to build their own understanding and curiosity to find possibilities (Taplin, 2011).

In the topic of tangent line to a circle, the problems may consist of visual interpretation. The problems sometimes are given in a wordy appearance. Students have to visualize this problem in their head in order to solve it. Teacher may ask them to draw the figure of the circle and visualize it into their own drawing. However, this method can be misleading if students does not accurately depict the figure being asked by the problem.

Besides that, some students also feel that solving problems is difficult. The problem lies on their ability to transform the context given into mathematical context. They need a tool to help them understand. Geogebra is one of useful software to help them transforming the problem into mathematical context. Geogebra can produce accurate interpretation of the circle and its tangent line. Students feel that Geogebra helps them to realize that there is more than one solution and to get the initial facts of the problem (Maricic, 2010). Moreover, it is flexible for the teacher to use it in accordance with studentsā€™ needs. Teacher may use it thoroughly, partially, or only to check the correct answer.

The combination between Geogebra and problem solving approach can improve studentsā€™ higher order thinking skill, visualization, and their reasoning skill.

Either teacher or students can look at the following lesson exampleĀ Tangent line & Geogebra – Lesson Example

Further readings:

Cai, J. n.d. What Research Tells Us About Teaching Mathematics Through Problem Solving. Reston, VA: National Council of Teachers of Mathematics. Retrieved from http://tlsilveus.com/Portfolio/Documents/EDCI327_ProblemSolving.pdf

Maricic, K. (2010). Problem solving using GeoGebra: the case of geometry. Lecture proposal for ACDCA.Retrieved from http://www.time2010.uma.es/Proceedings/Papers/A055_Paper.pdf

Rusczyk, R. (2005). What is problem solving?. Retrieved from http://www.artofproblemsolving.com/Resources/articles.php?page=problemsolving on March 17th, 2011.

Taplin, M. (June 12, 2011). Mathematics through problem solving. Retrieved from http://www.mathgoodies.com/articles/problem_solving.html on July 7th, 2011.

Tripathy, P. N. n.d. Problem Solving In Mathematics: A Tool for Cognitive Development. Oswego: State University of New York.

 

Note: All Geogebra related material is produced by Geogebra mathematics free-software.

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Teaching Algebra using Realistic Mathematics Education

Realistic Mathematics Education (RME) is firstly invented in Netherlands around 1970. This approach of teaching mathematics concerns on how mathematics is seen not as a subject to be learnt, but as a human activity. In another way, it concerns on reality, staying close to studentsā€™ daily activity, and connection to society (Sembiring, n.d).

Adapting RME, PMRI (Pendidikan Matematika Realistik Indonesia) is executed in Indonesia officially in 1998 (PMRI, n.d). Aiming the same dream, making mathematics is seen as human activity, PMRI is infused through schools in Indonesia to make a better mathematics lesson.

RME is one of innovation approach to teach mathematics. By implementing daily life aspects in teaching and integrating it into teaching and learning mathematics, high quality learning of mathematics is believed to be achieved.

The characteristics of applying RME in teaching algebra

RME is mainly characterized by:

  • Students should be given the opportunity to reinvent mathematics under the guidance of an adult (Gravemeijer, 1994, in Hadi, 2002); and
  • The reinvention of mathematics ideas and concepts should start from exposure

to a variety of ‘real-world’ problems and situations (de Lange, 1995, in Hadi, 2002).

  • Rejection to procedure-focused way of teaching that students can train by a lot of exercise (Heuvel & Panhuizen, 2001).

The first character shows that RME also wants students to find the concept by themselves. The second is the character that differs RME from any other approach. The content of the learning should be related to the real life context.

Why we should apply RME in teaching algebra

Realistic Mathematics Education (RME) appears to be a promising approach in the

teaching and learning of mathematics. RME is mentioned as the potential approach in increasing students’ understanding in mathematics (Streefland, 1991; Gravemeijer, 1994, 1997, in Hadi, 2002).

The approach of RME appears to stay as close as possible to the studentsā€™ life.

How we apply RME in teaching algebra

In using RME, teacher should relate every concept into real life problem. The context of real problem should not be only applied after students knowing the concept, but used for reaching the concept itself.

In algebra, the basic concept can be seen in elementary school for example, the concept of division. Usually, teacher first will give a very simple problem of division, like ā€œWhat is 6 : 3 ?ā€. After explaining the procedure, teacher will move to the more complex problem like, 72 : 6 and so on till the most complex problem.

However, if teacher uses RME approach, he/she should give the concept by giving real life problem first, like ā€œ342 match stickers are fairly distributed among five children. How many does each of them get?. In solving this problem, students are given chance to use any method they know in order to reach an understanding.

The point is that the lesson should stay close to studentsā€™ real life context.

Try the lesson plan hereĀ Realistic Math – Lesson Plan.

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Dream and Surroundings

A year agoā€¦

I have people I treasure most. And that is why I clash with them most. Somehow they simply want to protect me but it seems like they put me in jail. I cannot go out easily. I understand me as a girl and living in a metropolitan city. I understand there are so many dangers may happen to me. However, if they worry that much, why donā€™t they go along with me and make sure that I am okay. If they cannot, maybe they are too damn busy or stuff, okay, I understand. But can you trust me a bit? I have stuff to fix, too. I have business to do, too. And I cannot match it with everyoneā€™s schedule and make it into my second list. I still can and have to do it by myself. If you cannot trust me that much, how could I learn to trust anybody? Since you are my main source of learning. There are times when I can stand this matter and do what you want. I understand that all those restrictions are for my sake. There are also times when I cannot stand that. Making-up excuses just not to allow me to do something is frustrating. Why declaring as understanding and democratic people if listening opinion is a hard thing. At this point, I feel like I cannot pursue my dream. I feel like going further is not an option for me.

Nowā€¦

I have people I treasure most. And that is why I clash with them most. I try to see what is behind their action and sometimes stuck. I have dialogues with them and it is hard to understand their point of views. Yet, my dream is too good to be buried by my own depression. I feel like I should try harder. I should help myself. They see my stubbornness and feel guilty when leaving me alone. They start to ask my opinion. They help me without me knowing that. At the end, they support me. They understand me. I feel like I can do it. not because now they help me, but because now I understand I waste my time on complaining my surroundings.