AUTHENTIC TASKS IN A STANDARDS BASED-WORLD and OPEN – ENDED QUESTIONS AND THE PROCESS STANDARDS

Tugas Asli dalam Berbasis Standar Dunia

Standar kurikulum membimbing instruksi tapi tidak merepresentasikan semua dan menjadi akhir dari matematika sekolah.Masalah Pertemuan Makan Siang sebagai contohnya. Masalah Pertemuan Makan Siang ini mencari tempat dengan jarak yang sama dari tiga tempat lainnya.Mulanya siswa menjawab Colombus yang berada ditengah ketiga tempat tapi dengan sofware IGS mereka menemukan Paskerburg, Virginia Barat sebagai solusinya.Mereka memulai dengan memilih kombinasi berbeda dari kota – kota dan dokumen hasil kecurigaaan.

Karena keliling lingkaran merupakan solusi, banyak siswa menganggap titik tengah segitiga sebagai solusi dan mencarinya dengan IGS Geogebra. Lalu banyak sekali praduga yang bermunculan, seperti ketika tiga kota membentuk segitiga tumpul akan lebih baik bila salah satunya tetap dirumahnya.

Kami belajar bahwa jarak total dari puncak akan menjadi minimal mengacu pada teorema Fermat. Meskipun poin Fermat tidak diikutkan dalam Konten Standar Akademik Ohio, itu sangat penting dan berguna dalam pengapikasian seperti yang terjadi pada penyelesaian Masalah Pertemuan Makan Siang. Penggunaan sowfware IGS juga membuat kesempatan bagi siswa untuk mengetes praduga yang tidak praktis, jika dengan pensil dan kertas saja tidak memungkinkan. Sekali lagi dapat disebutkan bahwa Masalah Pertemuan Makan Siang mengingatkan kita untuk menginterpretasikan standar kurikulum secara luwes.

Sumber: Mathematics Teacher. Vol. 106, No 5 ° December 2012/January 2013

Pertanyaan Terbuka

Tiap masyarakat butuh orang yang mampu menyelesaikan permasalahan kompleks dan menerapkan pengetahuannya  pada konteks yang bervariasi sebagaimana dia dapat bekerja kelompok dalam menyelesaikan masalah dan mengkomunikasikan kepada penanggungjawab pendidikan matematika. Jadi kita harus mendidik siswa untuk hidup, bukan mendidik mereka untuk menyelesaikan tes.

Disini ada “Pertanyaan Terbuka” yang bisa digunakan. Pertanyaan Terbuka ini adalah pertanyaan yang bisa diselesaikan dengan berbagai cara yang fokus pada aspek konseptualnya. Penulisan pertanyaan open ended dapat ditulis dengan berbagai cara, beberapa disebutkan sebagai berikut.

1.       Apa yang salah dengan ini?

Pada cara ini kita menanyakan siswa kesalahan dan meminta menjelaskan kenapa bisa salah.

2.       Buat contoh atau situasi

Pada cara ini kita memberikan siswa parameter dan meminta mereka membuat contoh atau situasi yang sesuai parameter.

3.       Siapa yang benar dan mengapa?

Pada cara ini kita memberikan dua argumen yang berlawanan dan meminta siswa meidentifikasi mana yang benar serta memberikan alasannya.

Terakhir, Pertanyaan Terbuka ini dapat membantu guru fokus pada perintah dan penilaian dari standar proses NCTM dan pada pembuatan alasan dan kepekaan, yang mana merupakan esensi dari matematik. Lalu juga memberikan gambaran guru kesalahan konsep yang dipelajari siswa. Terakhir, terkait dengan menjawab pertanyaan tingkat tinggi yang diatur dalam standar proses dan fokus pada arti, siswa akan disisapkan untuk menyelesaikan semua tes, baik di sekolah ataupun hidupnya.
Sumber: Mathematics Teacher.  Vol. 107, No. 3 ° Oktober 2013

OPEN – ENDED QUESTIONS AND THE PROCESS STANDARDS

All societies need citizens who can solve complex problems and apply knowledge in a variety of contexts as well as citizens who can work collaboratively to solve problems and communicate solutions to mathematics education stakeholders. We must educate students to use NCTM’s Process Standards (NCTM 2000) and move beyond being able to work routine exercises on standardized tests. We are not educating students for tests; we are educating them for life. All stakeholders need to see this broader picture and support teachers in this broader purpose.

Open-ended questions, as discussed here, are questions that can be solved or explained in a variety of ways, that focus on conceptual aspects of mathematics, and that have the potential to expose students’ understanding and misconceptions. NCTM’s Process Standards—Problem Solving, Reasoning and Proof, Communication, Connections, and Representation—are difficult to assess with multiple-choice tests. For example, one aspect of the Communication Standard requires students to “communicate their mathematical thinking coherently and clearly to peers, teachers, and others” (NCTM 2000, p. 60). This standard cannot be assessed through multiple-choice questions. Although students need to rely on some procedural knowledge to answer this problem, they have to decide which procedures would apply to it.

WRITING OPEN-ENDED QUESTIONS

Open-ended questions can be written using various templates, several of which are discussed here.

Template 1: What’s Wrong with This?

The earlier question about expanding (x + 3)2 is an example of this type of question used to identify errors and misconceptions. We can ask students to identify errors and explain why they are errors. This template is useful for getting students to think critically about common misconceptions. Example question using this template : Bert was trying to graph y = (x – 3)2. He said that he could simply shift the graph of y = x2 three units to the left. Convince Bert that his method is incorrect.

Template 2: Create an Example or a Situation

This form of question is similar to the form of the questions for the game show Jeopardy™. We give students some parameters and ask them to come up with an example or situation that fits the parameters. We give them the answer and have them come up with the question. Example question using this template : Create a system of linear equations that has the solution (–2, 3). Explain how you determined your system.

Template 3: Who Is Correct and Why?

This form of open-ended question—Who is correct and why?—can be used to set up two opposing arguments. Then students can defend one or the other argument. Example question using this template : Lucinda thinks that the grades in mathematics class should be calculated using the mean. Norm thinks that the grades should be calculated using the median. With whom do you agree and why?

The templates presented here can be useful in giving teachers a place to start when writing openended questions, but teachers must be cautious when using them. Just because a question fits a template does not necessarily mean that the question is open ended or of high quality.

For example, we could ask the earlier question in this way:

Jasmine solved x + 3 = 5 and got x = 2. Stuartsolved x + 3 = 5 and got x = 8. Who is correct and why?

This form of the question is no different from askingthe traditional question “Solve x + 3 = 5 for x.” The formulation does not involve the conceptual underpinnings of equation solving.

The purpose of schooling from a broader perspective and about preparing students to solve the kinds of problems that they will encounter in society—not just about preparing them for standardized tests— so we need different strategies. Open-ended questions can help teachers focus their instruction and assessment on NCTM’s Process Standards and on reasoning and sense making, which really is the heart of mathematics. Moreover responses to open-ended questions give teachers so much more information about students’ ways of thinking and misconceptions, and these can provide important avenues for further investigation of mathematics. When students answer higher-order questions driven by the Process Standards and focused on meaning, they will be prepared for any test we give them—in school or in life.

Sumber: Mathematics Teacher.  Vol. 107, No. 3 ° Oktober 2013

AUTHENTIC TASKS IN A STANDARDS BASED-WORLD

Curriculum standards is guide instruction but not represent the be-all anda end-all of school mathematics.

The Meeting for Lunch problem is stated as follows : Three friends – one from Hamilton, one from Marrietta, and one from Fostoria – are making plans to meet for lunch. Each wishes to drive the same distance. Whera should they meet? Justify your solution mathematically.

The students surmised that Colombus, Ohio but with interactive geometry software (IGS), the sketch suggest that Parkersburg, West Virginia, is equidistant from these three city.

Students began to select different combinations of cities and document curious results. For instance, suppose three friends drove from Lorain, Hamilton, and Cleveland (marked as points A, B, and C in fig. 4).

Clearly, a lunch meeting in West Virginia is a less-than-optimal plan for friends living in Lorain, Hamilton, and Cleveland, Ohio.

We encouraged students to use the sketch to look for a point minimizing the total distance trav- eled from Hamilton, Lorain, and Cleveland.

When an angle of a given triangle ABC is greater than or equal to 120 degrees, the point that minimizes the total distance is the vertex of the obtuse angle.

Although the femat point was not included in the Ohio Academic Content Standards, it is important and usefulness in aplications such as the Meeting for Lunch problem. As students confronted the Meeting for Lunch problem, they explored a plethora of mathematics topics from triangle centers, angle measures, and types of angles and triangles to maps, distance, and scale factor. Mathematics can be engaging and creative for students. The use of Interactive Geometry Software afforded students opportunities to make and test conjectures in ways that would be impractical, if not impossible with paper and pencil. Our responsibility as educators is supporting students as we learn mathematics together, whether on or off the map.

Sumber: Mathematics Teacher. Vol. 106, No 5 ° December 2012/January 2013