Based on National Council of Teachers of Mathematics

(NCTM)

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Six principles for school mathematics:1. Equity. Excellence in mathematics education

requires equity-high expectations and strong supportfor all students

2. Curriculum. A curriculum is more than a collectionof activities: it must be coherent, focused onimportant mathematics, and well articulated acrossthe grades

3. Teaching. Effective mathematics teaching requiresunderstanding what students know and need tolearn and then challenging and supporting them tolearn it well

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4. Learning. Students must learn mathematics withunderstanding, actively building new knowledgefrom experience and prior knowledge

5. Assessment. Assessment should support the learningof important mathematics and furnish usefulinformation to both teacher and students

6. Technology. Technology is essential in teaching andlearning mathematics; it influences the mathematicsthat is taught and enhances students’ learning

Six principles for school mathematics:

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Structure of CurriculumNCTM

1. Pre- K -2

2. Grades 3-5

3. Grades 6-8

4. Grades 9-12

Indonesia

1. Pre-school/PAUD

2. TK (2 th)

3. SD (1-6)

4. SMP (7-9)

5. SMA (10-12)

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Standards for School Math

The Content Standards The Process Standards

1. Number and Operation

2. Algebra3. Geometry4. Measurement5. Data Analysis

and Probability

1. Problem Solving

2. Reasoning and Proof

3. Communication4. Connections5. Representation

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Instructional programs from pre kindegarden throughgrade 12 should enable students to:

1. Understanding numbers, ways of representingnumbers, relationship among numbers, and numbersystem;

2. Understanding meaning of operation and how theyrelate to one another;

3. Compute fluently and make reasonable estimates

1. Number and Operation Standards

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Grades 6-8Standard Expectation

1. Understanding numbers,ways of representingnumbers, relationshipamong numbers, andnumber system

• Work flexibly with fractions, decimals, andpercents to solve problems;

• Compare and order fractions, decimal, andpercents efficiently and find their approximatelocations on a number line;

• Develop an understanding of large numbers andrecognize and appropriately use exponential,scientific, and calculator notation

• etc

2. Understanding meaning of operation and how they relate to one another

Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computation with integers, fractions, and decimals, etc.

3. Compute fluently and make reasonable estimates

Develop and use strategies to estimate the result of rational number computations and judge thereasonableness of the result, etc

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Grades 9-12Standard Example ofExpectation

1. Understanding numbers,ways of representingnumbers, relationshipamong numbers, andnumber system

• Develop a deeper understanding of very largeand very small numbers and variousrepresentation of them

• Compare and contrast the properties ofnumbers and number systems, including therational and real numbers, and understandingcomplex number system as a solution toquadratic equations, etc

2. Understanding meaning of operation and how they relate to one another

Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities, etc

3. Compute fluently and make reasonable estimates

Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-pencil calculations for simple cases and technology for more complicated-cases. etc

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2. Algebra StandardsInstructional programs for PK to grade 12 shouldenable students to:

1. Understand patterns, relations, and functions

2. Represent and analyze mathematical situations andstructures using algebraic symbols

3. Use mathematical models to represent andunderstand quantitative relationships

4. Analyze change in various contexts

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Grades 6-8Standard Sample of expetations

Understand patterns, relations, and functions

represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; related and compare different forms of representation for a relationship; identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equation

Represent and analyze mathematical situations and structures using algebraic symbols

Develop an initial conceptual understanding of different uses of variables; explore relationship between symbolic expression and graphs of lines, paying particular attention to the meaning of intercept and slope; etc

Use mathematical models to represent and understand quantitative relationships

Model and solve contextualized problems using various representations, such as graphs, tables, and equation

Analyze change in various contexts

Use graphs to analyze the nature of changes in quantities in linear relationship

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Grades 9-12Standard Sample of expetations

Understand patterns, relations, and functions

Generalize pattern explicitly and recursively defined functions; understand relations and functions and select, convert flexibly among, and use various representation of them; etc

Represent and analyze mathematical situations and structures using algebraic symbols

Understand the meaning of equivalent forms of expressions, equations, inequalities, and relation; use symbolic algebra to represent and explain mathematical relationship; etc

Use mathematical models to represent and understand quantitative relationships

Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationship; draw reasonable conclusions about a situation being modeled

Analyze change in various contexts

Approximate and interpret rates of change from graphical and numerical data

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3. Geometry Instructional programs for PK to grade 12 should enablestudents to:

1. Analyze characteristics and properties of two-and threedimensional geometric shapes and develop mathematicalarguments about geometric relationships

2. Specify locations and describe spatial relationships usingcoordinate geometry and other representational systems

3. Apply transformations and use symmetry to analyzemathematical situations

4. Use visualization, spatial reasoning, and geometricmodeling to solve problems

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4. MeasurementInstructional programs for PK to grade 12 should enable students to:

1. Understand measurable attributes of objects and the unit, systems, and processes of measurement

2. Apply appropriate techniques, tools, and formulas to determine measurements

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5. Data Analysis and Probability standards1. Instructional programs for PK to grade 12 should

enable students to:

2. Formulates questions that can be addressed withdata and collect, organize, and display relevant datato answer them

3. Select and use appropriate statistical methods toanalyze data

4. Develop and evaluate inferences and predictions thatare based on data

5. Understand and apply basic concept of probability

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1. Problem Solving Standard Instructional programs for PK to grade 12 should

enable all students to:

Build new mathematical knowledge through problemsolving

Solve problem that arise in mathematics and othercontexts

Apply and adapt a variety of appropriate strategies tosolve problems

Monitor and reflect on the process of mathematicalproblem solving

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PB for grades 9-12 Problem solving seperti apa yang ada pada kelas 9-12?1. Problem solving merupakan salah satu strategi dalam proses

pembelajaran matematika untuk mengajarkan hal-halmendasar (PB sebagai alat)

2. PB sebagai tujuan utama : melengkapi siswa denganpengetahuan dan ketrampilan yang dibutuhkan untukmenyelesaikan masalah sesuai dengan konsep-konsep yangtelah dipelajari siswa

Bagaimana peran guru dalam membangun PB di dalamkelas?

Dalam menciptakan PB dibutuhkan beberapa persyaratandiantaranya:

Konten materi matematika; pengetahuan tentang strategipenyelesaian masalah; kemampuan monitoring diri yangbaik /refleksi; dan memiliki disposisi yang baik dalammembuat masalah dan menyelesaikannya

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2. Reasoning and Proof StandardInstructional programs for PK to grade 12 shouldenable students to:

Recognize reasoning and proof as fundamental aspectsof mathematics

Make and investigate mathematical conjectures

Develop and evaluate mathematical arguments andproofs

Select and use various types of reasoning and methodsof proof

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Penalaran dan pembuktian pada tingkat 9-12

Penalaran dan pembuktian seperti apa yang ada pada kelas 9-12?

1. Penalaran dan pembuktian secara alamiah merupakan bagian dariproses pembelajaran

2. Siswa dapat menjelaskan dan menjustifikasi kesimpulan mereka3. Membuat siswa dapat melakukan absraksi dan kodifikasi dari hasil

observasi4. Penalaran dibangun melalui pertanyaan-pertanyaan “mengapa?”5. Siswa terbiasa membuat argumentasi langsung untuk membuat

validitas dari sebuah konjektur Bagaimana peran guru dalam membangun PB di dalam

kelas?1. Memahami matematika dengan baik2. Menciptakan situasi yang kondusif agar siswa dapat mencari,

merumuskan, dan membuat kritik atas penjelasan dalam suatudiskusi

3. Mampu membuat pertanyaan-pertanyaan yang tepat

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3. Communication StandardInstructional programs for PK to grade 12 shouldenable students to:

Organize and consolidate their mathematical thinkingthough communication

Communicate their mathematical thinking coherentlyand clearly to peers, teachers, and others

Analyze and evaluate the mathematical thinking andstrategies of others

Use the language of mathematics to expressmathematical ideas

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Komunikasi matematis untuk tingkat 9-12

Komunikasi matematis seperti apa yang ada pada kelas 9-12?1. Membangun kemampuan berpikir runtut/terstruktur dan logis,

mengekspresikan gagasan secara koheren dan jelas,mendengarkan ide-ide matematis dari orang lain, dan memikirkankepada siapa mereka menyampaikan gagasan dalam berbicara ataumenulis.

2. Siswa harus dapat menggunakan bahasa dan simbol matematikadengan benar dan tepat dalam berbagai bentuk representasi

Bagaimana peran guru dalam membangun PB di dalamkelas?

1. Menciptakan iklim kondusif di kelas agar siswa dapat merasanyaman dalam berkomentar, membuat konjektur danmemberikan penjelasan

2. Mmebuat klarifikasi dari statetment mereka dan memperbaikinya3. Membantu siswa agar dapat menulis dan membaca menggunakan

bahasa matematis dengan tepat dan membantu merekamemahami literatur matematika dengan baik

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4. Connections StandardInstructional programs for PK to grade 12 shouldenable students to:

Recognize and use connections among mathematicalideas

Understand how mathematical ideas interconnect andbuild on one another to produce a coherent whole

Recognize and apply mathematics in contexts outsideof mathematics

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Koneksi matematis untuk tingkat 9-12

Koneksi matematis seperti apa yang ada pada kelas 9-12?

1. Siswa harus dapat meningkatkan kapasitas dalam membuathubungan antar ide-ide matematika

2. Siswa harus dapat menggunakanberbagai macampendekatan yang berbeda dalam menyelesaikan masalah

Bagaimana peran guru dalam membangun koneksimatematis di dalam kelas?

1. Memilih atau membuat masalah yang sesuai agara dapatmenciptakan berbagai kemungkinan koneksi antar konsep

2. Membuat inisiatif khusus agar dapat menyatukan berbagaikonsep aljabar, geometri,dan statistik atau yang lain dalamsuatu proses pembelajaran disesuaikan dengan kurikulum

3. Banyak memberikan masalah yang memiliki berbagaimacam pendekatan dan solusi

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5. Representation StandardInstructional programs for PK to grade 12 shouldenable students to:

Create and use representations to organize, record,and communicate mathematical ideas

Select, apply, and translate among mathematicalrepresentations to solve problems

Use representations to model and interpret physical,social, and mathematical phenomena

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Representasi matematis untuk tingkat 9-12

Representasi matematis seperti apa yang ada pada kelas 9-12?1. Siswa harus dapat menggunakan berbagai jenis representasi konsep

matematika dengan luwes2. Siswa dapat mengenali struktur matematik dari berbagai macam konteks3. Siswa dapat menciptakan dan menginterpretasikan berbagai model dari

fenomena dengan cara mengidentifikasi hal-hal esensial dari situasi danmenemukan representasi yang menangkap hubungan matematis darikonteks-konteks tersebut

Bagaimana peran guru dalam membangun representasi matematisdi dalam kelas?

1. Mengenalkan siswa pada bentuk-bentuk representasi matematis danmembantu mereka dalam menggunakan representasi tersebut

2. Menggarisbawahi bahwa bentuk representasi yang berbeda dari konsepyang sama dapat memberikan informasi yang berbeda pula

3. Membantu siswa agar dapat memilih jenis representasi yang sesuaidengan tujuan

4. Membantu siswa menghubungan bentuk representasi miliknya denganbentuk representasi yang baku

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Tugas 41. Buatlah analisis mengenai standar isi dari kelas 6-12

menurut NCTM dan bandingkan dengan standar isidalam KTSP! (individual)

2. Butlah situasi/konteks/masalah/soal yang sesuaidengan 5 standar pembelajaran matematikamenurut NCTM dengan menggunakan materi yang terdapat dalam standar isi KTSP kelas 7-12

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