Tuesday, May 12, 2020
Tuesday, March 17, 2020
Mathematical Power Dimensions
Mathematical Power Dimensions (Sahin & Baki, 2010 version) divided into 3 parts, as follows
1. Mathematical Knowledge (NCTM,2000)
Reasoning and Proof. Mathematical reasoning and proof offer powerful ways of developing and expressing insights about a wide range of phenomena. People who reason and think analytically tend to note patterns, structure, or regularities in both real-world and mathematical situations. They ask if those patterns are accidental or if they occur for a reason. They make and investigate mathematical conjectures. They develop and evaluate mathematical arguments and proofs, which are formal ways of expressing particular kinds of reasoning and justification. By exploring phenomena, justifying results, and using mathematical conjectures in all content areas and—with different expectations of sophistication—at all grade levels, students should see and expect that mathematics makes sense.
Connections. Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study. When students connect mathematical ideas, their understanding is deeper and more lasting, and they come to view mathematics as a coherent whole. They see mathematical connections in the rich interplay among mathematical topics, in contexts that relate mathematics to other subjects, and in their own interests and experience. Through instruction that emphasizes the interrelatedness of mathematical ideas, students learn not only mathematics but also about the utility of mathematics.
Communication. Mathematical communication is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. When students are challenged to communicate the results of their thinking to others orally or in writing, they learn to be clear, convincing, and precise in their use of mathematical language. Explanations should include mathematical arguments and rationales, not just procedural descriptions or summaries. Listening to others’ explanations gives students opportunities to develop their own understandings. Conversations in which mathematical ideas are explored from multiple perspectives help the participants sharpen their thinking and make connections.
Get more information on
https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf
Reasoning and Proof. Mathematical reasoning and proof offer powerful ways of developing and expressing insights about a wide range of phenomena. People who reason and think analytically tend to note patterns, structure, or regularities in both real-world and mathematical situations. They ask if those patterns are accidental or if they occur for a reason. They make and investigate mathematical conjectures. They develop and evaluate mathematical arguments and proofs, which are formal ways of expressing particular kinds of reasoning and justification. By exploring phenomena, justifying results, and using mathematical conjectures in all content areas and—with different expectations of sophistication—at all grade levels, students should see and expect that mathematics makes sense.
Connections. Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study. When students connect mathematical ideas, their understanding is deeper and more lasting, and they come to view mathematics as a coherent whole. They see mathematical connections in the rich interplay among mathematical topics, in contexts that relate mathematics to other subjects, and in their own interests and experience. Through instruction that emphasizes the interrelatedness of mathematical ideas, students learn not only mathematics but also about the utility of mathematics.
Communication. Mathematical communication is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. When students are challenged to communicate the results of their thinking to others orally or in writing, they learn to be clear, convincing, and precise in their use of mathematical language. Explanations should include mathematical arguments and rationales, not just procedural descriptions or summaries. Listening to others’ explanations gives students opportunities to develop their own understandings. Conversations in which mathematical ideas are explored from multiple perspectives help the participants sharpen their thinking and make connections.
Get more information on
https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf
2. Content (NAEP,1990,1992)
Number Properties and Operations.This content area focuses on students' abilities to represent numbers, order numbers, compute with numbers, make estimates appropriate to given situations, use ratios and proportional reasoning, and apply number properties and operations to solve real-world and mathematical problems. This content area also addresses number sense—comfort in dealing with numbers—and addresses students' understanding of what numbers tell us, equivalent ways to represent numbers, and the use of numbers to represent attributes of real-world objects and quantities. At grade 4, the focus is on whole numbers and fractions; at grade 8, the focus extends to include rational numbers; at grade 12, the focus extends to include real numbers.
Measurement. This content area focuses on students' understanding of measurement attributes such as capacity, weight/mass, time, and temperature as well as the geometric attributes of length, area, and volume. Students may be asked to select appropriate units and tools for measuring, to measure length with a ruler at all three grades, to measure angles with a protractor at grades 8 and 12, and to solve application problems related to units of measurement. At grade 4, the focus is on length, including perimeter, distance, and height. At grades 8 and 12, students are also expected to understand and demonstrate knowledge of volume and surface area. Knowledge of both customary and metric units is expected. Students may be asked to solve problems that require conversions between (with conversion factors given) or within systems of measurement.
Geometry. This content area focuses on identification of geometric shapes and transformations and combinations of those shapes. By grade 4, students are expected to be familiar with simple plane figures such as lines, circles, triangles, and rectangles as well as solid figures such as cubes, spheres, and cylinders. They are also expected to be able to recognize examples of parallel and perpendicular lines. As students move to middle school and beyond, increased understanding should deepen of two- and three-dimensional figures, especially parallelism, perpendicularity, angle relations in polygons, congruence, similarity, and the Pythagorean theorem. Students at all grades are expected to show knowledge of symmetry and transformations of shapes and to identify images resulting from flips, rotations, or turns. Justifications and reasoning in both formal and informal settings are expected at grades 8 and 12.
Data Analysis and Probability. This content area focuses on students’ skills in four areas: data representation, characteristics of data sets, experiments and samples, and probability. At grade 4, students are expected to use standard statistical measures such as the median, range, or mode, and to compare sets of related data; at grades 8 and 12, they are also expected to show understanding of other statistical concepts such as the impact of outliers and the line of best fit in a scatterplot. By grade 8, students are expected to have some knowledge of experiments and samples, such as being able to recognize possible sources of bias in sampling and identify random versus nonrandom sampling; by grade 12, students are also expected to make inferences from sample results. Students at all grades are expected to use statistics and statistical concepts to analyze and communicate interpretations of data. Students may be asked to solve problems that address appropriate methods of gathering data, the visual exploration of data, ways to represent data, or the development and evaluation of arguments based on the analysis of data. Probability is assessed informally at grade 4 and more formally at grades 8 and 12.
Algebra. This content area focuses on students’ understanding of patterns, relations, and functions; algebraic representation; variables, expressions, and operations; and equations and inequalities. At grade 4, students are expected to show knowledge of simple patterns and expressions; at grade 8, this knowledge extends to include linear equations; and at grade 12, it extends further to include quadratic and exponential equations and functions. Representational skills, such as students’ ability to translate between different forms of representation (e.g., from a written description to an equation), the ability to graph and interpret points located on a coordinate system, and the ability to use algebraic properties to draw a conclusion, are assessed in this area. Students may be asked to express relationships algebraically as number sentences, equations, or inequalities; manipulate algebraic expressions; or solve and interpret algebraic equations and inequalities that are grade-level appropriate.
Number Properties and Operations.This content area focuses on students' abilities to represent numbers, order numbers, compute with numbers, make estimates appropriate to given situations, use ratios and proportional reasoning, and apply number properties and operations to solve real-world and mathematical problems. This content area also addresses number sense—comfort in dealing with numbers—and addresses students' understanding of what numbers tell us, equivalent ways to represent numbers, and the use of numbers to represent attributes of real-world objects and quantities. At grade 4, the focus is on whole numbers and fractions; at grade 8, the focus extends to include rational numbers; at grade 12, the focus extends to include real numbers.
Measurement. This content area focuses on students' understanding of measurement attributes such as capacity, weight/mass, time, and temperature as well as the geometric attributes of length, area, and volume. Students may be asked to select appropriate units and tools for measuring, to measure length with a ruler at all three grades, to measure angles with a protractor at grades 8 and 12, and to solve application problems related to units of measurement. At grade 4, the focus is on length, including perimeter, distance, and height. At grades 8 and 12, students are also expected to understand and demonstrate knowledge of volume and surface area. Knowledge of both customary and metric units is expected. Students may be asked to solve problems that require conversions between (with conversion factors given) or within systems of measurement.
Geometry. This content area focuses on identification of geometric shapes and transformations and combinations of those shapes. By grade 4, students are expected to be familiar with simple plane figures such as lines, circles, triangles, and rectangles as well as solid figures such as cubes, spheres, and cylinders. They are also expected to be able to recognize examples of parallel and perpendicular lines. As students move to middle school and beyond, increased understanding should deepen of two- and three-dimensional figures, especially parallelism, perpendicularity, angle relations in polygons, congruence, similarity, and the Pythagorean theorem. Students at all grades are expected to show knowledge of symmetry and transformations of shapes and to identify images resulting from flips, rotations, or turns. Justifications and reasoning in both formal and informal settings are expected at grades 8 and 12.
Data Analysis and Probability. This content area focuses on students’ skills in four areas: data representation, characteristics of data sets, experiments and samples, and probability. At grade 4, students are expected to use standard statistical measures such as the median, range, or mode, and to compare sets of related data; at grades 8 and 12, they are also expected to show understanding of other statistical concepts such as the impact of outliers and the line of best fit in a scatterplot. By grade 8, students are expected to have some knowledge of experiments and samples, such as being able to recognize possible sources of bias in sampling and identify random versus nonrandom sampling; by grade 12, students are also expected to make inferences from sample results. Students at all grades are expected to use statistics and statistical concepts to analyze and communicate interpretations of data. Students may be asked to solve problems that address appropriate methods of gathering data, the visual exploration of data, ways to represent data, or the development and evaluation of arguments based on the analysis of data. Probability is assessed informally at grade 4 and more formally at grades 8 and 12.
Algebra. This content area focuses on students’ understanding of patterns, relations, and functions; algebraic representation; variables, expressions, and operations; and equations and inequalities. At grade 4, students are expected to show knowledge of simple patterns and expressions; at grade 8, this knowledge extends to include linear equations; and at grade 12, it extends further to include quadratic and exponential equations and functions. Representational skills, such as students’ ability to translate between different forms of representation (e.g., from a written description to an equation), the ability to graph and interpret points located on a coordinate system, and the ability to use algebraic properties to draw a conclusion, are assessed in this area. Students may be asked to express relationships algebraically as number sentences, equations, or inequalities; manipulate algebraic expressions; or solve and interpret algebraic equations and inequalities that are grade-level appropriate.
Get more information on https://nces.ed.gov/nationsreportcard/mathematics/contentareas2005.aspx
3. Mathematical Skills (NAEP,1990,1992)
3. Mathematical Skills (NAEP,1990,1992)
Monday, March 9, 2020
Mathematical Power for All Students K-12
by C.I.A.I. (Curriculum •Instruction •Assessment •Improvement)
Mathematical power divided into 3 parts, as follows
1. CONTENT STRANDS
• Number Sense and Concepts
• Measurement
• Geometry and Spatial Sense
• Algebraic Thinking
• Data Analysis and Probability
2. MATHEMATICAL ABILITIES
• Conceptual Understanding
• Procedural Knowledge
3. PROCESS STANDARDS
• Problem Solving
• Reasoning
• Communication
You can get more information on https://etc.usf.edu/fcat8math/resources/mathpower/fullpower.pdf
Sunday, March 1, 2020
Ontologi dan Epistimologi Daya Matematika
1. Ontologi
Ontologi adalah ilmu hakekat yang menyelidiki alam nyata ini dan bagaimana keadaan yang sebenarnya. Ontologi daya matematika atau makna (arti) daya matematika diperoleh dari orang yang menulis, meneliti dan lain sebagainya. Secara filosofis makna ontologi dibagi menjadi dua, idealis dan realistis.
Dikatakan realistis, apabila kompetensi siswa sudah sampai outcome (berdampak). Sebagai contoh, artikel berjudul “Pengaruh Pendidikan Matematika Realistik terhadap Daya Matematika Siswa”. Guru memberikan pembelajaran matematika yang berorientasi pada matematisasi pengalaman sehari-hari dan menerapkan matematika dalam kehidupan sehari-hari. Oleh karenanya, siswa mempunyai kemampuan untuk menghadapi permasalahan-permasalahan baik dalam permasalahan matematika maupun permasalahan dalam kehidupan nyata merupakan kemampuan daya matematika.
Monday, February 24, 2020
Pembelajaran Bermakna (Meaningful Learning)
Suparno (1997) mengatakan, bahwa pembelajaran bermakna adalah suatu proses pembelajaran dimana informasi baru dihubungkan dengan struktur pengertian yang sudah dipunyai seorang yang sedang dalam proses pembelajaran. Pembelajaran bermakna terjadi bila siswa mencoba menghubungkan fenomena baru ke dalam struktur pengetahuan mereka. Artinya, bahan pelajaran itu harus cocok dengan kemampuan siswa dan harus relevan dengan struktur kognitif yang dimiliki siswa. Oleh karena itu, pelajaran harus dikaitkan dengan konsep-konsep yang sudah dimiliki siswa, sehingga konsep-konsep baru tersebut benar-benar terserap olehnya. Dengan demikian, faktor intelektual emosional siswa terlibat dalam kegiatan pembelajaran.
Saturday, February 15, 2020
Romantisme, "Cinta Sejati"?
*Limbuk*: Lha cinta?
*Cangik*: Cinta itu plus minus sayang plus minus kuasa.
*Cangik*: Cinta plus sayang itu cinta. Cinta minus sayang itu benci.
*Cangik*: Jadi cinta itu merentang antara benci sampai cinta.
Monday, February 10, 2020
Pengertian Daya Matematika
Kemampuan untuk menghadapi permasalahan-permasalahan baik dalam permasalahan matematika maupun permasalahan dalam kehidupan nyata merupakan kemampuan Daya Matematis (mathematical power), menurut Mumun (2008).
Tujuan pembelajaran matematika dalam kurikulum di Indonesia menyiratkan dengan jelas tujuan yang ingin dicapai yaitu:
(1)Kemampuan pemecahan masalah (problem solving);
(2)Kemampuanberargumentasi(reasonning);
(3)Kemampuan berkomunikasi (communication);
(4)Kemampuan membuat koneksi (connection) dan
(5)Kemampuan representasi (representation).
