Fundamentals of Mathematics

• Math focuses on quantity, structure, space, and change.
• The philosophy of mathematics offers other perspectives.
• Mathematicians identify patterns and create conjectures.
• Mathematical proofs determine the truth or falsity of conjectures.
• Mathematical reasoning provides insight and predictions about nature when mathematical structures model real phenomena well.
• Abstraction and logic led to mathematics evolving from counting, calculation, measurement, and the study of physical object shapes and motions.
• Practical mathematics has been a human activity since written records began.
• Solving mathematical problems can require years or centuries of research.
• Rigorous arguments were first seen in Greek mathematics, notably in Euclid's Elements.
• Mathematics developed at varying rates until the Renaissance; mathematical innovations combined with scientific discoveries, rapidly increasing mathematical discovery, which continues today.
• Mathematics is vital in natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics has created new mathematical disciplines like statistics and game theory.
• Mathematicians study pure mathematics (mathematics for its own sake) without considering applications.
• Pure and applied mathematics are not clearly divided; practical applications for pure mathematics are often discovered later.

Areas of Mathematics

• Quantity includes numbers and arithmetic.
• Structure includes algebra.
• Space includes geometry.
• Change includes calculus and analysis.
• Mathematical logic and set theory contribute as foundations of mathematics.
• Probability theory, statistics, game theory, and mathematical finance are newer branches.

Quantity

• Deals with numbers, the best-known mathematical objects.
• Extends to number systems like natural numbers, integers, rational numbers, real numbers, and complex numbers.
• Arithmetic includes operations like addition, subtraction, multiplication, division, exponentiation, and root extraction.
• Number theory studies the properties of integers.
• Encompasses Fermat's Last Theorem and the prime number theorem.

Structure

• Focuses on studying relationships.
• Includes algebraic structures like groups, rings, and fields.
• Abstract algebra is a key aspect.
• Vector spaces are studied in linear algebra.

Space

• Geometry is the origin point for studying Space.
• Includes Euclidean geometry, trigonometry, analytic geometry, differential geometry, and topology.

Change

• Understanding and describing change is common in the natural sciences.
• Calculus was created to study change.
• Functions describe changing quantities.
• Differential equations, difference equations, and dynamical systems are studied.
• Chaos theory studies unpredictable systems.

Foundations and Philosophy

• Mathematical logic and set theory were developed to clarify math foundations.
• Mathematical logic establishes a rigorous axiomatic framework for mathematics and studies the results.
• Set theory studies sets, which are collections of objects.
• Type theory is another foundational system.
• Philosophy of mathematics addresses the epistemological status of mathematical statements.
• Poses questions such as: What is a number? What is a mathematical proof?

Discrete Mathematics

• Discrete mathematics studies discrete, rather than continuous, mathematical structures.
• Includes combinatorics, graph theory, and theoretical computer science.
• Used in computer science and operations research.

Music, Arts, Physical Education and Health (MAPEH)

• MAPEH combines Music, Arts, Physical Education, and Health as one subject.
• Aims to give students a complete knowledge and love for the arts, physical activities, and healthy living.

Music

• Music education studies musical elements, history, culture, and performance.
• Students learn about rhythm, melody, harmony, form, and other musical concepts.
• Common activities include singing, playing instruments, composing, and analyzing music.

Arts

• Visual arts education encompasses drawing, painting, sculpture, printmaking, and design.
• Students explore different art styles, techniques, and media.
• Art history and cultural context are studied.

Physical Education

• PE emphasizes physical fitness, sports, and movement skills.
• Students engage in physical activities, games, and exercises.
• They learn about physical health, nutrition, and safety.

Health

• Health education covers personal hygiene, nutrition, disease prevention, mental health, and substance abuse.
• Students learn to make healthy choices and promote well-being.
• Includes reproductive health, first aid, and environmental health.

Integration

• MAPEH is designed as an integrated subject to show discipline connections.
• Music and the arts connect through musical forms and visual art styles.
• Physical education and health connect through promoting physical activity and healthy living.

Mathematics explores quantity, structure, space, and change, seeking patterns and formulating conjectures. Proofs validate these conjectures, and when structures mirror reality, mathematical reasoning provides insights into nature. Developed from counting and measurement using abstraction and logic, practical mathematics has been a human activity from the earliest written records.