GE103 (Midterms)

• Mathematics reveals hidden patterns that help us understand the world around us
• Mathematics


  A diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems
• Mathematics


  • A science of pattern and order
  • Its domain is numbers, chance, form, algorithms, and change
  • It relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth
• The special role of mathematics in education is a consequence of its universal applicability
• The results of mathematics - theorems and theories - are both significant and useful; the best results are also elegant and deep
• Distinctive modes of thought offered by mathematics

  • Modelling
  • Abstraction
  • Optimization
  • Inference from data
  • Logical analysis
  • Use of symbols
• Mathematical power


  A capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives
• Mathematics empowers us to understand better the information-laden world we live in
• Mathematics gives us a way to understand patterns, to quantify relationships, and to predict the future
• John William Navin Sullivan: 'Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art: by informing us of the nature of our own minds, it informs us of much that depends on our minds.'
• Mathematics is at the center of history and development of culture, evident in ancient arts, Egyptian architecture, and music
• Analytical thinking


  Helps a person investigate and determine the truth around him or her
• Mathematics has played a central role in the physical sciences for centuries, and is currently being used by life scientists in analyzing patterns present in plants, animals, and humans
• Mathematics


  The abstract study of how the structures of systems relate and operate
• In its early forms, mathematics helped us quantify time, make measurements, and take records, especially during the development of agriculture when surpluses in food allowed trade
• Number systems of different civilizations

  • Everyday objects
  • Geometric shapes
• Ancient Greeks believed that numbers were both living entities and universal principles; numbers were active agents in nature
• Schools of thought on the nature of mathematics

  • Realism (math exists objectively and independent of human thought)
  • Mathematical anti-realism or idealism (mathematics is a product of the human imagination)
• Applied mathematics is the branch of mathematics involved in the study of the physical, biological, or sociological world
• Mathematical realism


  Mathematical concepts are disembodied in the universe and available for us to uncover and bring into practical use
• Mathematical anti-realism or idealism

  Mathematics is a product of the human imagination and is carefully engineered to make formal statements about nature in order to aid our understanding of the behavior of the universe
• People often wonder what relevance mathematicians serve today
• Applied mathematics


  The branch of mathematics that is involved in the study of the physical, biological, or sociological world
• The idea of applied math is to create a group of methods that solve problems in science</b>
• Modern areas of applied math

  • Mathematical physics
  • Mathematical biology
  • Control theory
  • Aerospace engineering
  • Math finance
• Applied math not only solves problems, but it also discovers new problems or develops new engineering disciplines
• Applied mathematicians


  • Require expertise in many areas of math and science
  • Require physical intuition
  • Require common sense
  • Require collaboration
• Common approach in applied math

  1. Build a mathematical model of a phenomenon

  2. Solve the model

  3. Develop recommendations for performance improvement
• Pure mathematics


  Driven by abstract problems, rather than real world problems
• Much of what's pursued by pure mathematicians can have their roots in concrete physical problems, but a deeper understanding of these phenomena brings about problems and technicalities
• Pure mathematics is abstract and based in theory, and is thus not constrained by the limitations of the physical world
• Difference between pure and applied mathematics

  Pure mathematicians prove theorems, and applied mathematicians construct theories
• Pure and applied mathematics are not mutually exclusive, but they are rooted in different areas of math and problem solving
• The solutions developed from the processes of pure and applied mathematics have affected and improved the lives of all
• The laws of mathematics govern everything around us, and without a good understanding of them, one can encounter significant difficulties in life
• Learning math is good for your brain
• Research indicates that children who know math can recruit certain brain regions more reliably, and have higher gray matter volume in those regions, than those who perform more poorly in math
• The brain regions involved in higher math skills in high-performing children were associated with various cognitive tasks involving visual attention and decision-making
• Math helps you with your finances
• Math helps you have better problem-solving skills