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