California Institute of Technology (Chemistry, Biology)
Carnegie Mellon University (PhD)
In order to build a strong understanding of any subject there are three things that are needed
a) a reason to learn
b) a willingness to try
c) a good teacher
These 3 things are clearly related. A good teacher helps provide a reason to learn, and willingness to try help makes a good student. In my experience of teaching from kindergarten to professional levels, the best teachers light a trail for the student to walk. I can do that, light a trail in chemistry and mathematics. I understand what these subjects are about, and can share my knowledge and enthusiasm for these subjects. I can't force or coerce a willingness to try. I can provide a reason to learn. Fundamentally that is because chemistry and math are some pretty fun things to play with. Make it fun, and the learning follows. That being said, these subjects are not everybody's cup of tea. I can and will make the tea sweeter, drinking the tea - that depends on the student. In order to build a strong understanding of any subject there are three things that are needed
a) a reason to learn
b) a willingness to try
c) a good teacher
These 3 things are clearly related. A good teacher helps provide a reason to learn, and willingness to try help makes a good student. In my experience of teaching
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Algebra is the cornerstone for advanced mathematics, physics, engineering, and just about anything that involves solving for an answer. A good basic understanding of algebra leads to a clearer and deeper understanding of all of the physical sciences, and is definitely a must have skill for this modern, increasingly technological world.
Advanced algebra moves from constants to variables. Generally variables serve as a means of parameterizing problems so they are easier to solve.
Good algebra skills have lots of real world applications, from the basics like how much lumber to buy to build a deck, to the more abstract such as how do unemployment rates affect the balance of trade. Regardless of the problem, algebra provides a clearly defined procedure for finding solutions.
I hold a B.S. in chemistry, a Ph.D. in inorganic chemistry, have taught chemistry at the graduate and undergraduate level, own and run a small chemical company, and just generally like chemistry and doing experiments with matter. Chemistry is the study of matter under relatively ordinary conditions. Fun with chemistry comes from understanding matter well enough to make new forms of matter, or study existing matter to reveal it's mysteries. If you can touch it - it's chemistry.
My background in chemistry starts in high school, when I really did not get it at all my first year. My second year I tried much harder to get beyond the cryptic symbols and somewhat dry experiments and began to understand that it is about achieving power and control over the stuff from which all things are made. If letters are the building blocks for making words, then the elements are the building blocks for making matter. Most people can use about 5,000-10,000 words, but chemists can use and understand literally millions of compounds. This is exciting stuff if you are willing to learn the rules and recipes that have been developed over about the last 400 years to actually master chemistry. Otherwise, it can be overwhelming. Like most science it is front loaded, meaning that you have to ante up and do the work first to get to the point where you understand enough to make it fun. Hard at first, easier later.
A good basis in math lays the foundation for not just surviving but excelling in this modern technical world. Whether it is adding up the change in your pocket, calculating how to launch a rocket to mars, or figuring out how to build a dam, math is literally the foundation on which everything else is built.
My background in math comes from taking advanced high school math, two years of college calculus, a year of applied math, from other math based subjects and from programming in Mathematica (a symbolic math language that interfaces with all types of computers).
Basic science forms the foundation for understanding at the most basic level how things work. From the seasons to fireworks to how to build a bridge, these are all basic science concepts that are both practical and enrich our lives.
As a practicing scientist, I have a special fondness for all things scientific. My favorite are experiments. When you don't know the answer and you can design an experiment or series of experiments to help find the answer, that is very gratifying!
Have you ever wanted to build something, or measure something, or look at the beauty of architecture, art, or a gemstone? Have you ever wanted to design something? Then geometry is the ticket for that.
Matrices, vectors and n-dimensional spaces are the bread and butter of linear algebra. This is the stuff you need to know if you want to do any type of 3D rendering, engineering calculations, physics, waves, materials science, or physical chemistry. It underlies most of quantum mechanics and forms a significant part of our understanding of topology and even the structure of the universe.
Most are exposed to linear algebra as part of calculus, or as part of advanced problem solving approaches required for instance to describe the hydrogen atom.
My background in linear algebra comes mainly from two years of calculus, and a year of applied mathematics that I took at Caltech followed by several years of physical chemistry in graduate school at Carnegie Mellon University. For me the hands-on application for linear algebra was NMR (Nuclear magnetic resonance) which most people are familiar with under a slightly different name - MRI - magnetic resonance imaging. Linear algebra is the means by which solutions to complex problems for describing the physical world can be expressed.
Mathematica is a very cool math language that allows the computer to do the heavy computational lifting. The list of programmers reads like a who's who of american mathematics. The syntax is odd--but logical--kind of like a cross between C and HTML. I've been programming in Mathematica since Mathematica 2.0, but the program has expanded continually. If you can phrase it as an equation, Mathematica can solve it, one way or another.
In terms of tutoring the basics are file IO, equation solving, and graphics display.
Prealgebra develops the rigorous thinking necessary to pursue more advanced algebra.
From a good beginning, most students can then excel in more advanced mathematics.