Whether you view mathematics as a tool for application to other investigations or as free-standing logical academic study, it is very important to realize that it is not a collection of hundreds or thousands of formulas, facts, and processes which are to be memorized.
Current mathematics instruction is such a mess that most students view it as a vast collection of disparate formulas and processes to solve a myriad of non-sensical problems.
In my blog postings, as in all previous writing and speaking, I wish to promote the philosophy that mathematics is about deductive reasoning using very general principles.
The purpose of high school and early college algebra courses is to introduce the student to the use of abstraction, generalization, problem solving techniques, deductive reasoning, and critical thinking while exploring the structure, patterns, and relationships of a variety of algebraic entities including, but not limited to, relations, unary operations, binary operations, and mathematics objects such as equations, inequalities, algebraic fractions, polynomials, and functions. Furthermore, I hope to instill a firm belief that it is better (and easier) to use a single mathematics principle to solve a multitude of problems rather than use what appears to be a random process with little apparent connection to mathematics properties for each of a multitude of problems.
I intend to use The Number Line, The Distributive Law, The Transitive Law, The Law of Trichotomy, and The Euclidean Algorithm to demonstrate how we can achieve the above stated goals in early algebra courses.
I strongly recommend that you read my essay “What is Wrong with Mathematics Instruction” found in the “Pages” section of this blog.