When taking a math course there are four things that a student should learn.
The Fundamentals
The fundamentals include the definitions, the rules of operations, and the tactics of manipulation. It is essential that you understand the definitions and can visualize them. The rules of operations need to be practiced until they are second nature but they
should never be divorced from a simple illustration that explains the rules. The tactics of manipulation are the sequences of steps needed to solve the types of problems that will be encountered.
“As strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations”. (Ernst Mach Paul)
The Applications to Problem Solving
Math is the language of science. Math was developed to solve problems. You are in this course because you have other courses that will require the problem solving...
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Algebra involves numbers, unknown numbers and operations between the numbers and unknown numbers. Algebraic expressions and equations are built with a series of operations. Each operation comes in pairs with a "forward" operation and a "backward" operation.
The forward operations in algebra are addition and multiplication because they are easier than subtraction and division. The progression of operations both forward and backward in high school math is as follows:
Addition/Subtraction
Multiplication/Division
Powers/Roots
Exponentials/Logarithms
Trig operations/Arc Trig operations
Differentiation/Integration
The first 2 groups are covered in Algebra 1; the third group in Algebra 2; the fourth and fifth groups in Pre-calculus and the last group in Calculus.
For example the equation 3x + 2 = 14 is a series of operations applied to x. The order of operations requires the first step to be multiply x by 3. The second step...
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An algorithm is a precise systematic method for solving a class of problems. It takes input, follows a set of rules and gives output that provides a conclusive answer. The accuracy of the answer depends on how expertly one applies the specific techniques
or algorithms. Students practice algorithms using exercises from a textbook.
Problem solving is the process of applying previously learned rules to a situation which for the student is new and different. The common view in high school is that problem solving is a word problem. But problem solving is not just a word problem. Math does
involve the translation of words into symbols and symbols into words, but if you have just taught students how to work word problem, that word problem is just an exercise. Problem solving must involve for the student something new and different never before
encountered by the student. It requires the selection of a technique among various approaches.
Math has been developed to solve...
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The typical secondary math education includes:
Pre-algebra
Algebra 1
Geometry
Algebra 2
Pre-Calculus
Calculus
This list is not very descriptive of what is being learned. For instance, why do we call it "Algebra"? The word carries no descriptive meaning in English, The word calculus (pebbles) is also misleading. Using the word "pre" is also non-descript. It makes
math educators seem unknowledgeable and uncreative. Yes, each subject comes with a variety of topics, but we should be able to characterize what is generally covered. So below I try to rename the above list to be more descriptive of their contents.
Pre-algebra mainly covers the dreaded "Fractions". The word shares the same root as "fracture", as in a broken bone. So fractions are broken numbers of the form a/b, for example 3/4ths. So we could simply call the first secondary school course "Fractions".
That includes ratios, proportions...
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