In Teaching vs Training 1 (2 Dec 2012) I identified ten flaws in a particular example of training which is common in mathematics classrooms. The third of these flaws was:
The student can become very proficient at these steps without any understanding of the underlying mathematics principles.
It is my contention that understanding mathematics principles is important. I will argue that point on another day. Today’s post will primarily serve to list some of the principles overlooked and ignored in the training example presented on 2 Dec 2012. For the sake of brevity, I will present my case today in outline form. Eventually I will write the book. Excuse the numbering–I don’t seem to be able to get WordPress to use the style numbers I want.
The mathematics principles directly at play in the solution of a simple linear equation in one variable of the form ax + b = c are:

The concept of equivalent equations.
 Two equations are equivalent if they have the same solution set.

The role the concept of equivalent equations plays constructing a strategy for solving.
 The strategy is to form a sequence of equations each equivalent to the previous equation (consequently equivalent to the first) that ends with a simplest equation of the form x = k.
 All the equations in this sequence of equations have the same solution set.
 The solution set of this simplest equation is clearly {k}.
 This set must then be the solution set for the original equation.
 The strategy is to form a sequence of equations each equivalent to the previous equation (consequently equivalent to the first) that ends with a simplest equation of the form x = k.

The dependence on the concept of equivalent equations.
 Suppose by some process we generate a sequence of equations which terminates in a simplest equation x = k.
 If we don’t know that each equation is equivalent to the previous one, we cannot conclude that the solution set to that simplest equation is also the solution set to the original equation.

Two methods for generating equivalent equations.
 If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation.
i. It is very important for the student to realize that this applies to all equations, not just linear equations in one variable.
ii. It is very important for the student to understand that any expression (not just numbers) can be added to both sides of the equation.
iii. It is important that the student NEVER think of moving an expression from one site to the other.
iv. It is important for the student to NEVER start believing that if you do the same thing to both sides of an equation an equivalent equation results.  If both sides of an equation are multiplied by the same nonzero real number the resulting equation is equivalent to the original equation.
i. It is very important for the student to realize that this applies to all equations, not just linear equations in one variable.
ii. It is very important for the student to understand that multiplication of both sides is restricted to multiplication by nonzero real numbers.
iii. Multiplying both sides of an equation by 0 need not generate an equation equivalent to the original equation.
iv. Multiplying both sides of an equation by an expression containing a variable need not generate an equation equivalent to the original equation. It is easy to demonstrate that multiplication by a variable expression does not yield an equivalent equation. Begin with the equation x=3 whose solution set is {3}. Multiply both sides by x to get the equation x^{2} = 3x whose solution set is {3, 0}.
 If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation.

Definitions of solution and solution set.
 A solution of an equation is a number which makes the equation TRUE when substituted for the variable.
i. Recall that a conditional equation is an equation which is TRUE for some values of the variable and FALSE for other values of the variable.
ii. In the equation 3x = 6 the variable x may be replaced with any real number. Some of those replacements make it true and some make it false. For example, if x is replaced with 2 a true statement results while a false statement results if x is replaced with 7.
iii. Solving an equation means to find all those values of the variable which make the equation TRUE.
iv. It is absolutely incorrect to believe x can only be one value in the equation 3x = 6.  The solution set for an equation is the set containing all solutions of that equation.
i. Solving an equation means to find the solution set for that equation.
ii. The strategy for solving an equation must permit the solver to find solutions and to know without a doubt when all solutions have been obtained.
 A solution of an equation is a number which makes the equation TRUE when substituted for the variable.

Distinction between solution and solution set and the value of such distinction.
 It should be easy for the student to understand the difference between a number which makes an equation TRUE and the set of all such numbers.
i. If a solution is found there is no guarantee there might not be other solutions.
ii. If the solution set is found, that guarantees there are no other solutions.
 It should be easy for the student to understand the difference between a number which makes an equation TRUE and the set of all such numbers.
Beyond these concepts which are directly involved in the solving process, other important and useful concepts are simply omitted from discussion when training to “get the answer” is the goal.

The graph of an equation.
 The graph of an equation consists of all the points, and only those points, whose coordinates are solutions of the equation.
i. So in simplistic terms if we draw the graph of an equation we are drawing a picture of its solution set.  Traditionally we discuss solution set for equations in one variable, but we do not discuss graphs of these equations.
i. We should discuss and produce the graph of equations in one variable.
ii. We should make an issue of the fact that equations in one variable are onedimensional and that their graphs necessarily must be onedimensional. Therefore their graphs must be on the number line. Not in the Cartesian Coordinate System.  Traditionally we discuss and produce the graph of equations with more than one variable, but we do not discuss the solution set of these equations.
i. We should discuss the solution set for equations in two variables.
ii. We should emphasize that each solution is an ordered pair of numbers.
iii. We should discuss difficulties inherent in any “solving” process.
iv. We should make an issue of the fact that an equation in two variables is two dimensional and that their graphs necessarily must be twodimensional. Therefore their graphs must be in the Cartesian Coordinate System.
v. We should discuss how the graph “shows” us the individual solutions.  We should be struck by the inconsistency demonstrated by b and c. Mathematics should be consistent.
 The graph of an equation consists of all the points, and only those points, whose coordinates are solutions of the equation.

The related inequalities.
 The = symbol in the equation ax + b = c may be replaced with < to obtain the inequality ax + b < c.
 The = symbol in the equation ax + b = c may be replaced with > to obtain the inequality ax + b > c.
 When considering an equation or an inequality we should always consider the other two siblings.
 In mathematics we solve only two kinds of objects; equations and inequalities.
 As will be seen a little later, the solution set of an equation and the solution sets of the corresponding inequalities are closely related.

The Law of Trichotomy.
 The Law of Trichotomy states that for any two real numbers m and n exactly one of the following is true:
i. m > n
ii. m = n
iii. m < n  The Law of Trichotomy has a very special interpretation in reference to equations and inequalities. Consider the simple linear equation in one variable. Then The Law of Trichotomy states that;
i. Each real number is a solution to exactly one of the following: ax + b > c
 ax + b = c
 ax + b < c
 This means that the solution sets of e1, e2, and e3 have no elements in common and the union of the three solution sets is the entire real number line.
 The solution set for an equation in one variable will consist of isolated numbers.
 The graph of an equation in one variable will be a collection of dots on the real number line. Hence that graph divides the real number line into intervals and rays.
 The solution set for an inequality in one variable will consist of intervals, rays, or unions of intervals and rays.
 The graphs of the inequalities corresponding to a particular equation consist of the rays and intervals formed by the graph of the equation.
 The equation is usually called the boundary equation because its graph forms a boundary between the graphs of the two inequalities.
 The Law of Trichotomy states that for any two real numbers m and n exactly one of the following is true:
It has always been interesting to me that in traditional algebra courses we make use of some of these principles when we discuss graphing linear inequalities in two variables. But even then we normally do not present the process as a very general relation between equations and inequalities. It just becomes another of thousands of isolated responses the student is supposed to memorize.
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