Archive for December, 2012

Get The Answer

December 20, 2012

It is not uncommon for me to have students in my College Algebra class who solve an equation like (3/4)x = 6 by multiplying both sides of the equation by 4 to obtain the equivalent equation 3x = 24.  In a second step the student then divides both sides of the equation by 3 to obtain the equivalent equation and solution x = 8. There is no error in the process and the solution is correct.  However, I feel sorry for the student who does this.

I sympathize with the student on two counts:
1)  The student has no understanding,
2)  For years the student has suffered under the tyranny of inept teachers.
It seems the student is not aware of the basic number fact that the product of a number and its reciprocal is 1 or at least is not able to use that fact in this situation, to simply multiply both sides of the equation by (4/3) to obtain the equivalent equation and solution x = 8.  The really sad part of this tale is that the student’s teachers did not teach mathematics.  They trained the student to “get the answer”.  In many cases I suspect the teacher actually instructed students to solve such equations involving fractions in exactly this manner.
They trained the student to “get the answer” because they knew no better.  They knew no better because in too many colleges, prospective teachers are not required to take sufficient mathematics (or any other) content  courses.
The entire mathematics education system from top to bottom is in shambles.

Teaching vs Training 4

December 18, 2012

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:

  1. The concept of equivalent equations.

    1. Two equations are equivalent if they have the same solution set.
  2. The role the concept of equivalent equations plays constructing a strategy for solving.

    1. 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.
      1. All the equations in this sequence of equations have the same solution set.
    2. The solution set of this simplest equation is clearly {k}.
    3. This set must then be the solution set for the original equation.
  3. The dependence on the concept of equivalent equations.

    1. Suppose by some process we generate a sequence of equations which terminates in a simplest equation x = k.
    2. 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.
  4. Two methods for generating equivalent equations.

    1. 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.
    2. If both sides of an equation are multiplied by the same non-zero 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 non-zero 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 x2 = 3x whose solution set is {3, 0}.
  5. Definitions of solution and solution set.

    1. 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.
    2. 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.
  6. Distinction between solution and solution set and the value of such distinction.

    1. 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.

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.

  1. The graph of an equation.

    1. 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.
    2. 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 one-dimensional and that their graphs necessarily must be one-dimensional.  Therefore their graphs must be on the number line.  Not in the Cartesian Coordinate System.
    3. 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 two-dimensional.  Therefore their graphs must be in the Cartesian Coordinate System.
      v.     We should discuss how the graph “shows” us the individual solutions.
    4. We should be struck by the inconsistency demonstrated by b and c.  Mathematics should be consistent.
  2. The related inequalities.

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

    1. 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
    2. 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:

      1. ax + b > c
      2. ax + b = c
      3. ax + b < c
    3. 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.
    4. The solution set for an equation in one variable will consist of isolated numbers.
    5. 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.
    6. The solution set for an inequality in one variable will consist of intervals, rays, or unions of intervals and rays.
    7. The graphs of the inequalities corresponding to a particular equation consist of the rays and intervals formed by the graph of the equation.
    8. The equation is usually called the boundary equation because its graph forms a boundary between the graphs of the two inequalities.

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.

Teaching vs Training 3

December 7, 2012

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 second of these flaws was:

No mathematics is involved in the process.

We perform mathematics, like solving an equation, by using mathematical operations.  Through the use of deductive reasoning with mathematics objects, operations, relations, and stipulative definitions we are assured of the validity of the work.  If some other form of reasoning (memory, emotion) enters the picture, the results are no longer assured. If non-mathematics operations are introduced, the results are no longer assured.  If non-mathematics relations are introduced, the results are no longer assured. If words are interpreted differently than the accepted stipulative definitions, the results are no longer assured.

There is no mathematics operation called move.  Consequently the idea of moving b to the other side of the equation has no foundation in mathematics and any results of such action cannot be trusted. 

The next step in the process is to divide both sides of the equation by the coefficient a.  It would be a weak argument to try to dismiss this idea of division although multiplication by the reciprocal of that coefficient might be more desirable.  So we will accept dividing both sides of the equation by the coefficient as legitimate mathematics.

Finally the student believes this result to be the correct “answer” as a result of his memory of a pronouncement by his teacher, not deductive reasoning.  Although memory is an amazing mental function, it is not a good mathematics tool and does not provide adequate justification.  Notice that in this limited example, the student might not realize what constitutes a solution of an equation.  I encounter this absence of understanding nearly every day.

The entire exercise has degenerated into training the student to perform a motor skill instead of obtaining the solution for the equation through intellectual activity.  If our only goal is the motor skill, we can train (program) a computer/calculator to perform that activity.  In fact we frequently do program calculators and computers to perform routine tasks.


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