Get The Answer

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.
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