All math teachers have heard the questions: “Why do I need to learn algebra?”, “When will I ever use it?”. In my calculus class this week both questions were answered.
We were finding derivatives using the power rule. An exercise in the book asked for the derivative of
At this point in the course the function needs to turned into a summation of powers of x and the power rule needs to be applied to each resulting term. When the authors created this problem, intentionally or not, they mixed two styles of algebraic notation.
The function definition mixes together both an exponential form and a fractional form. The best approach in solving this problem is to change to a common form and then distribute multiplication over addition. There are two options, 1) all-exponential form and 2) all-fractional form.
Once the function is written as a summation of powers of x the determination of the derivative is easy.
The students worked on the problem for a few minutes and most of them used Option 2, fractional form, to solve it. I used Option 1, exponential form. I wrote my solution on the board at the end of the exercise. Surprisingly, there were two students who could not understand why my solution worked. One student was rusty in his algebra skills (gleaned from earlier work with him) so his confusion was understandable. The other student had participated in class discussion and answered open questions on other sequences of algebra, so his confusion was more unexpected. Not understanding and misunderstanding in math can be difficult to correct. The student is seeing something different from the instructor, misinterpreting symbols, or missing unstated assumptions. The student, by definition, cannot tell the instructor what is not seen or is missing. The instructor, being fluent in the operations, does not see what the student is seeing. An inefficient dialogue can be the result. In this case the misunderstanding was resolved for both students by reminding them of the basic rules of exponents in algebra. A simple problem, but a problem that gives answer to the leading questions: “Why do I need to learn algebra?”, “When will I ever use it?”.