## It’s All Formulas

No, it’s not.  It is not all formulas.  Using ununderstood formulas is dangerous.  Using a formula is often unnecessary.  So learn as few formulas as possible and thoroughly understand the ones you do use.  Remember formulas are just summaries of a process of thinking and it is more important to know and understand the process than memorize a bunch of symbols.

Case in point.  On my last precalculus test I asked students to calculate the length of a circular arc given the radius of a circle and the angle in degrees swept out by the arc.  In class I taught this as a simple proportion.  If $s$ is the length of the arc, $r$ is the radius of the circle, and $\theta$ is the angle, use the simple proportion

$\normalsize \frac{s}{2\pi r}=\frac{\theta}{360^{\circ} or 2\pi radians}$

choosing $360 ^{\circ}$ or $2\pi$ radians naturally depending on the units of $\theta$.  The key concept is that similar shapes have proportional corresponding lengths.  This is the fundamental concept behind the definition of $\pi$ and the definition of trig functions.  When my students practiced on-line using the text publisher’s on-line help, they were shown the formula $s=\theta r$ with the same definition of variables above except that $\theta$ must be specified in radian units.  When the students got to the test they remembered the formula but forgot that $\theta$ had to be in degrees and got ridiculous answers.  They were using a formula that they didn’t understand and thus were liable to err.

Upon reading the last paragraph over I see that the valid objection could be raised that I am also using a formula to find arc length.  But this “formula” comes directly from understanding with no algebra or picky detail intervening.  Calculating the arc length  using my method (formula) emulates a reasoning process directly.

If I had wanted the students to really know the formula, $s=\theta r$ I would have asked them on a test to derive it with a complete explanation.  I didn’t because I wanted them to reproduce the reasoning process using the fundamental concept of geometric similarity.  Maybe I should have put some sort of essay question on the test that required understanding this principle, or a much harder question that required a deep knowledge of similarity and proportionality.

Many students say that they don’t like math because it is all formulas.  It’s not!  Behind most formulas are fundamental ideas.  That is what is important.  If I want students to memorize a formula, I tell them that they will be asked for it on a no calculator portion of a test.  On the calculator okay part of the test, if I want to see if they can use a formula in the right situation, I give them a list of formulas.  Still the fewer formulas the better.