One reason tricks become necessary is that students are rushed through concepts that they don’t yet fully understand or don’t have enough background math knowledge for. Tricks get them to the right answer without the understanding.
Another, and possibly more insidious, reason is the implicit value that school math places on speed. I didn’t even realize some of these tricks existed–ASTC (All Students Take Calculus) for which trig functions are positive in which quadrant, or ball to the wall for dividing decimals, or man on the horse for fractions.
ASTC got me thinking. It took me about 30 seconds to confirm that it was true (all functions positive in Q1, sine in Q2, tangent in Q3, and cosine in Q4). My honest initial reaction was, “cool I should remember that”. I feel like I might be justified, because I could figure out which function is positive in which quadrant again if I need to. It would save me a few moments when working with trig functions, and probably wouldn’t do much to my conceptual understanding–and I bet that’s the perspective of teachers pushing these tricks on students. But from the student perspective, it’s one more piece of meaningless mathematical trivia, devoid of reason or meaning. Students should be deriving the sign of the trig functions over and over again until it becomes second nature which function corresponds to which axis–the point isn’t to be automatic at signed trig functions, but to gain intuition and understanding of the unit circle and what it is used for.
That dissonance between the teacher perspective and the student perspective is, I think, at the root of where these tricks come from. Teachers find tricks useful because they save time working through concepts the teachers already know. And the value of tricks is underscored for students by the implicit (or sometimes explicit) emphasis we put on speed in school math, whether it’s through assessments, students that get praised by teachers, or stereotypes of good math students. This is a big, difficult push for teachers to make, to flip student thinking to value careful process over the number of questions that get answered. Jo Boaler has some awesome thoughts on this, and her MOOC was a goldmine of great ideas and resources.
I will make one caveat that working with trig functions made me think about. Not all mnemonics and tricks are terrible. I will stand by Chief Soh Cah Toa for the rest of my life. This is because Soh Cah Toa is a reminder of a definition, not a shortcut for a calculation or a concept that students should be deriving on their own. It’s an important distinction–definitions are often arbitrary, and remembering definitions is key to building understanding. Maybe I’m wrong about this, I’d love for someone to push back.