Math instructors, in public, private, and home schools teach many different methods of approaching mathematics. This is normal and, in fact, healthy, as the diverse methods of looking at a concept are each unique and work for different people better than others. However, there is one aspect that is detrimental and damaging to a student's education. This problem is shortcutting. Many teachers teach "shortcuts"-- methods to get the same answer by cutting across the process and plugging numbers into an equation mindlessly. Let me explain why this is a problem.
All of us like solving problems quickly. If we have an equation we can plug numbers into and get an answer, that is quite handy. However, there is a fine line between handy and destructive. Let us take the quadratic formula, for example.
This may seem like a random formula that popped randomly into some geeky mathematician's mind and he wrote it down and it works for all humanity. That's not true. All these formulas have an underlying story of what is actually being done. The problem here is that some teachers may teach the shortcut first. This leads most students to solving an equation by plugging into that formula but never understanding why it works. This can prove fatal later in their math career when they need to begin applying those skills in different ways. This is why the students must also learn the "story beneath the formula".
Now, there is a way to correct this and to compromise these two methods to make one strong and helpful method of learning math. That is, teaching what's actually happening in the process of solving the equation before teaching the shortcut. After the teachers embed the raw process in the students' brains, then the teacher can proceed to teach some "shortcuts" in solving the equation faster, still with an explanation of why the shortcuts serve as a valid way to come up with the same answer.
To conclude, shortcuts, in and of themselves are not evil. However, if they are taught too prematurely with no basis explanation of what's actually happening, they are harmful. For more information, tune in on Monday, May 29th at 12 PM Mountain Time on Facebook Live for a live video explaining more about this blog post. You can find the video here after the event.