In hindsight, maybe I overstepped the bounds of what is right and what is wrong.
Clearly, I knew the individual understood the question and answer and there's no doubt in my mind, he understands the formula. Having said that, I'll re-evaluate the whole scenario on Monday, God willing.
I have no doubt that the student in question would gain a lot of respect for you if you did. Especially if he knows that the incident was so important to you that you sought advice from this forum and had a change of heart.
Teaching is an art and most teachers just plain don't know how to teach. I teach amateur radio classes that prepare students to pass an FCC licensing test. In my mind, if a student fails, I have failed. So far, every one of the students that completed my classes passed the FCC exam on the first try. This includes my daughter, who was 11 years old at the time.
Where a teacher fails the most is engaging in a battle of egos. Traditionally, it's always been teacher vs. student. This is evidenced by the manner in which the typical classroom is laid out. Students on one side of the desk, and teacher on the other.
I don't do it that way. I mingle amongst the students and sometimes sit with them and put another student in front of the white board. I encourage an open dialogue and love it when a sharp student catches a mistake I have made. (Yes, I have made mistakes teaching classes).
FWIW, in high school I challenged my teachers every opportunity I got. One day in my physics class I found an error in my teacher's handout. I rode my bike down to the local college (about 6 miles) and found a college level textbook proving that I was correct. I checked the book out of the library and brought it to my physics class. When I challenged my teacher and provided the textbook as proof, she blew a gasket and screamed (In front of the entire class) "The book is wrong!!"
From that point on she became known as Mrs. The Book Is Wrong.
On edit, I finished #1 in my 40 student apprenticeship class and was the first person in the history of our local to get over 100 percent in our 5 year final exam. I got an extra credit question correct that was not taught in class. I learned it from a journeyman I worked with during my 1st year. I challenged my teachers there, too and even so, I am friends with all of them to this day. My first year teacher was a ham radio operator like me and the journeyman that taught me the answer to the extra credit question was a ham, too.