An epiphany after 17 years of teaching

Two years ago, two students sat side by side working on review problems for the upcoming test. Let’s call them William and Mary. William and Mary worked together on the problems, talking about them and calling me over to ask questions when they didn’t understand something. The next day, Mary got an A on the test and William got a D. This happened several times throughout the year. Both students had practiced and felt prepared for the exam, but the results were completely different. William was one of many students I have had over the course of my years of teaching who said to me, frustrated, “I studied and thought I knew everything, but I still bombed the test. Why?” Why indeed?

Then it came to me. Mary was not satisfied until she fully understand WHY her answer was correct and William was satisfied just to GET the correct answer, whether or not he understood why. William and I had a big “Ah-Hah” moment at the end of the year and I encouraged him to take the class again. He had made big strides in his effort; now he knew what else he needed to do to be successful. He became a different student, getting a B in the class, and grew confident in his ability to do math. (Precalculus, no less!)
Now I make a BIG DEAL out of telling my students that getting the right answer is not the goal; understanding the mathematics is the goal. I have found that students who work hard and study and then do badly on my tests, for the most part, are students who try to memorize procedures and formulas without understanding them. This year I have noticed a marked improvement in the overall quality and success of my students now that they are aware of what it means to  understand concepts deeply instead of just memorizing procedures. Students now diagnose their own poor test grades: “I didn’t really understand the concepts; I just memorized stuff” instead of wondering “Why did I do so badly on the test when I studied so hard?” Students are actually putting more thought into understanding their homework instead of just doing it
So spend time helping students with their study habits. Most students who give up on math don’t realize that they are sabotaging their own goals by not knowing how to learn math. We need to help them understand themselves as much as we need to help them understand the math.