Why Teach Integration in High School Calculus?
Does content really matter?
Most of the students in our school take calculus because they believe it is a necessary gateway to college acceptance. Few will remember any of the content presented, and even fewer will ever pursue mathematics at more advanced levels.
For over a decade I taught AP Calculus at both the AB and BC levels. Virtually all of my students passed and over 95% of the students at the BC level earned 5’s on this exam. With committed students and a knowledgeable teacher, it’s just not that hard. Students were happy, parents were happy, but I wasn’t always happy.
When we gave up the AP program at our school I didn’t change much about the content of the Calculus courses I taught at first. I actually don’t believe that content is all that important. I evaluated what I was teaching and why. I kept coming back to my belief that the most important thing I should teach students at the high school level is how to learn.
Integration is a big topic in calculus. If I teach method A on Monday, method B on Tuesday, method C on Wednesday, and assign homework each night most students will do very well with one method at a time. It’s when I give a quiz on all three methods that the questions begin. Students struggle with identifying when to use which method and why.
I identify and teach 14 integration techniques, far beyond those required for AP Calculus. Why learn 14 different integration techniques if only 2 or 3 will ever be used in applications?
Because what I actually hope to teach is not only how to integrate, but also how to identify, categorize and differentiate.
Teach ways to identify when to use problem-solving methods not just how to imitate their use.
When we teach specifically to the AP exam we can quickly pick the problems that include the integration techniques the students need to learn. The problem with the multiple-choice section of the BC Calculus exam is that it allows students to very easily identify the procedure used. If the answers have natural log, for example, chances are very high that the problem was integrated with partial fractions. The majority of my students used intelligent guessing and succeeded.
Our goal should be to have students truly understand why a problem-solving method works. Guess and check may be an acceptable technique for certain types of problems, but when a student congratulates himself for getting a problem right simply by guessing perhaps we should reevaluate the value of the problem.
Teach intelligent guessing, and ask students to defend their choices.
I tried asking students to explain what method would be most efficient and why, without actually completing the integration. Most students found this really frustrating. Many really wanted to know if they got the right answer or if they would get credit for the right answer no matter how they did it. (which they would) Bit by bit they started to think about how to articulate their reasoning; sometimes valuable debates occurred. On some occasions when a student got the correct answer using one method, she was convinced by a classmate there might have been a faster way to solve the problem.
Support the fact that here’s more than one way to solve a problem, and discuss what the most efficient way is.
Having students create their own problems is also a meaningful lesson. I strongly suggest trying this in class without internet help. Valuable lessons and good classroom debate happen when someone creates an unsolvable problem
The typical math class teaching method of I do, we do and you do isn’t enough. True thinking, not mimicry, is the way to expand your brain. Finding new ways to approach content and to help students learn to ask the right questions is vital to mathematics education.
When my students say they don’t get it or they are lost I insist that they ask a question. And I help them to articulate a good question. I encourage this by giving bonus points for really good questions.
Asking the right question is where true learning actually begins.
