I am teaching the dreaded calculus II this semester. I’ve known many students who flew through calc I in college (having taken calculus in high school) only to receive a reality check from calc II the next semester.
In the US, calc II often involves a significant section on “techniques of integration” where students learn techniques such as partial fractions, trig substitutions, integration by parts. Unlike much of differential calculus, which is taught in calc I, and unlike much of the math taught before college, integration is harder to do algorithmically. That is, a calc II professor cannot simply outline surefire steps guaranteed to give an antiderivative for any function.
The inimitable Robert Ghrist explains it this way in his “funny little calculus text“:
Of course, algorithms for computing many antiderivatives do exist (and are used in Maple, Mathematica, and Wolfram Alpha), but I’d be fired if I tried to take my undergrads through Symbolic Integration I: Transcendental Functions.
Instead, calculus II teachers teach a handful of methods and attempt to teach students intuition for where to use what technique. Even more important, I try to teach my students the skill of trying a method, seeing that the method does not work, then trying something else. Try-fail-try-fail-try again.
I do not think that high school students learn that skill—a skill vital to success not just in calc II, but in every discipline requiring analytical problem solving. Yesterday, my adviser and I were discussing the first big research problem that I’ll be tackling this summer. He noted that our first attempt at solving a massive problem would probably fail; they usually do. Fortunately, I’ve been failing at solving problems at least since taking number theory with Dan Krider in 2003. I know what Edison meant by, “I have not failed. I’ve just found 10,000 ways that won’t work.”
I hope my students are learning how to fail and how to try again. However, I think that kids need to learn earlier. High school assignments should not be set up for students to succeed the first time, every time. Somehow, teachers need to allow students to take risks, learn from their mistakes, and rebound.
I’d love to hear feedback from students who are learning these lessons and teachers trying to teach them.
I could not agree more. I see a similar law in teaching people to fly:
1. Let them try, and do it poorly, or fail.
2. Explain what they did wrong and how to correct it.
3. Let them try again.
The trick is knowing how far to let them go. Another trick is learning to see if the corrective measures you are taking are working. Another trick is leading questions.
I also see the try-fail-try law in my own life and endeavors as well.
Great post!
Yeah, I’m up against calc II this summer. It’ll be the only time I’ve ever only taken one class at a time. Precisely because people tell me calc II is the shakeout class.
I’ll be sure to tell Dan about his positive influence!
Really great points, and I hope your calc II teaching goes well. It’s going to be a tough row to hoe for a lot of people involved.
Unfortunately, part of the reason learning to fail happens at the post-secondary level and not sooner is because of the structure of public education in North America. Think of it: grade school and high school curricula are basically designed like big checklists of requirements. There’s almost no incentive to challenge yourself, and plenty of incentive to tow the line. If you get 65% in all of your courses in high school (ie scraping by) you’ll get your HS diploma; if you get 95% in all courses except for one required course, and fail it for trying to think “outside the box”, you’ll not only not get your diploma, you’ll have to figure out how to make up the requirement (could be night school, summer school, taking it the following semester, etc). This could set back a typical high school student at least a semester, if not a whole calendar year, and even cost them money in additional tuition, resources or foregone wages at work. Much better to just stick to the script, as it were.
I really agree with this. I think if you fall and then learn it the correct way is much better. By the way everybody make’s mistake’s. It’s the best if we learn from them.
I think that personality cannot be taken out of the equation, either. Some students (at any age) are great at picking themselves up and dusting themselves off after that first failure. They could care less how it looks. Others are so afraid to fail that first time that they sit and stare at a problem for an hour before putting pencil to paper. You have to ask whether this is a nature issue or a nurture issue. I know some students (I’ve taught jr high and sr high math for the last 13 years) who will never learn to fail “properly” no matter how the education system is changed to encourage trial-and-error-and-trial-and…