I was honored when asked recently to offer a topic for discussion on #mathchat.
My suggested topic:
Is it necessary for students to develop automaticity in their pencil and paper mathematics skills? Why or why not?
First some definitions and examples to clarify the intent of the question.
By automaticity, I also mean procedural fluency. A student that has developed automaticity is familiar enough with the mechanics of a particular task to not have to devote substantial thought to how to do it. It also is connected to retention over time - how well do the details stick with a student as more information is learned over time?
In an Algebra class, for example, do the details of arithmetic need to be automatic so that the student can focus on applying algebra knowledge to solving an equation? In Calculus, should students be able to apply the product and quotient rules efficiently when working on optimization or related rates? Or is it reasonable for them to figure out the derivative using basic principles or use a computer algebra system to take care of this step when it comes up?
I also refer specifically to pencil and paper skills because, for what I would guess is a majority of us that teach math, we tend to assess students by pencil and paper at the end of the day. A student can use a graphing calculator, Geogebra, or other piece of technology to explore a concept and check her/his work. The thing I often wonder about is how the use of activities and technologies help students perform mathematical tasks when these technologies are not available.
Is it necessary to do these tasks when these tools are not available? I don't know. I think that's open to interpretation and individual opinion. There are some cases, however, when that choice is not up to us. Standardized tests are one example. Given that they do exist (and independent of whether or not we agree with their content/quality/use), standardized tests are not typically electronic and are timed. These are often posed as opportunities for students to choose an appropriate method of finding answers to questions and then find those answers with a limited set of resources available.
Let me be clear - I am wildly inconsistent on this, because I don't have a good answer to the question. I emphasize understanding through the activities I do in my classes - very rarely will I directly tell students how to solve a problem, have them practice the skills with me, and then send them home to practice those skills in isolation from others. I really appreciate Conrad Wolfram's point about using computers to handle the calculating, and leave the thinking to us and our students. I have decided on occasion not to assign #1-30 for students to practice differentiation because my feeling at that time is that if they can apply it correctly several times, they get the point, and are ready to apply that knowledge to more interesting contexts.
But when these same students that complete the short assignment, later struggle in finding anti-derivatives, I wonder if I should have drilled them more. My decision not to burden them with repetitive exercises because they are repetitive often has implications for the future of the students in class. Do I need to drill this to automaticity so that next year's teacher doesn't come complaining to me about how "your old students can't XXXXXXXXXX" where XXXXXXXXXX = [arbitrary math skill that either (a) will mean the difference between getting into a top choice school during Senior year or (b)won't matter at all ten years after leaving the classroom]?
So I call upon the collective brilliance of the #mathchat community to help find an answer.
For those unaware, #mathchat is a Twitter based chat held every Thursday night at 8PM in which all respondents use the hashtag #mathchat in their post so that everyone else following that hashtag is updated with the latest responses. If you aren't up on using Twitter for professional development, you need to be. It completely changed my perception of how Twitter is useful and has put me in contact with some pretty amazing folks from around the world.