My feelings on this began a couple years ago when a couple of former students who had been Calculus BC students came back to visit me. They were juniors in college. One had become an English major, I believe, while the other was in an engineering school. After much small talk about how things were going, playfully berating the one student who had defected to English (he was a great math student!), I told them I wanted to try an experiment which they both readily agreed to.
I proceeded to give them three problems. The first was a factoring problem, easily Algebra 1, while the other two were a relatively basic derivative problem and an integral. Both struggled with the problems, the engineering student much less so. In fact, if I remember correctly, the engineering student got all three right, but would have failed the “quiz” in class because it took her way too long. The English major figured out the factoring problem, but only by solving it first using the quadratic formula and working backwards, did the derivative problem, and could not remember how to do the integral. When I had them in Calculus BC this would have all been standard fare, and at most would have taken 5 minutes – if they had taken a 2 minute nap to begin. It took them nearly 15 minutes, and the English major just threw up his hands at the integral.
The real question I had was why. It was clear that the English major had not taken mathematics in awhile (which brings up the question about why he rushed to get to our BC class – which can only be taken after AB – by the time he graduated). The engineering student told me that in the course of her classes if she need the answers to these questions it would be because it was in service to some larger problem and not the end in itself, thus they would use Mathematica (or similar software – I cannot remember exactly which one they had). Did they both still understand the concepts? Yes, but their facility to use them had clearly diminished.
I was impressed that the English major could still reason his way to a solution of the factoring problem, even though he could not remember the specific techniques to get their directly. A good example of understanding the concepts. That the engineering student understood the concepts was evident, she just did not waste her time on the “grunt work.”
It makes me think about tools we use in math nowadays – particularly things like dynamic geometry software, calculators, or CAS. There is something to be said about the notion that the more students practice the skills (drill and kill as it is ‘affectionately’ known in some circles) the better their facility with them. My concern has always been that the more they practice skills, the less time they have to work on understanding the concepts. I do not believe that merely practicing the skills leads to an understanding of concepts. That’s like saying teaching someone to hammer nails, saw wood, and put in screws will help them understand how to build a house.
CAS is inherently dangerous in a high school classroom because it will ‘do’ all the skills that we try to impart to our students in the curriculum. It requires the students to understand the concepts or they won’t know when they need to use what skills. At the same time, students still need to practice those skills, else how will they know when an answer their calculator gives them makes sense or not?
It is a balancing act.
What I took away though is that we don’t know what our students will be doing once they leave, and I think I can safely assume that, at best, only a small minority of them (and I am thinking of my Calc BC students here) will become math majors. So we need to prepare them for more than that. That includes how to use the tools that exist. CAS is one.
