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Drill for Skill

December 2011

Wow, has it ever been a long time since I've written one of these. Life just keeps getting in the way. Not that I want the alternative...

So what has finally inspired me to get back to this? I wanted to lend some support to an effort my brother Rob is involved in and this is a natural way. He and some other mathematicians are lobbying against changes in the local math curriculum. Here is their new website: Western Initiative for Strengthening Education in Math (WISE Math).

So why would I, a Physicist who now works primarily with Engineers think a battle over math curricula on the Canadian prairies is worth commenting on here? Enough to wake me from a long rest from producing these articles? Very simple, math is important, very important, and it seems to me this local battle is only one front in a long running battle. While I care about numeracy in the general population, I'll focus on those who aspire to science.

Even as a grad student 25 years ago I was getting dismayed about the poor basic skills of many of the students in my tutorial classes. In the mid 80's calculators were relatively new, but already a significant number of aspiring engineers could not spot simple errors in their work. When I was asked to help find their mistakes I could glance though their page and ask something like "here where you multiply 13x9 you get 52, is that right" or "when you divide a number of the order of tens of thousands by a number of the order of hundreds, should the answer be of the order of millions?", etc. and most of the time they would reach for their calculator to find out. These questions were not even asking them to have memorized times tables or to do two digit arithmetic in their head. A numerate person should be able to answer those kinds of questions in their sleep. In the time since I've only seen more of this, not less. There is a kind of laziness and complacency with very poor math skills that that has crept in to even groups that need good math skills.

Let's be completely frank about something, we all make mistakes. Even if our calculator or our Finite Element software is perfect, we are not. We will on occasion mistype numbers that we are inputting. We will as professionals obtain bad results from poorly setup calculations. In the world outside of the classroom the mark is not getting 9 out of 10, we cannot crosscheck our answer against a known answer. The whole point once you are out there working as a scientist or engineer is that you are solving things where the answer is not known yet. If there was already a known answer to compare with, why would anybody be paying you to maybe or maybe not reproduce it? In the world after graduation your assignments, exams, and grading are like this:

  • Does the bridge fall down when a heavy truck goes over?
  • Is the new motor design better than the one it is supposed to replace?
  • Will you be given a raise for good work, or fired for bad work?

Ergo, it is important that you are able to spot mistakes on the fly. A large part of this is having the math skills that enable you to hesitate at a very wrong computation and say "that's quite different than what I was expecting, I better double check". The more types of math problems you can do this for, the better you can do your job.

It is natural enough that people are getting lazy with the advent of calculators and don't do as much math in their heads anymore. Technology can do many great things to aid in the learning of math; it is a pity that some consider it to be essentially a replacement for the learning of math. More the pity that many of those people seem to have sway over curriculum development.

I'll leave the basic issues to my brother and the site referenced above. What I will comment on is the idea that we can teach math better by encouraging students to explore at the expense of learning and practicing standard algorithms and methods to develop our skills. Exploring is good, I love to explore math and try to find new ways of solving problems. However, let's not kid ourselves that this is a replacement for learning and practicing the standard methods. Rob introduced me to the phrase "drill and kill", used to denigrate basic practice. A quick Google search verifies that this is a common phrase used to hammer home the idea that practice leads to boredom, disinterest, and therefore little learning. It is a normal marketing tactic - because it is short and rhymes that is somehow supposed to make it sound true. Stupid word game, but two can play that game. I liked the twist in this page suggesting it should be "Drill and Skill". Skills Development—Drill and Kill? I would put a different twist on it and say we "Drill for Skill".

I would rather avoid the chosen vocabulary of those who essentially deny the value of practice. That is what we're really talking about here. A few years ago my son and I took up Taekwondo together. Today I am a red belt; he is a second degree black belt. We both "explore", but he practices more than I do, so he is better and further advanced. Isn't that simple? The page I cite above makes the same point about fencing. It is the practice of standard skills, the ones the masters of the discipline have learned are essential, which gives you the skill that ultimately sets you free to truly explore and see what all you can do with those skills. I don't think I have ever heard of a coach who built a great team by cancelling practice drills and instead giving the players a bunch of balls and sending them out to explore what they could do with it. In fact, I would say that until you have drilled some skills into them, sending them out to explore will not only be unproductive, but risks letting them ingrain many bad habits that they will eventually need to be trained out of.

Why would we expect learning math or physics to be different than learning anything else? If you don't have the skills of a subject, you have not learned it. When I taught I would often have students come to my office after an exam or assignment and say "I thought I knew this, it all made so much sense when you explained it in class, but I just couldn't solve the problem myself". Yes, I could say the same thing about my French and other things that made perfect sense to me in class, but which I just haven't practiced enough since. This is why I introduced my own set of practice problems a few years ago (another project I've got to go finish up some time), because I know the biggest problem for most Physics students is that they have not solved enough practice problems yet for the material to be fully learned.

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