Kerry> I'm a bit of two minds about drilling. I get how over-drilling can kill any creative spark a student may have for the beauty of math. Having taught it at the undergraduate level (which too often means, "at the middle-school/high-school level"), I've had to endure the scowls and sneers of those who didn't care a lick about math but had to do their calculations anyway, because that was part of the syllabus. However, I can also see how it certain facility with numbers and arithmetic operations is necessary to even begin to deal with math at anything more than a pure computational level. I had a buddy who also taught and was quite proficient in martial arts. In his class, he approached arithmetic drills as he would repetitive practice/warm-up in the martial arts context--do something relatively mundane so often that it becomes automatic, then move on to more sophisticated stuff. If there is an answer, it's probably the same one that it's always been--adjust the material with the level and goals of the student in mind. Kerry -- lkmitch@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=lkmitch%40gmail.com>www.kerrymitchellart.com --------------- Indeed, everybody's different. I think I benefited from drilling. Why wasn't I bored? Maybe I could fell my skills improving. Maybe I sensed a challenge in successive examples becoming trickier or otherwise harder. Maybe there was a pattern to the answers suggesting a little theorem/general formula. The latter seems like a cool way to motivate lots of concealed drilling: Give sequences of special cases of little identities whose pattern the student should eventually guess. --rwg And where there is no pattern, make the last few examples extra credit, with steeply rising difficulty. And yes, Richard, may I have a copy of your triangles paper, too?