>Let me digress a little with an analogy. Last night I went to play banjo with >some friends. I"m not particularily good and always out of practice. When I >want to learn a song, I need to have an overall idea of what the song will >sound like. In addition, my left hand must know where to go on the neck to >find the notes while the right hand must know which strings to pluck. For all >of this to happen requires practice and the overall concepts. If I don't do >some practice, I will hear the song in my head, but I won't be able to play >it - I won't be able to communicate the song to other people. If I practice >exercises night and day but don't have a song in my head, no one will want to >listen to my endless rolls. > >Learning math is much the same. Students must have some concepts in their >heads and practice at the basic skills. I think time must be spent developing >basic understanding and practicing skills in an integrated manner. To teach >one without the other is to do the students a great disservice. > >As a secondary teacher, I find it quite frustrating when students believe >that calculators are essential when doing basic arithmetic. Part of learning >to understand math is the ability to manipulate the numbers in their heads. >Since I learned arithmetic before the time of the calculator, I developed an >understanding of the basic algebraic properties before I was told their names >because they made it easier to do the arithmetic in my head > >marge >
I enjoyed your analogy (and now, can't get "Dueling Banjos" out of my head...). Indeed, understanding the underlying concepts and being proficient at the "basic skills" are important parts of mathematics. The concepts and the skills are, to a great extent, complementary -- the understanding of one enhances the understanding of the other.
It is unfortunate that so many students feel the need to turn to calculators for so many basic computations. Much talk has been given to decreasing the attention we give to paper/pencil algorithms. I personally think we should spend about 1/4 the time on paper/pencil algorithms than we currently do, and use the freed up time for:
Exploring the operations (models, properties, etc.) Learning to compute mentally Learning to use the calculator
Considering the three basic ways of computing -- mental, paper/pencil, and calculator -- it seems that the one to which we devote the most attention in the classroom is the same one we use least outside the classroom.
By the way, on a somewhat related note: an article appeared in JRME a few years ago that studied the habits of good mental calculators. From what I recall, good mental calculators did not necessarily have better memories or know their basic facts better than poor mental calculators. The big difference between the two groups was that good mental calculators understood the *properties* of the operations and the numbers. They may not specifically have known the names of the properties, but they knew how to use those properties to make the computation easier.