I have found after 37 years that rote knowledge of the multiplication tables through the 16s, the first 25 perfect squares, and the first 10 cubes makes much of the so called contrived hand calculations given in textbooks mental arithmetic especially once a student passes into grades above the 6th grade. Also, this level of knowledge makes the so called drudgery of long division not so difficult. The algorithms that the weather person have valid mathematical reasons for working, but those textbooks do not teach them. Nor is there significant emphasis in the teaching of the basic arithmetic properties of commutativity, associativity, and distributivity. Properties needed in Algebra and Geometry, this lack of mathematical understanding seems to be the real problem.
As a student, I was not exposed to the "New Math" of the mid-60's and by the time I started teaching, it was no longer an emphasis in schools. I did study it in pedagogy classes in graduate school as I recall there were two different trends used in the 60s. The methods presented in the video is just another set of methods that have come from the NCTM Standards efforts to dumb down the need of rote learning in the elementary schools. Calculators are a tool to be used once basic skills are conquered. I tell my students that calculators are for word problems; problems that are contrived for practice of arithmetic skills need to be practiced without a calculator. I would not assign columnar addition of 4-digit numbers of length greater than 3 without a calculator, multiplication problems with products greater than 1000, and division with dividends greater that 625. Word problems that are not contrived to be arithmetic practice and other problems, I have no problem with a student using a calculator. Some word problems are contrived to teach how to setup an arithmetic problem to be solved; here is where a calculator should be used.
The biggest problem that I have is the students that think that every problem requires a calculator to solve it, but the use of a calculator is a detriment to solving the problem. Other problems where the calculator is needed to solve, the student wants to use the calculator to early in the process and miss the problem completely. There is a skill need to decide when to use technology. This is one reason that the AP exam and other College Board exams have a non-calculator sections.
Jon Stark wrote: > > There could be immense value in dropping emphasis on boring > repetitious hand calculations, and it might well be true that too much > valuable class time is wasted on such things. The question unanswered > in the snippets we see is whether what replaces that drudgery really > contributes to mathematical depth of understanding and/or power to be > used in application. If students come better prepared to think > logically, to do reality checks on their work, to see new approaches > to problem solving, to understand the assumptions behind their work, > to find interest and utility in mathematics, to show creativity and/or > courage and/or stamina and/or curiosity, then it could be a good > trade-off. I haven't yet seen the benefit side in the students who > come my way, but the loss of computational skill is apparent, so this > juror is still deliberating. >