Technology Changes Curriculum - Or does it?

Technology changes things. That's sort of an understatement.

The first big curriculum changes resulting from technology occurred some 40 years ago. Four function calculators became available. I remember my first one cost $70. At that time something between 2 weeks and a month was spent in Algebra 2 classes teaching computing with logarithms.

For you young folk: back in "the old days" calculations involving multiplying, dividing, power, and roots we routinely done using logarithms. The logarithms were listed in tables. A  logarithm table for 3-digit numbers is two pages long, 4-digit numbers 10 pages and 5-digit numbers 100 pages. While in college I worked summers for a civil engineer; he had an even larger table that took up an entire oversize book -- and he used it. There were also tables of the logarithms of the trigonometric functions.

The were no calculators. We had in the office a machine to do multiplication and division. Huge thing; rows and rows of buttons. You entered one number and then to multiply you turned a crank for the other number one digit at a time. For the next digit you turned another crank which move the whole carrage over one place so you could crank in the next number. To divide you turned the crank backwards. Hence the term "crank out the answer." Someone good with logarithms could do the job just about as fast.

The other thing close to a calculator was the slide rule. But the slide rule is just an analog representation of logarithms. 

Students in Algebra 2 were taught all the tricks of calculating with logarithms: how to deal with the decimal point, how to interpolate between the values in the table to get more accurate answers, how to use co-logs to avoid subtracting and of course how to use the table in the other direction to change  the final answer back to a regular number (called finding the antilog).

Multiplication and division were done by adding and subtracting the logarithms; powers and roots by multiplying or dividing by a  single-digit number. I remember after teaching how to set up the problem and find the logarithms that one year the students did all the set up work, looked up the logarithms and then got out their calculator to do the adding and subtracting! That's when I know logarithms were on their way out.

There are lots of topics in mathematics that were invented to make calculations with numbers easier and more efficient. Logarithms was certainly one of them, but there are others. The advantages of four-function calculators and soon thereafter scientific calculators were obvious.

But, here's the thing, computer algebra systems, CAS, do not seem to have had the same effect on the curriculum. Why? There are many basic computations that we still require students, not just to know, but to be proficient at doing by hand. I'll list a few below, but so that I don't sound too far out, let me first say this:

Just as logarithms are and always will be important and students need to understand them, so too the things in the list are important and students need to understand them. But they don't have to be good at doing them! Just as very, very people who graduated high school in the last 30 odd years could come close to doing a computation using logarithms, students in the future don't need a by-hand proficiency with these thing. They need to understand what's going on; they do not need to be good at doing it.

Okay, here's my short list:

• Simplify radicals
  
• Rationalizing denominators (or numerators for that matter)
  
• Factorig quadratic polynomials (those few that actually factor)
• The Quadratic Formula and the "discriminant" (although completing the square is a keeper, but not for solving equations)
• Cotangent, secant and cosecant (these have not been taught in Europe for decades, if ever. Just noticed that the spell check on this Blog doesn't recognize "cosecant" as a word; maybe there is hope.)
  
• Newton's Method for approximating roots of any equation
• Antidifferentiating (there are tables of over 1,000 antiderivatives in print, these too can go)
• Synthetic division
• Descartes Rule of Signs
• The Law of Sines and the Law of Cosines
• Cramer's rule and the discriminant
  
• Matrix arithmetic and find the inverse of a matrix

 So that's my short list. Anyone care to add anything?

They were invented for a good reason; but that reason no longer applies.

Why hasn't technology changed these things? I think the major reason is that they appear on the state graduation exams -- now there's a good reason to teach them!