Dr. Darken is in the unique position of working in a field that has 
already battled over the "technology requirement" issue for decades.

A core issue in mathematics education is "number sense". Just as Dr. 
Darken mentions, students should just "know" that 1/8 is less than 1/3 
coming into the university. By the time they leave, I hope they also 
understand that x^2 is more than 2*x (that is, the rate of change for a 
power is greater than a fixed factor). None of this really comes from 
using a calculator or a laptop.

I dropped out of UTC in 1990. When I returned to UTC in 2002 as a math 
major, I decided to retake Calc 151/152 to get my mind back into 
mathematics. It was interesting to see how the use of technology had 
changed. In 1990, Dr. Kuhn was exploring the use of the HP-28S in 
calculus classes. It did graphing, symbolic algebra and definite 
integrals. I happened to have an HP-28C because I found a display model 
on clearance for $180! The idea of using calculators for calculus was 
pretty high tech. Now, it's standard - as is the use of Maple. But the 
Math department has done an excellent job of creating labs for calculus 
that both require the use of Maple and require that you understand what 
the computer is doing and why it's necessary. Additionally, you have to 
be able to do similar work without the aid of the computer.

BTW, after Calc 152, the only other time I needed a calculator was for 
Numerical Methods I - where again, the issues of mathematical problems 
that can only be solved with calculators and computers were explored. 
Ironically, in Numerical Methods II, we were dealing with more abstract 
problems that couldn't be done with a calculator and generally required 
custom programming (like how to manipulate matrices with millions of 
columns and rows).

As you can see, I have always been a technophobe. I used to love my 
HP-28C, especially since it was so advanced that most teachers didn't 
understand what it could do! In my office, I currently have seven 
computers. But I am also opposed to using technology inappropriately. 
Technology can be an excellent learning tool. Technology can be an even 
better research tool. But students, especially at the undergraduate 
level, need an understanding the fundamentals behind what the technology 
is doing.

As far as the issues of technology: students should be able to debate 
the benefits of technology. They should be able to decide when 
technology is good and when it's bad. As the generations come through 
the university that have grown up with ubiquitous technology, the 
greatest thing we can teach is how to best exist as human beings in a 
world of rampant technology.

-Eric

Betsy Darken wrote:
> Technology is a double-edged sword.  On one hand (to mix metaphors), we have
> a responsibility to determine what our majors and graduates should know
> about technology so that they will be prepared for their careers,
> intelligent citizenship, etc.  On the other hand, we have to consider what
> we think students should be able to do without technology (if anything).  
> 
> I believe that there are times when the use of technology is essential and
> times when it should be forbidden.   This is a big issue in mathematics.  I
> am involved in a research study of the use of technology in calculus
> courses.  One institution that had permitted its students to use TI-92's
> (symbolic calculators) on all of their tests decided, based on the results
> of the study, to develop two-part tests, one with calculator and one w/o.
> They came to the conclusion that there are some basics that students should
> know without having to look them up.  This is a mathematics education issue
> from K through college.  Should students know that 8 times 6 is 48,  know
> how to add decimals without a calculator, tell at a glance that 1/8 is less
> than 1/6, know the percent form of 1/3, etc.?  Many of our students do not
> know these things and are clearly overly dependent on calculators. I cringe
> when I watch students pick up their calculators to compute 84 - 40, 10*100,
> etc.  This is not just an issue for the Math Dept but also for the entire
> faculty.
> 
> Dr. Betsy Darken
> Professor of Mathematics
> University of Tennessee at Chattanooga