Major Error on Calculator involving Series

[Han]

I am not an employee for HP nor any company that makes calculators, nor any group associated with the college board or AP exams. I do teach mathematics, however, and encourage students to take advantage of whatever technology is available to them (whether it be a scientific or graphing calculator, or programs such as Maple, Mathematica, Matlab, etc). In fact, I love technology and enjoy finding ways of incorporating it into the classroom. When I was in high school, calculators were a novelty. Nowadays, even toddlers know how to use cell phones and tablets. I see technology as yet another means to connect with students who may not have been as interested in learning mathematics without technology. Now, with that out of the way...

Technology in the classroom, in my opinion, should always be used as a means for getting answers more quickly, as opposed to simply getting the answers. No matter how well a calculator is programmed, it will never be 100% mathematically correct. In fact, even with the most powerful hardware, calculators are limited by... well, their hardware. For example, when one graphs \( sin(ax) \) where \( a \) is a large value, the graph (on any calculator or even computer) will be wrong for sufficiently large \( a \) simply because the pixel density will eventually be insufficient if the frequency in the graph is high enough that several peaks or valleys of the sinusoidal graph occur within the width of one single pixel. And graphing is one of the most basic features of graphing calculators. I am of the opinion that this bug will never be completely eradicated since screens will always be physically limited by the number of pixels it can display.

If students are losing substantial amounts of points due to errors in their calculator, then the first issue is not the calculator. I would question whether students properly learned the mathematics needed for the problem as opposed to having simply learned how to use a calculator to get answers. There is a huge difference between learning, say, calculus and its principles and learning how to use a few commands to solve calculus problems. The former would enable students to at least have a strong suspicion that their calculator is giving them the wrong answer.

Consider the hypothetical (but very possible, as I have seen many such cases) example of a student who has poor mathematical skills that would normally prevent him from correctly computing a derivative (poor algebraic skills, or improperly applying various derivative rules). With a calculator that has a computer algebra system, he would be able to overcome that deficiency and solve optimization problems. So on the one hand, he can properly demonstrate that he is capable of solving applications that require derivatives, but on the other hand, he can only do so with a crutch. I have seen a very large number of students (over the course of many years, though) who regularly butcher a simple question of "find the derivative of the following function(s)" and yet can give me the precise sequence of steps of solving optimization problems -- including the proper definitions of terms such as "critical points" and "concavity" as well as properly explain what derivatives are and how they are computed and used, etc. It seems quite reasonable to consider such a student as having mastered optimization (provided he have a calculator as a "crutch"). And in the real world, knowing how to compute a derivative by hand would more often be atypical than not. So the deficiency does not really hinder him out in the real world. On the other hand, one could argue that the student has not mastered being able to apply appropriate derivative rules to compute a derivative. So if he cannot compute derivatives, one might question his ability to solve any problems involving derivatives.

Since I teach with technology, I find myself constantly reminding my students that technology is a man-made tool. Since man will always make mistakes, they must not rely on their tools to the point of not being able to discern whether the tools are working properly.

You may find this article an interesting read about a bug in Mathematica (which is an expensive commercial package): When a computer algebra system gives the wrong answers

Within the article is also a link to The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?

Again, I would like to emphasize that the bug is not only known, but has not been addressed fully, and exists in a commercial product that costs a lot of money -- and that real-world professionals rely on for their research.

Quote:Additionally, no where in my post have I compared competitors' products to yours. I find your response in regards to that curt, and unnecessary. Furthermore, your response didn't illicit a "Like" as it has with other members on this forum.

I respectfully disagree. The comparison actually had nothing to do with competitor's products as you suggest; Tim was talking about practices used by HP and its competitors with respect to communication with their respective consumers. I think the practice of open discussion with consumers and even heeding the advice (in the form of feature requests) of consumers shows that the HP calculator group has a sincere approach to making this calculator as great as it can be.