In his paper on Linearity in Calculus, Dan Teague wrestles with the question of knowing one's audience when teaching calculus. In particular, there is a growing dichotomy between those students who have a strong calculator-based precalculus experience and those who do not.

By giving students a visual view of a function, graphing calculators have made it relatively easy to determine whether or not a function is linear. The question with which Teague wrestles is this: How can such an approach strike the proper balance between providing enough rigor for practicing mathematicians and enough information for those students studying calculus as part of their general education?

The approach developed for use at the North Carolina School of Science and Mathematics takes advantage of the use of graphing tools in preparatory calculus courses. Students can use the graphing calculator results as launching point into differential calculus. Thus, this approach lays a good foundation for those students who will take more rigorous mathematical classes.

Should calculators in general, and graphing calculators in particular, be used in such classes? (Interestingly, this is one question I was asked during my job interview for my current teaching position.)

On the con side, if students are using calculators in class, are they truly demonstrating their knowledge of the mathematics being taught? For example, if I am teaching a section on simplifying radical expressions, when I ask students to simplify the square root of 48, I often will get 6.928 as an answer. The correct answer, based on the classroom instruction, is 4√3. The first answer is "correct" according to the calculator, but it does not demonstrate the mathematical knowledge which is being tested. In addition, a calculator will rarely return the negative root solutions to such a question. What is the square root of 49? Many students will say "7", but "-7" is also a valid solution and a calculator will not provide this result. As an instructor, I must become very specific in communicating my instructions and expectations when testing for such knowledge.

On the pro side of using calculators, almost every electronic device which one owns these days has a built in calculator. Cell phone, watches, computers, IPOD's ... calculators are everywhere! So, why not take advantage of them? After all, isn't 6.928 close enough to the square root of 48? 6.928 squared results in 47.997184 ... isn't that close enough to 48? For most of our students and for most of society, the answer is "Yes, this is close enough". And for most practical causes, it truly is close enough.

As a mathematics teacher, I have had to find the balance between acknowledging the technology is readily available and accessible while insisting that my students demonstrate math knowledge. In our state standard exams, students are not allowed to use calculators for the first few math section questions. Once they have completed that section, calculators are allowed to be used. I take a similar approach in my class by allowing calculator use in cases where it does not undermine the fundamental knowledge being tested.

Doug, why do students need 4√3 as opposed to 6.928? In other words, do they appreciate mathematical situations where that form is needed, and have assignments where that form (and not other forms) are necessary for further steps?

ReplyDeleteFor example, they may need to work with vectors with √3 unit lengths, or prove that the answer is irrational (which the calculator can't do).