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Sage in Calculus Classes
by Gregory V. Bard,
Author: Sage for Undergraduates, American Mathematical Society, 2015.
Associate Professor of Mathematics
Dept. of Math., Stat., and Comp. Sci.,
University of Wisconsin---Stout,
(Wisconsin's Polytechnic University)
Should computer algebra systems (like Sage) be used at all in Calculus?
After teaching calculus for ten years, and after speaking to many university professors about what they do in the classroom, I think it is very fair to say that essentially everyone teaching calculus uses computer algebra software (CAS) at some point or another.
As evidence for this, all major calculus textbooks have sections on computer algebra systems, and often mark a selected subset of problems (e.g. 2-4 per section of each chapter) as "CAS only." These are problems that are intractable by hand. Here, intractable either means absolutely impossible or alternatively, so difficult as to be spectacularly unpleasant.
Of course, there is variation. I have seen campuses where the calculus class has a lab section, where the students each have a computer in front of their seat, and use it to solve problems that are simply intractable by hand. Even for problems that are tractable, a CAS such as Sage can be useful---for example, to help visualize a volume of revolution or a multiple integral. Let's be fully honest here: most mathematics faculty cannot draw in 3D as well as a computer can. After all, visualization is very important for students.
The other extreme also exists. There are some faculty where the CAS will be reserved for the hardest problems. It will appear occasionally, only when needed.
It should be noted that very, very few faculty will allow the use of a CAS during an examination or major test.
In summary, I cannot think of any faculty that use the CAS 100% of the time, and I cannot think of any faculty that refuse to use computer algebra at all.
Why not just use graphing calculators?
Back in their day, graphing calculators were rather popular. I used one in high school in the 1990s.
Let's say you're working with a large data set. Using a system like Sage, a professor can upload data into a project, and distribute it to all of his students with a click. With a graphing calculator, the data would have to be entered by each student, by hand.
Students can take the images and outputs of their computations in Sage and easily add them to any document for their classes, or undergraduate research papers suitable for publication. A picture on a graphing calculator is idle and immovable.
The appearance of graphs and 3D plots on a computer is vastly more realistic and comprehensible than the display of a graphing calculator.
The "online help" systems available (such as web-pages) such as Sage's wiki can be a tremendous boon to the student who is new to Sage. A graphing calculator just has a manual.
Many faculty working with Sage have made online videos, to help new students learn Sage.
If a student learns Sage, then the student learns Python "along the way." Python is an extremely popular computer programming language, used in industry. (Ranked #3 in this article published in IEEE Spectrum.)
Can university freshmen calculus students handle Sage?
This is a very legitimate question. After all, calculus is somewhat intimidating and putting a complex piece of software in front of a student can increase that level of mathematical anxiety. Happily, the answer is that students can use Sage comfortably and effectively. A huge fraction of Sage developers are university faculty (or PhD students) and they use Sage in the classroom regularly.
This is how I would see the breakdown:
In a mid-level class like differential equations or a proof-based linear algebra course, the students can handle Sage with little or no coaching.
In a class like Calculus II and Calculus III, students can use Sage with moderate coaching. Using a book such as Sage for Undergraduates, published by the American Mathematical Society in 2015, can deliver that coaching.
In a class like Calculus I, a reasonable amount of coaching is needed. This can be handled by motivated and serious faculty (or experienced graduate students).
Why use Sage and not Mathematica or Maple?
It seems to be a shame for students to go through a great deal of effort to learn a programming language, like Maple's or Mathematica's language, and then never use it again after calculus class is over.
Unlike Maple and Mathematica, which have their own unique programming languages, Sage uses Python. The computer language Python is one of the most widely used programming languages in industry. (Ranked #3 in this article published in IEEE Spectrum.)
Moreover, the cost of Mathematica or Maple is enormous in comparison to the cost of using Sage. Large parts of Sage are free, and CoCalc is extremely inexpensive.