In the past months, ChatGPT gathered headlines for its ability to, if not write the truth, present its version of it to the point that NYC schools banned it. We have little precedent for this, but we have seen how mathematics adapts to technology like calculators and computer algebra systems (CAS). While many argue that CAS, like ChatGPT, represents a classroom threat and a potential weapon of mass cheating, this ignores that exams already can overcome this along with these technologies’ positive opportunities.
CAS makes short work of a typical algebra or calculus exam: with a few keystrokes, you can solve an equation, take its derivative, integrate it, or both, giving a result in seconds.. However, for calculus at least, the solution is simple and is on many exams: problems such as “f’gx*gx= ?(x=5),” with a chart of relevant values off to the side. These problems focus on theory and actual knowledge of the problem.
While this format is unusual, it can appear on homework to get students prepared for exams, and accurately tests if a student knows how to apply calculus concepts. Problems like these, along with ones that require a student to think through a situation to set up an equation, both remove the main advantage of CAS: solving simple problems. While this approach requires more thinking from teachers, it forces them to design exams that test not just if students solve integrals, but also if they understand and apply the theory.
Another complaint is an expensive CAS’s advantage over a cheap scientific calculator, placing students who cannot afford this at a disadvantage. This is resolvable by changing how exams are written, but this is also easier to solve by having students plug in numerical values. CAS’ cost soars into the hundreds, but more basic features are common on graphing calculators and are also found on $20-$30 calculators, along with algebraic solvers. While numerical differentiation, integration and solutions only work with problems without variables and give unclear answers, these limitations are easy to check or, if important to a type of problem, taught to a student.
Even if CAS is never used in an exam, the teaching tool can help students work through examples or gain an understanding of a problem by immediately plotting the differential of a function or quickly experimenting with how a concept like the chain rule behaves. Seeing how the chain rule would propagate over an increasingly recursive function would take too long and would be error-prone, while simply asking a CAS to do this would take less time than it would take to understand the results.
Desmos is a web graphing calculator and learning tool that has CAS, as teachers can use it to transform classrooms. However, Desmos is also just an advanced graphing calculator: it can show you the graph of a very complicated function without solving it. CAS can do this as easily for the user as it can calculate the cosine of a number.
We stand on a new frontier with tools like ChatGPT and already see its effects. In the future, this technology will propagate, leading to possibly better language models and forcing teachers to change how they teach and design assignments. While we have yet to see how language models will improve and regress, we already have a system that allows a removal of the user’s work to the point that assignments designed to be difficult are very easy. We have the opportunity now to see how students using assistance will change teaching, just like we did decades earlier with calculators and will in the next decade with ChatGPT-like programs. The question is if we will decide to simply build walls blocking out change, or if we will decide to redesign paths and add guardrails to accommodate it.