You love math and want to learn more. But you’re in ninth grade and you’ve already taken nearly all the math classes your school offers. They were all pretty easy for you and you’re ready for a greater challenge. What now? You’ll probably go to the local community college or university and take the next class in the core college curriculum. Chances are, you’ve just stepped in the calculus trap.
For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too. This is not to say that high school students should not learn calculus – they should. But more importantly, the gifted, interested student should be exposed to mathematics outside the core curriculum, because the standard curriculum is not designed for the top students. This is even, if not especially, true for the core calculus curriculum found at most high schools, community colleges, and universities.
Developing a broader understanding of mathematics and problem solving forms a foundation upon which knowledge of advanced mathematical and scientific concepts can be built. Curricular classes do not prepare students for the leap from the usual – one step and done – problems to multi-step, multi-discipline problems they will face later on. That transition is smoothed by exposing students to complex problems in simpler areas of study, such as basic number theory or geometry, rather than giving them their first taste of complicated arguments when they’re learning a more advanced subject like group theory or the calculus of complex variables. The primary difference is that the curricular education is designed to give students many tools to apply to straightforward specific problems. Rather than learning more and more tools, avid students are better off learning how to take tools they have and apply them to complex problems. Then later, when they learn the more advanced tools of curricular education, applying them to even more complicated problems will come more easily.