Pre-calculus is often presented as a slightly more advanced version of a good algebra II course, rather than a course which actually prepares students for and motivates the study of calculus. Topics such as trigonometry, rational functions and logarithms are revisited but not expanded upon in a way which is inspiring to strong math students ready for new ideas. Additional topics such as sequences and matrices are introduced, but no attempt is made to connect them to the rest of the course content. In the same vein, early parts of calculus such as limits are tossed in at the end. As a result, the course tends to have a scattered and disconnected feel, and the students have little sense of why calculus (and higher mathematics in general) may be relevant for careers they are interested in.

As a teacher at the Masters School in Dobbs Ferry, New York, I chose a new approach to teaching the honors-level pre-calculus course. I tied the content together with central themes of calculus – limits, rates of change and accumulation. I gave special emphasis to concepts such as increments, difference equations and sequences of partial sums, as they are the discrete analogues of derivatives, differential equations and integrals. Having the opportunity to explore the relationships between change and accumulation in the discrete setting helped students understand the critical role these ideas play in mathematical modeling and provided motivation for the study of calculus. Because of the discrete nature of computer data, understanding the interconnections between discrete and continuous models is more essential now than ever before. Lastly, I dedicated a significant portion of the course to applications in probability theory, a fundamental ingredient of virtually all applied mathematics (and a rich source of practical problems which require calculus to solve). Students who completed the course had a stronger sense of what calculus is all about than with previous approaches, and they were better- prepared to grasp the subtleties that AP Calculus students tend to struggle with most.

• Written by a passionate math educator and Ph.D. student at Northeastern University.
• Down-to-Earth exposition, presented as it would be spoken in class
• Completely rigorous definitions, statements of theorems, and proofs
• Margin side-remarks and historical references
• Hyperlinked table of contents, index, and cross-references
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