When mathematicians draw a distinction between pure and applied mathematics, they often mention the idea of “rigor.” However, the world of calculus education is rife with inconsistency regarding what counts as “rigor.” For example (as I describe in this post), calculus classes often teach the Mean Value Theorem in a way that lacks any consistent sense of what needs to be proved and what doesn’t. And most classes give no justification at all for separating differentials to solve a differential equation, yet expect students to cite the Mean and Intermediate Value Theorems to “justify their work.” Why is justification so important in one case and not the other? And can it really be called “justification” if students are citing results they haven’t proved?
In developing Living Calculus (the blog, the book and the problems), I have tried to focus on helping young people grow into strong mathematicians. Rigorous proofs and definitions are certainly a part of this larger goal, but so are identifying patterns and using intuition to make reasonable conjectures. I want to teach good mathematical habits and to help students see the mathematical enterprise for what it is.
The following list highlights some of the other mathematical practices I want to model for students. Although these principles are my own, I owe a great intellectual debt to the works of Imre Lakatos, George Polya, and Paul Lockhart, as well as to many conversations over the years with David Bressoud. These goals shape everything from how I write problems to how I sequence topics to how I talk with students in class.
- I want students to see that deductive and inductive methods work together in developing mathematical ideas; mathematicians often make a conjecture inductively (i.e., by making observations and spotting patterns) before attempting a deductive proof. For example, I want students to explore the power rule inductively before we prove it. Even in the course of working on a proof, mathematicians often use deductive and inductive reasoning simultaneously—for example, checking parts of a proof on specific examples and non-examples. In some cases, attempting a proof may even help a mathematician see how to construct a counterexample and therefore realize that the original conjecture was flawed. This process is a stark contrast to the presentation in most books, in which deductive proofs follow almost immediately after definitions, as if these theorems and proofs came into the world fully-formed and perfect. I want students to see some of the processes that led mathematicians to develop theorems and proofs in the first place.
- I want students to see that different parts of the mathematical process may demand a different balance between formality and intuition. Mathematicians might be intrigued by a conjecture based on highly informal reasoning, and they might subsequently celebrate an elegant and detailed deductive proof of that conjecture. Along the way, they might produce a partial proof in which some parts are worked out in great detail and others are not. They might even provide informal evidence supporting one part of a conjecture, coupled with an actual proof of another part. However, no mathematician would claim to have proved a conjecture when the “proof” relies on assuming the truth of claims that have not yet been proved (but calculus classes often do this with the MVT). Thus, even as we use different levels of formality at different moments, it is important to be self-consistent in the level of formality we bring to a given problem or proof at a given point in the book.
- I want students to learn that they should trust and develop their intuition. For example, I want the fundamental theorem to feel intuitively true even before students have proved it. Proof is hugely important in validating intuitive thinking, but the development of new mathematics often begins with messy, fuzzy ideas: a mix of inductive observations, reasoning based on intuitive versions of rigorous definitions, spotting analogies between similar systems, and just having a sense of what “ought” to be true. While we want to train students to turn intuitive thinking into something more robust, we also want to train them to nourish and trust the kernel of mathematical intuition they are beginning to develop. If a student says “isn’t it obvious that a function is increasing if f’ is positive, I want to say “yes! It certainly feels like that should be true. And if it weren’t, I’d have some serious doubts about the definition of the derivative. But proving it is surprisingly hard because…”
- I want students to see that the really important thing about proof is that it provides insight into why something is true and gives a framework for understanding a theorem. The point of a proof isn’t primarily to convince us that something is true, but rather to understand the truth and its meaning. This is why mathematicians often embrace and enjoy different proofs of the same theorem: even though only one proof is needed to establish the truth of a theorem, each new proof provides a different way of thinking about why, and gives us greater insight into the meaning of the theorem. As much as possible, I want proofs (even hard ones) to have a clear idea behind them.
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