Introductory proof writing is much like English as a Second Language. Developing written mathematical skills involves wrestling with issues of vocabulary, word choice, grammar, word order, punctuation, native expressions and (after enough experience) eventuates in fluency in the foreign language of mathematics. This analogy can provide a natural context for the unavoidable frustrations a student experiences in learning any (and therefore this) language, in addition to opening a window onto new possible pedagogical tools for teaching mathematics as a foreign language.
The basic building blocks for any language are vocabulary words. Every mathematics course comes equipped with a new set of vocabulary to learn—limit, continuous function, derivative, integral, sequence, series—all of which are meticulously defined to obviate even the tiniest shred of ambiguity. Many of these vocabulary words come with their own notation, adding a second layer of complexity to the requisite wrote memorization of vocabulary. Even regular English phrases frequently get symbolically compressed: “there exists” becomes 
The grammar of mathematics is deeply intricate in order to accomplish the goal of absolute precision, which requires the speaker to be infuriatingly meticulous. For example, if you change the word order in the phrase “
Mathematics as ESL is initially a strange analogy to make explicit: the student is sitting in your US classroom because he or she has already mastered the English language! (This can make learning the language of mathematics all the more frustrating relative to its other foreign language counterparts.) But mathematics requires a wider vocabulary, uses only a small fraction of familiar English words to glue things together, and insists on an entirely new, much less flexible grammatical structure.
Perhaps the experience of learning how to write proofs could be made less psychologically traumatic for some students if we make explicit the expectation that we are about to begin a foreign language. Saying on the first day “You’re not in the Kansas of Calculus anymore!” remains merely a philosophical utterance that is forgotten until the quagmire of frustration has swallowed the student whole. Rather, while the student is concurrently enrolled in French 201, we might draw upon the familiar imagery of recent success in a beginner’s level language.
Over the long arc of the semester, a realistic expectation is that a student would be able to write the mathematical equivalent of the standard paragraph on “What I did last summer”—which I recall as the first exercise in my Spanish II course. (Over time, we progress beyond issues of basic grammar and vocabulary to consider style and voice. And even then, there are moments where we find ourselves saying, “Yes, it’s grammatically correct, but a native speaker would never say it that way.”) Even this modest goal will require a semester rife with the standard sound practices of introductory language study, and here are some expectations that I make explicit to students in my Linear Algebra and Discrete Math classes:
- Learn actively. Many mathematics courses are taught in lecture format, but you can’t learn the language merely by passively taking notes and reading the text. Having someone explain the grammar to you and trying to formulate either written or spoken sentences yourself are two totally different exercises; the second of which is exponentially more difficult than the first! Be bold enough to feel natural and comfortable stumbling through attempting to write and speak yourself.
- Success requires regular practice speaking and writing. Mastery requires experience, plain and simple. At the beginning, a significant number of those hours should occur in the presence of the “native speaker” mathematician so that gentle, corrective guidance can be offered.
- Don’t let yourself suffer from writer’s block! If you have a writing response due in Italian tomorrow morning, you just write something, anything. Brainstorm, make a “to do” list, write out the definitions of any new vocabulary; just write anything to get yourself started. For example, have you ever written a paper and only as you were finishing that last paragraph realized what your thesis statement was? Often the practice of writing itself generates ideas.
- Expect to make mistakes—tons! It’s possible that this may be the first mathematics course in which a paper comes back to you bleeding red with a score of 60%. But this is what you’d expect during your first few months of Chinese, and that is the correct comparison to make.
And framing the psychology of the course is merely the beginning of what I believe to be a powerful analogy. If we as faculty reshape our own thinking about teaching introductory proof writing courses as teaching English as a Second Language, there is a well-developed garden of pedagogical fruit grown by our foreign language instructor colleagues, and the hybrids we collaboratively develop might be really juicy!