I Don't Know…Yet

Mr. McArthur (ZMC) “The limit does not exist. The limit does not exist!” – Cady Heron, Mean Girls. I taught calculus for the first time last spring to a class of twelve eager, mathematically-bent sophomores and juniors.  Less than two class periods into the subject, my understanding of calculus came into fundamental disagreement with Section 1.1 of our book. “The limit does not exist,” declared Anton, Bivens, and Davis via the bright yellow, 8-lb behemoth named Calculus, 10th Edition. “The limit is clearly infinity!” my brain confidently whispered, as I visualized the function accelerating northward on both sides of the vertical asymptote. I waited a moment, and then uttered three words that 1871 CEO Howard Tullman has no tolerance for. “I don’t know,” I say, three powerful words I use dozens of time each week in the math classroom. It’s a phrase that I think good teachers embrace. “I don’t know” doesn’t necessarily mean that I haven’t thought about a concept; I might know enough about that topic that two or more ideas feel in conflict. Given a question from a student that either has the answer “6.3” or “I don’t know,” I’d pick the latter every time. “I don’t know” is the beginning of a conversation, returning the pursuit of learning to the entire class. “Why don’t we all look into it tonight and compare ideas tomorrow?” Asking great questions in math is as much a skill to develop as methods to find solutions. Questions that don’t have clear answers, questions that make you work for their answers, questions that lead you to learn something new as you explore their answers, those are the questions I want students to ask. “I don’t know” doesn’t mean I won’t know. We’ve learned much about growth mindset—the fact that we can develop our abilities through care and hard work—at Latin over the last couple of years. Mr. Tullman’s banning and belittling of “I don’t know” may be his blunt way of communicating to employees to do the grunt work of seeking out answers, but it leaves a bitter taste in my mouth as an educator.  We teachers try to model lifelong learning, and saying “I don’t know” is a simple way to present that mindset. It’s important to follow up those words with actions that demonstrate ways to find difficult solutions: talking with colleagues, watching YouTube videos, reading books. To me, saying “I don’t know” most often means “I don’t know, but I will do my damned best to figure it out.” Though I respect the culture of grit and perseverance Tullman is trying to create at 1871, outlawing “I don’t know” and calling its users “lazy” feels too broad a stroke and symbolic of an aggressive, know-it-all culture. The fact that I don’t know something makes me human, and even though I love and am relatively good at mathematics, there is always more to learn. “I don’t know” is also a completely valid answer to many questions, even in mathematics, a subject in which many think there is always a correct answer.  As you reach higher levels of math, you focus more and more on investigating conflict and nuance. In my Algebra 1 class this fall, we had an animated discussion about the value of 0 divided by 0.  Does it follow the pattern of x/x = 1?  Some students focused on the numerator: “0 divided by anything is 0”, they confidently proclaimed. Or does the 0 in the denominator mean the expression is undefined? The students in the class that could hold these three individually persuasive yet incompatible ideas in their head at once came to what I believe is the best understanding of the value of 0/0: “I don’t know.” This conclusion is far from lazy, Mr. Tullman. It’s a product of number sense and years of math education. On a pre-calculus test last week, I asked students to “find the x-value which outputs the largest y-value.” The answer I was looking for was “I don’t know.” There wasn’t enough information to come to a conclusion. But because I phrased the question so definitively, some students spent half the time on their test searching for that x-value that was impossible to hone in on.  Though I mildly regret the wording of the question, it led not only to a mathematical discussion about symmetry in polynomials, but also a more significant conversation about knowing when you don’t know something. For many, including some of my Algebra 1 and Pre-Calculus students, it can be hard to write or say “I don’t know.” As Tim Krieder of the New York Times puts it, “To admit to ignorance, uncertainty, or ambivalence is to cede your place on the masthead, your slot on the program, and allow all the coveted eyeballs to turn instead to the next hack who’s more than happy to sell them all the answers.” This quote reminded me of watching ESPN’s Around the Horn sports talk show, a miserable half-hour of television replete with middle-aged men yelling their opinions on questions that don’t have answers. “Would Matt Ryan be more successful than Tom Brady if he were in the Patriots system the last ten years?” Woody Paige scored 7 points with his confident response, J.A. Adande 9 for his louder, surer, yet equally substance-less thoughts. How do you two have any idea? I don’t see admitting uncertainty as a weakness. In the context of my profession, saying “I’m not sure, and here is why” feels far better than brushing a student off with a hastily thought-out answer, an answer that might not only leave them with misunderstanding, but also with less drive to search for the answer themselves. If I can model curiosity and specific strategies on what to do when answers don’t come easily, then I think my students will be better prepared for the complex world beyond Latin’s walls. As for my fundamental disagreement with Anton’s Calculus book? “My very short answer is that both you and your book are correct,” wrote my college friend Matt, now a math professor at Holy Cross, whom I reached out to after class. A weekend-long email exchange later, my understanding of limits involving infinity had strengthened, and as a little lagniappe, I had scored an invitation to “stop by if you’re ever passing through Northeast CT.” The next Monday’s calculus class ranks as one of my favorite class periods in my ten-plus years of teaching. I printed out the email exchange and read it with my class. We debated if John Green abused the word infinity and ideas of cardinality in The Fault in Our Stars. We began exploring the formal delta-epsilon definition of a limit, and whether there are mathematical situations that warrant going rogue and ignoring it. Energy, mostly in the form of questions, filled room 419. “When are two quantities equal?” “What is a number, anyway?”. You can bet what phrase was said a lot that memorable morning.   And Mr. Tullman, the thirteen of us in room 419 are far from lazy people, and we all care deeply about the mysteries of infinity and the beauty that is mathematics.  ]]>