What Is Inquiry-Based Learning?
The term “inquiry-based learning,” or IBL, is rapidly approaching buzzword status in the novelty-crazed math-reform community. But it’s frankly not that novel. It’s just that folks are finally catching on. IBL is a style of education that emphasizes mathematical inquiry above all else. Instead of being presented with a theorem, students in an IBL classroom are presented with its raw materials — axioms, leading questions, and whiteboard markers. The teacher acts as a facilitator in this setting, guiding students but not pushing them. Students will form conjectures and learn to prove them just as mathematicians do.
A certain kind of collaborative IBL has been practiced at the Hampshire College Summer Studies in Mathematics for over forty years, where students work together in a non-competitive environment to build an entire mathematical culture from scratch in six weeks. In my three summers staffing HCSSiM, I have seen high school students prove the Law of Quadratic Reciprocity, Lagrange’s Theorem, the rank-null theorem, Chebyshev’s Inequality, and countless other lemmas, theorems, assertions, conjectures, and propositions — smiling all the way.
How I Teach Math
I try to cultivate an atmosphere of openness and creativity in educational settings. Through wackadoodle metaphors, absurd examples, and an occasional pinch of Socratic pedantry, I work to dismantle the unseemly wall of self-seriousness that keeps so many students out of the mathematical community. My methods are not silly for silly’s sake, though. By removing mathematics from the realm of the plausible, I remind myself and my students that mathematics is an abstract construction. In learning what their teachers call “mathematics,” students are actually developing problem-solving skills far more general than are applicable to any one worksheet. The sooner a student realizes this, the better she will succeed. Formulas, symbols and long Latinate words can all be confusing and intimidating. But peeling away the veil of mathematical language is liberating: behind it, there is no anxiety, no worry — just all the beautiful things you have in your head already.
Mathematics, as it is practiced by professionals, is a collaborative endeavor. The “lone-genius” image shows up in popular culture as the archetype of the mathematician, but it’s hardly accurate. More commonly, mathematicians work together in teams and research groups to solve problems and improve each other’s work. Even the few lone-genius types that really exist rely on the wider mathematical community to verify their findings, and to put them into context. The significance of all mathematical work comes from this process of collaboration.
It is vital to a student’s education — in my (strongly-held) opinion — that he or she understand the importance of collaboration early on, and what better way than to experience it first-hand? Probably the most obvious difference between a high-school algebra class taking a quiz and the same number of professional mathematicians solving a problem is the silence in the former. The opportunity to communicate and collaborate is both a problem-solving tack and an educational opportunity, and when we as teachers suppress it we suppress, for example:
– Student A filling in gaps in Student B’s knowledge, and ten minutes later vice-versa
– Student C presenting a brilliant proof to the class on the board
– Student D presenting a counterexample invalidating C’s proof entirely
– … Except for a special case that Student A proposes, based on her earlier discussion with B
– Student M suggesting, “We could express that more clearly if we…”
– Student Y coming to the board and presenting an alternative proof that X, Y, and Z devised in a corner of the classroom while A through W were arguing about the first one.
… et cetera.
I am at my pedagogical best when working with groups of students. I am experienced in leading and redirecting classroom discussion, and also stepping back when it’s time for the students to take charge. Much of my role as math teacher is as moderator, trying to preserve the diversity of the discourse so that one point of view or solution strategy does not dominate. Even on those occasions when I’m standing in front of a room and lecturing, I am in constant dialogue with myself and my students so as not to convey the sense that there is One Right Answer or One Way to Do Math. As students mature mathematically, they come to appreciate the mental flexibility they learned this way. Everyone who’s ever done math occasionally comes upon a roadblock, and so everyone who does math should be prepared with strategies for looking at their problem in a different way.
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