Successful Teaching Strategies
Strategy: Open-ended Presentation Option for Projects
Oftentimes when I want to encourage students to think deeply and/or take creative risks when confronting a philosophical aspect of something we’re studying, or build their own bridges between prior knowledge and current challenges, I construct an activity in which I try to keep the analytical and mathematical requirements fixed, but give the students free reign over both the content and mode of presentation for the task. The example student work I’ve provided in the session is from a project inspired by a passage from David Berlinski’s book A Tour of the Calculus. In his text, Berlinski refers to an “archipelago of functions,” implying that each function is its own island, complete with appropriately behaved flora and fauna. After discussing the nature of functions using this and other passages from A Tour of the Calculus, I challenged my students to develop their own metaphorical schema in which “each function is its own ________.”
In both years of assigning the Berlinski Project, I’ve been profoundly surprised and impressed with what my students have come up with - in regards to both the schemas themselves as well as the way in which they were brought to life. Each function has been its own dance move, bite of a cupcake, hairstyle, type of fast food, article of clothing, family member, traffic sign, style of music, and animal in the rain forest. I have graded projects that were turned in as interpretive dances, fashion shows, Power Point presentations, creatively altered snack foods, mix tapes, children’s books, and endlessly ingenious poster after poster after poster.
Artifacts
In the first year of assigning the Berlinski Project, I happened to have the school’s digital camera in my room for the presentations, and plenty of students volunteering to take pictures. Here are some of the results:
However, in the second year of assigning the Berlinski Project, I received a presentation that impressed me so much that I could do little else but award the group a test grade of 150%. Their creativity and professionalism raised the bar on what I could imagine the capabilities of motivated youth to be. Here it is, an homage to the often overlooked art of marching around a field with a tuba:
Connection to a Philosophy of Education
On my part, the most successful aspect of the Berlinski Project - at least in regard to a philosophy of education - would be the dedication to an element of choice in the project’s construction; on my student’s part, the most successful moments in their projects are those moments where I’m caught off-guard - moments where a student or group of students have suddenly made all of the pieces fit together in a delightfully unsuspected way (i.e. “of course each function is its own bit in cupcake - this is just a chalkboard away from conic sections”). In my Statement of Philosophy page, I make note that nothing can be learned without choice, and I have spent many stressful nights over the past two years trying to balance a commitment to student choice with the responsibility to move curricula forward. It is in these open-ended projects that I have found the most fruitful balance; I had to trust that students would fit their unexpected and spontaneous talents, if only I could make a reasonable skeleton. That being said, the most satisfying moment of looking at these projects is realizing exactly how many connections I could not have come up with on my own, and that it is never a bad idea to provide well-constructed windows to let fresh air into the classroom. As a teacher, it is one thing to be proud to have trained students very well, and it is another thing to be proud to have done a good job putting them in front of blank canvases.
Strategy: Interdisciplinary Connections
As a student myself, I always enjoyed finding my way to the center of the Venn Diagram of all things. Nothing exists in a vacuum, especially academic pursuits, and no avenue of intellectual discourse has reached its end, despite what some mid-90s Ph.D. dissertation would like to tell you. In this regard, many of my students often fuss at me because they have no idea what class I am trying to teach, integrating mathematics with the sciences is one thing, but (at least to them) trying to use Alice’s Adventures in Wonderland to teach word problems, Langston Hughes’s “Theme for English B” to teach linear systems, and Plato’s Allegory of the Cave to teach limits is another thing. Though integrating these sorts of connections into the classroom is definitely a characteristic teaching style, I am still at a point where its successfulness is far from perfect: Students whom I taught last year for Algebra II still poke fun at the idea that there is anything mathematical about Lewis Carroll, and my last attempt to integrate Berlinski’s writings into my Pre-Calculus class (again, to discuss functions) met with many disenchanted and reluctant stares.
Despite all this, I am a firm believer in the idea that anything can be taught using anything else, and it is often the case that you need to make as unexpected a connection as possible in order to allow the intellectual validity of whatever it was you were supposed to teach that day seem more believable. One of my signature “Mr. Molina done lost his mind” lessons is the one in which I use Plato’s Allegory of the Cave to try and show how the concept of a mathematical limit is embedded within the way we construct and/or interpret reality. It’s gone like this:
~After a short and often frustrating introduction to the idea of taking a function and looking at an impossible domain element by means of possible domain elements creeping closer and closer and closer to the boundary, my class and I read the Allegory of the Cave out loud together.
~We spend time drawing pictures of the cave, making sense of the wall and the fire and the chains and the shadows and the sun and all that other stuff.
~I explain to them that I am going to use the Allegory of the Cave in our discussion to prove that straight lines don’t exist. (If these students were taking me in my second year at Jim Hill, I would have secured by this time a microscope that can project its magnified image onto the wall of the classroom.)
~Without fail, some presumptuous young man or woman claims to - of course - be able to show me a straight line, at which point I find a flaw in its straightness and rigidity and ask the whole class to draw me the straightest line they can find. (If it’s my second year at Jim Hill, they do so on a strip of transparency.)
~After many claims and counterclaims, most of us end up convinced that straight lines only exist as limits of our experience of “lineness.” The same fate is accepted for circles, rectangles, inches, and if we’re lucky, Truth itself.
~In further discussions and applications of limits, we have a rich foundation of anecdote, imagination, and myth-making to return to in those moments where the computational process of limits (especially in the case of derivatives) seems a bit detached.
Download: plato's cave - text and questions
Artifacts: Excerpts from a Calculus Test
What follows is a list of excerpts from my 05-06 Calculus class's response to the question: "To what degree is our entire experience of mathematical truth - as it relates to concrete reality - an issue of dealing with limits?" This appeared on an Intro to Limits test.
"Limits have everything to do with mathematical truth. As we sit in class grasping concepts of a line or a circle and how perfect they are or the can be the reality of it is that we'd never find the perfect line. A line as we know it is not a line. A line is defined by having no depth or width, but in order for us to see a line without imagining it gives it width [sic]. Really everything is imaginary and what's real is fake."
"Philosophy says that everything we were taught is to be a lie [sic]. I believe [sic] this six years ago. Although if you think about it long enough everything will still be the truth... We believe only what is taught and have no proof. For if we were to search, ordinary people like us has [sic] made things up but we still are advised to believe what is taught no matter what."
"Our Mathematical truth to me really is no truth. The truth of all things doesn't ever exist it is just an allusion [sic] of all things. Nothing exists only just an example or explanation of them exists. We could draw straight lines all day long but actually we'll never reach the point of drawing a 100% straight line. The reason is because of the line is never precise it always has crooked and rough edges [sic]. The smallest edge or defect in the line makes it not a line."
"It's all a lie! You can never achieve a perfect circle or a perfect line or a perfect everything for that matter. Because no matter what, there will always be falacies [sic] in the construction so it could be said that the limit to the function of reality is perfection!"
Honestly, any indication I get that a student’s response to a prompt like this is “successful” comes in its ability to seem confident and direct in attempting to maneuver around such a philosophically daunting context. It’s reasonable to assume that in general most human beings would shudder in the face of a prompt like “To what degree does is our entire experience of mathematical truth... an issue of dealing with limits,” let alone a group of high-school students. But in many ways, this is exactly the point: making a successful (albeit out of left-field) connection between Greek philosophy and modern mathematics allows me to show my students in a single lesson how brilliant they are, and also allows me to confront any nay-sayers with a quick retort: “look at what they can do! Of course they can learn Calculus - that is not the hard part. What is more, you see here that they can understand what Calculus does, as long as you make it compelling enough.”
Connection to a Philosophy of Education
Content is arbitrary. I can’t say this enough. In my mind, academic disciplines are delineated not by the content that fills them but the attitude (or “faculty of attention”) by which they navigate. For these purposes, mathematics should be nothing more than an opportunity to participate in an atemporal, counter-anthropomorphic focus on the process of rational and analytical sense-making. To this end, any contrasting faculty of attention which can help return a student to the validation of a mathematical mode of inquiry is exactly what is appropriate for a mathematics course. I believe that it is just as valid to confront students with the Mad Tea Party as a way of discussing whether in mathematics we should “say what we mean” or “mean what we say” as it is to speak of one train leaving St. Louis traveling east and one train leaving Boston traveling west, and so forth. In fact, at times the effect upon a student’s appreciation of the isomorphic qualities of mathematical thought inspired by an unexpected setting can make this angle more valid than the same old way of talking about things. In my School District Project for Dr. Mullins’s class, I speak a little bit more about my convictions regarding the relationship between content and discipline:
“I want to cultivate a position structured around a firm advocacy of “power of process to create value” [...] [W]hat it means to be a teacher of mathematics has almost nothing to do with the content of mathematics itself (in fact, the content of high school mathematics is laughably arbitrary, as well as internally inconsistent). It has something more to do with the relation of this content to the actions of my students.”
“...[A]ll of this is more or less to say that content should be completely unfixed, and that a school should instead focus itself primarily on developing a rigorous attitude of process that can be infinitely adapted to any given content strain. To nurture this fluidity, academic disciplines need to be deconstructed, hybridized, and seen as isomorphic in intent... [T]he primary concern of the school should be helping students approach the moment of confrontation of the human mind to an arbitrary set of intellectual stimuli; not in the commodification and institutionalization of these stimuli as bounded objects that need to be painstakingly artificially valued for student consumption. Furthermore, it must be highly apparent that an appreciation and utilization of the preexisting modes of simultaneity and obsolescence be critical to this endeavor...”
“In the end, schools must be confident enough in the rigor of their methods to trust the creativity of minds, and the playfully chaotic idiosyncrasies of culture interacting within their controlled space.”
In looking back over the last two years, it’s been incredibly difficult to separate the trees from the forest. That being said, it’s rather uncomfortable to imagine that there are “strategies” employed in my classroom in any consistent or even explicit manner. However, there have been moments during my time at Jim Hill in which I’ve done things that I wish I had done more of the time or all of the time. That is, if I were developed enough as a teacher to say that there were particular strategies that defined my day-to-day practice, these would be two of them.
A step routine.
A board game.
A runway (again).
A schizophrenic monologue.
A batch of cupcakes.
A runway.
A hairstyle magazine.
A crown full of jewels.
A shoe collection.