What should students be able to do to demonstrate understanding?

By Benjamin Nockles, MƒA Master Teacher

Last year, a colleague of mine rhetorically asked me, "Can you imagine what our classrooms would look like if, instead of an exam, teachers themselves determined what students needed to do to demonstrate understanding?" This question struck such a chord in me, as it speaks directly into some of the greatest challenges I've faced as a teacher. 

I began my teaching career at a small theatre arts high school in Manhattan. Suffice it to say students preferred theatre over mathematics. As a new teacher, it took a few years to get my feet under me while I adapted my teaching to the needs of such a student body. I don't think a veteran teacher in New York City would be surprised to hear that some of my students needed a calculator to multiply a number by 9, were not familiar with the division algorithm, and couldn't tell time on an analog clock. The geometry standards at that time required a student to graphically solve a system of quadratic and linear equations. If I had any success at all, it was only proportional to the degree to which students were enabled to exercise rote memorization, which is no success at all in a course designed to develop genuine problem solving skills.

My students hated my class. What we called “exam prep” was really just a daily reminder of how far below grade level they were. I was most successful whenever I could pin a mnemonic on a procedure; I was least successful whenever I tried to explain how it worked. By my third year of teaching, after having forcibly plowed through two years of Regents recitation with abysmal results, I realized I had nothing to lose if I decided to change everything. I stopped letting the pressures of the exam dictate my teaching and I began to ask myself what I wanted students to be able to do and how their existing skills and interests could compliment their experience in my classroom. They needed a mathematical encounter, a tangible experience that could illustrate what math was truly about. When I taught about angles and bisectors, we studied origami folds. The "Parallel Lines" unit became the "Popsicle Stick Bridge" unit. In the similarity unit, we measured the proportions of Barbie and Batman dolls and compared them to our own, life-sized bodies.

It was time-consuming. We once spent four days in a row doing nothing but gluing popsicle sticks. I gave them an exam on similarity in which the diagrams of triangles were replaced with diagrams of Barbie and Batman. I dramatically reduced my emphasis (and associated remediation) on any standard that couldn’t be brought down to earth in a tangible way. We focused on what students could do, see, and build.

As a result, students reported a much more positive experience, came to class on time, completed homework and passed my class against the new requirements. Their Regents scores, on the other hand, barely improved. I did this for two years. In the third year, my principle encouraged me to focus more on the Regents, hoping for a more substantial increase in scores. The following year, my students were dismayed to hear there would be no bridge building unit. We focused on the Regents and, consequently, saw no improvement in Regents scores (nor declination). Interestingly, the passing rate more than doubled when students who had failed the exam after completing my project-oriented curriculum took a single semester of exam prep (their third consecutive semester of geometry.) The same was not true of students who took three semesters of the traditional Regents-focused curriculum.

I can't argue that my project-oriented approach is a more (nor less) effective method for improving Regents scores, but I am certain of one thing: it is a more effective method for teaching students to appreciate and engage in mathematics. None of my students will remember their Regents scores better than the day we stress-tested our popsicle stick bridges because a great education is not about performance on an exam, it is about experience. Today, I may work at a different school, but I continue to develop alternative ways to assess students that emphasizes genuine, real-world experience. While the course may culminate in a Regents or AP exam, what I really want to teach my students does not end there.