Constructing a New Geometry

Facing the roll-out of a laptop one-to-one program and dissatisfied with their current geometry textbook, the math department at The Episcopal Academy in Newtown Square, Pennsylvania, decided to build their own curriculum using a cult classic piece of software, Geometer’s Sketchpad, and Euclid’s own timeless logic. Grace Wingfield, upper school math chair, and Jim Farrell, veteran geometry teacher and coach, received a grant from the school to research and develop a new way to teach geometry that involved more building and less proof writing and arithmetic.

Take yourself back in time for a moment to high school geometry. Remember the straight edges, compasses, and protractors? The satisfaction of bisecting an angle for the first time using just a compass and a straight edge? The tactile sensation of the pencil being dragged around the sharp, fixed tip of the compass to produce a perfect circle?
 
Digital simulations may lack the tactile element, but Geometer’s Sketchpad is a beautiful educational tool in its own right. Developed in 1995 by Nicholas Jackiw, it has stood the test of time. Twenty years, in the digital world, is an eternity.
 
Grace and Jim agree that the rules of Sketchpad follow exactly the postulates set down by Euclid over 2,000 years ago: Draw a point and a line and upon request Sketchpad will construct one and only one perpendicular line for you, intersecting the line and the point. The tool is simple, but perfect. There are no shortcuts or dead ends. It is a reliable, rock-solid work horse of math education.

Creating a New Curriculum

Jim and Grace spent several weeks during the summer developing a curriculum in Sketchpad. Jim curated an extensive set of sketches and lessons. He became the support contact for the rest of the department.
 
Jim quietly describes himself as “non-techie.” A classroom veteran for more than 30 years, Jim had favored chalk, pencils, compasses and straight-edges, bringing along students at every level of ability and nurturing their success and happiness both in the classroom, and in cross-country and track.
 
Grace is more technologically inclined, but has always looked for substance in the tools she chose. In addition to being chair of upper school math, she acts as a learning support specialist in math skills and meets with students one-on-one whenever she is not teaching a class.
 
When a very traditional, classical, liberal arts institution decides to try something innovative with 11 sections of geometry, preparation and caution abound. The school paid four teachers for five full working days, but the dedicated team put in more hours than just that. This was to be an endeavor, not an experiment. Both tracks of geometry, honors and regular, rested on a foundation of decades of institutional practice. The textbook, however unsatisfactory it had become, was always present as a reference and roadmap. Overall concepts did not change but the methods were new.
 
The methodological goal was to leverage the newly instituted one-to-one laptop environment in order to immerse the students in the technology. Learning Sketchpad takes a few days; becoming an expert takes a few weeks; after that the technology disappears and the students become fluent in the syntax of Sketchpad. They observe through a clear lens the crystalline logic of this strange and ancient branch of mathematics.
 
The students picked up the new technology quickly and began providing support to each other and to the teachers very early in the course. Grace asserts that she has never been fast enough to answer students’ tech support questions: Another student beats her to the answer every time.
 
The students began taking all of their class notes in Sketchpad, too. They drew figures, of course, but they also employed the text tool to provide explanations. And they frequently shared their figures and notes with each other. The teachers saw this and encouraged it further by giving extra points for collaboration on problems.
 
Episcopal did not, however, abandon the classic, two-column geometrical proof. The department thought it important to preserve this rigorous verbal exercise.

Jump-starting Aha Moments

In addition to the change in methodology, the math department also worried about reaching a mixed population of ninth- and 10th-graders in this exceptional year, when geometry was moving its place in the sequence from 10th grade to ninth. There was a big gap in experience and stamina to deal with in class. But the teachers found that the new tool actually closed that gap, and that many younger students were not only keeping up with their more experienced classmates, but were actually surpassing them. Sketchpad leveled the playing field for motivated students.
 
Was the same amount of material covered? No more. The concepts are simply easier to grasp if one is able to build, measure and transform figures in class and for homework. Pages of calculations and tedious arithmetic can be dispensed with by a single sketch. When you build it yourself; you get it. The tool also lends itself to an exploration of alternative solutions. The teacher shows you one way to construct a figure, and then you show her another way.
 
Jim describes increased spontaneity, autonomy, self-motivation, and literal “aha” moments during class. At first, some students seemed reticent to jump in, preferring to let the teacher lecture and provide demonstrations. But the teachers made a deliberate point of making the classes more like labs, pushing the more passive learners out of their comfort zones.
 
The tool itself has the magnetism of a good, digital game. Sketchpad encourages exploration, cultivates non-linear thinking, and takes the class off script. Even students who insisted on step-by-step elucidation of proofs turned back to Sketchpad to test the steps and to reproduce the results.

Watching Students Do Geometry, Differently

I watched a lesson from the back of Jim’s class, furtively looking around at the students’ screens from behind. The lights go off, and Jim Farrell says, “Sketchpads out.” Jim constructs a circle on the projector. I notice only one distracted student. (She shoots off a few messages to friends before pulling up Sketchpad.) For the rest of the 40 minutes, everyone is doing geometry.
 
But none of them seem to be doing the same geometry. As Jim proceeds, I notice that some students are trying alternative solutions; some linger and interrogate the properties of a figure; others race ahead and solve the problem on their own. But on some level they are all together. As Jim constructs a line perpendicular to the radius of the circle, they all say in unison, “tangent.”
 
The class is very quiet except for the tapping and clicking. The student in front of me has created three or four desktops on his MacBook Air, with several different sketches running at the same time. He swipes back and forth, comparing them.
 
Constructing figures in Sketchpad requires rigorous, algorithmic thinking. Computer science trains us to build “test sets” various cases used to test the validity of the algorithm, ensuring that code will run for every case. We do the same with Sketchpad: What happens when X goes to zero? What happens when it goes to infinity? The test cases appear before us in the construction and we can slide, rotate, flip, and measure.
 
Jim, Grace, and the math teachers of Episcopal executed a deliberate, well-considered shift in how one class was taught, exploiting the advent of the school’s one-to-one laptop program and a trusted piece of software. The idea grew from classroom practice with the support and enthusiasm of teachers. This is sustainable innovation: mindful of the past, embracing the cultural legacy of math education, and adapting pedagogy to take advantage of new opportunities.

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