"At a fundamental level, everyone needs to understand rate of change," says BMCC math professor Jason Samuels.
"You go to the gas pump, it's dollars per gallon; rates are often constant but in more complicated contexts, they're always changing. Also, in the real world, almost everything—from air masses to financial trends—changes and has some kind of irregularity; some sort of curvy trajectory. So how do we analyze all that?"
The answer, he says, "is to reduce the situation to a straight line, and how you do that, is what calculus is."
Starting with a visual: mathlets, models and more
Facilitating that process of finding the straight line in an irregular context, Samuels has created for students a series of free mathlets, downloadable computer applications that perform a small set of tasks.
He opens one of these, on his phone.
"On the top, you see the graph of some function," he says, "and then there's this black box around a section of it, and that part is presented on the bottom."
As the box on the top half of the screen zooms in on the graph, the bottom half of the screen responds accordingly, applying whichever function the user chooses, "and you can identify the slope, or the rate of change at that point," Samuels explains.
"I like to introduce calculus visually," he says, "as opposed to introducing it algebraically, or with formulas, which is the more traditional method."
He redesigned his calculus courses around this strategy, and students not only use the mathlets as a resource, they also start with 3D models and other tactile, physical representations.
"I've assessed the outcomes, to see if students were actually learning calculus, doing it this way," he says, "and those outcomes make it clear that students who first explore calculus visually have a much deeper, more connected understanding."
An NSF grant to develop instruction
Recently the National Science Foundation's Division of Undergraduate Education awarded Jason Samuels of BMCC; Brian Fisher of Pepperdine University; Eric Weber of Oregon State University, and Aaron Wangberg of Winona State University over $225,000 through the Transforming Undergraduate Education in Science, Technology, Engineering, and Mathematics (TUES) program.
Their goal is to investigate innovative new methods for teaching and learning multivariable calculus, or Calculus 3.
The project, "Raising Calculus to the Surface," is sponsored by Winona State University in Winona, Minnesota, and will teach students by starting with a visual exploration—they'll draw, measure and grasp concepts geometrically, using three-dimensional, clear plastic models on which they can write with a dry erase marker.
"It's like thinking about topography," says Samuels.
"In fact, one of the earliest uses of Calculus 3 was to create topographical maps that could show you things like, all the mountains 5,000 feet high are at this line; all the mountains 10,000 feet high are at that line."
Students using surfaces of the 3D models, he says, "will explore the questions, 'What is a level curve?' and 'What is a contour map?' They'll get the concepts from the actual physical model itself."
Less struggle, more meaning
Samuels loves playing the popular logic puzzle Ken Ken. He competes annually in the Google U.S. Puzzle Championship, is a "huge, huge Yankee fan," and leads the BMCC Math Team. He also encourages students to take part in CUNY-wide Math Challenge and contests sponsored by the American Mathematical Association of Two-Year Colleges.
He's always enjoyed math himself, he says, and when students start with a graphic or physical representation and then move to formulas, they can enjoy math, too, he says, and internalize deeper meaning.
Still, visual-first calculus instruction has its skeptics.
"A lot of teachers think, 'Oh, if I try to do all this experimental stuff, I won't have time to actually do the math'," says Samuels, "and nothing could be further from the truth."
He explains that in the classic instructional presentation; definition, theorem, proof, example, "often students struggle. They try to memorize and copy what you're doing, and they devote so much mental energy and time trying to understand what it means, it actually takes away from the learning process."
On the other hand, he says, "If you have them explore the idea first through discussion and tactile or visual activities, then cap it off by applying a formula and maybe even a theorem, you don't have to explain what it means, because students already understand the full concept that goes with the formula. It actually takes less time, and you can cover more meaningful mathematics."
Research using classroom outcomes
In summer 2014, Professor Samuels—who has written books on the use of visualization in calculus instruction—and the NSF-funded project team will host a series of workshops on how to teach Calculus using the 3D models and their surfaces.
The participants, dozens of faculty now being recruited from high schools and two- and four-year colleges around the country, will then take that methodology back to their own calculus classes.
Eventually, Samuels and his team will compare the outcomes of those classes, with the outcomes of control classes using more traditional methods.
"This large-scale implementation could provide compelling evidence and convince other Calculus 3 instructors to try using their surfaces to strengthen student learning," says Professor Samuels.