### Interactive Calculus at a Distance

DR. JERRY UHL, Professor University of Illinois at Urbana-Champaign Champaign, Ill. In 1988, Dr. Horacio Porta and I, along with professor Bill Davis of Ohio State University, wrote Calculus&Mathematica (C&M), a three-semester, interactive calculus course that combines textbooks with Mathematica-based instructional software. Developed by our colleague Dr. Stephen Wolfram, Mathematica is a sophisticated, highly graphical math program and programming language. We designed C&M to exploit Mathematica's unique capabilities and today, C&M is used on campuses across America and abroad. The C&M Course, in Brief The course is directed squarely at students who envision a career in calculation, measurement and modeling. Of all the new calculus courses, C&M leans on technology the most, exploiting students' natural interest in technology to slip in the math. It is based on entirely interactive text in which each example is, literally, as many examples as a student wants. The courseware consists of four types of Mathematica files (called notebooks): Basics, which present fundamental ideas; Tutorials, samples of the basics; Give It a Try, actual student work; and Literacy Sheets, questions that the student answers away from the computer. Four textbooks accompany the software. The courseware always puts the mathematics in the context of measurement, and puts the programming in the context of mathematics. We put new ideas into students' heads by having them interact with Mathematica graphics, and by having them explain what the graphics mean. Thus, students get a vivid image of things. We wrote C&M in informal American English, a vernacular that students will read and, most importantly, imitate in their own writing. The heart of the course is the Give It a Try section, in which students use Mathematica's calculation, graphical and word-processing capabilities to work and write assigned problems and projects. The goal of C&M is to help students make a smooth transition from the classroom into engineering, science and mathematics. In a nutshell, C&M gives them: Work that is relevant to the real world; Professional tools; New ideas, communicated visually and experimentally; A chance to organize their thoughts by explaining themselves in writing; and The opportunity to learn the math, programming and writing in context. Calculus&Mathematica gets instructors out of the way and ends the one-size-fits-all calculus course, giving students freedom to think for themselves. The real-world issues and models that appear in C&M's first semester include: Japanese economy cars versus big American cars; data analysis of the Challenger disaster; living off an inheritance; oil slicks; drinking and driving; war games, including a model of the Battle of Iwo Jima; and grading on the curve. From These Roots. . . When we first wrote these lessons and offered the course, an alarming number of students began dropping the class. We soon realized, however, that they hadn't dropped it, they simply weren't coming to the lectures. At that point, we saw that the students understood the course better than we did. We changed it, and today students are in the classroom just one day a week&emdash;for a class discussion, not a lecture. Then in 1990, Porta and I gave a presentation on C&M to the Illinois Council of Teachers of Mathematics. A few high school teachers lingered afterwards, and one said, "If we take you seriously, then I have a question. Do your students even have to be on your campus?" After some thought, we decided that they didn't, although they'd miss the discussion section. We've since realized our approach is really distance learning (which many have defined only as two-way video). Technology shrinks distance and that's the key. This, too, is distance learning&emdash;by computer. The year following our presentation, that same high school teacher, Shirley Treadway, organized some of her colleagues into a Distance Education Program. Two students were signed up at each of four high schools, and Apple gave each school a computer. The next year, since we were teaching a course that used college material, we convinced the University of Illinois' Extension Office to offer credit for it. This year, we have 70 students. Each is provided with a university mentor, who serves more than one student. Our mentors are all undergraduates at Urbana-Champaign. Curiously, out of 20 mentors, only one is a math major; most are enrolled in science or engineering. Sponsored now by the Illinois' Scientific Literacy Program, the 1993/94 program operated on a budget of just $90,000. With 80 students at 24 high schools, it may turn out that teaching this way is cheaper than having the students on campus. Our approach may also offer a means for survival for small, isolated school districts, which must try something new and innovative. We look for schools that have only two or three calculus students. We're not using distance education to displace high school calculus teachers; we seek students who can't get calculus instruction any other way. Of the high schools now participating, most are rural (one of our districts has just 180 students, K-12), although we do also have three inner-city Chicago schools. District superintendents have quite a grapevine among themselves, so new schools to participate are often neighbors of older members. Usually, the coordinator is a math teacher, but in one case it's a chemistry teacher and in another, a guidance counselor. We're happy with that because the ideas behind Calculus&Mathematica have never been bureaucratic. We don't have rules that say you must do things a certain way. The course has to be infinitely flexible. Role of Mentors and More Our high school students are primarily seniors, but some are juniors. Most complete the course with rather high grades. The duties of the undergraduate mentors are straightforward and absolutely vital. Mentors grade students' homework, check their explanations of that work, give advice and encouragement, and keep in contact by phone and e-mail. Some hold telephone review sessions before the tests, and many have met their students during on-campus visits. Since mentors are only a year or two older, they understand the pressures and frustrations of high school students. Distance education students set their own learning pace. Normally, high school calculus is taught at half the speed of a college course. While most C&M students follow that, some prefer the faster college pace. Trisha Mills, our head mentor and a sophomore computer-science major, likens a mentor's role to that of an older college buddy who has already taken the course. "If you give the students a little bit of freedom," Mills says, "that's when you succeed. If you stand up and lecture, they'll sleep." The mentors, she adds, are all good friends, which sets the mood of the course. "The whole point is to keep it relaxed," she notes. "Computers are cold. And without people like this, this course would be cold." Few university professors would give so much responsibility to undergraduates. But the brains behind a lot of what we do in the C&M program are young&emdash;teenage or just barely out of it. For example, the C&M network administrator is 21 and he's getting old for this group. Many of our mentors first participated in the program as high school students. (According to a university study, the C&M Distance Education Program brings students to our campus who wouldn't come here otherwise.) The mentors effectively increase the size of the teaching corps and act as role models for the younger students. The university now makes every semester of calculus available to the high schools. We may also offer a Differential Equation course in the future. Hardware & Software Components Our program is based on Macintosh technology; once equivalent Windows software becomes available, we also plan to support those machines. (Mathematica is already available for Windows, UNIX and NeXT computers.) Although it isn't ideal yet, our hardware configuration works well. For grading and communicating online, we use two fast machines, a Power Macintosh 7100 and a Mac LC 475. Older Mac SE 30s answer the modems and attach callers to our network. Fast modems and AppleTalk Remote Access allow connection over standard telephone lines. Students dial a single, toll-free 800 number that rolls over to other lines if it is busy. Once connected, students gain access to our Apple Workgroup Server 95, on which they store their homework and transfer lessons. Turning in homework is a simple "drag and drop" process with a mouse. In addition to Mathematica, we rely on three other software packages: Eudora, Timbuktu Pro and PacerForum. Eudora is a communications program that makes electronic mail easy to send and receive. Once on our roster and on our network, students and coordinators receive individual e-mail accounts. Timbuktu allows our staff to lend live assistance with lessons by sharing control of a student's computer (a process mentors call "grabbing"). This software permits a mentor's virtual presence across many miles. Finally, with PacerForum, no C&M student ever has to be alone. This intuitively designed software creates "virtual forums" in which students and staff leave and respond to public messages&emdash;with graphics, text or sound. PacerForum allows people to gather virtually and discuss ideas. Think of it as similar to a news group on the Internet. For the most part, our current system works well. One problem we do encounter, however, is with the line quality at some rural schools. Occasionally, students turning in homework lose the connection because of static on the line or because rain affects a small town's old PBX hookup. We'd like to see the state sponsor ISDN lines&emdash;very fast, digital data lines&emdash;for connections to the Internet though a community college or other source. On our main campus, the computer network is superb. Should a rural student call with a technical problem that a mentor can't resolve, our young network administrator can remotely control the high school's computer and tackle the problem himself&emdash;all from his dorm room. In the future, I don't want to see a huge distance education industry running out of our labs on the Illinois campus. I call our operation here "the hub." Rather than running a centralized operation, I'd like to see hubs all over the country. We also are currently developing a method for long, long-distance education over the Internet and setting up a sub-node at The Ohio State University. A Complete Model for All Disciplines? Is Calculus&Mathematica a complete answer to what ails calculus education today? Probably not, either at the university or the high school level. C&M has not proved to be ideal for all students. Those who don't see a clear need for calculus in their futures or students who want to approach calculus as a "liberal art" are often unwilling to make the necessary commitment. Those who believe that a teacher's job is to teach passive students often don't succeed either. According to the mentors, students who drop out of the Distance Education Program are often simply too busy, or not in the right frame of mind to accept the responsibility for tackling such a challenging course. But most of our high school students do succeed, and earn university credit for their efforts. I think a lot of science learning could happen this way&emdash;certainly in math and geometry. After all, C&M is full of experiments. The state is looking at other applications for Mathematica courseware, including physics and industrial-technology courses. I would also like to see C&M extended even deeper into the high schools, perhaps to the freshman level. But is this a model for teaching history or literature? I don't know; I've spent more than 30 years learning how to teach math. However, one state official believes the potential is there. Don't be surprised to see new grant proposals bubbling up from the high schools. Success with distance education in rural high schools is certainly not limited to our state. If it works in rural Illinois, it can work in Kansas. On the other hand, a direct translation of the current popular text to the Internet will fail. The overwhelming reason for our success is not the technology itself, but how we use it in our courseware. In that regard we are unique. The entire Calculus&Mathematica course can be downloaded free of charge from the C&M World Wide Web site. The URL is http://binkley.math.uiuc.edu. The C&M course can also be downloaded from Wolfram Research's MathSource, an electronic resource containing Mathematica-related packages, notebooks, examples and programs. To obtain the C&M course via anonymous ftp, connect to mathsource.wri.com. Log in as anonymous and give your e-mail address as the password. Go to /pubs/Applications/Education/Calculus/0207-638 and retrieve the files in that directory. Or send the e-mail message send 0207-638 to mathsource@wri.com. Jerry Uhl is a professor of mathematics at the University of Illinois-Urbana Champaign. Products mentioned: Calculus&Mathematica, Addison-Wesley, Reading, Mass., (617) 944-3700. Mathematica, Wolfram Research, Inc., Champaign, Ill., (217) 398-0700. PacerForum, AGE Logic (acquired Pacer Software in early 95), San Diego, Calif., (619) 755-1000. Timbuktu Pro, Farallon Computing, Inc., Alameda, Calif., (510) 814-5100. Eudora, QUALCOMM Inc., San Diego, Calif., (800) 238-3672.This article originally appeared in the 06/01/1995 issue of THE Journal.