"One of the things I've become convinced of, in my reading and thinking on the subject, is that in order to truly understand mathematical ideas, you have to be able to visualize them."

--Bruce Carpenter

In this section, we examine the learning environment ^a that the UIUC bricoleurs^b have created to achieve their goals for student learning. Specifically, we will look at how BioCalc works--the tools that are used, the learning activities ^c assigned, and the learning processes that are enabled by this type of environment. But first, the bricoleurs shared with us their thoughts and ideas about how students learn.

The Learning Process
In our talks with Jerry, Bruce and Brad, it became quite clear that the ideas informing their classroom practices do not focus on teaching per se, but on learning. They described to us a learning environment based on the principle that learning best occurs when participants are engaged in an active and contextual manner.

That explains why Jerry is so adamant about why the BioCalc and other C&M courses work: they are designed to, in his words, "actively engage the students in their own learning." He believes that students don't and won't learn math by sitting passively and listening to lectures; effective learning depends on student engagement and interaction--with ideas and concepts, with instructors, and with each other.

The bricoleurs believe that the most effective way to engage students is to introduce mathematical ideas visually--through graphs and pictures--before lengthy and complex descriptions enter the picture. They call these visuals "math kitties," in reference to an observation about learning by famous mathematician Ralph Boas:

Suppose you want to teach the "cat" concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractile claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying 'kitty' each time, until it gets the idea.

The Learning Environment: Tools and Activities
Ideas about how students learn are an important foundation in the creation of a learning environment, but how do such principles translate into actual practice? Let's take a closer look at the BioCalc learning environment to find out.

As stated earlier, BioCalc is a section of Math 120 (Calculus and Analytic Geometry I) offered specifically to life science students. It uses the Calculus&Mathematica courseware in conjunction with Mathematica software and heavily emphasizes life science examples and applications in order to help students learn calculus concepts in a meaningful context.

BioCalc classes meet in a computer lab four days in week, where students spend their time working through lessons presented in an electronic notebook format. One day a week, the class meets outside the computer lab for a discussion period and for practice doing hand calculations. (Class periods are 90 minutes each day.) All Math 120 students earn five credit hours. Students in BioCalc earn an additional credit for learning the Mathematica program.

Instructors (primarily graduate students) are responsible for assigning lessons, preparing exams, and conducting discussions. During class, they act as guides to the students, often walking about the room, discussing problems and helping students find the answers to their own questions. Tim Braun, a C&M student and lab technician who has observed many classes, described the role of the instructor in typical C&M courses:

In the lab, the faculty are there to guide their students more, to make sure that they're getting the basic concepts. The best instructors generally wander around the lab looking at what each group of students are doing. If they see students stuck, they try and help explain them through it without just handing them the answer. They're there just to see how their students are doing--more than anything else, to just kind of guide them along. It's a lot of calculus. Mathematica is self-discovered through the lessons, but you do need guidance. That's what the faculty are there for. They keep you on the right track if you're missing a concept or just not getting what the lesson is trying to teach and getting really frustrated. They'll help get you right back on track.

There is also a classroom assistant (a veteran C&M student) who is available to help students, as well. The CAs are equipped to handle the more technical questions about Mathematica but also field student questions that arise from the lessons. Brad explained:

An undergraduate assistant is assigned to each class. The undergraduate assistant is someone who has experience with the Calculus&Mathematica program and with the mathematics for that course, so they can answer questions. They're sort of a front-line resource for the students, who can ask them for help with the Mathematica code or with lower level mathematics questions.

There are no textbooks in the BioCalc courses. Instead, students use the Calculus&Mathematica courseware, which features a series of electronic "notebooks." The notebook format is a unique attribute of the Mathematica software; it allows the combination of text, commands and graphics to function in a single interactive electronic document. The C&M notebooks are divided into discrete, thematic lessons. The lessons presented in BioCalc (as well as other C&M sections of Math 120), for example, include:

1. Growth	7. The Race Track Principle
2. Natural Logs and Exponentials	8. More Differential Equations
3. Instantaneous Growth Rates	9. Parametric Plotting
4. Rules of the Derivative	10. Integrals for Measuring Area
5. Using the Tools	11. The Fundamental Formula
6. Differential Equations of Calculus	12. Measurement

A C&M Math 120 syllabus is online, and a syllabus for a traditional, lecture-based section of Math 120 is also online.

Each C&M lesson contains four sections. The Basics section introduces students to the key concepts of the lesson through visual examples and explanations. In the Tutorial section, students uses those key concepts to explore problem-solving techniques and applications, working with interactive examples (the math kitties) as many times as they like. Answers are provided so that students can check their own progress. The third section, Give it a Try, presents problems that students solve on their own. Here, they not only work the problems and find numerical answers but often must give written explanations of their findings, as well.

An example of a C&M homework problem is online.

The notebook lessons--the backbone of the C&M courseware--are designed to more fully engage students by communicating new ideas visually and experimentally. When students open the notebooks, they are immediately presented with graphic interpretations of mathematics concepts. In the Basics these are presented for students to view, but in the Tutorials students interact with and manipulate the data by varying expressions and input data to see how such changes effect output. In this manner, students "are forced to learn," one teaching assistant told us, "because they have to look at the information on the screen and do something with it."

The emphasis on visualization is a key feature of the C&M courseware. In every part of the lessons, mathematical ideas are presented in visual formats. Jerry explained that "instead of overwhelming students with definitions before solving problems, the notebooks are designed to take the student through a series of calculations and graphs, giving the student an actual feel for what the algorithm does before giving them a text definition" (Rhee, 1994).

Typically, a class completes one notebook lesson per week. Students are free to work through the Basics and Tutorials at their own pace in class, reviewing concepts and reworking problems as many times as they like. When they feel they have a grasp on the material, they move on to the Give a Try section. At the end of each week, they turn in problems from this section to their instructor for grading and comments.

The homework is submitted electronically, through a website known as "Course Space." Tim Braun, the C&M lab technician, explained that students log in to Course Space to submit and retrieve their homework and view announcements and assignments.

Course Space has announcements from the instructors and the TAs. It has a syllabus with all the assignments and when they're due and when the tests are going to be. Students also use Course Space to turn in their assignments--they upload them to that server. Then, their class assistant downloads them from the server, grades them and then uploads the graded files again. There are no disks being passed back and forth; it's all done through networks now.

In addition to the Basics, Tutorials and Give it a Try homework sections, each lesson includes a "literacy sheet"--a set of printed problems (not graded) that students complete by hand. Literacy sheets signpost the concepts a student should know and be able to discuss after completing each lesson. They also indicate to the students the level of pencil-and-paper calculations they should be able to perform. Twice during the semester, students take one-hour "literacy tests." These are pencil-and-paper tests (calculators are allowed) and, Jerry explained, are often based on the literacy sheets.

The Role of Technology
Technology, when employed in ways consistent with the Seven Principles, can have powerful and lasting effects on student learning (Chickering & Ehrmann, 1996).

For an in-depth discussion of the BioCalc learning environment and Chickering and Gamson's Seven Principles, see Discussion 1.

In BioCalc and the other C&M courses, the learning technology (e.g., the computer, the Mathematica program, and the C&M courseware) is critical to student learning. It is not used to complement the learning process but rather to initiate it. Remember the math kitties? The technology allows students to see and interact with ideas and concepts previously deemed "too hard" to introduce at the start.

Bruce: [The C&M courseware] really does rethink the approach to mathematics from the ground level. When you first start to study a topic, you start at a very elementary level traditionally. Then you introduce concepts, etc., etc., and then basically the semester runs out before you really hit the broader stuff, the stuff that's really applicable. With the addition of technology, what you can essentially do is take those traditionally hard to reach subjects and put them at the very beginning. Get students feet wet at the very beginning with the really major stuff because they're not disenfranchised by the inability to do the hard calculations.

Students in BioCalc, for example, can sometimes be heard complaining about having to do differential equations. Their counterparts in standard Math 120 courses, however, have minimal exposure to differential equations. In this case, the use of technology gives students exposure to key concepts and topics they might not otherwise see. It also allows them to begin using those concepts in meaningful, interactive ways.