The Connected Curriculum Project (CCP) is a coordinated effort to create interactive learning environments for a wide range of mathematical topics and mathematics-based applications. Our materials combine the flexibility and connectivity of the Web with the power of computer algebra systems. These materials may be used by groups of learners as an integrated part of a course or by individuals as independent projects or supplements to classroom discussions.

David Smith and Lang Moore are Associate Professors Emeriti of Mathematics at Duke University.

[Editor's Note: All CCP materials and related articles in Issue 3 of Voluime 1-- including this article -- have been under the editorial supervision of Associate Editor Jerry Porter to avoid the obvious possibility of conflicting interests when the Editor is also an author. This article is about the entire CCP site, not specifically about the selected CCP modules published in JOMA. Publication of this article is not an endorsement by or publication of any of the CCP materials other than those specifically listed in JOMA. DAS]

Copyright 2001 by David Smith and Lang Moore

Published December, 2001

CCP is a collection of learning materials primarily for lower-division undergraduate mathematics. Each CCP learning unit uses at least some of these powerful tools:

• hypertext links,
• Java applets,
• sophisticated graphics,
• a computer algebra system,
• realistic scenarios,
• thought-provoking questions that require written answers,
• summary questions that enable students to see the forest as well as the trees.

Our purpose is to challenge students to engage with important mathematical concepts in meaningful contexts -- to experiment, to conjecture, to test their ideas, (sometimes) to discover relationships, to check their work until they are convinced that it makes sense. In particular, we do not (in general) provide answers -- rather we provide tools by which students can decide for themselves whether their answers are correct.

Most of our experiences using the CCP materials have been in interactive classroom environments -- a setting in which students begin the work under the eye of the instructor. However, the materials can also be used wherever students have access to the Internet on a computer with one of our supported computer algebra systems.

One of us has recently published an article on the interactive classroom environment in which we currently use the CCP materials. The article also links design features of the materials to recent research on teaching and learning.

There are many tools available for creating interactive learning environments -- and hence many ways to do the same thing. Our principle for deciding whether to create the interaction in a web page (with a Java applet, say) or in the CAS worksheet is this: If we don't need to give the student total control over the interaction -- if we can get the job done from a menu of possibilities, say -- then we do it in the web page, and it doesn't matter which CAS is preferred or available. (A few modules are entirely in web pages and don't need a CAS at all.) But if we want to go beyond the limitations of applets and let students explore in directions we have not completely prescribed, then the CAS environment, guided by a web-based common set of instructions, gives us and the students the freedom and flexibility to do this. A side benefit of this approach is that students learn to work with a modern general-purpose calculational tool. This capability will always be a part of their working environment. We discuss our uses of CAS at greater length in Section 4.

Most of the CCP units are modules, that is, single-topic units that can be used for a two-hour lab, or for a shorter supervised period with follow-up on the student's own time, or for self-study. Most of our modules are class-tested with students working in two-person teams in a lab environment. Some modules use an application to stimulate learning of mathematics, and others go straight to the mathematics.

CCP also contains two projects (and more to come), longer units that may be completed by small groups of students over a period of one or more weeks. Each project involves a significant application of mathematics and is intended to demonstrate the process by which mathematics is used by working scientists and engineers.

The site has additional resources for teachers and developers. The teacher resources will be described in a later section of this article. An important resource for developers is the textbook Design Principles for Interactive Texts, by Julie Jacobs and William Mueller, which we will not discuss here.

The modules -- approximately 100 in number -- that make up the majority of the learning materials at the Duke site were developed for use in undergraduate mathematics courses including

• precalculus,
• differential calculus,
• integral calculus,
• multivariable calculus,
• ordinary differential equations,
• linear algebra,
• engineering mathematics, and
• mathematical finance.

Each module consists of several HTML pages containing discussion and guidance for exploring a topic, generally using Java and/or a computer algebra system. Nearly all modules have downloadable worksheets currently available for Maple, Mathematica, and Matlab, and some also have Mathcad worksheets.

In addition to the undergraduate materials, there is a collection of modules designed specifically for a guided or independent study course for high school students who have completed a year of calculus. These modules are part of the Post CALC Project.

The Duke component of CCP is the successor to Project CALC: Calculus As a Laboratory Course, which was supported by the NSF Calculus Reform Initiative (1988-93). Most of the precalculus and calculus CCP modules are based on laboratory materials originally developed for Project CALC. A follow-on NSF grant in 1993 supported development of modular lab activities for courses beyond calculus: linear algebra, differential equations, and engineering mathematics. These modules were created as interactive texts in specific computer algebra systems: Mathcad, Mathematica, and Maple.

In 1995 the authors joined forces with a group led by Mike Colvin  and Don Hartig at California Polytechnic State University, San Luis Obispo, and with Frank Wattenberg, then at Weber State University, to form what is now CCP, with the goal of developing a Web-based library of reviewed, edited, and tested interactive materials. This library effort received substantial support from NSF in 1998. In addition, NSF has funded the related PostCALC project to develop interactive mathematical modules for high school students who have finished a course in calculus.

The scope of available CCP materials and the family of CCP authors both continue to grow as additional units are submitted. Our news page gives regular updates on new materials. In addition to materials found at the Duke site, some of the CCP materials are housed at Cal Poly and some at Montana State University. In this article, we discuss only the materials on the Duke CCP site.

NSF Grant Support for CCP

• Interactive Modules for Courses Following Calculus, Duke University, NSF DUE-9352889, 1993-97
• Interactive Connected Curriculum Project, Cal Poly San Luis Obispo, NSF DUE-9555407, 1996-98
• The Coordinated Curriculum Library, Duke University, NSF DUE-9752421, 1998-2001
• Web-Based Interactive Enrichment Modules for Mathematics, Duke University, NSF ESI-9730714, 1998-2001

Most of the CCP modules require that the user have one of the following helper applications installed on his or her computer:

These buttons appear on the Contents page of each module and link to a corresponding CAS worksheet designed for use with that module. In most cases, the user may allow the downloaded file to launch the CAS, but saving the file first is an option. Matlab is a little different -- since multiple ".m" files are needed, they are delivered in a variety of zipped archive file formats, and the user can choose an appropriate format.

Most CCP modules are designed to take advantage of both the Web and the preferred CAS. All discussions and assignments take place in the web page, and students make calculations and record their observations in the CAS worksheet. Most users find that the materials are easiest to use when both the web page and the worksheet are visible side by side, as shown here (121K gif file), with a module web page on the left and a Maple worksheet on the right. (Maximize the browser window to see the full-screen display. Close the window to return here.)

A few of the modules do not use a CAS because all of the necessary interaction can take place in the web pages. An example is our Introduction to the One-Dimensional Heat Equation.

Each of our subject areas contains tutorial modules for the relevant computer algebra systems. The basic parts of these tutorials are the same across subject areas and very similar across systems -- they just get the student started with operations that will have to be used in any CAS, such as simple algebra, function graphing, writing mathematical text, and file management. Then each tutorial also has a few subject-specific sections on matters such as differentiation in calculus and matrix operations in linear algebra. The tutorials do not aim at expertise but rather at getting started. As additional CAS commands are needed, they are introduced in the context of modules in which they will be used in a meaningful way. To see a sample tutorial, visit the Maple Tutorial for Differential Equations.

We have a design principle for what to put in a CAS worksheet, as opposed to what to leave out. If it's something we don't want students to spend time thinking about, we put it in. If the whole point is to think about it, then we leave it out. For example, if the task is to graph simultaneously a set of data points and an approximating curve -- after figuring out what curve should approximate the data -- we will provide

• the complete data structure,
• a complete set of graphing commands that refer to a named function, and
• an incomplete instruction to define a function with the proper name,

and we will leave out

• the formula for the function.

But all of the provided code refers to tasks that students had to learn to do for themselves in earlier modules (including possibly the tutorial), such as entering data and defining and graphing functions.

Another design principle is to provide students a lot of help at the beginning of each module, but less and less as the exercise progresses. In later sections of the module -- and especially the Summary section -- there may be nothing but a text line to mark the section and some empty command lines. Students quickly learn to copy, paste, and edit code (from earlier in the worksheet or from some other exercise) to avoid having to do everything from scratch. That is, they learn without much prompting some of the skills we use all the time.

Over time, students become quite proficient with the commands they use a lot, and they learn where to look for names and syntax of commands they may not have used or have used only occasionally. This addresses our secondary goal of having students become confident that a CAS is a tool they can use in new situations when they need to. (It also provides an occasional shock for a physics or engineering professor in some subsequent course.) Our primary goal, first, last, and always, is learning mathematics.

To illustrate how CCP modules can fit into a conventional course as weekly activities, we include here the syllabi from two recent courses. (Each will open in a new window -- close that window to return here.)

Multivariable Calculus, Fall 2001

Engineering Mathematics II, Spring 2001

We and our colleagues have taught many courses using the CCP materials. From the table below, you may link to our course homepages, from which you will find textbook choices, course outlines, homework assignments, grading schemes, and other information on courses involving the CCP materials. Some of these courses have been taught multiple times over the past four years. Only the most recent home page for each course is linked in the table.

The more recent course pages are in Blackboard, which will ask for a login. You can log into any Blackboard course page as guest with password guest and see everything except the student-specific areas.

Editor's note, 12/6/04: Duke University has made the Blackboard pages inaccessible to guests. The remaining links are to personal course pages constructed in the 1998-99 academic year or earlier. These pages include weekly plans that illustrate the context in which the CCP modules were used.

Links to Home Pages for Courses using CCP Modules