Welcome to the Montana State University site of the Connected Curriculum Project. Be sure to visit our sister sites at Duke University and California Polytechnic University, San Luis Obispo.
If you are new to this site and want to get an idea of what is here now and our plans for the future this page will begin a guided tour. If you want to use this material more seriously you will eventually need some free helper applications and one of the computer algebra systems -- Mathematica, Maple, MathCad, or the TI-92 -- and you will need to configure your browser to work with these helper applications. This link takes you to a page describing how to configure your browser and the helper applications we use with links to Web sources for the free ones.
We'd like to invite you on a short guided tour of our project. This window will be your guide. Along the way you will open other windows and you will go back-and-forth among the different windows. This is typical of the way you should use our material. We think of windows like this one as guides or conductors orchestrating the work that you do -- sometimes in other browser windows; sometimes using other computer-based software; and sometimes using other equipment like the Texas Instrument CBL, a graphing calculator, TinkerToys, or pencil-and-paper.
Because so much is going on, you need to make efficient use of the limited "real estate" or space on your monitor. You will often have two or more windows open -- for example, one window like this one with narrative material and another window with a computer algebra system or a Java applet.
At the end of this paragraph we will ask you to click on a link that opens another window. When you do so you should arrange the two windows so that they overlap as shown in the pictures above and you can move easily back-and-forth between the two windows by clicking on the exposed portion of the inactive window to make it active. After you have opened the new window and arranged the two windows as we just described, click on the exposed portion of this window to make it active, so that we can continue quiding you through the tour. Now click here to open a window into The Connected Curriculum Project.
The new window is divided into two frames. The frame at the top is the navigation frame and, together with your browser's navigation buttons, will enable you to move around The Connected Curriculum Project. Moving from left-to-right in the navigation frame we see the following icons or links.
Now look at the second frame -- the content frame. In the upper right hand corner of this frame you will see the icon below.
This icon indicates that a printed guide is available for this particular module. In a classroom or laboratory, the instructor will usually hand out the guide but it is also available online using the Adobe Acrobat Reader -- free software that reads pdf files. The pdf format is particularly well-suited for material intended to be printed. Your browser may already be configured to use the Adobe Acrobat Reader and, if so, you can click the guide icon in the other window to see how it works. We use pdf format in other situations -- for example, long proofs -- where printed copy is better than screen copy.
You will also see blue triangles like the blue triangle above throughout The Connected Curriculum Project material. These triangles can help avoid some "scrolling." Clicking on one of these triangles brings it to the top of your screen. Click on the one in this window now to see how it works.
The Connected Curriculum Project uses the usual three perspectives -- numeric, visual (graphic), and algebraic -- to help understand phenomena involving mathematics. In addition, we use hands-on experimentation. We chose this module as the starting point for our tour because it illustrates the use of hands-on experimentation and our emphasis on using mathematics as a language to express, communicate, and reason about real-world phenomena.
One of the central themes of this modeling course is Newton's Model of Cooling. As with all models, the limitations of Newton's Model are as important as its successes in describing real-world phenomena. We begin with an experiment, described in this module, intended for high school or middle school science or mathematics classes. Read the introductory material in the other window up to the point where you see a row of graphing calculator icons. Then return to this window.
The graphing calculator icons link to material for the Texas Instrument CBL (Calculator-Based Laboratory) and the TI graphing calculators. The TI-CBL is a powerful, flexible, robust, and inexpensive data collection device that is used together with a variety of probes or sensors and one of the TI graphing calculators to collect data. In this experiment we use the TI-CBL together with three temperature probes to compare two cooling cups of hot water -- one in calm air and the other in windy air.
Click on one of the TI graphing calculator icons to open a third window. You will need to arrange your three open windows so that they overlap and you can move easily back-and-forth among them.
Skim through the new window to get an idea of what it contains. This window explains how to use the TI-CBL together with the calculator that you chose. Notice this window has buttons that link to additional help -- for downloading programs and for uploading data -- and it has a link to the TI Graph Link program. This link uses free software from Texas Instrument that enables us to transfer data and programs back-and-forth between the computer, the calculators, and the Web. If your browser is already configured to use the TI Graph Link programs and if it has a TI Graph Link cable ($35.00) then you can see how this works. Click on one of the TI Graph Link program buttons. The browser will launch the TI Graph Link program and hand it a TI graphing calculator program that came in over the Web. If you plug the Graph Link cable into the appropriate calculator and click the send button in the Graph Link program window then the calculator program will be transferred to the calculator. These instructions vary for the different calculators, so you may need to click one of the help buttons for additional help.
When you are done looking at the TI graphing calculator material close the Graph Link window and the calculator instruction window. You should be back to two open windows, this one and the module Prolog for a Winter Day.
Now look through the rest of this module. Although we highly recommend collecting your own data using the TI-CBL, this module and other modules include data that you can use if you don't have the necessary time or equipment.
Notice the postscript at the very end of the module. Click the link interesting email from the real world to see an example of connections between the classroom and the workplace.
We continue our tour by looking at a sequence of modules on differential equations. These modules are intended for use in a calculus class or a differential equations class. We choose these modules in part because they illustrate one way in which we use Java applets. Click here to begin with the second module on differential equations. Notice that the navigation frame at the top looks a little different than the previous navigation frame. The row of buttons or links is very similar but it enables you to navigate easily around the Calculus "book" rather than the Modeling "book."
At this point in the course, students have some experience with Newton's Model of Cooling and they have completed a general introductory section on differential equations. This module introduces slope fields and begins the graphical or qualitative analysis of differential equations. Skim this module until you see the picture below.
In the actual module (but not in the picture above) this picture is a Java applet. This applet provides very limited interaction. We want to convey the idea that a differential equation describes a slope field -- that is, at each point (t, y) the differential equation gives us the slope of any solution passing through this point. Users can see this by clicking at various different points and seeing a line with the appropriate slope appear. You can even construct a rough "Euler's Method" approximation to the solution of an initial value problem by clicking on the initial value and then clicking at the end of the slope line that appears and then clicking on the end of that slope line, and so forth as shown in the picture below.
We think of the slope field for a differential equation as describing the currents in a lake and we think of solving an intial value problem as dropping a cork in the lake and determining its trajectory as it is carried along by the currents. Immediately below the first Java applet, another Java applet illustrates this idea. You should try these two Java applets before going on.
Click here to skip the rest of this module and go on to the next module.
This module is the first of two modules on Euler's Method. The previous module built up the picture of a differential equation as a slope field and in this module we exploit this picture to visualize Euler's Method. Skim this module until you come to the words -- " Click here to open a new window ... " -- beginning a paragraph. Click on this link to open a new window with a Java applet. Arrange your three windows so that they overlap and you can move easily back-and-forth among them by clicking on the exposed portion of an inactive window to make it active.
Now work through the rest of this module using the Java applet as described in the module. When you are done looking at this module close the window with the Java applet and return to this window.
The next module in this series introduces the formulas for Euler's Method.
Click here to continue the guided tour with an example illustrating how we use the Internet to obtain data. This module is from the modeling course we looked at earler. In this module we study exponential models. Our first example is population growth. At the end of the module we look at another example using the Texas Instrument CBL.
Skim through this module until you come to a series of links taking you to different data at the United States Census Bureau site. Students compare actual data with predictions made by exponential models. Notice one of the links takes students to general instructions for creating abstracts from the 1990 United States Census. When you are done looking at this module be sure to close any extra windows.
The next stop on our guided tour looks at the way we use both Java applets and a computer algebra system AND it looks at one of the most important ideas we are trying to get across -- functions are not always formulas. We work with functions that are defined by data, by initial value problems, by physical experiments, and other non-formulaic means. In the module we look at next, we look at functions defined graphically.
Click here to open a section from our integrated course Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World. This section introduces Fourier Series as a means of analyzing periodic functions.
Skim through this section until you reach a link for Nancy's blown speaker. This link leads to an interesting application of Fourier Series. If you have Mathematica and have configured your browser to use Mathematica as a helper application, you might want to follow this link and look at the associated Mathematica notebook. Mathematica has a procedure that Plays a function. Using this procedure, users can hear the high frequency component introduced into a signal by "clipping." This phenomenon often causes blown speakers. This feature of Mathematica is especially nice because we can find the high frequency component mathematically and then confirm it with our own ears.
Continue skimming through this module until you reach the paragraph beginning "Click here to open a new window with a Java applet ..." Click the link to open a new window with a Java applet and then arrange your windows so you can move back-and-forth among them in the usual way.
Play with this applet as described in the module. This applet enables users to easily experiment with the Fourier series for step functions and other functions that are easily described graphically. Other applets throughout The Connected Curriculum Project encourage students to work with graphically defined functions in this way. This is a good example of one use of Java applets. But Java applets are not the answer to everything. Students working with Java applets like this one don't see all the details; they are limited to the kinds of interaction programmed into the applet; and they are not learning the same kinds of generally useful skills as they would with a computer algebra system.
When you are done with the Java applet, close its window. You may want to look at the material for this module for one of the computer algebra systems. You can se the material for the TI-92 by clicking on the TI-92 icon in the navigation frame. If you have Mathematica or Maple and your browser is configured to use one of these computer algebra systems then you can look at the appropriate material by clicking on the appropriate icon in the navigation frame. We have not yet written the MathCad material.
Notice that the links to the computer algebra systems are usually in the navigation frame at the top of the window. Because the navigation frame is fixed, users can launch their computer algebra system whenever they want. We expect students to use their CAS at their own initiative whenever it is appropriate. We often supply notebooks or worksheets with material for a particular module but students are also expected to use the more general tools of their CAS without specific help.
The next stop on our guided tour illustrates the use of animations. We believe animations should be used sparingly because motion on the screen, while initially striking, is distracting. In this module we use animation to help students understand the differences between multiple integration and iterated integration. Skim though this module to see how animation is used. Note that you will need to follow the MORE link at the end of the first part of this module.
One of the most exciting developments in education is Virtual Learning Environments -- computer-based simulations that exploit the power of virtual reaility. You can get some idea of the potential of this technology by looking at the newest electronic games. More importantly, NASA, the University of Houston, and others are developing very striking simulations. But virtual reality is NOT reality and we will need to maintain a balance between real reality and virtual reaility. Furthermore, it is extremely important that students know the differeence between the two and that they understand the modeling behind virtual reality. The next stop on our guided tour looks at a module on Boyle's Law. This topic is frequently used in courses and there are many Boyle's Law applets. In fact, at a recent meeting one participant said "If I see one more Boyle's Law applet I'll ... " We include our Boyle's Law module because we think it illustrates two of the ideas that permeate The Connected Curriculum Project.
You may want to skim through this module.
Where we are and where we are going
Our goals are very ambitious. Eventually we would like to write a massive amount of interconnected material covering mathematics and the physical, life, social, and applied sciences. Our strength is in mathematics and most of our current material is in mathematics.
We have enough material for a semester long course in Mathematical Modeling aimed primarily at preservice and inservice teachers. This course has been taught twice and the second time several students took it off-campus via the Web. We have roughly one-third of the material for a course integrating Multivariable Calculus, Linear Algebra, and Differential Equations, and we are in the third draft of a book for that course. We have some material for calculus, although that material was written over two years ago and we have learned a lot since it was written about how to use this new medium. The calculus material is scheduled for drastic revision. We also have a few sample modules for a course entitled Connected Before Calculus.
In addition, we are developing a library of supporting material -- help modules of various sorts for students. We plan a series of "case studies" -- extension applications that cut across disciplines and that students return to at various times during their academic careers. Finally, we have material written for instructors who would like to add material of their own to The Connected Curriculum Project, either for their own use or for broader use. Ultimately we hope hundreds of authors will contribute to the project.
This concludes the guided portion of the tour. You should close the other window now and then click on the "web" icon below to return to our home page. We encourage you to browse and also to send us your comments by email. Be sure to visit our sister sites at Duke University and California Polytechnic University, San Luis Obispo.
