Excerpted from the file Calculus and Mathematica Teaching Guide.
"C&M has brought to me something I thought I would never have-- a bit of understanding of the important relationships in mathematics. Some people (say in my [traditional] calc I class) were able to look at the texts and seem to understand, but I have been frustrated very many times by not being able to. I guess that's why I signed up for this class. I used to not like math. But I was just being a product of the way I was taught."
" This course ... has an advantage over textbook courses in that I am given a lot more freedom in how I want to learn." - PreviousCalculus&Mathematica students
Calculus&Mathematica presents a complete rethinking of:
Unlike any other mathematics course, Calculus&Mathematica attempts to actively involve students in their own learning by putting them in a position to acquire mathematical ideas visually. This means that instead of hanging technology on the end of the learning process, Calculus&Mathematica uses technology to initiate the learning process. It accomplishes this through its electronic interactive text, as described above, and it invites students to write about the mathematics they are doing as they are doing it. Typically, Calculus&Mathematica students learn by doing in the lab and attend no lectures.
Prime characteristics of a successful teacher of Calculus&Mathematica are:
Prime characteristics of a unsuccessful teacher of Calculus&Mathematica are:
In four years of teaching of Calculus&Mathematica at Illinois, four teaching assistants have tried unsuccessfully to teach Calculus&Mathematica. Each tried to lecture heavily and each felt the need to supplement Calculus&Mathematica with material from the traditional course. Each seemed reluctant to use Mathematica in the presence of their students. Their students complained that their Calculus&Mathematica experience did not match the experience of students in other sections, that the lectures imposed on their time unjustifiably and that they were really taking two courses-one in the lab and one in the classroom. Instructors who want to teach the traditional course should try to teach it instead of trying to patch up Calculus&Mathematica.
First year mathematics teaching assistants have proved to be a particular problem because many of them want to teach the calculus course they took. Seasoned teaching assistants who have become aware of the futility of trying to teach the traditional course are usually good bets to teach Calculus&Mathematica.
Actually, the best teachers of Calculus&Mathematica are the students themselves working together in the lab.
You should be spending some of your spare time in the lab familiarizing yourself with Mathematica and the Calculus&Mathematica interactive text which resides on the machines. Take time to get into the Calculus&Mathematica philosophy of writing and plotting with empirical and heuristic explanations. You should not expect to walk into your first class cold and to be able to keep up with your students. And you will have no students unless you have some sort of lab. For more on this, see the next section for advice on setting up a lab.
Some new Calculus&Mathematica students and instructors are convinced that the students must study Mathematica as a programming language before they can begin their study of calculus.These students have condemned themselves to needless misery. Calculus&Mathematica is written so that students learn as much Mathematica as they need on a "just in time basis." They learn it gradually and always in context. The going might be a bit rough for the first couple weeks as students adapt to an unfamiliar environment, but about 90% of Calculus&Mathematica students in the field tests who adjusted to Mathematica this way reported that they were comfortable with Mathematica by the third week.
Students should copy, paste and edit as much as they can. Encourage students to look in the electronic Basics and Tutorials for some Mathematica code or text that they can adapt to their current work and then encourage them to copy it, paste it, and edit it to suit the problem they are working on. Let the students know that they don't have to apologize for this. There is no reason for students to "tough it out" when they can copy, paste, edit and then go on to a new challenge.
Students who have learned to use Mathematica this way in Calculus&Mathematica have never come close to participating in a revolt similarly to the massive revolt in 1993 by calculus students at the University of Pennsylavania against the Maple System.
Many Calculus&Mathematica Give It a Try problems lend themselves to the copying, pasting and editing approach. These problems are present to encourage the students to become familiar with the ideas in the Basics and Tutorials. Many other Give It a Try problems don't lend themselves to copying, pasting and editing. After the student is finished with the copying, pasting and editing problems from a given lesson, the student is likely to have enough experience to tackle the Give It a Try problems that don't lend themselves to copying, pasting and editing.
Remember Calculus&Mathematica is a math course and not a programming course.
The Give It a try problems are the heart and soul of Calculus&Mathematica. Great care has been taken in the design of these problems. You won't find problems copied out of traditional calculus texts. In fact, you won't find many problems that emphasize just routine manipulation. There are some of these, of course, because there is a minimal core of hand calculational competency required of calculus practitioners.
The overriding issue in the design of a problem is the calculus content of that problem. In this day of revisionist calculus courses, one is tempted to design problems that are exceptionally interesting to professors, can be done with the calculus at hand, and which might play well to audiences watching calculus instruction change from the outside. There are such problems here, but Calculus&Mathematica has one primary rule: Calculus content must be the primary feature of a Calculus&Mathematica problem. In fact,many problems in Calculus&Mathematica are present solely because they set up visualizations and calculations that point the student in the direction of underlying ideas.
Perhaps the most amazing quality of Calculus&Mathematica students is their ability to discuss calculus and to write about calculus. This does not come by accident. Included in Mathematica is a word processor that students use to explain reasons for an upcoming calculation or plot and that students use to assess the meaning of the result of a calculation or a plot.
Tell your students to explain what's happening in their own words and not in the rigid style they perceive as mathematical discourse. In other words,encourage your students to talk about calculus the way that mathematicians and scientists really talk about calculus and not to try to talk about calculus the way that most authors of calculus traditional calculus books try to write about calculus.
Often Calculus&Mathematica asks students to announce and explain what's going on. At first, some students resist this because, as one new C&M student said, "In mathematics courses, students are not supposed to have opinions." Once the students find out that you are interested in what they have to say and that you are not going to insist on the precise phrasing, they get into the swing of things. If you don't make the mistake of demanding precision and rigor before an idea has crystalized in the students' minds, then you'll be very pleasantly surprised with what you get in return.
The students seem quite excited to get answers and explain them in their own terms. With a little guidance (found in Basics and Tutorials) they soon enough announce correct results based on their experience. Granted, they cannot always give mathematical proofs of these facts, but they are convinced, being the originators of the statements and fully committed to their validity.
Insist on good descriptions of the work done. On the other hand, Calculus&Mathematica students understand concepts first visually and computationally. So don't insist on precise writing, but do insist on clear writing. A student who can write completely coherent descriptions of what has occurred in a computation or plot has done what you want — learned the material.
This has been surprisingly easy — mainly because Mathematica Notebooks set up a natural environment for writing. To get students to write, just tell them that on each assignment they will get two grades — a math grade and a presentation grade. You'll be surprised at how good the writing turns out to be.
Calculus&Mathematica students want some tests. They want some comparison of their "standard" calculus skills with those of their peers. They want to be reassured that they will be adequately prepared for courses which follow and use calculus.
Students have plenty of chance to show ingenuity in their work on the Give it a Try assignments. Tests and quizzes should deal with matters of calculus literacy. Quizzes are also useful for the students to help in perfecting their hand skills. (Usually hand skills come on line after machine experience.)
Problems for tests and quizzes should be lifted off the Literacy Sheets. Light modification of these problems is fine. Short quizzes should be constructed to signal to the student serious gaps that must be filled. Some instructors teaching a five hour course find it benefical to give quizzes on a day other than Wednesdays, thus reserving Wednesdays for discussions only.
At Illinois, we give two one hour "Literacy Tests" each semester. These tests are given in classrooms away from the machines. Calculators are allowed. Oral final exams are a real possibility and have been used successfully by several mathematics instructors.
NEVER SHOULD STUDENTS BE TESTED ON THEIR KNOWLEDGE OF MATHEMATICA INSTRUCTIONS.
Leave this to the computer science department.
Almost every Calculus&Mathematica class currently taught is run as a lab course. Instructors don't lecture. Calculus&Mathematica is treated as a shared challenge for faculty and students. Students are given lessons and assignments. They spend their time in the lab working on the lessons, asking lab instructors and instructors questions and sharing their insights with their peers. In the lab, the teacher functions as a shopkeeper, ready to answer questions and to volunteer help. The result is students working during their assigned class time and for another hour or two each day. Once a week the instructors meet their students outside the lab in a regular classroom to talk generally about what is going on in the current lesson, the Literacy Sheets and, perhaps, to give an on-paper quiz. Here the instructor behaves as a coach. New material is never presented in the discussion hours.
The center of the course is the Give It A Try problems. Everything is here: the experiences introducing ideas and topics, the challenges to intellect and patience, and the excitement of beating the course and solving a really difficult problem. This is where the students spend most of their time and energy. That's what you want.
The pace for the course is just about one C&M notebook per work week in a three semester hour course. In classroom testing, we have learned the hard way that regular homework submission dates are very important. Making the length of assignments reasonable pays big dividends. There is a temptation with a nearly perfect problem set like the one provided here to ask students to do all of the problems. The result, though, when students encounter long assignments, is that they get frantic and concentrate only on finishing the assignments rather than savoring them. They do savor the problems when they come at a reasonable rate. You will probably need to discover how many problems you can assign each week. You should also be flexible at first.
By the way, it's sometimes difficult to strike the appropriate balance of making the length of assignments suitable and giving enough of a challenge. Whatever you do, keep them a bit strained and working hard. That's when they have the most fun and when they learn the most. But don't break their backs.
At the current writing, no school that has initiated Calculus&Mathematica has considered it to be a failure. What's remarkable is that at nearly all locations, the teaching of Calculus&Mathematica has evolved in the same way as it has evolved at Illinois and Ohio State. Here's how it goes:
Introduce yourself in the classroom assigned, call the roll, bring the students to the lab and get them off and running with Lesson 0 (The Feel of Mathematica). Circulate around the lab helping students individually.
Meet in the lab and make an assignment from the Give It a Try section of the first lesson to be due in one week. Tell the students that they should bring up a copy of the Give It a Try, cut out the problems not assigned and save under their own name. The students work on the resulting files until they are ready to turn them on the network. Have lab instructors show them how to turn in their work.
One tip: Tell students not to save from Mathematica directly to a floppy. Instead they should work on a file on the machine's hard drive and copy it to a floppy after they have quit Mathematica.
Hang out in the lab during your scheluded class meeting time.
Act like a good shopkeeper. Encourage questions by circulating around the lab, looking over the shoulders of the students and asking them how they are doing. If they seem to need some help, volunteer some. If you see a student staring at error messages, provide help immediately. Be on the lookout for a student who is stuck. Remember that when you answer a question that a student asks, you have that student at the ultimate teachable moment.
Some students will choose not to work in the lab even on class days. Many of these students have their own machines or have access to better machines than found in the lab. Worry about them only if they are not turning in their work.
Meet in the classroom for a discussion centered on student questions. If there are not enough questions, call on students to answer questions from the Literacy Sheets. Don't be afraid to bring the computer on a cart into your discussion sections. If you are uncomfortable running the machine, call on an eager student to run it for the class. Resist the temptation to interfer with the students' learning by giving "introductory lectures." Instead, discuss material that the students should already be familiar with.
When the environment for learning is so enjoyable, it's impossible not to try your best and to build a relationship with everyone around you.
Techs support both the lab machines and the software used in this program.
In the event of a problem, send an e-mail to tech@cm.math.uiuc.edu.