| Philip B. Yasskin | Michael Stecher | |
| yasskin@math.tamu.edu | stecher@math.tamu.edu | |
| G. Donald Allen | Maury Rahe | |
dallen@math.tamu.edu
| |
rahe@math.tamu.edu |
|
Three important parts of an on-line math course are exploration, practice and testing. All of these involve presenting the student with problems, providing an appropriate level of guidance as the student solves the problems, receiving intermediate results and/or answers and grading them either for self evaluation or course evaluation. As part of our WebCalC, Calculus I course,[1,2] we have found that the more ways the students interact with the material, the more they pay attention to the web pages and the more they think about the material. So we have developed many types of interactive examples, exercises, tutorials, drills, quizzes and exams. Some of these are presented in this paper. In the process, we will discuss some lessons we have learned from teaching the course.
The main portion of our WebCalC, Calculus I course may be viewed only using Scientific Notebook, which is a combination of (1) a WYSIWYG scientific word processor with excellent mathematical typesetting which stores its files in LaTeX, (2) an interactive computer algebra system using the Maple kernel, (3) a web browser which displays LaTeX files within SciNotebook and calls a standard browser for other files and (4) a quiz and exam generating program called Exam Builder. After installing SciNotebook on a Windows 95, 98 or NT machine, click on File and Open Location and type
The rest of the paper is divided into sections which discuss individual types of student activities. These are:
In this form of activity, the students are asked a question and expected to give an answer. This is typical of the exercises which appear at the end of a section. In a book, the answers to the odd problems are usually at the back of the book. Often, the students complain that they don't have the answers to the even problems and cannot tell if they are right. In our web text, we also have exercises at the end of each section, but we decided to give the answers to all the problems since the students may not have an instructor to ask for help. At first, we put the exercises on one page and the exercises with answers on another page. For example, the exercise page for derivative rules might contain the problem:
Differentiate:and the corresponding answer page would include
Differentiate:However, we soon found that many students linked directly to the answer page without first trying to do the problems themselves. So we decided to combine the two pages, putting the answers into pop-up notes. Now the above problem would appear as:
Differentiate:When the student clicks on the
, a small note pops up which contains the answer:
The students may still not be doing the problem but at least they need to pause for a second and perhaps think as they click on the answer.
Eventually, we would like the above problem to appear as
Differentiate:Clicking on the...
would pop up a hint such as
Clicking on the
would pop up the answer and clicking on the would pop up the complete solution as
The pop-up notes are a substantial benefit of a web book over a paper book.
These simple question and answer problems appear not only on the exercise page, but also in the regular text, directly after many of the examples. We find it useful to immediately follow an example by an exercise on the same technique.
The main drawback to the simple question and answer problems is that they are fixed; there is only one problem. If the student gets it wrong the first time, there is no second problem for the student to try again. It is the same problem when the student reads the page the first time and for review. Similarly, the exercise page has a finite number of problems. This well-known problem with printed texts is easily remedied on-line. One can use a large database of problems or use randomly generated problems. We have opted for the latter because it provides more questions and because it is easy to produce random questions with a small amount of programming using SciNotebook's Exam Builder. Thus, after many of our examples, there is a pop quiz which usually consists of a single multiple choice problem on the same topic as the example. For instance, the above problem on the quotient rule might be coded as
Differentiate:where a, b, c, d and n are random numbers which take on different values each time the student looks at the pop quiz. If a, b, c, d and n each have 10 values, there are 100,000 versions of this quiz. The multiple choice answers are also coded in terms of a, b, c, d and n. When the student selects an answer, the quiz is immediately graded: ``Instant gratification for our TV viewers.'' We can also include hints and a full solution.
The randomly generated multiple choice quizzes are not limited to a single question. After some examples and at the end of some sections there is a multi-question multiple choice quiz with randomly generated questions and choices. Every time the student opens the quiz, there is a new set of problems. After answering the questions, the student can click on a button to have the quiz graded. If an answer is wrong, the student can go back and try again or ask to see a hint, the answer or a full solution. We can also have the student's grade report emailed to the instructor. As an added bonus, the instructor can print an instance of the quiz to administer to the whole class. An answer key is also generated. In fact, the security minded teacher could print a separate quiz and answer key for each student in the class. More realistically, in a distance learning situation where the students work at their own pace and come to a testing center to take their quizzes, it may be necessary to have each student take a different quiz.
The simple question and answer format is fine for a quiz but it doesn't give much guidance when the students are just learning a concept. For that the students need a tutorial where they are guided through each step of the solution. For example, when the students are learning the method of substitution, the following series of questions are useful. Clicking on each gives a pop-up note with the answer, as shown below each question:
To compute the integralby the method of substitution, you should define u as: ...
With u defined as above, du is: ...
With u defined as above, the integral
The above u-integral is: ...
Substituting back, the integral
The above fixed tutorial can also be given in a randomly generated form by coding it as:
To compute the integralwhereby the method of substitution, you should define u as:
With u defined as above, du is:
With u defined as above, the integral
The above u-integral is:
Substituting back, the integral
X is randomly chosen as one of the variables x, y, z or t;The answers can be coded in pop-up notes or in multiple choice form in terms of the random variables X, M, a and N.
M is one of the numbers 2, 3 or 4;
a is a random number between 2 and 9; and
N is a random number between 4 and 99.
A remaining drawback in all of the above randomly generated quizzes and tutorials is that all of the answers appear either in pop-up windows or as multiple choice answers. SciNotebook's Exam Builder has the ability to allow students to enter extended answers. However, at present, these must be graded by hand.
To remedy this situation, we have written randomly generated drills which appear as fill-in-the-blank forms which may be viewed using any browser (Netscape works best). (Technical Details: The drills are written as Perl CGI scripts which generate HTML code to display the questions and answers in the browser and which call Maple on the server to generate and grade the problems.) These drills are essentially the same as the random quizzes and tutorials discussed above, except that the students are expected to type the formulas for their answers, and further, the drills are more interactive.
For example, in the drill on Shifting Functions, the students are shown a plot such as the one at
the right (which is
)
and told it is the graph of one of the basic functions
(x,
,
,
,
,
,
)
which has been shifted left or right and up or down. The students are asked to type in the formula for the function. The drill is more interactive than the quizzes and tutorials because (1) the student can click in the plot to get the coordinates of the vertex of the parabola and (2) if the student gives the wrong answer (say
),
the graph of the wrong answer is added to the plot in a different color, as shown. The students quickly learn how the signs control the direction of the shift, much more quickly than reading it or hearing the instructor say it.
In summary, a wide variety of interactive quizzes, tutorials and drills help students learn in a web-based course, and we feel they are essential tools for a successful on-line course.
[1]Allen, Stecher and Yasskin, ``The Web-Based Mathematics Course," Syllabus Magazine, Vol. 12, No. 4, Nov./Dec. 1998, to appear.
[2]Allen, Rahe, Stecher and Yasskin, ``WebCalC I," this volume.
Philip Yasskin