## Abstract: WeBWorK -- Math Homework on the Web

WeBWorKis an internet-based system for generating and delivering homework problems to students. It increases the effectiveness of traditional homework as a learning tool by:These are important educational advantages, and WeBWorK is a tool designed to conveniently deliver these advantages to large numbers of students. It is currently be used by more than 1200 students at the University of Rochester and at Indiana University.

- Providing students with immediate feedback on the validity of their answers and giving students the opportunity to correct mistakes while they are still thinking about the problem. As one student said,
"I can fix my mistakes while [the] problem is fresh in my mind."- Providing students with individualized versions of problems which means that instructors can encourage students to work together; yet each student must develop an answer to his or her own version of the problem.
Readers can try WeBWorK for themselves by connecting to actual courses at http://www.math.rochester.edu/webwork.

Authors:

Prof. Michael E. Gage,

University of Rochester

Department of Mathematics

University of Rochester

Rochester, NY 14627

E-mail: gage@math.rochester.edu

Phone: 716-275-9424

Prof. Arnold K. Pizer,

University of Rochester

Department of Mathematics

University of Rochester

Rochester, NY 14627

E-mail: apizer@math.rochester.edu

Phone: 716-275-7767

## WeBWorK -- Math Homework on the Web

WeBWorK is an internet-based system for generating and delivering homework problems to students. It increases the effectiveness of traditional homework as a learning tool by:

- Providing students with immediate feedback on the validity of their answers and giving students the opportunity to correct mistakes while they are still thinking about the problem. As one student said,
"I can fix my mistakes while [the] problem is fresh in my mind."- Providing students with individualized versions of problems which means that instructors can encourage students to work together; yet each student must develop an answer to his or her own version of the problem.
These are important educational advantages, and WeBWorK is a tool designed to conveniently deliver these advantages to large numbers of students. The pilot version of WeBWorK has proven to be very popular with faculty and students at the University of Rochester. First used in a single course with 29 students in the fall of 1996, WeBWorK currently serves 750 students in seven courses ranging from pre-calculus to second semester calculus to a physics course. WeBWorK has been used successfully at Indiana University and is being evaluated for use at several other universities including SUNY Stony Brook.

At the University of Rochester, WeBWorK has proven to be a highly cost-effective way to help students get the most out of their homework assignments. It has the potential to become an important tool for high school and college science, mathematics, and engineering . Readers can try WeBWorK for themselves by connecting to actual courses at http://www.math.rochester.edu/webwork.

Each WeBWorK problem set is individualized; each student has a different version of each problem. Using a standard web-browser, students view or print a copy of the assignment, do as many problems as they wish and then log onto the internet and enter their answers. The WeBWorK system responds by telling them whether an answer (or set of answers) is correct or incorrect but does not reveal the correct answer. However, at the instructorÕs discretion, students are allowed to attempt a problem as many times as they wish until the due date. WeBWorK doesnÕt explain mistakes, but it does reassure students who are doing problems correctly and lets other students know when they need to seek help from fellow students, the TA or the professor. Not all of the problems in a homework set need to be done at once. Students can log off, think about the homework for awhile and log back in to finish the remaining problems.

The immediate feedback and the opportunity to correct mistakes are keys to WeBWorKÕs educational effectiveness, and students are quick to appreciate this fact:

"I can fix my mistakes while [the] problem is fresh in my mind."

"... it gives us a chance to correct our own mistakes which helps me learn better."

"... it makes no sense to do a problem wrong and think it is right!"

Students have a strong incentive to, as one student wrote,

"work on the problem until [they] get it right,"and our experience is that this is just what they do. We consistently find that more than 50 percent of the students continue to work at their WeBWorK problems until they have answered every problem correctly. Before the use of WeBWorK, we would assign standard homework problems, but we did not have the manpower to grade more than a small percentage of the assigned problems. This sometimes reduced studentsÕ incentive to complete homework assignments. With WeBWorK, every problem is automatically graded, and we now find that almost all students do almost all the WeBWorK problems. In this way, WeBWorK encourages mastery of the material, not just familiarity. Independent of anything else, this is a major improvement.WeBWorK does not seek to replace conventional teaching; rather, it increases the effectiveness of the assigned homework. Students using WeBWorK know when they are succeeding and when it is time to seek help. The individualized nature of the problems means that students can work together, and yet each of them still has to come up with an answer to his or her own version of the problem. Instructors and teaching assistants also report that students now ask more focused questions and collaborate more effectively with their peers.

Using WeBWorK as we do at Rochester (e.g., usually giving students unlimited attempts to do a problem correctly) promotes mastery of the material. By limiting the number of allowed attempts and/or presenting the student with a different version of a problem each time they attempt a problem, WeBWorK becomes more of an evaluation or testing engine. Most web-based authoring/testing systems are of the latter type. WeBWorK, however, gives instructors the flexibility to use either or both approaches.

Innovative features of WeBWorK: Although the last four years has seen the development of many web based authoring systems for educational purposes, WeBWorKÕs innovations make it uniquely appropriate for authoring mathematics problems, particularly problems for calculus, trigonometry and upper level secondary mathematics courses. The following innovations stand out:In addition to these features which make WeBWorK uniquely suited for mathematics education, WeBWorK provides the standard features found in most CGI/database education solutions.

- Advanced mathematics problems can be authored and displayed and printed with typeset quality.
- WeBWorK produces a similar but individualized problem for each student. This makes WeBWorK particularly effective in a group learning setting, since students can collaborate without copying.
- Flexible mechanisms are available for handling numeric, symbolic, and string answers. Numeric answers may (at the instructorÕs option) allow elementary functions such as 3*sin(pi/2)+ln(e^2) which WeBWorK will evaluate, or the instructor can require that the student enter a numeric answer such as 5. Symbolic answers allow for questions such as: enter an anti-derivative for x2sin(x3). Some correct answers are .3333*sin(x^3) or (1/3)*sin(x^3) + 7; however (cos(x))^2 + sin(x^3)/3 + (sin(x))^2 is also correct and WeBWorK will accept that too. WeBWorK will accept any correct answer. String answers allow for T/F, matching, multiple-choice, and short answer questions.
- Graphs of functions can be generated "on the fly" by a single statement, enabling one to ask individualized questions involving graphs of each student.
- Problems can also include Java applets and javaScript code.
- The PG language in which most WeBWorK problems are written is built on the popular scripting language Perl. Mathematical formulas can be written in TeX (or LaTeX), the mathematical typesetting language, and as with TeX, ease-of use- can be added in the form of macro packages. Even complicated numerical programs can be included to help check the answers to problems. Novice problem writers will use these macro packages to write problems, while expert problem writers can extend the capabilities of the language by writing new macro packages. As the new HTML standard syntax for specifying mathematics emerges, it can also be used in writing problems. No change to the WeBWorK system will be necessary.
The flexibility of WeBWorK allows its use by instructors with very different teaching styles. For the first year calculus courses at Rochester we have set up WeBWorK so that:

- Students receive immediate feedback about the accuracy of their answers.
- Students can access WeBWorK from any computer connected to the internet and instructors can use any computer and browser for management of the assignments.
- Students with disabilities can benefit from immediate feedback outside of the traditional classroom setting. Our pilot WeBWorK has already proved useful in this way.
- The instructor or TA can view the precise version of the problem seen by each individual student, making it easy to answer specific questions from a student via e-mail.
- All pages have a feedback button which sends an e-mail message directly to the instructor(s) (or who ever the instructor designates). Students find this a convenient way to communicate with their instructor usually requesting help on a particular point.
- Problem sets are graded automatically and the results are easily exported to and imported from spreadsheet programs such as Excel.
- The instructor can easily find out the current progress of a class or an individual in correctly completing any assignment.
- Instructors can send e-mail to an entire class reporting individual homework grades, exam grades etc.
- WeBWorK allows great flexibility in administering individual homework. For example an individual student can be granted an extension on an assignment without granting an extension for the entire class.

- Students can attempt a problem as many times as they wish until the due date (unless the instructor desires to place a limit on the number of allowed attempts).
- Problems can have individualized solutions and/or hints (e.g. solutions can use the same individualized constants each student sees).
- After the due date, students can review the homework, including the answers expected by the instructor. Solutions to problems, if provided by the instructor, are also available after the due date.
## Bibliography

"New software improves teaching of calculus," NewsLinks, the newspaper of International Schools Services, Vol. XVI, No. 4, 1997.

"Technology Across the Campus, University of Rochester," Syllabus Magazine, March 1998 (http://www.syllabus.com/CS03_02_98.htm)

"WeBWorK -- Web based homework delivery" by M. E. Gage and A. Pizer, talk delivered at the Special Session on Mathematics Education and the Internet at the AMS meeting in Manhattan, Kansas on March 28, 1998.

"Rochester Four Years Later: From Crisis to Opportunity" by Douglas C. Ravenel, in Notices of the AMS, Vol 46 Number 8, September 1999.

## Funding

Profs. Gage and Pizer are partially supported by NSF grant DUE-9950567.