The literature shows that the speed of feedback is very important in the effectiveness of homework. As Grant Wiggins has pointed out, if you went to a batting cage to take some swings, but only found out a week later which times you hit the ball and which times you missed, it would be very difficult to improve. Online homework provides a means to give students feedback on their work immediately.
Because of the special nature of mathematical questions, it is possible to design online homework with many more features than the usual multiple choice questions. Dr. Andrew Bennett developed an online homework system specifically for mathematics (though Steve Warren and his students in Electrical Engineering have now adapted it to their needs). This system has the following features:
- Answers can be entered in a variety of forms, including not just multiple choice or numerical values, but formulas and graphs as well.
- The system can automatically grade the homework, including recognizing multiple correct ways of writing a formula and even properly handling formulas with arbitrary constants such as indefinite integrals.
- The system can generate tens of thousands of different assignments with similar questions, where the specific numbers and functions randomly varied. Each student can take as many attempts on an assignment as they wish. The current record is 200 attempts on a single assignment, which seems to have been a student quickly submitting wrong answers to see how long it would take for the assignments to repeat. The record number of attempts for a student genuinely trying to solve the problems is in the 40s.
- The system provides significant feedback to help students learn. We have experimented with including videos in the feedback as well as written feedback. (See the research page for more details.)
The system is now available for Math 100 College Algebra, Math 150 Trigonometry, Math 220 Calculus I, Math 221 Calculus II, Math 240 Elementary Differential Equations, and Math 320, Math for Elementary School Teachers. After several years of experimentation, we have found that the following format is best for student learning.
- Students sign in and are given an individualized problem set, which means they cannot copy off of other students as on typical written assignments.
- Students may print out their problems, and if they have entered work that they are not yet ready to grade, they can save it for later. They will return to the same problem set when they sign back in.
|Problems can have numerical, graphical or algebraic solutions.
- The first time students submit their answers for grading, they are told what they have right and wrong, and they are given a chance to correct their errors.
- The second time students submit their answers for grading, they are told the correct answers and are also given links to see how to work the exact problems on their assignment (with the exact same numbers, not just a generic example from the text).
|If students get a problem wrong twice, they get feedback on how to solve their specific problem.
- At this point, the student can go back, generate another problem set, and try again. They receive their best score over all attempts on each assignment. Hence, students are graded on mastering the material, whether they do so on their first attempt or their fifth.
|Students can work another problem until they get it right.
By giving students different problems and changing the problems after two attempts, the class focuses on the general technique and not a specific problem. By providing students with a chance to fix their errors for full credit, we are prompting students to reflect on their work. By providing answers and full feedback after a second wrong answer, we give students who are lost substantial help.