Better Homework Through Technology
By Mark Schmitt
As a high school algebra teacher my job is to help students learn the mathematical skills they will need to be successful in later mathematics and science courses. To support student learning, I assign homework regularly.
Homework might simply be problems taken from the textbook or it might be original problems assigned on worksheets. There are times, however, when I use computer-based homework because it adds an important dimension to the homework process.
I expect a number of different benefits from every homework assignment. First, I expect that my students will practice the skills demonstrated in class and through that practice, become more adept with the skills. I also expect homework assignments to highlight, for both students and me, areas of proficiency and deficiency. Ultimately, I expect that homework will help instill a sense of personal responsibility in my students.
Redefining Practice
Many students think that "practice" merely means attempting the problems. While it is true that a student cannot practice without attempting the problems, it is not the case that merely trying the problems constitutes practice. Practice also involves assessing and refining the attempts that are made.
By reflecting on their work, students are able to discover where their strengths and weaknesses lie. The self-awareness that results will help them avoid their personal common errors and help them recognize those errors when they make them. When this reflection is done among peers, students learn to recognize more and less efficient strategies for problem solving.
While it is true that a student cannot practice without attempting the problems, it is not the case that merely trying the problems constitutes practice. Practice also involves assessing and refining the attempts that are made.
Reflecting on solutions also yields thought-provoking questions both in class and outside class. Students notice similarities and differences between problems that can lead them to deep insights about mathematics. When these questions come up, everyone benefits.
When students recognize their common mistakes, it provides me with a starting point to reemphasize correct methods. By reflecting on the types of questions I receive, I am able to assess the efficacy of classroom presentations. Problems that generate a large number of questions necessitate additional class time and a revamping of the original presentation.
The effort required to perform the type of meaningful practice I have described is substantial. While the effort requirement is sometimes an obstacle and students resist doing it, they are rewarded with a built-in benefit. When students’ efforts on an assignment have tangible effects on their understanding of the material, the value of hard work is reinforced. By holding students accountable for completing assignments, students learn lessons in personal responsibility.
Making a case for computer-based assignments
I know that these are lofty ideals for homework assignments and it is true that there are many impediments to achieving these goals. Research suggests that any break in the practice cycle of attempt, assess, and refine will decrease the benefits of the assignment. And the earlier the break occurs, the larger this decrease will be. Students who attempt and assess but do not refine will see fewer benefits than those who do all three steps but more than those who only attempt and do not assess.
The first place that the cycle can break down is at the initial attempts of the problems. For a traditional textbook or worksheet assignment, this could happen because the student does not bring home the necessary materials. If the textbook or worksheet is not with the student, the problems cannot be attempted, and the practice cycle stops before it even begins.
By putting assignments online, a student has access to the assignment anywhere he or she can get on the Internet. A computer at home or in a local library, an Internet café, or the school’s computer lab can all offer up the assignment without the need to carry home heavy books or keep track of loose paper.
Another way that students break the practice cycle before it starts is by copying someone else’s work. While a student will gain some benefits from seeing a complete solution, the benefits will not be as significant as if the student generated the solution for him or herself. Copying work is easiest when every student has the exact same set of problems, as happens with book and worksheet assignments.
Generating a different assignment for each student could be an overwhelming burden for a teacher if done by hand. But by using algorithmic problem templates and random number generators, a computer can create thousands of unique assignments in moments. When each student has a unique problem that shares a common framework with everyone else’s problem, students can be encouraged to use classmates’ solutions as examples. The students can study those solutions and then apply the methods shown to their own problems.
Once we have students attempting the problems, we need to ensure that they can assess their attempts. With textbook assignments, this often means assigning odd-numbered problems, whose answers appear in an appendix. For worksheet assignments, this may mean posting an answer key for students to look at. Both of these methods will allow a student to determine immediately if his or her answers are correct. Unfortunately, this is accomplished by giving the correct answer out, which can undermine the refinement step.
If a student notices that his or her answer is the opposite of the answer provided, there is a temptation to merely change the sign of the answer, without looking for the error that led to the incorrect sign.
To avoid the problem of giving out all the answers, teachers may make use of even-numbered problems, or give worksheets without an answer key. In these cases, students may use their classmates’ answers to assess their own. This method results in a small delay in the assessment step but is fine if their classmates have correct answers.
If a group of students all make the same error, resulting in the same wrong answer, they may not know it if they only compare answers among themselves. They may reinforce misconceptions without even knowing it. To help battle that possibility, many teachers grade parts, or all, of the homework assignment to which students do not have access to the answers.
This is a fine solution, but it can take a great deal of time for a teacher to grade the papers, and that results in a large delay in the assessment step for the students. Often this delay is large enough that students do not continue on to the refinement stage. If every student is working out unique problems, to prevent copying, grading by hand can take an especially long time.
Encouraging Refinement
I believe that homework is most effective if we can minimize the delay at the assessment step while encouraging the refinement stage by not giving out answers. It is here that computer assignments have a great advantage. Just as computers can use algorithms to generate unique problems, they can use algorithms to generate answers.
I believe that homework is most effective if we can minimize the delay at the assessment step while encouraging the refinement stage by not giving out answers.
Once the computer has calculated the correct answer, it can compare that answer to a student’s answer and provide immediate feedback. The computer does not have to report the correct answer, it can just report back whether the answer was correct or not. In fact, it is possible for the computer not only to check for a correct answer, but also for any number of common mistakes. If a student’s answer matches an answer that results from a common mistake, the student can be alerted to that fact and partial credit can be awarded.
If the answer is not correct, the student will need to review and refine the work until arriving at the correct answer. Because the feedback is immediate, the problem will still be fresh in the student’s mind, and the student will still be in "homework mode," which makes an attempt at refinement highly probable.
Using the computer to check homework has other benefits as well. Computers are great at storing and compiling data. When the computer checks a student’s answer, it can record for the teacher how many wrong attempts each student made, how many students eventually got a problem right, and how many attempts on average it took for students to solve a problem.
I can get that information without spending a great deal of time grading and use it to determine where my students are having difficulty. That frees up my grading time for planning ways to clear up misunderstandings or actually meeting with students to talk about the material. In fact, I can get this information in real-time as my students are working, without having to wait for students to complete and turn in a full assignment.
The trade-offs for computer-based assignments
The immediacy of this information is a boon for me as a teacher, but it comes with a price. When I grade papers by hand, I see all the steps that were taken to arrive at an answer. Those steps allow me to pinpoint errors that must be addressed. The computer cannot give me that level of detail. Because of this, I use traditional textbook and worksheet assignments, as well as quizzes and tests, to give me a fuller view of my students’ progress. Also, I require that my students record their work for all computer assignments in a composition notebook. I use this written work to help students identify mistakes when they come in for help. I can also use it to see how my students’ thought processes evolve from attempt to attempt.
There are other trade-offs, too. The most obvious drawback is the need for a computer. Not all students have regular, easy access to computers away from school. The digital divide is real and can limit the practicality of using computers for homework. Luckily, I teach at a "laptop school," so this is not an issue for me.
Another drawback is the need for specialized formats for the answers. While I can ask students to write "y" for infinity on their papers, a computer will require some other form of input, since "y" is not an ASCII character.
It is also difficult for students to draw a graph for the computer to grade, although some early attempts have been made by others. And finally, developing materials or converting existing materials for use on the computer is a time-consuming process. As more educators begin to use computer-based homework, I hope the work can be spread out among many.
Options for Computer-Based Homework
In my own classes, I use WeBWorK. WeBWorK is open-source, free software that was originally designed for use as Web-based homework in college calculus classes. A large number of math problems, from algebra through multivariate calculus, probability, and statistics are available freely from the University of Rochester. I have worked with my colleagues at Detroit Country Day School to write several hundred problems in physics, chemistry, and anatomy. I have also worked with colleagues to put WeBWorK to use in English classes to support vocabulary building and reading comprehension.
The LON-CAPA project is described as an open-source distributed learning content management and assessment system. The LON-CAPA software is also free. There are more than 20,000 pages of content and nearly 19,000 problems available, with a focus on physics, astronomy, chemistry, biology, geology, and math.
WebAssign is a commercial online homework and quiz system. WebAssign has problems in astronomy, biology, chemistry, engineering, math, and physics available.
How to integrate computer-based assignments into your course
My colleagues and I have used many different strategies to integrate WeBWorK into our courses. The most popular is to make one computer-based assignment for each test, effectively replacing more traditional review sheets. Generally, the assignment is opened to students at the start of a unit and due either the day of the test or a day before the test. This gives students an opportunity to work the problems as the class covers the material, and to apply their time management skills over a period of two to three weeks.
The second most popular option is to replace one book assignment each week with a computer-based assignment. These assignments are generally short, with five to 10 problems. They open and close on the same day each week, and serve to review just one week of material. The teachers who use this method generally do not add computer-based review before tests. The honors physics teacher has actually moved nearly all her homework assignments to the computer. She only uses book or worksheet assignments for problems she cannot effectively ask on the computer.
In the math department, my colleagues and I have also used computer-based assignments in place of worksheets when we have needed to miss class. We have designed assignments to cover just one topic in depth. By using the information collected by the computer, we are able to assess our students’ understanding of the material before we see them. This allows us to be prepared for our students’ needs as soon as we return.
Science faculty members have used WeBWorK for homework and quizzes. The computer has been particularly useful when detailed diagrams that do not photocopy well are required. English faculty members have used WeBWorK for vocabulary quizzes and also plot summary quizzes.
The most important of these benefits are pedagogical and they far outweigh the additional effort it takes to prepare such assignments.
My colleagues and I are continually exploring additional uses of WeBWorK in our curricula because we have found real benefits to using computer-based assignments. The most important of these benefits are pedagogical and they far outweigh the additional effort it takes to prepare such assignments. Computer-based assignments are one more tool to help students learn both content and valuable work habits. The success we have experienced encourages us in our ongoing development efforts.
Mark Schmitt is a mathematics teacher at Detroit Country Day School, an independent school in Beverly Hills, Michigan. He has used computer-based homework assignments in his classes for five years and is involved with the continuing development of the WeBWorK program.
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