By John Mays

One would think that the task of writing down a few thoughts about homework would be relatively simple, but I find it isn’t. The fact is, everyone—and I mean everyone—is confident they know what homework is for and how it works. After all, we have all had homework all our lives, have we not?

Like I said, not so simple. Education in America today can hardly be called a success story. And the educational failures of recent decades call into question all the methods typically used in classrooms today, including conventional notions about homework.

To keep our subject manageable, in this article I limit the discussion to homework assignments based primarily on computations, whether in science or in math. By computations I mean problems in science and math that have specific, quantitative, correct answers.

My core philosophy of pedagogy, which I summarize with the trilogy *Wonder—Integration—Mastery*, entails a number of implications for homework. In this article I develop these implications as six basic principles, some of which may seem radical or even patently unworkable. But I have shared these principles with many teachers, at different schools, in seminars and consulting sessions around the country. I have been gratified to learn from many of them that they have found these principles to be just as powerful and effective as I have.

Is the goal for our students *learning*, or is it something else? When I think of learning, I really have in mind another trilogy: *learn*, *master*, and *retain*. I want students to learn the principles and concepts associated with a given topic, gain proficiency to the point of mastery in applying them, and then retain that mastery by frequent reuse, practice, and rehearsal. This learning-oriented framework implies the following first principle:

**Always supply the answers along with computational assignments.**

The learning moment is when a student is actually working the problems. For many students, the next day in class is too late, and the teachable moment is gone. It is when a student is engaged in working on an assignment that he cares the most about the assignment. The immediate knowledge that he is working the problems correctly gives confidence and makes working through the assignment a lot more enjoyable. Conversely, immediate knowledge that a solution is incorrect will stimulate the desire *right then* to try to find out what is wrong, to dig in and seek the understanding that the student apparently does not have yet.

Just try watching what students do when they are working problems, and they check the answer on one and find it is wrong. For most, the last thing they will do is ignore the red flag and keep working. They recognize that to continue, when one is not doing problems correctly, is futile. Instead, they become gripped with interest and cannot rest until they figure out the error. (They may give up too early and come running to the teacher for help, but that is a separate issue.) This level of engagement is exactly what teachers seek to create in their classes. When the answers are provided, the engagement tends to happen by itself.

Put simply, if the student doesn’t have the answers, how will she know if she knows what she is doing? I can still remember when one of my own daughters came home in tears because she had gone to her class (in *elementary* *school*), math assignment cheerfully in hand, only to be crushed by the discovery that she had done half the problems incorrectly, and thus received a 50 on her homework paper. Having the answers doesn’t guarantee that a student will be able to work every problem successfully. But it does make it more likely that the next day in class the student will have an accurate understanding of where she has been successful and where she needs additional instruction.

There is one occasion when answers should not be given, and that is when answers are to be written down more or less by inspection. This happens with problems such as mental math, when students are estimating an answer, or in balancing chemical equations. With these, students can be tempted to look at the correct answer and think, “Well, that’s not what I got, but I see it now and I get it.” Often, they do not see it or get it. With these kinds of problems, it is better to have students take turns sharing answers aloud in class from their completed papers. Then the teacher can identify students who are having trouble getting it and arrange to work further with them as needed.

A more subtle challenge is to effectively establish the best type, quantity, and length for assignments. Our temptation as busy teachers is to assume that working all the problems in the text, or working the “odd problems,” is an appropriate assignment.

Not necessarily so. The appropriate assignment is the assignment that will accomplish the pedagogical goal. For me, the pedagogical goal is always *mastery* *of* *specific content objectives*, so I develop the assignment with mastery in mind. Accordingly, my second principle is this:

*Carefully design every assignment to accomplish mastery of specific objectives.*

When I teach freshmen how to calculate density, I know that using the density equation is usually the easiest part of the task. The more difficult parts are mastering volume calculations and unit conversions (especially between cubic centimeters and cubic meters—it’s not 100!), and correctly navigating between weight and mass. So, when designing the assignment, I assign quite a few problems involving these preliminary skills. Once students have mastered these, somewhat fewer problems are needed for mastering the density equation.

A big challenge here is that typically textbooks are not very helpful in this area. There is a standard set of problems at the end of every chapter, and the problem set is often the same length regardless of the difficulty of the material. Problem sets also tend to go easy on practice of auxiliary skills, or worse, present students with computations that do not require practice of auxiliary skills at all.

Crafting assignments to lead students to mastery of particular objectives also means that some exercises are simply unnecessary, while other exercises may need to be added to an assignment. Years ago, I realized that some of the assignments I was giving were pretty much just busy work. This occurred because I was uncritically using some sheets out of a workbook (a practice I have long since abandoned). Worksheets are notorious for keeping students occupied with busy work without solidly supporting any particular mastery goal. If an assignment is not doing its part to lead students to mastery of the specific learning objectives specified for the topic under study, get rid of it.

The next issue I take up may sound scary at first, but in the end it can dramatically improve a teacher’s quality of life:

*Don’t spend your time grading and correcting student problem sets. Students should do this work themselves.*

I hasten to add that for everyone except seniors, you *do* need to collect homework papers, review them for completion, and hold students accountable for turning in fully completed work. (I exempt seniors because seniors should have already learned to be responsible to do their homework. Seniors are old enough to know that if they fail because they are not doing their assigned work, it is their own fault.)

My conversations with other teachers who have discovered this principle have confirmed that this idea should be applied right down into elementary grades. There are many ways to do this, including the old “trade papers and call out the answers” method. But if students have had the answers already while doing the problem set, as I have already recommended, then all you really need to do is collect the papers and assess them for completion. This task only takes a few minutes for a class set of papers. You can tell students in advance that since they already have the answers, your completion check focuses on verifying that the work they have done on the problems supports the correct answer. Tell them also that if they come to a problem that they simply cannot work, they should show *some* kind of work—work that would allow you in one-on-one tutoring with that student to figure out where the student is having trouble.

Years ago, in frustration over the poor quality of the work some students were submitting, I decided to make a bold new policy: the math and science teachers in our department were simply no longer going to accept shoddy, messy, hastily prepared work. The principle here is:

*Require students to show a standard of care and thoroughness in all their assignments.*

There is no reason why students should be permitted to get by with submitting messy, sloppy, or torn papers, or papers with doodling on them, or graphs that look like they were drawn freehand with a broken pencil while jogging. This principle is not complicated and having a policy students must adhere to is not really any stroke of genius. All it requires is the will to enforce discipline.

Most teachers I have talked to about this are similarly put out by students who submit sloppy work, but without a policy that has teeth in it all we can do is nag. Putting a departmental policy in place takes the teacher out of the uncomfortable role of nagging or seeming unreasonable to repeat offenders. A policy places all classes on an even footing and establishes quality work and a standard of care as an expectation the entire institution has for every student. Quality work is no longer the pet peeve of one crank teacher.

To implement this new discipline, I wrote up a specification for neatness and thoroughness that would apply to assignments given in all science and math classes for grades seven and higher.

We rolled out the new standard the next year and trained the students at every grade level on the new requirements. In my classes ever since, I have had little trouble with students submitting messy work, or turning in freehanded graphs without axis scales, or doing their math homework in ink. Such work is unacceptable—as in, we do not accept it. Nowadays, if a student turns in a messy paper, I simply hand it back and inform the student and his or her parents that the student did not submit an acceptable paper.

While I was writing the new standard, I decided to address some recurring issues with classroom preparedness at the same time, and so I specified essential materials that students should regularly bring to class. The standard I developed is reproduced at the end of this article for anyone who would care to use it or adapt it.

If you have read any of my books, you know that one of the major principles of my pedagogical philosophy is that we should design our courses and teach our students with the goal of mastery, as I mentioned above. Running classes this way suggests another consideration for homework assignments:

*Devote significant time in class to students working on new assignments. Save time outside of class for practice and review.*

When teaching for mastery, the time students spend practicing old material will rival the time they need to spend working on new material. Rather than doubling everyone’s homework load, which is neither practical nor desirable, I simply devote a lot of class time to letting students work on problem sets.

I do this in grades 9, 11 and 12, in both science and math classes, and have found it to work very well. In my freshman science classes, an industrious student who works efficiently can get about 80–90% of the year’s daily assignments done during class. Students who choose to waste their work time daydreaming or chatting have more work to do at home, but that is their choice. In my junior Pre-Calculus class, my regimen has been to allow the students one whole day out of every three or four (not counting exam days) to work on their problem sets. This pattern obtains throughout the year, and students can depend on having the time as scheduled to work on their math assignments. They do need to work at home on other days as well, because 55 minutes is not enough time to complete the work assigned for two or three lessons.

In my senior physics class, I found I could present the new material for most chapters in about two days, followed by another day or two working example problems.^{[1]} I typically do this in bulk at the front end of a new chapter. Then I give the students the rest of the time allotted for that chapter (typically four to seven days) to work through problem sets in class. We take out a day here and there for laboratory work and group discussions. As with Pre-Calculus, this class time is typically not enough for students to complete the problem assignments in physics; additional time is needed outside of class. But it gives students enough class time that the home-work part of the assignment is greatly reduced, and time at home can be devoted more to practicing material from prior chapters to assure continued retention and proficiency.

There is one final principle I want to put forward, and it may be the most radical of all. I introduce this principle with an axiom of mine: *homework assignments are a necessary learning activity; they are not a valid assessment of whether learning has occurred*. Yes, everyone knows students must work through a lot of exercises to become proficient at new computational skills. But turning in a paper does not indicate how much a student has learned, nor does it signify that the student has reached the desired level of proficiency.

There are simply too many ways to get an assignment completed for the assignment to be valid as an assessment. A student can work with friends, letting the friends do most of the work. A student can work with a parent or a tutor, writing down what the tutor says to write down. A student can simply transcribe another student’s work (i.e., cheat).

The point is, that there are a lot of ways to get an assignment done that do not entail proficiency with the material. Since, in my view, grades should reflect achievement, which necessarily includes proficiency, my axiom implies this final principle:

**Give little to no grade credit for routine assignments or test corrections.**

Now, hold on. I know what you are thinking. If the assignment doesn’t count for any credit, how do we compel students to do the work? I am asked this question all the time, and there is a very straightforward answer, at least for those involved in some form of private schooling, which is most of my readership.

If you teach at a private school, your students’ parents are spending thousands of dollars to have their children taught at your school. The last thing they want is an email from a child’s teacher stating that the child has not turned in or completed his or her homework. When the email further says that the student cannot expect to succeed in the class without completing all the assignments, the parents instantly go into the mode of correcting their child’s behavior. The problem, which is disciplinary and behavioral, has now been placed where it belongs—between the student and the student’s parents.

The most legitimate assessments of what students have learned are quizzes and tests. (Again, I am speaking in this article about computations. The same applies to verbal questions in science. For other subjects, other considerations may apply.) Since this is the case, the most legitimate way to form the students’ grades is with quiz and test grades, not with homework credit.

In this regard it is important for the entire science and math department to work together to send a coordinated message to students. This message is that the goal of our work is to *learn*, not to get grades on homework. Students should be taught—from an early age—that we do homework to learn, and not because someone gives us a reward for it. The reward comes on quiz or test day, when the students are able to demonstrate what they have learned (and receive a grade they can be proud of, because they know it is a legitimate assessment).

Occasionally, when discussing the no-credit-for-homework policy with parents I have received some initial push back. (How do you expect him to turn his assignments when you don’t give him any credit for them?”, etc.) I explain to them my goals for homework assignments, and the difference between activities necessary for learning and valid assessments of learning. I also explain how the real issue is behavioral and disciplinary, and that I need the parents’ help to enforce this discipline with their child. Most parents are understanding and supportive once they understand this.

A further benefit of removing homework from the grading is that the grade is no longer padded with a 25% or 30% homework cushion. It is then a lot easier to determine from the grades that students are properly placed into advanced, grade-level, or remedial classes. Students who are properly placed, and who are doing their work, should be earning grades in the A or B range without any padding from a homework grade cushion.

When I wrote “little to no credit,” I left room for a policy that changes a bit from year to year as students mature. In elementary grades, work is not divided cleanly into homework and exams, the way it typically is beginning with middle school. When homework is treated more as a continuous learning process, students should be graded on their progress through that process.

But beginning in middle school, students should encounter the distinction between homework as a time to develop and practice learning, and quizzes and exams that are assessments of learning (and usually teaching tools as well). For seventh and eighth grade, I recommend that homework in math and science count should for 8% of the overall semester grade. This percentage should drop to 4% for ninth grade, and then drop to zero after that.

Each homework assignment can either help us bring students toward the goal of learning, mastery, and retention, or it can steal time from our students by filling an hour with another inefficient, meaningless, or maddening hoop for them to jump through. Our practices pertaining to homework assignments are a key part of an effective, mastery-oriented instructional program.

*Feel free to adopt and use the following standard:*

**Uniform Standards for Upper School Science and Mathematics**

The following requirements are intended to foster in our students an appropriate regard for the meticulous care, order, and neatness necessary for excellence in science and mathematics. These requirements apply to students in all science and math classes in grades 7–12. Following these standards positively affects a student’s grades. If a student does not comply, then points may be deducted from the semester grade, or the student may be required to resubmit the work.

Required Materials

Students must bring the following materials to class each day:

- One 3-ring binder for notes, handouts, and written assignments.
- Graph paper, 3-hole punched, for taking notes and for written assignments.
*Regular line-ruled paper is not used at all in science and math classes*. Paper torn out of spiral notebooks is not acceptable. Graph paper may be either loose leaf, or from a pad which has fine perforations for cleanly removing the sheets. - A scientific calculator.
- A ruler scaled in centimeters (15 cm minimum length).
- At least two mechanical pencils with No. 2 (HB) lead and erasers.
- One quadrille composition book for a lab journal (Mead 09127 or equivalent) (applies to science classes).
- A compass and a protractor (applies to all 7–8 grade students and those in Geometry).

Organization of papers

All papers, including homework, tests, and quizzes, are to be maintained for the entire year in a well-organized binder. Teachers may allow students the freedom to keep their notes and practice problems or assignments in a spiral pad separate from their quizzes and tests.

Use of pencil

All work, except typed reports and lab journal entries, is to be completed in pencil. Lab journal entries are to be completed in ink. There are no other exceptions.

Neatness

All work must be neat, clean, and orderly. All writing must be clearly legible. Papers must be free of extraneous marks, smudges, and doodling. Papers must not be wadded, torn, or otherwise mutilated. Papers torn from spiral notebooks with rough edges are not acceptable. Multiple-sheet assignments must be stapled together.

Showing Work

Students must show their work on all problems unless the answer may be written down by inspection or from memory. Teachers will advise students on how much detail is required for problem solutions in specific courses of study.

Paper Formatting

Work is to be written only on the front side of the paper unless otherwise instructed by the teacher. Students must leave neat margins, must space problems neatly (one complete blank line between each problem), and must identify individual problem numbers. If an exercise has a single correct final answer, students must clearly mark it. On written assignments from the text, students must identify at the top of the page the text section number and the problems in the assignment.

Algebraic Formatting

When performing algebraic manipulations, students must write each successive step on a lower line and must vertically line up the equals sign (=) for each new line of work.

Expressions in Final Answers

Unless otherwise instructed by the teacher, final answers must be simplified and written in a standard mathematical format. Correct units of measure must be shown, wherever applicable.

Graphing

All graphs must be large enough to suit the purpose of the problem. All straight lines, including axes, must be drawn with a straightedge. Coordinate points must be clearly identified. Labels and scales for axes must be clearly identified. Asymptotic curves must be drawn with reasonable care so that they appear asymptotic. Curves must be drawn carefully between points and with appropriate shape.

^{[1]} This modest time requirement is primarily due to students having mastered basic principles in their freshman introductory physics class. They still retain most of what they learned several years before.

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