The role and amount of homework to be assigned is the most controversial topic of discussion among educators: teachers, parents, administrators, psychologists, and researchers. Even politicians get into the fray.

Researchers have been trying to figure out just how important homework is to student achievement. The Organization for Economic Cooperation and Development (OECD) looked at homework hours around the world and found that there was not much of a connection between how much homework students of a particular country do and how well their students score on tests (OECD, 2009). However, in 2012, OECD researchers drilled down deeper into homework patterns, and they have found that homework does play an important role in student achievement *within each country*.

They found that homework hours vary by socioeconomic status. Higher income 15-year-olds, for example, tend to do more homework than lower income 15-year-olds in almost all of the 38 countries studied by the OECD. Furthermore, the students who do more homework also tend to get higher test scores.

An important conclusion of the study is that homework reinforces the achievement gap between the rich and the poor. For example, in the United States, students from independent schools do more homework than students from Christian/parochial and other religious schools. And students from suburban public schools do significantly more homework than those in urban public schools except the urban public examination schools. It is not just that poor children are more likely to skip their homework, or do not have a quiet place at home to complete it. It is also the case that schools serving poor children often do not assign as much homework as schools for the rich, especially private schools. Other findings from this study are also instructive. For example,

- While most 15-year-old students spend part of their after-school time doing homework, the amount of time they spend on it shrank between 2003 and 2012.
- Socio-economically advantaged students and students who attend socio-economically advantaged schools tend to spend more time doing homework.
- While the amount of homework assigned is associated with mathematics performance among students and schools, other factors (teacher competence in subject matter and classroom management; higher expectations from students, parents, and teacher; amount of classroom time allotted to content, etc.) are more important in determining the mathematics performance and achievement of school systems as a whole.
- Homework patterns among 15 year-olds, revealed that the children in western countries get much less homework than children in eastern countries. For example, students in United States and UK are assigned an average of five hours of homework a week compared to nearly fourteen hours in Shanghai, China, and nearly ten hours weekly in Russia and Singapore.

There are many other studies about the role and utility of homework with conclusions ranging from assigning no homework to students should be assigned substantial daily homework. However, most such studies are survey types that describe the state of homework and opinions about homework. There are some correlational studies where students (or their parents) are asked about the amount of homework they do and the status of their mathematics achievement. These studies are also suspect as the responses are purely subjective. The problem with such studies is that the quality and nature of homework vary and is self-reporting. These are not causal studies.

Most research indicates that it is not necessary to assign huge quantities of homework, but it is important that assignments are well thought-out—systematic and regular, with the aim of instilling work habits and promoting autonomous, self-regulated learning.

Researchers emphasize that homework should not exclusively aim for repetition or revision of content, as this type of task is associated with less effort and lower results. Research has consistently found that students who work on their own on their homework, without help, performed better—score higher than those who ask for frequent or constant help. Most studies show that self-regulated learning is aligned to academic performance and success. Self-regulation, organization, and perseverance are important components of the complex of executive functions.

When it comes to homework, how is more important than how much.

**The Purpose of Homework
**Meaningful homework is a means to reinforce classroom learning in the home. Homework transfers learning from formal, socially guided learning to individualized responsibility and accountability. The impact of supervised and independent practice using effective classroom instructional techniques and well-organized homework is well known to teachers. Teachers know that both provide students with opportunities to deepen their understanding and skills relative to content that was initially presented and practiced in the classroom. Most teachers and parents know them as important factors in student achievement; however, what and how to make them real and useful is a problem. The objective of teachers’ assignments should always aim to have impact. Effective teachers, therefore, plan activities in such a way that they have the most impact. For assignments to have impact, students need to practice (a) choosing strategies and (b) have retention.

The objective of homework is to:

- Communicate to students that meaningful learning can continue outside the classroom;
- Help children to develop study habits and foster positive attitudes toward school;
- Reinforce and consolidate what has been learned in the classroom;
- Helping students recall previously learned material;
- Prepare, plan, and anticipate learning in the next class;
- Extend learning by making students responsible for their own learning.
- Practice to achieve fluency by initiative, preparation, reinforcement, preparation, and discipline of independent learning.

For these reasons, daily homework assignments should not be busy work but should always be well thought out, meaningful, and purposeful. To achieve the stated goals of homework, it should have three components: *cumulative items*, *current practice exercises*, and *challenge tasks. The integration of these tasks adds a key element of learning—reflection on one’s learning*. Reflecting on one’s learning aids in the development of metacognition—a major ingredient for growth and achievement.

**Principles Guiding Homework
**Research shows that homework produces beneficial results for students in grades as early as second. I remember, my daughter wanted to do her Kindergarten homework too when her older brother was doing his homework. A routine was set. The earlier these routines are set, the earlier the formation of life time habits.

There are three parties to homework: teacher, parents, and the student. When homework compliance does not take place, we need to work on all the components and find ways of removing any hurdle. Teachers design the homework; parents support the homework completion, and students complete it, alone or with someone’s guiding support. The following principles can guide teachers and parents:

- The school, with teachers, should establish and communicate a homework policy during the first week of school. The policy should be uniform across grade and subject levels. Students and parents need to understand the purpose of the homework, the amount to be assigned, the positive consequences for completing the homework; description and examples of acceptable types of parental involvement should be provided.
- The amount of homework assigned should vary from grade to grade. Even elementary students should be assigned homework even if they do not complete it perfectly.
- Research indicates, to a certain limit, homework compliance and mathematics achievement are related. The curve relating the time spent on homework and mathematics achievement is almost an inverted “wide” parabola. For about every thirty minutes of additional homework a high school student does per night, his or her overall grade point average (GPA) increases approximately half a point. In other words, if a student with a GPA of 2.00 increases the amount of homework he or she does by 30 minutes per night, his or her GPA will rise to 2.5. On the other hand, oppressive amounts of homework begin to reduce its benefits. Homework is like exercise, difficult to start and keep up, but the more we do it, the better we get at it and, within limits, we can do more.
- Parents should keep their involvement in homework at a reasonable level. At the same time, parent involvement in the classroom should be welcomed. Parents should be informed about the amount and nature of homework, and they should be encouraged to have moderate involvement helping their children. Parents should organize time, space, and activities related to homework. Parents should be careful, however, not to solve content problems for students; they can give hints, or explain the method, but not give a method, which the student does not understand. Giving “tricks” to solve problems is not useful in the long run. There are no tricks in mathematics only strategies. An efficient strategy for others looks like a trick because they may not have the reason why it works.
- Not all homework is the same. That is, homework can be assigned for different purposes, and depending on the purpose, the form of homework and the feedback provided to students will differ.
- All assigned homework should be commented on and responded to because the benefit of homework depends on teacher feedback.Homework with the teacher’s written comments has an even greater positive effect on students. It provides a formative assessment, information how the student is doing. This also offers information for parents about standards, pedagogy, and methods of assessment. When homework is assigned but not commented upon, it has limited positive effect on achievement. When homework is commented on and graded, the effect is magnified. In addition to teacher corrected homework, homework can be self-corrected by the student with the teacher providing the answers. The homework can be peer corrected. Some homework is corrected publicly under the teacher’s guidance. Still, at least once a week, the homework is commented upon by the teacher. These comments should address common problems – lack of concept, misconceptions, poor language, inefficient procedures, poor organization, and misunderstanding of standards – as well as the efficient and elegant methods and concepts used by students.
- Homework is practice. Students should practice at least 30 minutes a day on their academics just as they would an instrument or a sport. If one plays multiple instruments or multiple sports, does one give only 30 minutes of practice for both? Of course not! The same goes for reading and math, science and social studies. Research shows about 1 to 1 hour per day (7.5 hours a week) of homework, on consistent basis, can achieve the goals of homework.
- Parents should be active participants in their child’s academic career. However, that does not mean doing the homework for their child because it would be counterproductive. They can make sure to remind their child to do the homework and that it gets completed. They can give suggestions when necessary and review completed homework. Homework is a child’s academic practice. He/she needs rewards and consequences and a great deal of encouragement.
- Administrators and teachers should do everything to impress upon parents to make sure that they, in turn, make learning a priority for their children and practice every day. However, schools should not make children’s achievement solely dependent on this variable. They should make sure that all children get enough practice in the school itself. The lessons should be planned and delivered in such a way that there is enough practice in the classroom so that children feel confident in tackling the homework themselves.

A teacher should always ask: “Does the completion of homework have any impact on her instruction? Does it inform her instruction? Does it contribute to the teaching and learning of the new material? Does she learn something about the child and/or her teaching from it?” If the answer is affirmative to any of these questions, the homework is worth assigning. Can the goals of the curriculum and her instruction be achieved through some other means? If so, then there is no need to assign homework. However, if there is no homework, we have to find more time for instruction in the day or reduce the allotted time for regular instruction to be redirected to practice, reinforcement, and reflection. Both situations are costly. Therefore, we should always look for ways to improve homework compliance.

**Composition of Homework
**Most teachers assign homework at the end of each section in the book: “OK, now do problems 1-25 on page ____.” Does this work for students? Not based on what I have seen and heard in my 56 years as a mathematics educator. To help students develop competence and confidence in math, teachers should be concerned with the quality of problems they assign in the classroom and for homework rather than the quantity.

Most mathematics assignments (homework as well as practice activities) consist of a group of problems requiring the same strategy. For example, a lesson on the quadratic formula is typically followed by a block of problems requiring students to use that formula, which means that students know the strategy before they read the problem. Most times, they do not even read the instructions before solving a problem. Problem sets made up of only one kind of problem deny students the chance to practice choosing a strategy—that means thinking about the problems (reflection). When faced with a mix of types of problems on an exam, such students find themselves unprepared. These classroom or homework problem sets are called **blocked **assignments. The grouping of problems by strategies is common in a majority of practice problems in most mathematics textbooks.

The framers of CCSSM (2010), recommend thatstudents must learn to choose an appropriate strategy when they encounter a problem. Blocked assignments deny such opportunities. For example, if a lesson on the Pythagorean theorem is followed by a group of problems requiring the Pythagorean theorem, students apply it before reading each problem. If all the problems for practice are direct application of the Pythagorean formula (a^{2}+ b^{2 }= c^{2}, where a and b are the legs of a right triangle and c is the hypotenuse), then this direct “blocked” practice is a practice in algebraic manipulation, not a practice in understanding and applying an important geometrical result about right triangles and its role in higher mathematics.

An alternative approach to practice is when different kinds of problems—varying concepts, procedures, and language, appear in an ** interleaved **order (mixed and uncategorized) problems. Such problem sets require students to choose the strategy on the basis of the problem itself. Such problem sets are also referred to as

**or**

*distributed***practice. For example, consider the problem:**

*spiraled**A bug flies 6 meters east and then flies 14 meters north. *

*Her starting point is at point A and her final destination is represented by a point B, represent her flight by a diagram on the coordinate plane. **How far, in terms of tenth of a meter, is the bug from where it started? Give reasons for your choice of solution approach. How much distance did it travel? Why are these two distances different? (No calculators)*

This problem is ultimately solved by using the Pythagorean theorem. The distance travelled by the bug is different than the distance between points A and B—the hypotenuse of the right triangle, they drew, with sides 6m and 14m. To find the length of the hypotenuse, they used the formula:

The bug flew 20 m to reach point B and the distance between A and B is about 15.3m.

In this problem, students first draw a diagram. The diagram suggests a strategy. Then, they choose a strategy (Pythagorean theorem) to apply and then they execute the strategy.

The choice of strategy means that a student is observing a pattern (*mathematics is the study of patterns*), recalling a theorem or formula suggested by the situation (*learning is the residue of experiences and recall shows its presence)*, or noting the presence of certain conditions or language that suggest a concept, or a procedure (*integration of learning*). The choice of a strategy is dependent on understanding language, concept, and procedures involved in the problem situation.

Learning to choose an appropriate strategy is difficult, partly because the superficial features of a problem do not always point to an obvious strategy. For example, the word problem about the bug does not explicitly refer to the Pythagorean theorem, or even to a triangle, right triangle, or hypotenuse. This kind of assignment is called *interleaved *** practice **where a majority of the problems (practice, homework, assessment tasks, etc.) are from previous lessons, current work, mixed problems (new concept mixed with previous concepts and procedures) so that no two consecutive problems require the same strategy. Students must choose an appropriate strategy, not just execute it, just as they would be required to choose a strategy for a problem during a cumulative examination or high-stakes test. Whereas blocked practice provides a crutch that might be optimal when students first encounter a new skill, only interleaved practice allows students to practice what they are expected to know.

**(a) Cumulative Homework
**One-third of the homework assignment must be cumulative in nature. It should include representative problems from previous concepts and procedures. Whatever has been covered in the classroom during the year should find its representation in daily practice and assigned homework. In other words, what was covered in the months of September or October should continue to appear in the month of March or April. Such an assignment makes connections and achieves fluency. Consider, for example, the connections between multiplication and fractions, fractions and ratios, and equivalent fractions and proportions. Or, the relationship between algebra and arithmetic. There is such a close relationship between algebra and arithmetic that algebra is often referred to as “generalized arithmetic.” Using the distributive property in multi-digit multiplication procedure, combining like terms, applying the laws of exponents, and other rules and procedures are the same for algebraic expressions as they are for arithmetic expressions (e.g., long division for whole numbers and division of a polynomial by a binomial; short division for whole numbers and synthetic division for polynomials).

This part of the homework plays an integrative role in learning the material in the curriculum and provides opportunities for reflection. This part is to improve fluency and smoother recall of learned material. Familiarity and success on these problems emphasizes and meets the need for structure and success of the R-Complex and the limbic system. Another objective is to consolidate learning and connect concepts, procedures, and language. The topics, skills, and procedures mastered must be revisited on a regular basis. The memory traces of the learned skills must be retouched regularly because *knowledge atrophies over time if not maintained.*

** (b) Practice Problems**Another one-third of the homework must be a true copy of the work done in the classroom that day. The objective of this component of the homework or practice problems is to consolidate the material learned in the classroom and continues the learning outside the classroom. It also helps to remain current in the material. If the teacher has covered the odd problems in the section of the book, then she can assign the even problems for homework. When homework is assigned for the purpose of practice, it should be structured around content. Students should have a high degree of familiarity with the material assigned. Homework relating to topics that have not been clearly understood and a level of competence has not been achieved should not be assigned. Practicing a skill with which a student is not comfortable is not only inefficient but might also serve to habituate errors and misconceptions, and high probability of non-compliance.

True mastery requires practice. But again, quality often matters more than quantity when it comes to practice. If students believe they can’t solve a particular problem, what is the point of assigning them 20 more similar problems? And if they can solve a problem in their sleep, why should they do it again and again?

The objective of this segment is to develop procedural fluency. Both fact fluency and procedural fluency can be developed in the class and through homework. Procedural fluency builds conceptual understanding, strategic reasoning, and problem solving. It involves applying procedures not only accurately but also efficiently and flexibly and recognizing when one strategy or procedure is more appropriate than another. To help students develop procedural fluency, teachers must therefore assign problems that are conducive to discovering and discussing multiple solution strategies. And once again, this doesn’t require elaborate problems. Sometimes it’s just a matter of recognizing the learning potential within straightforward problems. Here’s a simple problem that generates rich discussion and helps students develop procedural fluency and number sense related to fractions: *Find five fractions between 2/5 and 4/6. Describe your approach and reasoning for it.*(This problem can be assigned as you’re practicing the ordering of fractions. It makes students think about all the ways of approaching it: common denominator, common numerator, converting to decimals, and comparing with a benchmark fraction such as 1/2).

You do not have toavoid using or assigning problems from textbooks. Make thoughtful, intentional choices that help students learn and like mathematics and feel good about themselves in the process.

**(c) Challenging Problems
**The problems in the last one-third of the homework should be (a) moderately challenging or (b) one or two-word problems from a previous topic. These problems are not mandatory but for those who want to solve these problems. Students can trade one problem from this set with two in other parts. These problems should add some nuance or subtlety to the problems of the type done in the classroom, or application of the concepts and procedures discussed in class during previous topics. Or, this component may introduce a related concept or procedure.

This component helps to prepare students for new content or to have them elaborate on content that has been introduced. Through these problems, even when students have demonstrated mastery of a skill, students can gain a deeper understanding of the math involved. For this to happen, teachers must assign the right problems and be prepared to scaffold students’ understanding. Here is one such problem that stretches students including those who have mastered or memorized the laws of exponents: *Which is greater: 2*^{40}* + 2*^{40}*or 2*^{50 }*? Can you prove your assertion? * Assign problems students are likely to mess up, and then help them learn from their mistakes so that they don’t make the same mistakes again. Mistakes make us learn more. Do not prevent students’ mistakes, prepare them for learning through them. Discuss student mistakes, misconceptions, and lack of understanding them in class. Help them to find mistakes and their causes. When they do make a mistake, give a counter example and create cognitive dissonance in their minds.

The problems, in this section, become the starting point for the next day’s lesson. In that sense, they are a kind of preview of the next lesson. For example, a teacher might assign homework to have students begin thinking about the concept of division prior to systematically studying it in class. Similarly, after division of whole numbers has been studied in class, the teacher might assign homework that asks students to elaborate on what they have learned and how this will extend division of whole numbers by simple fractions.

In both situations, it is not necessary for students to have an in-depth understanding of the content. The objective of these problems is to further the learning. It doesn’t matter if students do not solve any of the problems from this part of the homework as it will become the introduction to the next lesson. This part of the homework is to challenge the student and should be of a moderate level of novelty. It might invite participation from other members of the family. These problems are assigned so that students who need a challenge get it. These problems satisfy the needs of the neocortex.

Every quiz, every test, and examination are set from the problems (or a very close replica) assigned in the homework throughout the year.

Homework with these components is an example of interleave practice homework. The students take a while to warm to this new type of homework because it has been so long since they have actually seen how to do a particular problem. But once they get used to it, students like the new homework. When they are reviewing the old concept or procedures, there is an aha moment — “oh I remember that.” This increases confidence and compliance.

This interleaving effect is observed even though the different kinds of problems are superficially dissimilar from each other. Interleaving of instruction and homework improves mathematics learning not only by improving discrimination between different kinds of problems but also by strengthening the association between each kind of problem and its corresponding strategy.

Interleaved practice has these two critical features: Problems of different kinds are intermixed (which requires students to choose a strategy), and problems of the same kind are distributed, spaced, across assignments (which usually improves retention). Spacing and choosing strategies improves learning of mathematics and performance on delayed tests of learning.

The interleaving of different kinds of mathematics problems improves students’ ability to distinguish between different kinds of problems. Students cannot learn to pair a particular kind of problem with an appropriate strategy unless they can first distinguish that kind of problem from other kinds and interleaved assignments provide practice to learn this discrimination. In other words, solving a mathematics problem requires students not only to discriminate between different kinds of problems but also to associate each kind of problem with an appropriate strategy, and interleaving improves both skills. Aside from improved discrimination, interleaving strengthens the association between a particular kind of problem and its corresponding strategy. The abilities to discriminate and associate strategies are critical skills for doing well on cumulative examinations, such as standardized tests, SAT, achievement tests of different kinds. Since most of these examinations are cumulative and different kinds of problems are organized in it, students need to have mastered the critical skill of discriminating between the different types of problems.

Research on error analysis showsthat the majority of test errors take place when students have practiced using the blocked assignments but much fewer when they have practiced interleaved conditions. The errors occur because students, in blocked practices, are accustomed to choosing strategies corresponding to the assignments; they have learned identifying strategies in isolation, so when they encounter the problems in combination they mix them up. Through practice in interleaving situations, fewer errors are possible because interleaving improves students’ ability to discriminate one kind of problem from another and discriminate one kind of strategy from another.

Blocked assignments often allow students to ignore the features of a problem that indicate which strategy is appropriate, which precludes the learning of the association between the problem and the strategy.Blocked problems also lack subtleties and nuances. Blocked practice allows students to focus only on the execution of the strategy, without having to associate the problem with its strategy.

Helping students develop the discipline of completing homework is key to becoming independent and lifelong learners. Ifa student is not able to complete the homework on the first try, the teacher should ask the student to complete it after the material has been covered in the class. Although the teacher may collect work to record students’ progress, it should not detract from the responsibility given to the student for successful completion of all problems.

References

CCSS (2010)

OECD (2009)

OECD (2012)