### Homework: A Math Dilemma and What To Do About It

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The issue of assigning homework is controversial in terms of its purpose, what to assign, the amount of time needed to complete it, parental involvement, its actual affect on learning and achievement, and impact on family life and other valuable activities that occur outside of school hours. I have encountered all of those controversies in my years of teaching mathematics. Math homework is usually a daily event. Unfortunately, many teachers assign most homework from problem sets following the section of the text that was addressed that day. There is little differentiation. For the most part the entire class gets the same assignment. (In fairness, teachers do take into consideration the nature of those problems, which are often grouped by difficulty, deciding which to assign based on the general ability level of students in the class: below average, average, above average, or mixed.)

Often teaching involves a "one size fits all" strategy. Elaborating on Joseph Simplicio's (2005) thoughts, the student who "didn't get it"--that basic understanding of how to solve the problems--can't complete one homework problem, let alone 20. The better or gifted student gets it after a couple of problems and doesn't need to complete 20. Then there is the student who completes all 20 to build confidence that he/she does get it. The first gets frustrated and quits, the second gets bored and quits, and the third might get frustrated and bored by all the time it takes to get done or hastily complete the work with errors. Some might copy each other's work along the way, too. These scenarios are not all-encompassing, and you might be thinking, "You exaggerate." But homework has and most likely will continue to pose a dilemma.

It is time to treat math homework a little differently.

The Dilemma
Arguments against homework are becoming popular in the press. Critics, such as Alfie Kohn, question the need for homework and circumstances under which homework should be given. Certainly, it should not be expected to be the "default state" (2006). Parents find themselves more involved with helping their children to complete it, and there is more of it for younger students than in past, adding to everyone's frustration (Wallis, 2006).

I recall in my early years as an educator moving to a new state and one particular interview for a middle school teaching position. The principal asked for my views on homework, and I responded that, yes, I believed in the concept. He was adamant, however, that no homework be given at his school because students' home situations and expectations after school meant that, for the most part, they would not do it. All assigned work needed to be completed during class time. It was a lesson learned about the importance of knowing about the home-school relationship in the learning process. Today, given the hectic and often overfilled schedules that are part of so many students' daily lives, I'd consider Simplicio's (2005) solution to the dilemma, which "lies in setting aside time at the end of the school day to coordinate and supervise homework activities in school" (p. 141).

Without accommodating learner differences, we set students up for failure or boredom when, in fact, we can do something about that in the design of experiences outside of school that are meant to reinforce learning. As educators, we can also recall the many reasons a particular assignment was not turned in even by our best students. One of the more original I received was from a high school student who accidentally spilled soda on the homework paper. She said it burned it up when she put it into the microwave to dry.

I am not saying to eliminate a homework requirement. In spite of research design flaws, a synthesis of research from 1987 to 2003 on homework reveals "generally consistent evidence for a positive influence of homework on achievement." We do need to consider grade level and student characteristics, however, as "simple homework-achievement correlations revealed evidence that a stronger correlation existed (a) in Grades 7-12 than in K-6 and (b) when students rather than parents reported time on homework" (Cooper, Robinson, & Patall, 2006, p. 1).

What I also suggest is taking a closer look at current literature on teaching and learning, which calls for differentiated instruction and attention to learning styles, thinking styles, and multiple intelligence theory. When it comes to math homework, differentiation does not seem to carry over, and it should be considered beyond assigning the problems out of a text by level of difficulty.

Dimensions of Learning Math
As Richard Strong, Ed Thomas, Matthew Perini, and Harvey Silver (2004) indicate, student differences in learning mathematics tend to cluster into four mathematical learning styles. Those with a mastery style tend to work step by step; individuals with an understanding style search for patterns, categories, reasons. Students prone to an interpersonal style tend to learn through conversation, personal relationships, and association. The self-expressive learner tends to visualize and create images and pursue multiple strategies. Students can work in all four styles but tend to develop strengths in one or two of the styles.

So where does homework fit into this? Each of these styles tends toward one of four dimensions of mathematical learning: procedural, conceptual, contextual, and investigative.  "If teachers incorporate all four styles into a math unit, they will build in computation skills (Mastery), explanations and proofs (Understanding), collaboration and real-world application (Interpersonal), and nonroutine problem solving (Self-Expressive)" (Strong et al., 2004, p. 74). If you have ever solved Sudoku puzzles, you can appreciate the motivational value of options. Everyone might be solving a puzzle of the same size but has selected an easy, medium, hard, or evil challenge based on his/her understanding of how such puzzles are solved and a self-determined ability to do so. But, I'd soon lose motivation, if that was the only puzzle type I ever attempted. By providing options, adding variety, and differentiating homework into those categories, as well as in instruction, learning math might be better achieved for all.

Options
Mastery. Compacting, which is giving students credit for what they already know, makes sense when you consider that students learn at different rates. Technology resources can help with mastery, individualizing homework, and enabling teachers, students, and parents to monitor the time spent on homework and degree of accuracy that students achieve on a particular problem set for developing computation skills and procedural knowledge. (Wouldn't it be great if that online textbook had that monitoring capability?) Those sets can be randomly generated or developed from large databases of similar problems geared to a specific benchmark of a lesson.

For fixed problem sets in which students appear to have attained little mastery after a certain number of attempts (perhaps two), students might write or even record audio about what they do and don't know so that the teacher might better address the concept during the following class lesson. This latter promotes thinking about mathematics, tells the student that his/her thinking is valued, and would also minimize the frustrations that parents experience when attempting to "do" homework for and/or with their children. Given that honest effort, the student would have flex time to continue working on the assignment to develop the required skills. Of course, there are sites for homework assistance, and teachers should make appropriate sites available. Some include tutorials; others will generate the answer to a problem submitted. However, consider some of those a short-term solution to just getting homework done.

Understanding. An assignment might include addressing an essential question of the day or the unit, which the student would write about in a journal. Students can also develop a variety of thinking maps (e.g., see mapthemind.com) as a way to visually express their content understanding (Lipton & Hyerle, n.d.). An ability to explain a concept to oneself or others shows understanding, as would an ability to develop examples and non-examples of the concept or show how it visually relates across other themes.

Interpersonal and Self-Expressive. Using online resources and virtual manipulatives at home or in a school-based after-school homework setting can further assist with concept development, collaboration, and peer discussions. An at home investigation might involve discussion with parents and friends about a real world event in which the topic of the day would apply, thus enhancing the interpersonal style of learning math. These discussions and such activities as virtual field trips and Web quests might also help establish the link of mathematics to other subjects: science, literature, art, music, sports, and daily life (e.g., cooking, shopping, investments, savings, home decorating, construction). I've also had students use e-mail or instant messenger to discuss homework with each other.

Homework might involve an open-ended task or a unit long independent investigation, which serves to synthesize content and develop literacy skills. Short- and long-term projects and performance tasks, which include options for oral, visual, or written response modes, allow students to test their interpersonal and self-expressive styles of learning math. Again, technology can play a role. For example, the Partnership for 21st Century Skills developed Information and Communication Technology (ICT) Literacy Maps for core subject areas and included representative ways that technology can be used in mathematics at grades 4, 8, and 12.  Consider the following "tip of the iceberg" in relation to NCTM standards:

• Newspapers, books, spreadsheets, graphing programs, calculators, computers, Internet, films, TV programs, websites, databases, Internet, and digital libraries can help students gain information and media literacy. These would be useful sources for the study of number sense, statistics, and data analysis.
• Digital cameras, laptop computers, multimedia presentation software, graphing calculators, probes, and Web development software can be used to enhance creativity and intellectual curiosity.  For example, students might take photos showing geometric representations in their surroundings and create a math slide show or Web page. Taking pictures of road signs, buildings, and nature can be used to illustrate measurement and geometric concepts and key vocabulary and would be far more beneficial than just a worksheet on angles, parallel lines, polygons, symmetry, patterns, and so on. The investigation, sharing, and discussion of those bring out the self-expressive style of learning and certainly is a non-routine way of learning.
• Internet, presentation, word processing, and desktop publishing software can be used to communicate with students in other communities or countries, participate in national math competitions, or to discuss concepts with outside experts in online bulletin boards. Some of those experts might relay how they apply key benchmarks in patterns, functions, and algebra in their work. What fun it might be to report on those in the school newsletter.

These become tools for accountability and adaptability. Further, student reflections on their math learning and putting examples of procedural, conceptual, contextual, and investigative learning into a paper-based or e-portfolio provides evidence of growth and dimensions of math learning that go far beyond results of a single standardized test. Chances are that such a portfolio would also assist with test preparation, as Prince George's Public Schools (MD) finds for their MSPAP preparation process.

Getting HW Done
Robert Marzano and Debra Pickering (2007) provided the following research-based homework guidelines to help ensure that homework is completed and appropriate:

• Assign purposeful homework.  Legitimate purposes for homework include introducing new content, practicing a skill or process that students can do independently but not fluently, elaborating on information that has been addressed in class to deepen students' knowledge, and providing opportunities to explore topics of their own interest.
• [E]nsure that homework is at the appropriate level of difficulty.  Students should be able to complete homework assignments independently with relative high success rates, but they should still find the assignments challenging enough to be interesting.
• Involve parents in appropriate ways (for example, as a sounding board to help students summarize what they learned from the homework) without requiring parents to act as teachers or to police students' homework completion.
• Carefully monitor the amount of homework assigned so that it is appropriate to students' age levels and does not take too much time away from other home activities. (p. 78).

A rule of thumb for homework might be that "all daily homework assignments combined should take about as long to complete as 10 minutes multiplied by the students' grade level" and "when required reading is included as a type of homework, the 10-minute rule might be increased to 15 minutes" (Cooper, 2007, cited in Marzano & Pickering, 2007, p. 77).  Other tips for getting homework done can be found in Helping Your Students with Homework, a 1998 booklet based on educational research from the United States Department of Education.

Differentiation, Reality, and Student Satisfaction
Yes, varying homework, creating a tiered assignment structure based on student interests and abilities, and being flexible in homework completion schedules are among differentiated strategies that add to the complexity of teaching. Planning differentiated homework takes more time than assigning problems out of a text, just as planning for differentiated instruction takes time with its different class management techniques, flexible grouping, and teaching beyond the traditional lecture to a group of students sitting in straight rows. Planning is best done in collaboration with teaching colleagues who also develop a corresponding enhanced grading scheme, including rubrics.

Realistically, differentiation of any kind is difficult to sustain when math class sizes tend to be large and when there are so many demands on teachers for preparing students for standardized testing. Implementing differentiated homework means getting to know your students better than you might now, and having them and parents understand a different view of "fairness," particularly for grading purposes. Everyone might be doing the same amount of homework, but that homework might not be the same for all, all the time. Imagine the difference for students who will no longer need to complain, as did the 11-year-old daughter of the editor of this site, "Twenty-nine annoying exponent problems are 29 too many."

References

Cooper, H., Robinson, J., & Patall, E. (2006). Does homework improve academic achievement? A synthesis of research, 1987-2003. Review of Educational Research, 76(1), 1-62.

Kohn, A. (2006). The homework myth: Why our kids get too much of a bad thing. [Audio interview]. Available: http://www.alfiekohn.org/books/hm.htm

Lipton, L., & Hyerle, D. (n.d.). I see what you mean: Using visual maps to assess student thinking. Thinking Foundation. Available: http://www.thinkingfoundation.org/research/journal_articles/journal_articles.html

Marzano, R., & Pickering, D. (2007). The case for and against homework. Educational Leadership, 64(6), 74-79.

Simplicio, J. (2005). Homework in the 21st century: The antiquated and ineffectual implementation of a time honored educational strategy. Education, 126(1), 138-142.

Strong, R., Thomas, E., Perini, M., & Silver, H. (2004). Creating a differentiated mathematics classroom. Educational Leadership, 61(5), 73-78.

Wallis, C. (2006, Aug. 29). The myth about homework. Time. Available: http://www.time.com/time/magazine/article/0,9171,1376208-1,00.html

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About the author: Patricia Deubel has a Ph.D. in computing technology in education from Nova Southeastern University and is currently an education consultant and the developer of Computing Technology for Math Excellence at http://www.ct4me.net.

Proposals for articles, news tips, ideas for topics, and questions and comments about this publication should be submitted to David Nagel, executive editor, at dnagel@1105media.com.

Patricia Deubel has a Ph.D. in computing technology in education from Nova Southeastern University and is currently an education consultant and the developer of Computing Technology for Math Excellence at http://www.ct4me.net. She has been involved with online learning and teaching since 1997.