Mathematics education

Hmework – A Key to Improvement in Mathematics Education

Introduction

The United States has a serious problem in mathematics education. Test results show that beginning in middle school, the United States student proficiency declines in comparison with other developed countries throughout the world to near the bottom by 12th grade [PISA]. The Workforce/Education Subcommittee of the President’s Council of Advisors on Science and Technology identified two principal reasons for this: too many math teachers are not trained in math, and too many math textbooks are inadequate [Herbold].

This article suggests a third important contributing reason: low homework emphasis. The United States lags far behind in time spent by our students on homework [TIMMS]. Thus, improving homework completion at the secondary level may be a significant opportunity for US math proficiency improvement at that level. This article suggests that new Internet math homework-help resources may be very helpful in getting more students to complete and understand their homework. As an important by-product, more class time will be available for more in-depth teaching than is currently possible. We identify some of the available resources, discuss their application and the background research that supports their use, present comments from several contributing authors, and present a case study of the use of one of the Internet resources.

Homework Not Being Done

Without homework practice, middle school or high school students cannot fully master the concepts presented in class. Homework provides an opportunity for the students to extend their understanding and directs the students to concepts that need further explanation. Homework provides an opportunity for students to make, discover, and correct mistakes so they can learn from them.

Students of teachers who emphasize the importance of homework score higher than students of teachers who do not [House]. However, teachers are finding that getting students to actually do assigned homework is increasingly difficult. Some students, and often their parents, express that homework is a useless burden. Furthermore, homework competes with TV, video games and other activities. A teacher in Northern California had this to say [Bradley]:

I am a good teacher as evidenced by my experience, by peer, student, and parent feedback, and by advanced certifications that I have earned. Yet, many of my students fail. Most of those who failed will say it was because they did not do the work.

Worked Examples

The use of worked-out examples is a standard practice in classroom lectures and textbooks. Some students understand the daily lesson after observing just a few worked-out examples. Others need more examples than can be presented during a class period or provided in a textbook section. These latter students would benefit significantly from additional worked examples, but they might not know where to find them or have the motivation to seek them out. The research described below shows that providing worked examples for actual assigned homework problems is beneficial. The websites described below allow students to view worked out solutions for their assigned homework.

Internet Resources for Math Homework Help Via Worked Examples

A website that provided math homework-help by showing tutorial (worked out) solutions for actual math textbook homework problems was first introduced in 2001 [Beall]. Other websites with the same general capabilities have since become available. Students needing assistance on assigned homework problems now can find immediate homework help using an Internet-connected computer, at websites such as http://www.aolatschool.com, http://www.encarta.com, education.yahoo.com, http://www.calc101.com, and http://www.hotmath.com. Each of these websites provides immediate tutorial explanations for math problems. Thus, teachers may assign homework problems for which there are explanations (worked solutions) available on the Internet. In the case of Encarta and Hotmath, the explanations are correlated to the actual problems assigned from popular math textbooks.

Even students who may have fallen behind in math may be attracted to such websites, as they provide a new avenue for catching up. The help, via computer, with their assigned homework relieves them of the fear of peer or supervisory embarrassment. Math Teacher Jane Monson noted that more of her students are completing homework since they began using Hotmath.com because it answers the question, “What is the next step?” so that students can continue on their own. According to Math Teacher Lisa Winer, “I love that this website gives worked solutions to assigned homework, because students who want it can get instant help right away, on their own. If a student says they didn’t understand the homework, my response is that they should have logged on to get help.”

Scientific research has investigated the use of “worked out” examples in algebra, and the results show that this increases effective learning [Carroll, 1992]. A study was done in Texas: two groups of students, one of poor performers and the other of good performers, were taught together with only one difference; the poor performers were given their homework assignments with 50% of the problems accompanied by worked solutions. The good performers were given the same homework assignments without worked solutions. Interestingly, the poor performers achieved higher scores on the final exam than the original good performers. In related research, students in the worked examples group completed their work more quickly while perceiving the work as less demanding and displayed better performance on tests [Carroll, 1994]. The researchers suggest that the reduced cognitive load allows the students to process the underlying similarity of problems and integrate the methodology with existing knowledge [Grillmeyer, 2001]. They also note that less “wrong learning” results.

The availability of worked solutions also benefits advanced students. They can tackle the more challenging problems with more success and move forward with less outside help. Advanced, motivated students can work ahead in their textbook knowing that an instant tutor is available.
Homework review in class is an important part of math teaching. Teachers need their students to ask questions about processes, rules, and properties as a part of assessing both the students and their own teaching.

Yet, routine questions of interest to only a few students can be a very inefficient use of class time. According to Math Teacher Marty Atkins, “As our students have begun to use Hotmath.com ? I am beginning to get more ‘Why did they take that step?’ questions rather than ‘How do I start?’ questions.” A survey of math teachers using one of the homework-help sites concluded that about 20 minutes per class time was freed up [Grillmeyer, 2004]. According to Math Teacher Paula Evans, “We saved class time, which we immediately reallocated to activities which allowed students to develop insight about the material? We have used this time to develop in-class activities which ask students to extend their homework.”

When teachers assign homework problems with solutions available on any of the listed Internet homework help websites, their students can receive step-by-step explained solutions to their actual homework problems. Students are able to see their mistakes and learn from them, and parents are in a better position to see the methods being taught so they can amplify them as needed. Use of these sites is not simply checking or getting answers, but may be considered a directed, self-paced, tutorial experience.

Teachers justifiably want to balance the amount of available homework help so that students are certain to be challenged. Some students might mindlessly copy down solutions if they are available for every problem. The research concludes that 50% of assigned problems should have the available help. Two of the websites that provide solutions to actual textbook homework problems (Encarta and Hotmath) only explain the odd-numbered problems for which numerical answers are already available in the back of the textbook.

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