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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 www.aolatschool.com , www.encarta.com , www.education.yahoo.com , www.calc101.com , and 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.

Jesuit High School Case Study

I am a mathematics teacher at Jesuit High School in New Orleans , LA. This is a private Catholic urban school with approximately 1500 students. The student body is primarily middle class students from educated, success-oriented families. Our class size averages 25 students and 99% of the graduates attend a university.

When our math department introduced Hotmath.com to our students, only the most diligent took advantage of it. As individual teachers and as a department, we learned to approach the homework review in the classroom differently. Instead of asking, "What problems gave you trouble last night?" we began to ask "For what problems did you not understand the solution given on the website?" and "At what point did you not understand what Hotmath did?" Eventually, homework review went from over 30 minutes a class to under 10. Even better, we saw students' grades improve!

With the extra class time, our teachers were able to delve into problems or topics that focused on higher-level thinking skills. We could use open-ended problems to stimulate thinking both in a small group setting and to individual students. Even the lower track students became more able to think, rather than just repeat steps, to attain a solution.

As I listened to and read the experiences of the contributing authors, I realized that our school's experiences were not unique. Almost universally they achieved additional instructional time in class. They found that they could limit the amount of time spent on homework review, maintain or increase their rates of success, and introduce creative classroom activities to stimulate mathematical thinking.

Conclusion

In addition to the recommendation that teachers consider these Internet math resources, we feel strongly that our school and political leadership should actively and repeatedly stress the importance of homework as well as the importance of mastering math and science classes, so as to effect an attitude change over time that will benefit everyone.

References

Beall, R., Grant, C. & Grillmeyer, O. (2004). "Superior Improvement for Underperformers: Worked Homework Solutions are one Answer," ComMuniCator, California Math Council Quarterly Journal, March 2004

Bradley, Laura (2005), Letter to the Editor, San Francisco Chronicle, January 10, 2005

Carroll, W. M. (1992), "The Use of Worked Examples in Teaching Algebra," Annual Meeting of the American Educational Research Association, Apr. 1992.

Carroll, W. M. (1994). "Using Worked Examples as Instructional Support in the Algebra Classroom." Journal of Educational Psychology. v86, n3, 360-367.

Grillmeyer, O. (2004). "A Teacher Survey about Online Worked Solutions to Math Homework Problems," www.hotmath.com, 2004

Grillmeyer, O. & Chance, S. (2001). "Educational Foundation for Hotmath.com", Hotmath.com, 2001

Herbold, R (2004), "K-12 Establishment is Putting America's Industrial Leadership at Risk," Speech at Hillsdale College National Leadership Seminar, May 25, 2004

House, J Daniel (2004), "The Effects of Homework Activities and Teaching Strategies for New Mathematics Topics on Achievement of Adolescent Students in Japan : Results from the TIMSS 1999 Assessment," International Journal of Instructional Media, 2004,

TIMMS (1999), "Trends in International Mathematics and Science Study, Assessment Report," National Center for Educational Statistics, 1999

PISA (2003), Program for International Student Assessment, Organization for Economic Co-Operation and Development, 2003

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TECH CORNER
	21st Century Classrooms By Mark Benn Middle School Teacher

About three weeks ago, I had the chance to attend the MACUL conference in Detroit. This is an annual conference that deals with integration of technology within the classroom for the state of Michigan. After spending three days listening to many great speakers and seeing what others are doing in their classrooms, I came away with an even greater sense of what a 21st century classroom should look like.

The first thing to understand is that it won't look anything like the classrooms of the 20th century. Today's students are wired differently. They literally can't sit and listen to a lot of chalktalk. They aren't interested in textbooks, or a lot of pencil and paper activities. Worksheets bore them. The students go through the motions to get a grade, but check out when it comes to learning. They need to be involved in their learning in an interactive way. This is why technology is so important to use as a tool to get them to check back in.

A classroom of the future was set up at the conference. Every student had a computer. The teacher center had a projector, computer, and a multi-media projection system which allows the teacher to project anything from the internet to manipulatives on the screen. Also, the teacher could use student response systems to get instant feedback from the students on whether they understood a concept. The screen and the computers become the center of learning, not the teacher.

Some would say the teacher becomes outdated. This isn't true. In the 21st century classroom the teacher becomes the coach and supporter of the students' learning. The teacher must organize the skills the students are to learn and what tools to achieve this goal.

You may look at this and say that's impossible. It's true that the equipment might be a problem with budgets being tightened. But the key is our attitude. How we approach teaching and learning must change. Our job is to engage the student, not disengage them.    

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	Randomizing Class Choices:  Breaking Up the Monotony

One technique I use is drawing from a hat (or mug, box, basket, or other container). You can choose anything to put in the hat, and decide if you or the students will do the drawing. You can draw, or let your students pick. I try to keep the 'hat' above the chooser's head so there is no possible way to cheat on the draw.

In the hat I like to use different colored poker chips: white, red, and blue. We will use these for many applications, or at least any that involve three different outcomes. When grading freewrites, for example, drawing a blue chip means I take an immediate grade on the assignment

A white chip means "thank you for writing today", but we aren't going to grade it, just file the writing into your folder. A red chip indicates I'll collect the papers, read over them, grade them, and select a few to write comments upon. By drawing a chip, the students don't know if the assignment will be graded or not, so they must do their best. However, for the teacher, the students are writing more but you don't have to grade every paper!

We will also use the chips for minor homework assignments. Same idea - white is a no grade, blue goes immediately to the grade book. But on red chips, I'll allow a minute or two to fix mistakes before I collect them. It depends on the situation. It's that simple. And the students never know if the assignment will be graded or not, so they have to do their best just in case. Another technique is to use strips of paper in a coffee mug for completely random choices. This is great for games like charades where students draw random words, topics, or choices.  This could be used to randomly discuss class topics or answer questions.

I like to use this for choosing project topics. Put slips of paper numbered 1 through however many students are in the class. Fold the slips and then have students draw their own place in the waiting line.  Whoever has the slip #1 gets first choice of topics, #2 chooses second, and so forth. No one can claim a biased order of selection!  This is great for research paper topics, where you don't want students choosing the same topics. We will also use small slips of colored paper to form random groups of students. If I want four different groups, figure how many students you want in each group and tear that many small slips of colored construction paper. Do this for each group, using different colors. I find this is a good use for scraps of paper left over after an art project (the thick paper holds up better). Then go around the room and let the students 'choose' their group. Collect the slips back after recording the groups & names so you can re-use the slips again.

You could use all sorts of everyday items to get random choices. Flip a coin in a two-choice situation. A die or pair of dice can give you even more choices. You could even use a deck of playing cards.

To randomly call upon students, we utilize note cards filled out with student names and personal information. At the beginning of the year, students write their name, parents' contact info, text book numbers, hobbies/interests, and other information on a regular 3 x 5 index card. I then collect these and pull them out, shuffle, and select a random card (with the student's name on it.) Voila! Random selection of students.

And if you want to ensure you call upon everyone equally, just don't shuffle the cards, and place the used card at the back of he deck. You can cycle through the card deck over and over, ensuring you're calling upon every student equally.

Cards, dice, coins, poker chips and simple slips of paper can be easily used to make random selections in class. We'd love to hear any other 'random acts' ideas and techniques you may have. We'll add them to this article and post them on our website with credit to you!

Fast Rabbit Software is a great website with downloadable software, programs, and games for teachers and students.  All of the downloads are free for a trial basis, and if you like the program, you can just register for the remainder of the disabled features.  The free trial versions are very good and you can use them in class even without the full-featured versions.  

The website includes games for math, practice with money, coloring books, word problems, flashcards for music, spelling, and problem solving, interactive reading activities, and test and quiz generators.

This is a user-friendly website with quick links to the various parts of the site.  It is a great resource for elementary teachers.  

Check this site out, you'll be glad you did.  Simply click the link below:

http://fastrabbitsoftware.com

 

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"The Guy In The Glass" By Dale Wimbrow, © 1934 Themes on Life

In This Week's Issue 
(Click the Quick Links below):

Homework: A Key to Improvement in Mathematics Education

Tech Corner: 
21st Century Classrooms

New Teacher's Niche:
Randomizing Class Choices:  
Breaking Up the Monotony

Website of the Month

Themes on Life:  
"The Guy in the Glass"

10 Days of Writing Prompts

Winter Book Sale for Teachers

Book of the Month Club

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