THE RESULTS FROM THE THIRD INTERNATIONAL Mathematics and Science Study (TIMSS), which tested a half million eighth-grade students in forty-one countries, have recently been publicized. Students in the United States ranked below average in mathematics, whereas students in Singapore earned top scores. Examining how students in Singapore study mathematics should provide useful information to mathematics educators on how to improve the performance of students in the United States. Problem solving is emphasized in Singapore, where students are expected to struggle with problems that have real-life implications.

THE ACTIVITY THE PACKETS PROBLEMS, DEVELOPED BY Educational Testing Service (1995), is a collection of mathematical activities that allow students to develop several ideas as they struggle through real-life situations. The problems are aligned with the NCTM's Standards and facilitate linking instruction with assessment. Students can practice skills that they have already acquired and develop new skills and concepts. In each problem, a specific task requires mathematical thought and organization. The multiple acceptable solutions are based on the parameters that the students themselves design in interpreting the problem. Providing justification for their solutions is an important element of the activities, and a rubric helps to assess the strength of the student's argument. The following is "The Million Dollar Getaway":

Main Bank was robbed this morning. A lone robber carried the loot away in a big leather bag. The manager of the bank said most of the stolen bills were ones, fives, and tens. Early reports said the robber took approximately $1 million in bills. The Channel 1 News team thinks this sounds like more money than one person could carry. What do you think? Channel 1 News needs help in determining how hard it would be for one person to carry $1 million in small bills.

Please investigate this question to determine if this is possible. Prepare a report for Channel 1 News describing and explaining your findings. Your report should help the investigative reporter at Channel 1 better understand the situation to plan for tonight's broadcast.

The goal is for students to create a report to submit to the news station. The problem involves such mathematical strands as geometry; measurement; whole-number fraction and decimal computation; volume, area, and weight; and estimation. In addition to "content" strands, such "process" strands as communication, reasoning, problem solving, and making connections are developed.

A RATIONALE FOR INCLUSION THE PROBLEM WAS IMPLEMENTED IN A FIFTH-grade class at Magen David Yeshiva in Brooklyn, New York. Five of the students, who were weak in basic mathematics skills and concepts, regularly left the room to work on a modified mathematics curriculum. This practice seemed to run counter to the newly instituted inclusion in a regular classroom setting for language-arts students who had previously been in a reading-resource room. The concept of inclusion has attracted international attention. The 1997 RISE (Restructuring for Inclusive School Environments) conference featured a program on "International Perspectives on Inclusion." The rationale behind inclusion is that rather than leave the room for a modified program, students should have a teacher who is trained to implement the program. We decided, then, to use the inclusion model by having the five students work within the classroom with the regular teacher present. The goal was to provide these students with a problem-solving experience within a larger community, as well as to build their self-esteem by having them participate as an integral part of the regular class. All students received the same information and task, and the problem-solving approach was open-ended. No steps were specified for the task. Sylvia Bulgar taught the regular class, and Lynn Tarlow taught the modified group. Although Tarlow remained primarily with the modified group, both teachers were available for all students.

ASSESSMENT CRITERIA AN ASSESSMENT OF THE WORK WAS BASED ON examining the final product (the letter to the news station) as well as observing the processes that the students used to develop their ideas. The categories for assessment included problem-solving strategies, communication while solving the problem, demonstrating understanding of mathematics concepts, and completeness of the answer. An evaluation of the written product, in this case, the report to the news station, was based on the criteria set forth by the PACKETS program. Written products are assessed as being one of the following: noteworthy (not only achieves the client's immediate objectives but also addresses larger objectives), acceptable (achieves the client's objectives, needs minor revision), closely (resembles what a complete response would look like but fails to provide in a usable form everything that is necessary), needs major revision (demonstrates a basic understanding of the problem and a reasonable start but needs substantial work before it will be useful), and needs redirection (indicates a lack of understanding about the problem or a lack of useful ideas about how to solve it.)

COMPOSITION OF THE GROUPS NINETEEN STUDENTS, PLACED into four homogeneous groups, participated in the activity. Creating homogeneous groups encourages all students within each group to participate in developing the solution. Students learning in a constructivist environment typically work in homogeneous pairs or groups. Although such groupings remain controversial, the teachers involved decided to implement this methodology. Students were grouped according to past performances in mathematics-problem-solving activities. Group 1 comprised five students who had demonstrated the strongest ability in mathematical problem solving. Group 2 consisted of five students who had previously formed the second tier. Group 3 contained four students who had demonstrated average ability in problem solving. Group 4 consisted of five students who were regularly taken out of the classroom for a modified mathematics program.

GETTING STARTED ALL STUDENTS HAD DIFFICULTY GETTING started. They had to be reassured that they were not expected to solve this problem in one session and that they would struggle to find a solution. Many complaints arose about the problem's being too difficult, especially from group 2.

GROUP 3'S RESULTS Group 3 began by using Unifix cubes to represent the money. They had found this manipulative to be useful in solving several previous problems and apparently decided to use the cubes without considering whether they would be appropriate. After several attempts to create a model of the money, they discussed what $1 million looked like. Some group members did not really understand the concept of one million. Some could represent the number as being one thousand thousands; however, group members still were unable to map this representation to a concrete model. Apparently, rote knowledge that one million is equal to one thousand thousands was inadequate for understanding the value of the number.

After a discussion about how much the number 1 000 000 actually was, the group decided to cut out paper to represent the money. They would take one student's leather knapsack, empty it, and see whether all the money would fit inside. On the second day, this group discussed the weight and the volume of the money. The students theorized that if all the money would fit into the knapsack, then it would not be too heavy to carry. They did not, however, attempt to weigh the money to justify this theory. This group spent a lot of time trying to make $1 000 000 in fake dollars, later deciding to make only half of the $1 000 000 and see how much space it used in the bag. They reasoned that they could estimate whether the entire amount of money would fit in the bag. They saw the significance of estimation as a useful tool to make problem-solving tasks more manageable.

This group was the first to complete the product (i.e., the report to the news station). Their product would fall under the category of "needs minor revision," since it could be improved by adding specific information regarding dimensions. The solution alludes to an understanding of volume but does not refer to specific measurements. A discussion of volume would provide more substantial justification for the group's result. In addition, this group interpreted the assignment to mean that many large bills could be included as long as most of them were of small denominations. Their product would have been better justified had they explained the breakdown of the bills. The product, however, is clearly written and well organized.

GROUP 2'S RESULTS Group 2 had the most difficulty getting started. They complained about having no idea of how to begin and sat idle, seemingly overwhelmed by the task. They then approached a student from group 4 to ask for help. The students in group 4 are not generally consulted for advice on problems. Eventually, group 2 saw other groups cutting out paper in the size of actual money. They planned to cut and measure bills totaling $100 000 and then multiply their results by 10. This strategy shows an understanding of the concept of 1 000 000.

The group members used computation accurately and understood that they could use estimation. They understood the need for accuracy in depicting the size of the bills. Once they developed a strategy, they worked enthusiastically. At home they photocopied bills that they cut out to use for their simulation. This group's product needed major revision. They included a breakdown of the various denominations and totaled the amount of money for each denomination. They made accurate use of computation, emphasizing that they had accounted for the complete $1 000 000. However, at no point did the group members consider the weight or the volume of the money nor the total number of bills required to reach $1,000 000.

GROUP 1'S RESULTS Group 1, the first group to begin work, immediately developed a plan to solve the problem. Members decided that the money would have to be carried out in a suitcase, so they drew an area model of a suitcase to represent how the money would be placed in it. They measured a dollar to figure out how wide and how long the suitcase would have to be and then figured out how many bills would be needed to make $1 000 000. Initially, they said that they would make false money totaling a half-million dollars and multiply it by 2. One member suggested that they could make a quarter of a million simulated dollars and then multiply by 4, discussing the fact that a quarter of a million dollars was equal to $250 000. This group worked most efficiently, originating an idea and continuing to modify it until it was workable.

This strategy conforms with Papert's (1980) model of debugging, wherein students constantly modify their work until satisfied with the result. Although this group used a very thoughtful and detailed process by which it found the solution, it is not reflected in the product. Their product was acceptable but with revision could be classified noteworthy. They gave the dimensions that the bag would need to be, omitting the measurement of the length of the bag. This group began with an area model and built it to be a volume model. No discussion of this or any of the measurements of the money appeared in the written product. The group members stated that the volume of the bag would be too great for one robber to carry and alluded to the fact that a bag of that volume would also be too heavy to carry. However, they did not justify their hypothesis about weight.

GROUP 4'S INCLUSION-GROUP RESULTS After struggling to get started, group 4's students disagreed on procedure. For example, Jerry (students' names are pseudonyms) attempted to take charge and to convince everyone of his ideas. Jerry usually lacks confidence and is reluctant to initiate conjectures, but in this task he participated enthusiastically. He believed that the size of the money did not matter and that the group should concentrate on the weight. Isaac, however, believed that the model money had to be the exact size. Eventually, the group members decided to work together to make models of the entire $1 million. Their representations were arbitrary in size, and they used the calculator to total their stacks of "money." They marked the bills as $1, $5, and $10 in sequence to simulate an equal distribution of small denominations. They also included several large bills, since they reasoned that since "most" of the bills were in small denominations, some large bills could be included as well. It is significant that this group was highly motivated to pay careful attention to the details of the given information, since these students typically have trouble understanding written communication. They were determined to produce a reliable and convincing product.

Katie was absent the day that the problem was presented. When she returned, she found it hard to accept her group's ideas. She thought it sufficient to just measure a dollar bill, figure out the number of bills needed, and see what size bag would hold this amount. She concentrated on bag dimensions, but considered only width, not an area model. She insisted on working alone, confidently completing her product quickly. Her report needed "major revision." Although attempting to support her ideas using actual computation is a good approach, she needed to explain clearly where she got her numbers to justify her theory.

Ultimately, other group members decided not to hand in one collaborative product because of their disagreements. These students all have substantial language deficits, making it a taxing chore for each to produce written work. They each made a concerted effort to provide details in their individual products to make their reports more convincing. Jerry worked alone to prepare his product and created the most convenient scenario to conform with this hypothesis, assuming that one bill could be worth $500 000 and another worth $1 000. He reasoned that any large bill would weigh the same amount as a small bill and, therefore, even if the rest of the money were in small denominations, the money would not be too heavy to be carried by one person. His work needs "major revision." To be convincing, his argument should have contained a breakdown of the number of bills that would comprise the $1000 000. In addition, using weight in his argument, he should have discussed the number of pounds that the money would weigh.

Roger and William worked together and logically explained their conclusion, although they used the words pound and ounces interchangeably. They weighed the bills on a scale and determined that about thirty bills weighed an ounce. They knew that sixteen ounces are in a pound and were able to use this information to multiply 30 by 16 accurately to conclude that a pound of bills would contain about 480 bills. They then divided 480 by 1 000 000 to find the weight of 1 000 000 bills, making two distinct mathematical errors. They misinterpreted their calculator results because they had limited experience with technology and they confused the divisor and the dividend. Misled by this error, they believed that 1 000 000 bills would weigh 48 pounds instead of 2 083 pounds, and did not consider whether this result was reasonable. Their work needed minor revision; they did an admirable job of thinking through the problem and devising a sound plan to find a solution.

Isaac decided that only $2 000 of the money had to have been in ones, fives, and tens and the rest could have been in large bills. His work needed redirection in that he apparently did not understand that $2000 was not most of the money. This misconception reveals a weakness in number sense, as he did not seem to understand the value of the number 1000 000.

CONCLUSIONS

FIGURE 1 SUMMARIZES THE group's work. The dissent observed in group 4 is positive. Although they participated in group discussions, none of these students wanted to concede to one another; each believed in her or his own process and solution. This modified-curriculum group was able to function well in this setting for this activity. At times, students from this group competently mingled with other groups to discuss issues related to the problem.

Group 2 called on Katie for help when she had handed in her product, and Jerry discussed his ideas with other students before class. The support of their regular teacher encouraged them to conduct discussions about their ideas within the group. In so doing, they extended their knowledge and became more confident in justifying their problem-solving strategies within the larger community.

The week after the problem was completed, Tarlow conducted informal interviews with her students. After the project was completed, Jerry discussed the problem with his private tutor. The tutor told him that his solution was wrong and spent a great deal of time explaining the "correct" solution in detail. Jerry was unable to relate anything about the tutor's solution to Tarlow. He still believed in his own product and was able to recall every detail of his work, which confirms that students who construct and take ownership of knowledge retain and can apply it. Katie said that she enjoyed the task because she thought that she had done "good work." Isaac liked the task because it was "a very thinking problem."

The students in the modified-curriculum group expressed their desire to do other Packets problems that involve creating products for clients. To see whether these students could apply what they had learned, they were told that a fourth-grade class was trying to collect 1 000 000 soda-can tabs. They were told that in six weeks the students in the fourth-grade class had already amassed 1 000 tabs and believed that they would reach their goal by the end of the school year. Group 4 recognized the enormity of the task and protested the unreasonableness of the expectation. Jerry immediately and vehemently stated, "One hundred percent they can't do it by the end of the year; 1 000 isn't even close to 1 000 000," indicating increased number sense as a direct result of working on this problem.

In general, students' oral explanations were far superior to their written work. Several supportive details omitted in writing were clearly expressed verbally, which reinforces the importance of using different means to assess students' understanding. Therefore, teacher observations became an integral part of assessing students' solutions. This situation raises two issues: (1) it is crucial that the teacher observe carefully what processes the students use to solve the problems and encourage them to explain their work as they strive toward a solution and (2) students need to be given the opportunity to discuss their written products, to be questioned about them, and to make revisions that will reflect the thoroughness of their work. Every group demonstrated a need to map the concepts to concrete models, such as by cutting out and labeling bills.

Students were motivated by the problem. They remained on task through several class periods and continued to discuss related issues outside of class, during recess, and after school. They eagerly anticipated continuing their work. Although the students did an admirable job with the problem, discussion and revisions are still needed. Students need an opportunity to share their products with other groups, to rethink their ideas, and to make revisions. The big ideas in this problem will continue to be explored in other contexts; after that, the problem can be revisited.

ADDED MATERIAL

BIBLIOGRAPHY

Educational Testing Services. PACKETS Performance Assessment for Middle School Mathematics Teacher's Program Guide. Boston, Mass.: D. C. Heath & Co., 1995.

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.

Papert, Seymour. Mindstorms: Children, Computers, and Powerful Ideas. New York: HarperCollins Publishers, 1980.

Project RISE. "International Perspectives on Inclusion." Program presented at the annual conference sponsored by the College of Education at the University of Memphis, Memphis, Tenn., October 1997.

Wingert, Pat. "The Sum of Mediocrity." Newsweek 23, 2 December 1966, 96.

SOURCE: Mathematics Teaching in the Middle School 4 no7 478-83 Ap '99. Reproduced with permission from Mathematics Teaching in the Middle School, copyright 1999 by the National Council of Teachers of Mathematics. All rights reserved.

AUTHOR: SYLVIA A. BULGAR AND LYNN D.TARLOW

SYLVIA BULGAR, SBulgar@aol.com, is a mathematics curriculum coordinator and teacher in Piscataway, New Jersey. She is a doctoral student in mathematics education at Rutgers University, New Brunswick, NJ 08903. Her professional interests include teacher development and the use of problem solving as an assessment tool.

LYNN TARLOW, LTarlow@bigfoot.com, is a mathematics curriculum coordinator and teacher in Brooklyn, New York, and has taught all levels of mathematics, including special programs for gifted students and for students with learning disabilities. She is completing her doctorate in mathematics education at Rutgers University, New Brunswick, NJ 08903. Her professional interests include children's construction of mathematical ideas and teacher-development reform.