The superintendent of a local school district recently called for special education reform because of tremendous growth in enrollment during the past six years. The superintendent blamed the expansion on relaxed eligibility criteria for special education services. To be sure, eligibility decisions are sometimes murky, especially when attempting to distinguish students with learning difficulties from other low-achieving students. However, I can suggest to the superintendent another reason for the increasing number of referrals to special education: Curriculum and teaching methods too often do not meet the learning needs of many students.
Many decision makers use test scores to determine whether the discrepancy between a student's ability and achievement is large enough to meet the local or state criteria for special education services. This approach to special education focuses on what is presumed to be wrong with students rather than on developing instructional practices that make use of these learners' strengths (Ginsburg, 1997; Means & Knapp, 1991; Poplin, 1988).
For several years, I have worked on short-term interventions that have shown that students with identified learning disabilities do not exhibit much, if any, disability in problem solving when the problems are interesting and engaging. In fact, these studies show that adolescents with suspected learning disabilities can match the performance of students without such disabilities on complex math problems.
Engaging Problems, Interesting Formats
In the earliest study (Bottge & Hasselbring, 1993), a math teacher delivered short-term, video-based instruction to groups of 10th grade remedial math students. With help from the faculty and staff at the Learning Technology Center at Vanderbilt University, I developed a seven-minute video, Bart's Pet Project, which portrays a middle school student who wants to build a cage for a new pet. To figure out how much wood Bart needs to build the cage, students must add several combinations of 2-inch-by-2-inch lengths of wood for the cage frame. The cage dimensions are mixed numbers, so the students have to figure out how to add fractions, such as halves, quarters, and eighths. Students with learning and behavioral disabilities were able to solve the problem in three different ways, explain how they arrived at their solutions, and answer related text-based problems.
In the next study (Bottge, 1999), I wanted to see whether the program could reinforce math skills by asking students to transfer skills learned from the video problem to hands-on projects. The special education teacher joined the effort; then we asked the technology education teacher for help in designing the hands-on projects. He developed a set of problems in which students created two skateboard ramps that were each 7 feet long and 2 feet high. After the students solved the video problem, they transferred the math skills they had learned to planning and building the skateboard ramps.
After observing the success of these two short-term studies, the principal provided the three teachers with time to plan an integrated math curriculum for the remedial classes. She also revised their schedules to allow them to teach the class together. The next year, after solving the video problem, the students figured out the most economical way to build two large compost bins for the new high school. The names of the students who made the bins were engraved on a plaque attached to each bin.
More Complex Problem Solving
With the structure for a collaborative teaching team in place, the next challenge was to find ways to help students apply more complex math skills to challenging problems. We (Bottge, Heinrichs, Chan, & Serlin, 2000) used Adventures of Jasper Woodbury, a videodisc-based, problem-solving series developed by the Learning Technology Center at Vanderbilt University (1996). Students solved one of the most difficult Jasper problems, "Kim's Komet," which focuses on measuring and predicting natural world occurrences related to the functions of distance, rate, and time.
The video portrays two girls who compete to figure out the lowest point at which they can release their model cars on a 6-foot ramp to achieve a desired speed on a 10-foot straightaway. They want to predict these release points so that their cars can travel at the speeds necessary to negotiate challenging courses—including a loop, a banked curve, and double humps—that will be placed at the end of the straightaway on the day of the derby (see fig. 1). The cars will be able to negotiate each stunt only if they can travel within the specified range of speeds that will be announced at the derby. Because the girls want to predict accurately the straightaway speeds they can achieve for each height on the ramp, they construct graphs that show ramp heights on the x-axis and straightaway speeds on the y-axis. Students try to help one of the girls win the contest by calculating the speeds, completing the graphs, and predicting at which height she should start her car for each challenge. The problem introduces such concepts as linear function, line of best fit, variables, rate of change (slope), reliability and measurement error, and acceleration.
Figure 1. Car Derby Course
After students solved "Kim's Komet," they applied what they had learned by taking part in their own derby. They went to the technology education classroom, where the teacher had replicated the video's ramp, straightaway, and stunts. The teacher first showed students how to make their own model cars out of blocks of wood and explained the exact measurements that the cars had to be to fit the ramp. Students drew their designs on graph paper, traced them on the wood, and cut them out with a band saw. Several students took their cars home to paint and decorate them.
The next day, the teacher showed them how to operate the infrared detector he had fashioned to measure how long each car took to cross the 10-foot straightaway. Students took turns timing their cars from several heights on the ramp and entered their ramp heights and straightaway times into a table. From this information, they calculated the car's straightaway speed during each trial (see fig. 2). Finally, they constructed a graph to show the relation between the release points on the ramp and the car speeds on the straightaway.
Figure 2. Height and Speed Table
Students record the height from which the car is dropped and the time it takes to travel the distance of the strightaway. they can then calculate the car's speed on the straightaway when the car is released from that height.
Using Intriguing Problems to Improve Math Skills - table
Height | Length | Time | Speed (feet per second) |
|---|---|---|---|
| 24 inches | 5 feet | .759 | 6.6 fps |
| 42 inches | 5 feet | .484 | 10.3fps |
| 60 inches | 5 feet | .392 | 12.8 fps |
| 84 inches | 5 feet | .309 | 16.1 fps |
On the day of the contest, the technology education instructor attached courses for the stunts to the straightaway, and the math teacher announced the range of straightaway speeds necessary to negotiate each stunt. Students received points for starting their cars at the lowest point on the ramp and for completing the stunts successfully. Students consulted their graphs to calculate the lowest possible starting point on the ramp and recorded on their spreadsheets their points for the short jump, long jump, double hump, single loop, and banked curve.
Findings
We were pleased by the results of the research for several reasons. First, compared to students who received traditional instruction on the same topics, the students who solved video-based and applied problems did significantly better on post-tests and on tasks requiring an application of these math skills. In most instances, differentiating the work of students in the remedial and regular math classes was difficult.
Second, student behavior improved dramatically over the course of instruction. Our field notes were confirmed by a graduate student from Taiwan who accompanied me on classroom observations. Before the intervention, he had been shocked by the boisterous behavior he observed. As students began to work on the video problems, he was even more surprised by how quickly their behavior improved. He asked, "How can this happen? They are working like other students," referring to an algebra class he had observed earlier that day.
Third, the students were proud of their ability to solve the problems. When they discovered an answer, they voluntarily shared their findings with the rest of the class. As I approached the school one day, a student shouted, "Hey, Bottge, I solved your problem." One student with a history of underachievement in math earned a perfect score on the post-tests. She whispered to me, "Don't tell my parents about this. They will faint."
A repeating 8th grade student with a history of discipline problems spent the first few days of video instruction drawing pictures. Gradually, he became interested in the problem and figured out how to graph the video character's times to help the character win the derby. In the hands-on project, he carefully built and timed the speeds of his car and constructed a graph. He placed third in the competition. He also improved his score from 11 to 29 (out of 30) points on the problem-solving test and earned perfect scores on later tests that measured his retention of these math skills.
What Accounts for These Results?
Students were successful for several reasons. We asked them to solve problems that were challenging and meaningful to them. In cognitive science, interest in solving a problem is an important criterion of a "problem" (Polya, 1962; Schoenfeld, 1989). The level of interest in these problems contrasts sharply with attitudes toward traditional word problems, which students described as "tricky" and "dumb" in our follow-up interviews. Misunderstanding the nature of problem solving may certainly contribute to students' lack of motivation, one of the defining characteristics of students with learning disabilities (Deci & Chandler, 1986; Ellis, 1998).
By using video-based instruction, we also bypassed the difficulties that students with learning disabilities often have deciphering text-based problems. In addition, working on the single problem created a central concept and focal point for the class.
Like most teachers, the teachers in our studies were committed to helping all students learn. Collaboration among teachers and integration of subject matter were key elements in the success of the projects. The projects also tapped the ability of the technology education teacher to link math concepts to increasingly interesting applications.
The principal at the middle school encouraged and supported the teachers' involvement in the studies. Without her modifications of the schedule for planning and class time, the math students could not have moved easily between the classrooms, which were located in separate wings of the school. She also led the effort to restructure the math curriculum to make it more interesting and meaningful—without additional staff and cost.
The Future
Although the results are promising, our research is still in its infancy, and many questions remain unanswered. Next year, we will bring together math, special education, and technology education middle school teachers from three school districts in two states to study the impact of these practices on students. These studies will help us document different implementations of the program, an important step because education innovations are notoriously difficult to replicate across classrooms and schools. And perhaps we will be able to respond to the superintendent with more confidence than we can now.
