Time and Time Again, Again
In general, a case is a written or video description of a problem or situation requiring analysis and decision-making. Most cases depict real situations, and are based on actual site visits and data collection, though the characters, situations and data frequently may be disguised. Typically, a case focuses on a problem or a set of problems that are intended to stimulate discussion and analysis. Taken together, the case and its discussion comprise the "case method of instruction." Cases are not intended to be comprehensive or exhaustive; indeed, most cases are "snap-shots" of a particular situation within a complex environment.
One purpose of the case method of instruction is to develop and enhance skills in:
This is usually achieved by efforts on the part of the participants and the discussion facilitator to integrate theory and practice, and to build on the knowledge and experiences of the participants. The case method relies on the honest and open exchange of knowledge and opinions by the participants and facilitator, and benefits enormously from the multiple perspectives of the participants. Differences among participants and faculty typically arise, and conflicting recommendations emerge as participants with varied perspectives, experiences, and professional responsibilities consider the case.
The case method of instruction may be characterized by three "C's:" cases are Complex; cases build on a Constructivist notion of learning; and cases depend upon a Community of learners.
The preparation of a case for discussion generally involves the following steps:
The role of the facilitator is to engage many participants in presenting and defending their analyses and recommendations. The facilitator moderates the discussion, can guide its direction, and often may push participants to examine their thinking more critically. The discussion is intended to develop and produce a more sophisticated and nuanced understanding of the situation or problem.
As you read "Time and Time Again, Again," you may find it helpful to keep track of the issues that are raised by making a character web. Draw a circle for each major character and label it (Sally Elmore, Alison Tansey, Barbara Gross, Jack Mann, etc.). Inside the circle, list the issues or concerns that preoccupy each character in the case. Further, you might like to identify your ideas about the following questions before we begin the discussion process:
"Dear Dr. Tansey,
At the beginning of the school year when I agreed to participate in the NSF Making Sense in Middle School Math Project, I didn't realize how hard it would be to teach in this new way. I really don't think I can do this, and besides, the next session will be during the day. I simply can't afford to miss more teaching time. If I leave my algebra classes any more, I won't have time to prepare them for the final test before the holidays. Every class is precious, especially with the block schedule and the eighth grade test coming up in the spring. I am afraid I can no longer participate in this project. I am sure it is a good program, but it just isn't for me. Sorry. -Sally Elmore"
Alison Tansey peered at the email in disbelief. She scowled and even growled in frustration. Then she leaned back her in chair and took a deep breath. An angry response would get her nowhere. Still, she was incredulous that a teacher would back out now, only three months into the Making Sense in Middle School Math Project. She had enlisted teachers in the design of the Project and hired the best professional trainers that she could find. "So much for a fail-safe process," she thought to herself.
She typed a response: "Sally, I realize that instructional time is precious. Our own learning is too! I really hope you will reconsider. It'll make a difference for kids at West for years to come. How about leaving your regular plans for the sub, and asking Michelle, your department chair, to duck in to check on how things are going? Please give the course a chance and hang in there. I know you can be a wonderful middle school math teacher. It just takes a lot of hard work and time. Best, Alison."
Michelle was the math department head at West Middle School who had a strong background in mathematics. Alison hoped that by making this suggestion and not letting Sally off the hook easily, that she would change her mind and decide to continue the course. Alison picked up the phone and punched in the West Middle School number. While waiting for the secretary to find Jack Mann, the principal, she reflected on the history of this math initiative, which had come together in a rare burst of synchronous events. Would the plan unravel just as quickly, she wondered?
Alison Tansey and the Making Sense in Middle School Math Project
Alison Tansey had assumed the position of Assistant Superintendent for Curriculum and Instruction for the Barberton Public Schools just over three years ago. The Barberton Public Schools served a medium sized mid-western city that enjoyed both a thriving downtown as well as several close in suburbs. The demographics of the district were 67% Caucasian, 29 % African-American and 4% Asian. There were some 36,000 students in 30 elementary schools, six middle schools, and three high schools. Compared to state averages, the district was well funded, spending over $6000 per student per year.
Back in May, when Alison was weighing priority areas for the district's work, middle school math seemed an appropriate area to work on. For the previous six or seven years there had been a lot of district work in reading and writing in terms of curriculum alignment with state standards, professional development, assessment development, and material purchases at all levels. And while math had received attention in the elementary grades for the last three years and nearly all of the elementary schools had adopted new inquiry and standards-based curricula, like MathLand and Everyday Mathematics, it now seemed a propitious time to extend this work to the middle schools.
Math was a logical focus at the middle school level for several other reasons, too. Ever since the elementary schools focused on the new math curricula, several middle school math teachers had expressed dissatisfaction with their textbooks. In part it was a response to low level computational skills of the entering sixth graders, but it also was based on a feeling that the elementary schools always got more resources and better materials. Now, it was the middle schools' turn. In addition, a few teachers, as well as all of the middle school math department chairs, had expressed an interest in using more application-based materials. They hoped updated materials would engage many of the apathetic adolescents, as well as increase their computation skills and conceptual knowledge.
Several middle school teachers and department heads had attended an information meeting about the new mathematics programs developed with support from the National Science Foundation. They returned to the district genuinely excited about what they had seen. This enthusiasm, coupled with rave reviews from the state math association and the fact that these materials were all on the restrictive list of acceptable materials, clinched Alison's decision to go ahead and focus on the introduction of these new "standards-based" materials at the middle school level. It was also very clear to her, however, that many of the teachers would need a great deal of support through various forms of professional development in order to be able to use these new materials effectively. Simply buying new materials wouldn't do it.
Another reason to focus on middle school math was test scores. Alison felt chagrin at the district's test scores on norm-referenced tests that showed that many of their youngsters had weak conceptual understandings in mathematics. Their computation scores were no better. On the state test, 57% of Barberton 8th graders scored at the "Does not meet the standards" performance level in mathematics, compared with 36% of 8th graders statewide. In fact, there was fairly strong evidence that many kids had no sense of why mathematics worked the way it did. Since Alison had an undergraduate minor in mathematics and had started her career as a high school math teacher, she knew that there was more, much more, to mathematics than rules, right answers, and the memorization of algorithms.
In the late spring, when the National Science Foundation posted a grant opportunity for professional development of math and science teachers on the Internet, Alison was elated. She wrote a proposal for a LSC that focused on middle school mathematics teachers. She called the Project Making Sense in Middle School Math. The grant was to provide support for workshops and other professional development opportunities during the year as well as pay summer stipends. The stipend from NSF, coupled with Alison's commitment of district funds to begin buying new materials, was attractive to the teachers who were required by the state to accrue a certain number of professional development points every five years to maintain their certification status.
When Alison wrote the NSF proposal, she checked with the middle school teachers via a survey to see what they preferred with regard to the scheduling of the professional development sessions. The response was such a mishmash that Alison decided to schedule a range of different timeframes: school time, after school hours, evening meetings, summer days and even two all-day Saturday sessions were folded into the mix. It was impossible to find a schedule that everyone liked. But overall, the teachers accepted the idea of losing only two Saturdays, a few instructional days, and battling fatigue on a few after-school sessions. The teachers seemed to appreciate being asked about the schedule, and Alison heard through the grapevine that there was great excitement when the grant was awarded.
West Middle School
West Middle School, one of the six middle schools in the district, had some 1,200 students and 80 classroom teachers. It was under a spotlight for rather lackluster test scores, higher than average absentee figures, and significant staff turnover. The prior spring, the district management team had moved several principals to different schools, primarily to invigorate West. Jack Mann was assigned to West with a mandate from Dr. Barbara Gross, the Barberton Superintendent, to "shape the place up." Jack Mann had worked in the district for ten years as an elementary principal and was well regarded as a manager who could "make the trains run on time." His first priority was making sure his building was operating smoothly, but a close second was to lead his school into the "Twenty-First Century." He and Superintendent Gross wanted to keep ideas percolating in the school. Jack was the kind of principal who never met a good idea he didn't want to immediately implement in his school.
The management team's strategy appeared to be working beyond everyone's expectations. In just a few months (Jack was appointed in June and worked all summer), the new principal's fingerprints were on myriad projects, from granting flexible scheduling to teams to bedecking the halls with students' academic projects and artwork. Staff learning was almost as important to Jack as student learning. He had been around long enough to realize that teachers armed with adequate knowledge and support can make all the difference in the world and so was eager to create opportunities for intensive staff training in content and instructional strategies connected to cognitive research and adolescent development. He also was very interested in new forms of performance assessment and was launching a service learning requirement for his eighth graders.
However, West was plagued with a high staff turnover. By the time school opened in September, six of the twelve math teachers were new to West Middle School, either starting their careers or moving up from elementary schools in the district. So in one form or another, half of the math department was just learning the ropes. Of the six remaining veterans, two teachers were expected to retire at year's end and were doing little more than holding out for their pensions. However, with Jack's infectious enthusiasm for change, any change, most of the department seemed willing to participate in the new Making Sense in Middle School Math project that was slated to begin on September 1. Alison was relieved that Jack was at West with his strong administrative style and that the six new teachers seemed willing to give her professional development program a try.
Finding Coverage for the Teachers' Time
As Jack Mann picked up the phone, Alison realized that wanted Jack's support to ensure that Sally Elmore would stay in the program. Any teacher's departure at this early stage could signal to others that there was an exit option, which Alison definitely wanted to avoid.
"Hiya, Al," Jack said. He waited for her to proceed.
"Hey, Jack," Alison responded. She described the email from Sally Elmore. "Who is she anyway? I don't think I remember her."
"Oh, sure you do, Al." Jack replied. "Sally's one of those kind, sweet elementary teachers who moved up here this fall. Nice lady who has a lovely encouraging way with the kids. In fact, just yesterday, I visited her classroom for about fifteen minutes. It's wild how I can see the elementary school background of some teachers here. Anyway, Al, it isn't all rosy here in the Land of Oz. I had a flash of déjà vu, watching her. Déjà vu from that video we saw last year What was that thing called?"
"The TIMSS tape?" Alison offered. TIMSS was the Third International Mathematics and Science Study, the largest and most comprehensive study of math and science education ever conducted, spanning 41 countries. Earlier, district administrators had viewed a videotape depicting typical TIMSS classrooms in Japan and in the United States.
"Yeah, TIMSS, that's it," Jack said. "She just didn't seem all that comfortable with the math. She was fine with the kids, but the math ah, that's something else. She just accepted whatever the kids said without asking for explanations. And with the students who had wrong answers, well, she said something like, 'Not quite,' and moved on to another student."
Jack continued, "Frankly, Al, I worry about Sally's math knowledge. She seems to use slogans and rules rather than pressing for understanding. She just kept repeating something like 'a function is a rule' and that 'slope is rise over run,' rather than having the students figure out things for themselves. I'm no math teacher, but it sure seems different from what we saw in that TIMSS tape. Shouldn't she try different ways of checking students' understanding and going deeper into the math? This new math program of yours will help her learn more math, right, Al?? Boy, is she ever gonna need it!!! This program better work!!"
"There's got to be a way to keep her in the program," Alison said.
"Yeah," Jack replied. He paused. Alison could hear high-pitched laughter in the background, and the rush and rumble that signified a lunch session was beginning. She appreciated that Jack was concerned about this problem. "Yeah, yeah, yeah. Hey, gotta run. It's lunchtime," he said. "Can we talk about this after school?"
"But Jack," Alison said, now feeling more pressured and anxious. "The workshop is tomorrow!" It was no use. Alison would have to come up with a solution herself or wait until Jack had another minute to think about this. It was clear he was already onto the next issue. It sure was different being in the central office rather than in the schools, Alison thought to herself.
Alison's pressures stemmed not only from the LSC requirement that all teachers participate, but also from her Superintendent. She wanted to show the Superintendent the power of developing professional learning communities since it seemed that all the Superintendent really cared about was that there was a research-based, inclusive, and defendable process for curriculum adoptions in the district. In the Superintendent's view, the district had the responsibility to purchase the best materials, and the principals had the responsibility for ensuring that teachers were teaching well.
Alison recalled what Supt. Barbara Gross had told her. "Go ahead with the workshops, if you're so keen on them. But your job is to buy the absolute best materials for our kids. Get the research studies, find the evidence, and buy the best." Now it seemed that the professional development that Alison so deeply valued was about to go down the tubes if she couldn't even get teachers to attend the workshops. Alison felt that the introduction of new reform materials without strong professional development was a recipe for disaster, both for the students of Barberton and for her career in that district.
Sally Elmore's First Professional Development Class
Three weeks before she sent her email to Dr. Tansey, Sally Elmore had attended the first session of the Making Sense in Middle School Math training. She felt a little trepidation and had lots of questions over her six o'clock cup of coffee that morning. Reviewing the new inquiry based curriculum, Sally noticed that the materials were written for heterogeneous classrooms. While she believed that all students could learn lots of math by the time they graduated from high school, in her heart of hearts, she felt that there should be at least three levels of classes, geared to the understanding they had already achieved. "Yes, three levels at least," she thought. Some of the students were still concrete thinkers, she knew, and some were capable of abstractions such as understanding the concept of a variable, with a lot of help. And then there were the truly gifted students who understood a problem before she had left her lips. How could anybody teach all of these students together?
Most of Sally's teaching experience had been at the 4th and 5th grade level and in social studies and language arts. Even though she had nearly twenty years of experience, it was no secret that she was a novice when it came to mathematics. However, the few math classes she had taught in the past few years piqued her interest in having a fresh challenge. The idea of requesting a voluntary transfer had evolved quickly. One of her best friends was a veteran teacher at West. He told her that he'd never enjoyed teaching so much as he had in the past year. He suggested she try something new so the next week she put in her request, and a month later, it was granted. Simple as that. She knew students would feel cared for and safe in her classroom and she figured she could help her middle school colleagues learn some strategies for inclusive classrooms. But would she be a good math teacher?
Two hours later, Sally Elmore pushed open the doors to the conference room. The teachers were seated at one large table, listening attentively to a single speaker. Her heart sank, thinking she must have missed the beginning of the session. She heard Nicole Wittershein exclaim, "But that's not the most serious problem!" In response, a slim young man with a shank of dark hair leaned back and smiled calmly. It was the kind of calm that accompanies certitude, Sally thought. She watched him as Nicole continued, and she felt uneasy. Nicole's tone made her uneasy. So did the young man's too-cool reaction.
Nicole went on and on about the TABS test, the "Test of Abilities of Basic Skills" which the district asked middle school teachers to administer each spring in order to support placement decisions for the following year. Though Sally was new to the middle school, she had heard Nicole's plaint before. It had to do with students' woeful computation skills. The kids never were taught their math facts in elementary grades, the story went. Sally was amused by the inevitable slip between "teaching" and "learning" in such stories. Of course the elementary teachers had taught students their math facts. Sally could even attest that the students had learned them, but more frequently than she wanted to admit, it was clear that the students did not remember what they had learned just a few months before. Information was memorized, then summarily forgotten.
In another minute, Sally realized that she was not late, but that the meeting had not yet begun. The gangly young man, who had been coolly listening to Nicole, turned to Sally and then back to the assembled teachers and said, "I wonder if someone can tell me, are we all assembled? We're already in a dispute, and I don't even know your names. Please, come in," he gestured to Sally.
By the end of the second hour of the workshop session, Sally had experienced a huge range of emotions. She was exhilarated by the opportunity to use the graphing calculator, but grew terribly frustrated when she would try to have the calculator draw the graph of a linear equation and the screen was blank. She had no idea why this was happening. Despite her technological handicap, Sally tried hard to concentrate on the instructor's explanations.
As she listened and watched, the atmosphere in the room struck her. The instructor - whose name, improbably, was Igor - frequently persisted in questioning teachers who did not seem to understand the math. He sternly questioned them, and had a nasty habit of asking, "Are you sure? Are you sure?" even when they had the correct answer. His ferocity had even the most sanguine teacher saying, "Yeah, I think so," with a slight quaver. Sally was deathly afraid that Igor would call on her.
Igor Strome looked as though he were in his mid-twenties, but he was thirty-four. He had always looked younger than his years so he compensated for this by donning outfits he deemed "professorial," such as tweed jackets and corduroy pants. He did not realize that he came off as a geek, and an arrogant one at that. He deliberately put on a patient smile while people were working out problems, but the smile came off as an insolent sneer. The fact was, he had always been a whiz at math and he didn't really understand how anyone could find math difficult.
Igor had endured three years of teaching algebra 2 and trigonometry in a high school. He tolerated numerous annoyances with high school: lunchroom duties, advisories, and the adoption of block scheduling despite his reasoned protestations. A major disappointment, however, was his students' lackluster interest in mathematics. Almost all of the students took math to fulfill graduation requirements and college admissions expectations. Igor found the classroom to be intellectually dead and thus, for him, joyless. When a summer job of curriculum writing opened into a full-time position, he left teaching altogether. For almost ten years, he had worked with NCTM standards developing curriculum materials that aligned with the standards, planning teacher workshops, advising on state frameworks documents, and drafting questions for assessment materials. He loved NCTM's focus on mathematical thinking, and he agreed that all students should be as conversant with mathematical language as they were with the English language.
Late in the morning, Igor fixed on Iris, a new teacher whom he had asked to offer a rule for the sequence: 2,6,10,14,18 Iris wrote x+4 on her paper, and seemed genuinely shocked when Igor challenged this answer. He asked her what the 23rd number in the sequence would be and to explain her thinking to the class, emphatically adding that it was NOT 27. As she stammered, it became clear that x+4 by itself wouldn't work; however, Iris had no idea what to do. Igor kept pressing. "Think about it." he implored her. Sally was so uncomfortable watching Igor glare at her colleague that she did not follow Iris' explanation when she finally discovered the formula.
At lunchtime, Sally approached Iris in the buffet lunch line and asked her, "Are you okay?" Sally pretended to focus on the display of tuna, turkey, and veggie roll-ups in front of her.
"Well yeah, sure." Iris responded. "And how about you?" Iris returned the inquiry.
"Oh me? I'm fine." This was a patent lie, as Sally was getting knots in her stomach from feeling lost in the content and from the anxiety that Igor would call on her. "I was just wondering how you're liking the class."
"I think it's great!" Iris said. "I mean, Igor is tough, but he's good. I feel like I'm learning a lot. Nothing like being on the hot seat to activate the gray matter!" Iris laughed heartily, then moved away to join another first-year teacher at a small table by a window.
Sally was puzzled by this interaction. She stood lost in thought for a moment, holding the sandwich tongs, until someone elbowed her and said, "Hey, Sally." It was Nicole, who reached around Sally and took a turkey roll-up and a pickle with her bare hands. "He just doesn't get it, does he?" Nicole continued, oblivious to having startled Sally. "I mean, he really thinks that all of our kids are going to be able to do algebraic functions in heterogeneous classes. Its clear he's never taught at this level!! The eighth graders here don't have control of bodily functions yet, never mind algebraic functions! Ha! I'll be happy if some of our kids learn their math facts before they leave us." Nicole spun out of Sally's orbit just as quickly as she had spun in.
Sally with her Fourth Period Algebra I Class
Sally Elmore's shoulders sank. "Why is this so hard for them? They're eighth graders in the above-average track," she thought to herself. "And this is something we've talked about a zillion times. They sure knew how to do linear functions from the old textbook." In hopes of reaping success, Sally had selected an activity from a standards-based curriculum that would reinforce lessons students had already had. She was startled to see some initial confusion from her students, especially those whom she considered to be bright and capable.
Even though the math department still tracked their students, Sally knew that the plan was to move toward heterogeneous grouping over the next two years. The math department was the only department that still tracked students at the middle school level. Many members of the math department thought it was a mistake to move to heterogeneous grouping, citing the different social backgrounds and the range of abilities and knowledge of the students. One teacher, Abe Morgan, even went to a local state college library to see if there was any solid evidence about what was best for kids. Unfortunately, the only studies he could find were filled with statistical equations, t tests, and education jargon. He declared the articles "Worthless and unintelligible!!" He continued, "Even if you could even understand that stuff, I'm pretty sure there are just as many studies on both sides about tracking!!" After a few discussions in the teachers' room Sally learned that it was best for her to keep her mouth shut about this issue since feelings ran so high, especially among the "veterans." Though her background as an elementary teacher had led Sally to believe that valuable social skills were learned in heterogeneous groups, the range of knowledge, skills, and ability in math seemed much greater in her middle school classes. Perhaps the range, she reasoned, was not due to different academic abilities, but rather to the fact that children came from different elementary schools where some teachers had not taught certain topics.
"Ms. Elmore. Ms. Elmore! OVER HERE!" Tom was bellowing and waving his hand as if he were waving down a train from across the room. "Just a minute, Tom," Sally said with a bit of exasperation in her voice. "I am still helping Andy and Sam."
A few minutes before, Sally Elmore had placed a problem from one of the reform curricula on the overhead for her fourth period algebra I class. "The Raven," was an experiment intended to strengthen student understanding of linear functions. Essentially, the investigation involved starting with a clear cylindrical vessel, pouring in an arbitrary amount of water, depositing marbles of uniform size into the container, measuring the height of the water level, and plotting the height as a function of the number of marbles in the container. Today Sally wanted her students to actually experiment and record their observations, hoping that the experiment would help them move from a graphical to a functional representation of the relationships between marbles and water height. The activity took its name from an old Native American legend in which a raven, dying of thirst, drops pebbles into a deep well until the water level is raised to the point where it may be reached by the bird's beak. Sally had divided her class of 28 students into several cooperative learning groups. Though the students at the middle school often socialized in these groups, she still felt it was important to group students this way. The groups each received ordinary glass jars of undetermined volumes.
Sally moved over to a group with Tom, Veda, and Becca. "So what's the problem?" she asked the group. On their papers, she could see that the trio had made a table with columns labeled "marbles" and "water." It looked something like this:
Sally looked at their glum faces. "Okay, so now what? What do you think about when you see sets of points like that?"
"A line?" said Becca reluctantly, after a long silence.
"Yes," said Sally, "and what does a line always have?"
Tom looked confused. "Length?" he said with a tentative expression on his face.
"Yes, but what else?" Sally said, "What does a line always have?"
"Oh I know!" piped up Veda, "SLOPE!"
Sally felt relieved that at least one of the students seemed to remember that they were in the midst of a unit on slope, intercept, and linear equations. "So," Sally continued, "if this line has a slope, what is it?"
Becca made an ugly face. "What's the matter, Becca?"
"We already talked about that, but we don't agree.
"What do you mean you don't agree?" Sally inquired, peering at the threesome.
"We get different slope answers and Tom says that's OK because we used different points. I don't think it's OK." Becca responded.
"If you want to get the same slope, you haf't use the same points." Tom affirmed. Becca sighed, "But Veda and I say it don't matter which points we use. Who's right?"
Remembering what Igor had said about it being best for students to discover their own ideas, rather then being told, Sally returned the question with a question: "Well, why don't you try again using different sets of points. Tom, why don't you use (0, 10 1/2) and (10, 13), Becca you use (2, 11) and (6, 12), and Veda you try (4, 11 1/2) and (8, 12 1/2). See what you come up with."
After a couple of minutes' calculation, Becca and Veda both computed values of 1/4. Tom, on the other hand, found a slope of .35.
"So," said Becca. "I guess Tom is right. You gotta use the same points to get the same slope."
Seeing the two answers of .25 and .35, all of a sudden Sally wasn't sure. Did it make a difference, she thought to herself? Caught up with the question, and suddenly feeling quite uncomfortable, Sally was jolted out of her confusion by Veda's giggle.
"Hey Tom!! Where'd ya learn to subtract?" Veda said with big grin on her face.
"What do you mean?" Tom shot back, a bit defensive. "13 minus 10 1/2 is 3 1/2."
"Since when?" Veda continued. "13 minus 10 is 3. 13 minus 11 is 2. What's 13 minus
"2 1/2," interjected Becca before Tom could say anything.
Sally realized what had happened and said, "Maybe you forgot to borrow one from 13 and rewrite it as 12 and 2/2, Tom. Do you see that?"
"Yeah," he said with a deep sigh.
"So what's 2 and 1/2 divided by 10?" Sally continued.
Tom still looked puzzled. "Do I have to invert and multiply or is it one of those where you look for common what are they called?"
"What do you think?" Sally responded, following Igor's suggestion.
"Which one do I flip--or can I just cross multiply?"
Now it was Becca who looked exasperated. "Geez," she said, "2 and 1/2 is the same as 2.5 and when you divide by 10, you just move the decimal point one place. So it's .25. Okay?"
Tom's looked down at his paper and quietly said "Oh yeah, I get it now."
"Good, 1/4, the same as Becca's and Veda's calculations. Now do you think it matters which points you use to find the slope?" Sally continued.
"I guess not," answered Tom with a certain resignation. "But I still don't see why not," he added under his breath.
Ignoring Tom's comment, Sally pressed on, aware that the period was nearly over. " So now, what does slope mean in this investigation? Just what is slope?"
Dead silence. Nothing.
Becoming a bit impatient, Sally said, "Remember, slope is rise over..."
"RUN," exclaimed Tom, now looking somewhat vindicated.
"Yeah," Becca exclaimed, "One marble makes the water go up 4 spaces."
"Yup, that's right," said Veda. "It's got to be right because it's rise over run, that' s what you always say, huh, Mrs. Elmore? So since it makes the water rise 4 spaces, its 4 over 1 RISE OVER RUN."
Sally felt herself getting confused again. Veda certainly sounded convincing.
"But what's that .25 that we got?" Tom asked looking back at his paper.
Sally's head was spinning, but she had no choice except to stick with it. Now it wasn't entirely clear who was doing the investigating, Sally or her students.
"Let's see if we can make some sense of this." Sally tried to sound excited and confident about the search for understanding, even though it belied her lack of knowledge. "How much water did you start with? 10 1/2, right?"
By now Tom had lost both interest and confidence in Mrs. Elmore and was looking out the window. Veda, however, nodded in agreement.
"Now when you added two marbles, how high was the water?"
"11 cm," Veda said.
"So if 2 marbles made the water rise 1/2 cm., could 1 marble make it rise 4 cm, Tom?"
Jolted back to the reality of the classroom, Tom looked first at the girls and then at Mrs. Elmore, and said "No," though it was clear he didn't have a clue what the question was.
"You sure?" Sally inquired. Tom said nothing.
"OK, how about you two?" Sally said glancing at the girls. Both nodded their heads in assent.
"So it takes 4 marbles to make the water rise 1 cm. Explain to me what slope means," Sally said, returning to the original question.
"It means that the height of the column of water goes from 10 1/2 to 14 1/2." Veda quickly responded.
"What do you think, Becca?"
"Yea, that might be right, but I also think it means that 4 marbles make the water go up 1 cm."
Not too sure herself, Sally tried to clarify, "Is that the same as if someone said that 1 marble made the water level rise 1/4 cm. Is that the same??"
"Yea," said Veda, in a totally bored tone of voice.
"And you agree, Becca?"
"I think so I guess well, I'm not really sure. I think I'm getting confused about all of this. Maybe I'll come see you after basketball practice. Would that be OK?" Becca asked.
"Not today, Becca, I have one of those math workshops right after school. How about tomorrow?"
"Okay," Becca replied as the bell rang.
Reflecting on the class, Sally realized that some students who wrote linear functions perfectly well using the algorithms in the old textbook were now floundering, and she wasn't sure why. Did the activity only confuse rather than help her students understand the concept? In her plan book, she juggled classes around to give the raven problem one more period, in hopes that Tom and Becca would gain the same understanding and confidence that Veda had. She wished she had some of Veda's confidence herself!
Storm Clouds at State, Moving toward Barberton
The storm clouds were brewing some 200 miles away from Barberton in the state capital. Approximately 14 months ago, the state Board of Education appointed a mathematics curriculum framework committee whose charge was to revise the 1995 Curriculum Framework to provide greater guidance to school districts on matters of mathematics curriculum development, implementation, and assessment. The framework was to be the basis for the development of state assessment tools as well. The specific directive was to provide greater clarity to the Learning Standards, collapse artificial separations between topics, assure articulation among grade levels, and highlight key ideas and their development within strands, across strands, and between grade levels. This draft document called for the judicious use of calculators as well as a stance of inquiry and investigation in the study of mathematics.
The Framework Committee was charged with making recommendations to the state's Commissioner of Education, Dr. Donald Drew, who in turn would make a recommendation to the state Board of Education about what children should know and be able to do in mathematics in the state's K-12 schools. The Framework Committee was made up of 13 individuals, including two mathematicians in major research universities in the state, several parents (from urban, suburban and rural districts), a number of business representatives from private industry and the state business alliance, two classroom teachers (one from the elementary level and the other from the secondary level), and a middle school principal (who was named Middle School Principal of the year in the state).
The NCTM standards influenced the writing of the state standards with a clear emphasis on mathematical understanding in its introduction: "Mathematical skills must extend beyond the ability to calculate to the use of mathematics to investigate, analyze, and interpret." The document went on to describe a "math classroom" as a place that "should provide practical experience in mathematical skills that are a bridge to the real world, as well as explorations which develop an appreciation of the beauty and value of mathematics." There were nine content standards in the document. One was computation. The others were: numbers and number sense; data analysis and statistics; probability; geometry; measurement; patterns, relations, functions; algebra concepts; and discrete mathematics.
One member of the State Board (who would formally review and vote in February whether to approve the framework) had immediately lambasted the proposed math framework revisions. Mike Reynolds, a wealthy political contributor to the governor, and an outspoken critic of the performance levels of the state's public schools, frequently made known his opinions in the state-wide newspaper, The Star. In a recent article, Reynolds had been quoted as saying, "This math document only reinforces the confusion among teachers and the public about student performance levels. Words like explore, investigate, and develop are not measurable verbs. The document needs to eliminate all those 'explores' and 'develops.' When it uses 'constructs' or 'creates,' it's even worse." Reynolds made many speeches around the state and seemed to be politically influential.
One of Reynolds' favorite targets was a theory of learning called constructivism, which focused on children making their own sense of mathematical concepts through exploration, investigation, and inquiry. He expressed concern that kids "would just make things up" and that all the schools cared about was that "kids were happy, even if they got the wrong answers." About six weeks ago, there was a picture of Reynolds in The Star holding his nose, with the caption: "Jack Reynolds, local business leader and chairman of Governor Kearns' re-election campaign, says the new mathematics framework 'reeks with a constructivist philosophy.'"
The Barberton School Committee Meeting
Alison usually didn't pay much attention to the political maneuverings in the state; she only occasionally skimmed this section in the newspaper. But on the topic of mathematics curriculum, she felt she understood children and their learning needs better than many of the politicians and had a very firm grasp on issues of assessment since she had written her doctoral thesis on this topic. Because of Barberton's work in mathematics, Alison had ordered and read the draft of revisions from the state. She liked the general thrust of the document. She felt it recognized that many students, especially those at the middle grade level, could learn concepts in math from concrete models and manipulatives. So many of them, it seemed to her, were not yet at the abstract level in their thinking or functioning in mathematics. She agreed with the framers of the document that it was important for teachers to understand how math worked.
Reynolds' outbursts against the document worried Alison. She wondered if some of Reynolds supporters in the community would agree with his views and challenge her choice of curricular materials and Barberton's math curriculum.
When a group of parents attended a school board meeting in November to talk about mathematics, Alison's fears were realized. Now the political maneuvering of Reynolds and others at the state level became more than a distant distraction. During the public comment period, the leader of a group called Mathematical Accuracy Now! read a statement and asked several pointed questions about the system's mathematics curriculum. The statement, which they also submitted to the local weekly newspaper, stated in part:
"We, the undersigned citizens of Barberton, wish to express our deep concern about the academic competencies being developed by children in Barberton's Public Schools. The district's tests scores are poor in comparison with state rankings and are particularly troubling in mathematics. What will happen if our children fail the new Statewide Competency Test (SCT) and cannot graduate from high school? The problem is particularly acute in the Middle School where 57% of our youngsters scored at the 'needs improvement' level. We demand that the Superintendent and the School Board launch an immediate review of the Barberton mathematics curriculum. While we applaud district efforts to emphasize children's backgrounds and cultural heritage (Alison had submitted and won board approval for an ambitious multicultural curriculum in September), we seriously question what this has to do with passing the SCT?"
After the leader read the statement, the meeting turned into quite a free-for-all. One citizen, who no longer had children in the schools, declared, "Last week at the Shop Now supermarket, the check-out clerk couldn't make change for a dollar. The bill was for $1.98 and I gave her two one-dollar bills and three pennies. She looked at me like I had two heads! If kids can't make simple change at the supermarket, why should I pay taxes to support a school district that is failing? This is a public embarrassment! What's all this new stuff about anyway? Why can't kids learn math the old fashioned waywith hard work??"
The Chair of the School Committee glimpsed Alison Tansey's raised hand, and called on her to speak. Alison informed the group that Barberton administrators had been aware of the need for improvement in math performance for over six years, which is why they had initiated curriculum revisions and material purchases for the elementary schools. "Now we are turning our attention to the middle schools and are piloting a middle school math program deemed "exemplary" by the U.S. Department of Education," Alison continued. "To be rated exemplary, a program had to provide evidence that it worked, that it raised students' mathematical achievement," she said. But her assurances did not stave the tide of criticism. Even Superintendent Gross, who had strongly advocated for the use of evidence, avoided eye contact with Alison as she regained her seat. Alison knew her comments were unconvincing and wished she had brought the specific evidence of achievement to the meeting.
The public commentary continued. "But I don't think the elementary program is working either," said a man who identified himself as the father of a third grader in the Harrington School commented. "In my work as an engineer, my company doesn't pay me to sit around and make up plausible answers and then check it with my colleagues and on the Internet. I have to know how to solve the problem correctly or else I lose my job." A mother of a fifth grader chimed in, "Martin doesn't enjoy school any more. He used to love to play math games and do card tricks. We thought he might go to medical school. But with all of the emphasis on group work, cooperative whatever you call it, he keeps getting stuck with the slow kids in the class. He hates it and says he hasn't learned anything new in months. Tell me this," the mother continued, "they don't all go to medical school together, do they? When are you going to teach my son the skills and knowledge he wants and needs?"
After the Meeting
Some of the criticisms didn't sound right to Alison. For example, one parent questioned the use of calculators. This mother publicly stated that a mathematician told her that other countries do better than the United States, in part because they forbid the use of calculators below the ninth grade. The parent didn't mention the name of the study. Another said she had read in the newspaper that many of these new "reform" curricula didn't teach paper and pencil computation skills because now that everyone would have a calculator and a computer, such skills were out-dated.
There were so many allegations and unsubstantiated comments at the meeting that Alison went home with a pounding headache. Even though she couldn't wait to get to bed, she did check a document that she had read recently to see what it said about calculator use. There it was, on page 65 of the document, The Third International Mathematics and Science Study 'Pursuing Excellence' Booklet-12th Grade: "US students' use of calculators was similar to that of other countries." But at the bottom of page 65, the report stated: "About three-quarters of US twelfth graders reported using a computer at school, home, or elsewhere, which is higher than the international average."
Alison clenched her teeth in frustration. "I don't know if that parent deliberately misrepresented what the mathematician told her, or if that math person read a different report than this one," she thought to herself. "But that message is what people heard, so now what can be done to get this evidence out there?" She wondered if she should have foreseen that this would come up at the School Board meeting. But how could she have? It always seemed like the school district was playing catch-up, reacting, and responding rather than taking the initiative. It took her a long time to get to sleep that night even though she had a 7 am breakfast at the Harrington School the next morning.
Alison knew this group of parents and citizens was well organized (she had noticed the presidents of several PTA's in the audience) and that they would not go away. So she was not surprised when the Superintendent asked her the next afternoon to craft a response for the next board meeting and the public.
Workshop #2: Despair for Alison and Redux for Sally?
Alison Tansey happened to be at West in the afternoon and decided to pop into the Making Sense in Middle School Math workshop to see how things were going. As she walked over to the room where Igor and the math teachers would be meeting, she thought about the standards movement, math curriculum, parent concerns, and those long-term projects of hers that always seemed to end up on the back burner. One of those projects was called "Reporting Student Progress (RSP)." She wondered if it would be OK to raise some of the issues from that project with the math teachers who would be at the meeting. She had undertaken the "RSP" project because teachers complained that the current report cards did not allow them to report on the most important aspects of student learning. Alison could see that the report cards did not align with the standards, for one thing. And she agreed with the superintendent that the time was ripe for a wholesale revamping of the report card, since these forms had been in use for about fifteen years.
Everything she read pointed to danger, however, and she knew that report cards were hot issues for teachers, students, and parents. When she saw a conference on report cards was in the offing, she took a precious day away from the office to attend. The conference was led by Graham Haire, a national expert on standards, led the conference and confirmed Alison's own cursory research. Dr. Haire predicted that the standards movement would soon rub up against reporting traditions, with the consequence of lots of confusion, conflict and acrimony. For example, didn't it make more sense to try to report the best of a student's understanding and knowledge, recognizing that students would make mistakes early in their learning, rather than reporting an average for the term? But teachers typically average grades, thus rewarding the facile learner, the quick study. And what about printing the content standards on the report card itself, so that there was direct and unequivocal alignment between the standards and the reporting mechanism? But how would parents interpret a report card that had blanks for certain standards that had not been addressed during that marking period? Then there were the difficult issues of students who received A's and B's on their report cards, but scored in the "Needs Improvement" range of the SCT's.
When Alison arrived at the math workshop, she briefly met Igor Strome for the first time and then took a seat in the back of the room. As a warm-up exercise, Igor asked the teachers to carry out a small activity about standards: "Which of the following statements reflects what you believe about standards, A, B or C?
Once you've made your decision, go to the corner that best reflects your beliefs about the standards movement," Igor said.
Sally Elmore was sitting with Iris in the middle of the group. She could see that large red letters had been posted in three corners of the room. She knew what she believed. She believed in differentiation, even if she was not sure how to manage it in the classroom all the time. She was not the first to stride to her corner, but she did walk confidently to corner "B." In a few moments, most of her colleagues had joined her. Nicole and two others stood in corner "C."
"Okay, let's talk about this before we go any further," Igor told the group. He became even more bossy and vociferous than he had been before, perhaps because Alison was there. In the ensuing discussion, people disagreed vehemently with one another and with Igor; two people left the room. Nicole Wittershein felt that students must master the basic skills before they could grapple with concepts such as variables. Igor lectured the teachers about the importance of standards in the context of a rapidly changing world and that the standards were a response to a growing reality that kids won't make it unless they all have strong skills and understandings. He told them that the State Competency Tests (SCT's) measured those skills and understandings. It was clear he felt the teachers had a responsibility to prepare kids to do well on this important test. Furthermore, he told them that their commitment should be to provide instruction in problem solving and framing skills that were emphasized on the SCT's.
As Alison Tansey watched the warm-up exercise and listened to the ensuing discussion, she decided to drop her questions about report cards. If the teachers were in disagreement about the meaning of standards, how could she ever hope to get them to agree on a symbolic representation of the standards? She agreed with the workshop leader's position, but raised her eyebrows at his holier-than-thou tone. She hoped that teachers would interpret her presence for what it wasagreement with holding all students to the same high standards, realizing that it would take hard work on the part of staff to achieve this goal, but she also hoped that some key teachers noticed her raised eyebrows, so that her presence wouldn't be seen as condoning the style of delivery.
Sally was upset at the way Igor set up the exercise. Almost everybody had gone to her corner, and Igor basically told them they were wrong. He said, Let's talk about this," but didn't let anybody get a word in edgewise. It made Sally angry that Igor asserted his instructional strategies would help kids, when all he was doing was humiliating teachers and modeling strategies that certainly were not what students needed, not in her class anyway! She heard Igor state definitively, "Tracking slower students into different ability math classes doesn't help them. They'll never be exposed to the math they need to be successful citizens in those dummy classes." It was too much for Sally to bear so she got up and left for the bathroom. She intended to take a breather in the ladies' room, but after a few minutes of reflection about her classes and what she needed to do tomorrow, she grabbed her coat in the hallway and left.
An Integral Part
It was not dedication to the ideas promulgated in the workshop but commitment to her students that made Sally stick with the problem about marbles and water. In her next meeting with her 4th period Algebra I class, she found herself moving toward Andy and Sarah who were busy talking about the dance being planned for Thanksgiving. As she approached, they feigned attention to the task at hand. Andy and Sarah had begun their experiment with a different sized beaker from Tom, Becca and Veda's group. It had about 8 cm of water in it when they began.
Looking at their papers Sally noticed that both Andy and Sarah had the same table of values but their graphs didn't look the same: Andy had connected his points and Sarah hadn't. Again remembering to help the students inquire, Sally asked, "Do you think your graphs are the same?"
"Sure, Mrs. Elmore," Andy replied. "We used the same points. See?"
"You don't always connect 'em when ya graph," Sarah said, her brown eyes sparkling with confidence as she wrote with one hand and twirled her shoulder length hair with the other.
"You don't, Sarah? Can you tell me why?" Sally inquired, thinking she was wrong.
"Nope. I just know ya don't." She unconsciously snapped her gum as if to add emphasis.
"Just think a minute." Sally continued. "Don't you remember that two points determine a line? You should connect them, Sarah."
"I'm not sure, Mrs. Elmore," Sarah said. "Look here on Andy's graph. What's the water level at this point?"
Andy squinted and drew his finger horizontally across the graph to the vertical axis. "I dunno. About 11?" he answered questioningly.
"That's right, just about 11 centimeters," Sarah replied. "Now how many marbles are in the jar when the water is at this level?"
Sally put her finger on the same point as Andy had, but instead of sliding her finger across the paper, she drew it straight down to the horizontal axis. "About four and a half," she responded.
"But that doesn't make any sense!!" Sarah replied. "Who ever heard of a half of a marble?"
Sarah and Andy could scarcely contain their merriment at seeing Mrs. Elmore so completely flummoxed. Sally vaguely recalled a problem that Igor had asked the teachers to do in a workshop session where they had to find the number of buses needed to carry students to a football game. Was this the same kind of problem, Sally wondered? In the workshop when Sally answered 2 and 1/2 buses, Igor had responded in a disdainful tone, "That's just plain wrong, Sally. How can you have a half of a bus?"
But the embarrassment that Sally had felt when Igor pressed her at the workshop did not hold a candle to the feeling she was having now, looking down at Sarah and Andy's graphs. Why did she ever agree to move up to the middle school anyway? She knew she didn't know this stuff. Why she didn't even like it! How in the world would she ever gain a strong and deep enough understanding to teach it? Let alone teach it in this new inquiry-based way? Why couldn't she just tell the students the rules?
Time and Time Again, a Gulf between Theory and Practice
As soon as Alison got back to her desk after the workshop, she decided to look more closely into the students' computation and problem-solving skills herself. As Nicole had mentioned, the district used an in-house test that featured basic math skills in computation and use of algorithms, as well as several problems that required a good handle on functions and mathematical reasoning. The TABS test had been used for years in the district. Two different members of the math department scored each open-ended problem, using a rubric. With the new state legislation, this test took on new importance. It was considered part of the district's "Comprehensive Assessment System." Even though sanctions and rewards had not yet been attached to the local assessments, just the fact that results were reported to the state raised the visibility and hence the importance of this test.
By happenstance, the most recent TABS test scoring had just been completed, and the raw results were on Alison's desk, waiting for tabulation. However, in the space of an hour, Alison found herself in another project, having to do with inter-rater reliability. In an effort to support local districts in the creation of local assessments, a group of statisticians, professors of statistics, and education department personnel had co-authored a technical manual. This manual contained recommendations for checking the validity and reliability of local assessments. Alison, who prided herself on creating processes for high-quality products, was very interested in these recommendations.
She decided to check the inter-rater reliability of the TABS test scores. The manual contained two appendices on this issue, "Calculating Inter-Rater Agreement" and "Calculating the Pearson Correlation Coefficient (r)." She recalled the importance of "r" from her statistics course in graduate school, and so turned to that appendix first. While the calculations looked complicated at first glance, on closer examination she saw that the coefficient could be derived with simple computation. However, the example presumed that there were only two scorers. She deliberately involved the entire math department in scoring the TABS test, and she was not certain how to accommodate that many scorers in the model. She called up one of the principal authors, a professor at a branch of the state university system. She was in luck: he was in.
"Hi, Professor Carlson. I need help! I just read your paper on reliability, and I want to use the process for determining our inter-rater reliability on our district math test. Can you walk me through it?"
"Sure," Professor Carlson answered agreeably. "To set up the spreadsheet, have each column correspond to a rater's score. If you have 9 raters, you'll have 9 columns. Many of the cells will be blank for each student, of course, because only two raters rated each student's open ended questions, right? You'll also want to have some kind of student id column. To look at inter-rater reliability, simply note the percentage of students for whom the two ratings agreed.
"How do I do that?"
"Oh, just by looking. Do it manually. To look at rater effects, compute a mean rating for each rater. Let me know how it goes. I have to run to class now."
After speaking with Professor Carlson, Alison was less enthusiastic about calculating inter-rater reliability. She was incredulous that with technology at hand, the recommendation was to do manual computation from a spreadsheet. Surely there had to be a better way. And was there any way to note if a scorer had "drifted" high or low? She sighed, then chuckled, not knowing if the situation were sad or funny. What a difference there was between recommending that districts compute their "Pearson Correlation Coefficient" and actually pulling it off.
Time and Time Again
After her interaction with Andy and Sarah, Sally turned to look at the work of a third group of studentsLiz, Greta and Andrew. She noticed they had yet a third set of points because their container was a different size from the other groups.
Sally felt better about working with this group because they usually had the right answers and seemed to be strong in their computational skills. She could see that they, at least, were making the transfer to functional notation in their work. "What'd you get for your equation of the line?" Sally asked.
"We got something like y = (.45/7)x + 4 2/5 for the rational form and y = .064x +4.4 for the decimal," Andrew replied.
"Good. So what do x and y represent?" Sally asked.
"x is how many marbles you put in," answered Liz.
"And y, Greta?"
"How much the water goes up when you put the marbles in." Greta responded.
"How much did the water rise when you added 7 marbles?" Sally pressed a little bit, remembering Igor's instruction to push students until you were certain they really understood.
"4.85 cm," Greta answered.
"Sounds right," Sally accepted the answer but continued, "Are you all sure of Greta's answer?" for confirmation.
"No," replied Liz honestly, "but I think so. I don't see what else it can be."
"Well, Greta's right," Sally stated cheerfully and with relief that at least one of the groups in her class seemed to be getting the point of the exercise. "x is the number of marbles and y is how much the water rises."
Back at the Ranch
Alison stared out the window of her office at the gray November day. The sky matched her mood. "I need strong research studies as well as local data on math achievement in order to prepare my report for the Superintendent and the School Committee. I need to look at the new math program objectively and acknowledge that a few single-school studies find it does not work so well, but weren't those studies subsequently shown to be flawed?" she asked herself. "I've got to present information in a way that everybody can understand--and not take too much time doing it. I need a teacher who loves the program, and a teacher who has been won over. Should I invite Igor Strome? He certainly knows the state and national scene " She was mulling over how his presence might put off some parents when a phone call came in from Jack Mann. It didn't make her feel any better.
"Al, we gotta talk about the math workshops," he said. He recounted what he had learned in the last hour. At lunchtime, Jack asked Iris and Nicole how things were going in the math workshop sessions, and he had gotten an earful. Part of what he heard worried him. Nicole blurted out that the new math materials they were adopting were far worse than the old ones in terms of covering basic math skills. She predicted students would fare even more poorly on the TABS test as well as the Iowa test in the eighth grade. "Maybe they'll know how to think mathematically about probability and statistics," she told him, "but the problem is, they won't be able to double a recipe. Ha! This spells disaster. It's a terrible curriculum," she said. "Watch my words, this is a mistake. Parents won't stand for it," and she wagged her index finger at him for emphasis.
Iris, on the other hand, seemed delighted with the potential of the graphing calculator to convey the meaning for many students. "I've already seen the light bulb go on for a bunch of kids in my classes," she told him, excitedly. "I'd never have gotten this far if Igor, that's the workshop leader, hadn't pushed me. But he's a tough guy. Honestly I think he pushed Sally Elmore over the edge in the last session. He doesn't have a lot of patience, you know, and he just assumes that we know it all already. Don't tell Sally I told you, but I think she was crying in the bathroom during the workshop. She left at lunchtime because of Igor."
Jack pointedly asked Alison. "Is it true, Al? Is it true these materials don't teach the basics and won't help us get our test scores up? GAWD! I can just hear the parents now, let alone Superintendent Gross And where in the HELL did you find this Igor guy?" Jack's voice was rising, he was furious. "Who is he? I don't want Sally to be upset. She's new to West and I want her to stay to convey that caring ethos of the elementary schools that we need up here." She had been strong-armed into attending yesterday's session, by himself and by Alison. Professional development was great, but not at the cost of losing a caring teacher. "Time and time again, Al, we do the wrong thing for what seems to be the right reasons. We're not helping but hurting teachers."
Alison's heart sank. She didn't know what to do. How would she ever convince Sally and the other teachers, let alone her Superintendent and Jack Mann, that this new math program was best for kids? Furthermore, how realistic was it to think that a few in-service workshops would help prepare math teachers teach a math they never had, in ways they never experienced? Were these new standards-based curriculum ideas really worth it? She thought about a statement she once read in graduate school "Teachers are both the objects and the agents of reform." Could be both in Barberton, she wondered?
(1) Katherine K. Merseth and Molly V. Schen prepared this case for the National Science Foundation Teacher Enhancement Program and the Merck Institute for Science Education. Any viewpoints expressed in this case do not represent the views of NSF or the Merck Institute for Science Education. The names and places are fictional, though based on real experience. Arthur Proulx, Darien Lauten, Karen Graham, Melisa Collins and Steve Bensen of the University of New Hampshire developed an earlier version of sections of this case. A complete version of their case will appear in Windows on Teaching: Cases in Secondary Mathematics, edited by Katherine K. Merseth and published by Teachers College Press.