(Vol. 1, No. 1 - Fall 1996)

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Of Penguins and Problem Solving:
Working Through New Math Standards


By Anne C. Lewis


Jeremy and his two partners have just discovered they have a big problem. After perusing a heaping pile of research material, Jeremy announces with surprise in his voice: "Penguins eat four kinds of food!"

It's back to their calculator and their graphs for three of the aspiring marine biologists in Sandi Machit's sixth-grade math/science class at Hughes Middle School. Costing out a daily food budget for emperor penguins at Sea World is not the simple task it seemed at first.

All over Machit's crammed classroom, and spilling out into the hall, small groups of students are calculating the annual cost to feed 5 sea otters, 10 bottlenose dolphins, 19 sea lions, 8 killer whales, 8 emperor penguins, and 6 sharks. Once they figure up the costs for squid, shrimp, codfish and other delicacies from the lower end of the ocean's food chain, they must transfer their information to a circle or bar graph.

Other students are taking the average measurements of one of Sea World's marine celebrities and converting them from centimeters to inches in life-size drawings. Their miscalculations could lead to misshapen penguin bellies or whales with too-long tales. Still other students are busy gathering information from different sources for a written report about one of the sea creatures.

Everywhere they look, the young students can see math and science coming alive. A number line stretches across two sides of the room so they can visualize negative and positive numbers. Geometric mobiles made by students hang over their heads. The rules for their classroom "economic society" are posted on the door, describing their daily pay and average expenses (as well as fines and taxes).

Out in the hallway by the door hangs a large poster listing the Long Beach Unified School District's math and science standards for sixth grade. And on a table underneath the poster is an album of photos from Family Curriculum Night showing parents hard at work solving problems their children encounter in class.

It's fun but it's tough, too

No student in Sandi Machit's class is tediously working only with a textbook, alone and bored. Most obviously, her students have fun learning. Most importantly, they learn a lot.

This is a standards-based classroom, as are all of Machit's classes at Hughes. A member of the committee that developed new math standards for Long Beach, Machit began using them last fall -- well ahead of many of her colleagues. Most schools and teachers are still getting acquainted with the standards adopted by the school district in 1995 in math, science, English/language arts, and history.
In part because of her committee work, Machit has already made the leap that every teacher in Long Beach will be expected to make over the next year or two. She has figured out how to link the new standards to her everyday teaching and how to determine exactly what content in the standards is being covered by student work.

The Sea World assignments, in which her students pretended to be marine biologists applying for a job at the marine center and wishing to impress interviewers, was--as educators would put it--"rich in content."

Under Standard I for sixth grade math, for example, Machit's students were fulfilling an algebra requirement: "analyze tables and graphs to identify properties and relationships," part of an overall standard that says students should learn how to generalize concepts of variables, expressions, equations, inequalities, and graphing. Her students were also:
Scanning the old content standards for math, published by Long Beach's Instruction Department in 1988, one finds no mention of using skills in real-life situations. Nor do the old standards require students to explain what they know through writing. And there is only slight mention of algebra skills (starting in the seventh grade).

The old math standards were long, abbreviated lists, dry and pedantic, repeating many of the basic arithmetic skills that ought to have been learned in earlier grades. The lists looked more like textbook indexes than a well-thought-out, well-stated vision of what students should be learning.

The new standards are much more descriptive. Not only do they detail the skills and knowledge students should have, they describe different ways the students should be able to use the skills. Knowledge once expected only of advanced students or in high school classes now is woven into middle school math standards beginning in the sixth grade.

And students and teachers agree: the new standards are tough.

Long Beach writes its own standards

In 1994, the Long Beach schools decided to make the move to higher standards. Sandi Machit was among the group of more than 30 teachers, parents, business representatives, and college faculty who came together that summer to develop new math standards, drawing on the work of national groups and their own expertise. (See "Why do we need new standards?")

"We had elementary, middle, and secondary teachers in the same room for the first time," recalls Machit. Being in the middle "was like watching a tennis match," she says, with elementary and high school teachers going back and forth about what was and was not being taught.

By dividing up into student age groups, the committee shaped new standards that first described the all-important basic skills, then built on them by adding higher level skills and describing ways that students should apply what they learned.

"Our most important objective was to make math more meaningful" for every student, Machit explains. To help do that, the committee emphasized more creative teaching techniques, including the use of math "manipulatives" -- solid objects like blocks and rods and coins that help students get a "hands-on" understanding of math ideas. The committee also stressed the importance of new technologies like computers, and described ways to use writing to make sure that students understood complicated math processes.Most importantly, perhaps, the committee designed the standards so that they could be combined into complex lessons like Machit's Sea World project, where math, science, and other subjects might be combined, as they so often are in real life.

The process of setting standards begins at the end. You start by deciding what knowledge and skills your high school students need to succeed in 21st century America. Once the Long Beach schools made these decisions, the standards-setting teams began working backwards.

The Committee's draft proposal for math standards went to each school for comment. Every copy came back with red marks, says Dixie Dawson, math curriculum leader for the district."If anything, the comments set the bar even higher."

The process of setting standards begins at the end. You start by deciding what knowledge and skills your high school students need to succeed in 21st century America. Once the Long Beach schools made these decisions, the standards-setting teams began working backwards, drawing a curriculum "map" that stretched from 12th grade back to kindergarten, with standards set for each grade along the way.

For Superintendent Carl Cohn, the absolute minimum mathematics standard for all students is to graduate with strong skills in algebra. The foundation for that goal, says LBUSD curriculum director Chris Dominguez, must be poured in the middle grades.

Why should algebra be the basic standard? Although most parents--and many teachers--don't realize it, the reasoning skills that students develop in algebra classrooms are becoming the minimum skills for good jobs in the future.

Algebra is no longer just the "gateway" course for college-bound students who plan to take advanced mathematics in high school. Employers are talking more and more about building a bridge between school and work -- and raising the awareness of students, teachers, and parents about the kinds of basic and advanced skills students will need to operate computerized equipment, make decisions about quality control, and so forth.

"Algebra for all" is not an impossible standard. As Dominguez points out, when high schools eliminate general math and require all students to take algebra, failure rates don't go up. The percentage of students able to learn algebra is about the same as those able to learn general math (although it should be pointed out that several levels of algebra are taught).

Moving to higher math standards also required the district to select a textbook that closely matched the district's objectives. Teachers who examined various textbooks found that the district's previous math book matched the new standards only 43 percent of the time. So the district changed to a math series published by Prentice-Hall with a 90-plus percent match.

At the middle-school level, the new math book assumes students have learned elementary math and can read well, says math consultant Dixie Dawson. With the old text, "sixth-grade teachers were teaching fifth-grade math for a good part of the year." Sixth-grade math used to start with whole numbers; now it begins with graphing data.

It does not surprise Dominguez that teachers are saying the new textbook is hard. "Our standards are much higher than before," she says. And the new standards move teachers and students beyond textbooks, she says. To succeed, teachers need a strong background in math -- they can't rely on the textbook to carry them through the year.

Many middle school teachers have elementary school degrees and lack a strong background in the subjects they teach. So the issue of teacher knowledge is a big one for the school district as it pushes for higher standards. There is so much to learn, and so little time.

Getting standards off the shelf

More challenging math standards are in place. A textbook series has been matched to them.
What next?

Stanford University professor Nel Noddings says that "we educated adults...should establish some standards for ourselves before demanding that our kids make us world-class."

LBUSD's challenging new standards in mathematics (and science, language arts and social studies) won't mean much unless teachers have the skills and knowledge they need to help students meet the standards. (See "Helping Teachers Teach to High Standards".)

In the last school year, "we were in an awareness phase," says Dawson. In the fall of 1995, every teacher received a day-and-a-half of training on the new standards. Gradually, schools are shaping their professional development activities around meeting the standards. One way they hope to do this is by spending more time studying the work students do now and asking questions about how that work helps (or does not help) meet particular standards.Eventually, district leaders hope and believe, teachers and principals will begin to develop a common language and vision.

If this seems like a slow start to outsiders, they need to know that Long Beach is coming out of what one principal calls a "curriculum void" -- a time when schools and teachers were pretty much left on their own to decide what they would teach.

Under previous leadership, says former Hughes Middle School Principal Daniel Woitovich, the central office curriculum staff was eliminated. A new, standards-oriented staff is now in place, and the swing back to a more centralized curriculum makes some teachers feel like "here we go again," he says.

Many administrators and teachers are also waiting for
the next all-important piece of standards-based reform. It's the piece that will make it possible for district educators and the public to know whether students (and schools) are actually meeting the new standards.

"We are still in the growth period on new standards, on making them workable and understandable," Woitovich points out. Principals and teachers used to educational fads are waiting to see whether the push for standards is something more than that -- and just how serious the district is about making standards a top priority.

Many administrators and teachers are also waiting for the next all-important piece of standards-based reform. It's the piece that will make it possible for district educators and the public to know whether students (and schools) are actually meeting the new standards.

Most educators call the kind of standards the district has developed over the last two years "content standards." They describe for teachers what they should be teaching. The next phase -- "performance standards" -- tell teachers and parents whether kids are learning what the district expects them to leaern. Performance standards are like questions on a test. If a student can answer the questions, he or she meets the content standard.

What if a student answers some of the questions? Most states and local school systems are setting several levels of performance -- "basic," "proficient," and "advanced" are typical labels they use. For example, the Long Beach schools have said that, at a minimum, they expect at least 80 percent of all their middle school students to reach a "proficient" level by the year 2000.

While teams complete the district's performance standards, the Long Beach middle schools are attending to the new math standards in different ways. At some schools, the standards are still very much in the background, while faculty deal with what they believe are more urgent curriculum concerns, such as basic literacy. At others, standards are emerging as the framework for professional development.

"Content is what my teachers are good at," comments Karen DeVries, former principal at Marshall Middle School. Even so, she and her department heads decided that their goal last school year was "to keep the standards off the shelf and in teachers' hands, getting teachers to look at them carefully and be aware of what's in them."

Teachers were talking about and looking at standards on a regular basis. This kind of sharing was particularly helpful to teachers without a strong math background. They began sharing hands-on activities and helping each other learn how to use the new "math manipulatives" in their everyday instruction.

Each month, Marshall teachers organized reports on how their instruction was linked to standards. They were asked to give at least four examples and to submit student work as proof.

Teachers may have felt like "this is just one more thing to do," says DeVries, but they did it. As a result, she observed that the content standards "began to intrude on the traditional isolation of teachers."

Teachers were talking about and looking at standards on a regular basis. This kind of sharing was particularly helpful to teachers without a strong math background. Teachers began sharing hands-on activities and helping each other learn how to use the new "math manipulatives" in their everyday instruction.

This year, Marshall's teachers will be analyzing more carefully what their documentation reveals--where the strengths and weaknesses are. Again, student work is the most important tool they are using. DeVries already knew that the writing process in all subject areas would be a priority; this will be especially important in math instruction because the standards expect students to explain--in writing--the processes they are using. (For an example, read this letter to "Future 6th Graders.")

Still, DeVries says what her faculty needs most are the performance standards. As one teacher put it, "Once we know exactly how performance will be judged, we will have a much better idea about what we must do to get the job done."

Meanwhile, back at the Penguin House....

The aspiring marine biologists in Sandi Machit's class who grappled with scales and graphs in their Sea World project last spring had been solving real-life problems all year.

Another activity, their "Fantasy Baseball" game, played by the students frequently, required them to select teams and trade players using statistics taken from baseball cards, and then schedule teams to play each other twice. This activity was kicked off by an assembly with real baseball players as guests; parents served up hot dogs for all.

Machit teaches both math and science at Hughes -- twice each day during a large block of time in the morning and afternoon. Both block classes are set up as miniature economies -- students receive a daily salary of $20 but must pay certain overhead expenses, including utilities and rent. Committing a felony (cheating or being disrespectful to others) could cost them $150. A misdemeanor--not paying attention or littering--docked them $50. Students serve as the bankers and payroll clerks; there's even a "chairman of debts."

To introduce parents to the idea of standards, Machit invited teams of parents to compete on several assignments,then learn which standards they had been practicing. After this "Family Curriculum Night," a few parents sent in ideas for other activities.

Machit says the Sea World project most exceeded her expectations. In fact, she wrote an apology to parents, explaining that she had intended to confine the project to in-class time, but "the students refused to be so limited."

Actually, Machit told the parents, "the project seemed to take on a life of its own." Anticipating final group reports of perhaps 20 pages, she was presented with reports that averaged 50 pages. "You should be impressed by, and proud of, the determination, self-motivation, and perseverance of your child," she assured parents who had worried about the frenetic pace their youngsters exhibited at home.

For sixth-grader Scott Kazen, the strategies used by Machit kept him from being bored with math. "If you stare at a math book for 30 minutes doing math problems, you get bored and tired and you don't work as efficiently," he said.

"I like the new standard better because your brain really has to think about many different things and you aren't fixed on one idea, so you don't get bored."

Using only a textbook, chimed in Jeremy, "makes it so you don't like learning." For Melissa Flores, as well as many others in Machit's class, "the most valuable lesson I learned was how to work in a group."

Machit admits that many teachers are wondering if the standards-based changes are just a pendulum swing, "if they are here to stay or will we go back to computation?"

"No one," she argues strongly, "is saying you don't need computation. The new approach is to learn to compute, but to make it come alive, to go from hands-on, to representational, and then to the abstract, not start with the abstract and never learn how to use math out in the world."

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