Children raised their hands often in a recent fourth-grade math class at Middlesex Elementary School in Essex.
They lifted both arms straight up if they agreed with a suggested answer. If they disagreed, they waved their hands side to side, as if drying them in the air. And if they weren't sure, they gave an exaggerated shrug.
But there were few shrugs as a facilitator for Project Seed showed them how to manipulate exponents. The program - which uses directed questions to introduce algebraic concepts to elementary and middle schoolers - is one of several new mathematics initiatives planned for next year.
Baltimore County school officials will spend $172,000 next year to train elementary school teachers to use Project Seed techniques. The school system has also approved a two-year, $1.4 million contract for Algebraic Thinking, a program that presents subjects such as fractions and decimals in concrete ways for children who struggle with math.
New elementary math textbooks will also be used, at an estimated cost of $4 million over the next five years. And the school system began using a new Algebra I curriculum this year.
School officials say the programs will help them meet their goal of having students complete algebra by the end of their freshman year. All children who will graduate in 2009 or later must pass a state algebra exam to earn a high school diploma.
"We want to make sure our students pass Algebra I on the first time around," said Patricia C. Baltzley, the school system's director of mathematics for pre-kindergarten through 12th grade.
About 20 percent of 15,700 Baltimore County fourth- and fifth-graders received low scores on the state's math tests in spring 2005, the most recent data available. The school system used that information and classroom performance to determine whether to enroll the children, who will be sixth- and seventh-graders next year, in an hour and a half of Algebraic Thinking, Baltzley said.
Some who are also taking 90 minutes of daily reading intervention will have 45 minutes of the new program daily, with an additional 45-minute class every other day, Baltzley said.
Sixth- and seventh-grade teachers who will teach Algebraic Thinking will take a weeklong training course during the summer to prepare them. A coach from the program will also visit each middle school once a month during the school year and send lesson reminders and other tips and suggestions to teachers via e-mail, Baltzley said.
School system officials observed Algebraic Thinking at work in Charlotte, N.C., and in Anne Arundel County, Baltzley said. All Anne Arundel middle schools started using it last fall, said Joy Donlin, coordinator of mathematics there.
Anne Arundel school board members approved a three-year, $1.2 million contract for Algebraic Thinking in September, according to school board documents. A third of the sixth-, seventh- and eighth-graders there are taking the classes, Donlin said.
The program presents traditional content, using physical objects such as strips of paper to reinforce abstractions such as fractions and decimals, she said. Pupils use scales and cups to represent balancing equations and often work in groups.
"This is the grade-appropriate curriculum," Donlin said. "It's not a watered-down version of anything."
Students who scored "basic" on the state math exams were enrolled in 86 minutes of Algebraic Thinking daily, Donlin said. Standard methods of instruction weren't helping them, she said, and "they needed a different way of learning."
The curriculum also previews some of the concepts presented in Algebra I, she said. The goal is to get pupils to enter standard Algebra I classes by ninth grade - not complete the course over two years or take a remedial version, Donlin said.
Baltzley said the new elementary math textbooks and curriculum also incorporate some of the hands-on, visual methods to teach concepts.
By contrast, Project Seed is a supplemental program. Trainers worked closely with individual teachers at Stemmers Run Middle, Hawthorne and Middlesex elementary schools last month; they will work with them longer during the next school year. Other teachers in those buildings can also train and observe, Baltzley said.
In addition, science, technology, engineering and math resource teachers at 30 other elementary schools will be trained and receive additional resources, she said.
"Our intention is to have them start working and modeling in the classroom," Baltzley said.
In the future, the school system would like to teach middle school department chairs, she said.
Project Seed facilitator William Glee used the Socratic method of directed questioning to explore incorrect answers during a Stemmers Run sixth-grade class. For example, Glee showed the children 2-1 x 2³. One boy said, incorrectly, that this equaled 2 -4.
"What did he add to get 2-4? What was he thinking about?" Glee asked. He then reviewed the rules for adding and multiplying positive and negative numbers, leading them to get the correct answer: 2².
Later, when calculating 2³, students came up with three different answers: 6, 8 and 9. Glee wrote "2 2 2 = 6" on the board and asked students what operator was missing (plus signs, not the multiplication called for by exponents). He showed how another got nine by squaring three.
Glee said after class that incorrect answers are opportunities to build "number sense" - an understanding of how numbers work and recognizing patterns. "Most of the answers the kids are giving are not random," he firstname.lastname@example.org