All students can and should learn math — every good teacher of mathematics believes this. But how individual students learn, and what fires their imaginations, is a complex and challenging question.
Among the math teachers who appear here, none found one "right way" to approach the subject. These standout teachers agree that educators must be aware of the bigger picture in math even as we are teaching the basics. We have to constantly challenge ourselves. We must keep learning. That takes imagination and, as eighth-grade Massachusetts teacher Marcie Abramson says, the ability to "never lose the sense of being a student."
Picture Book Math
Math, like an exciting piece of literature, has many unexpected twists, turns, and shifts in the plot. And, like a story, math does not have just one possible outcome. Kathy Shultz, a fifth-grade teacher at Holmes Elementary School in Darien, Connecticut, knows that students sometimes see math as a one-way street. "They feel that there is only one answer, and when they find it, they move on to the next problem and solve that," she says. Her goal is to get children to see math as affording numerous possibilities — as fascinating as any tale they could ever read.
A large part of Schultz's math program incorporates manipulatives, writing, and, of course, literature. To teach attributes of geometric figures, she uses The Important Book, by Margaret Wise Brown (HarperCollins, 1999), encouraging students to follow the format of the text as they describe uses for favorite geometric shapes. After reading The True Story of the Three Little Pigs by a Wolf, by John Sciezka (Penguin, 1997), Schultz challenges students with questions such as "If the pigs' lawyer needs to be paid, which of the following payment methods works better?"
After the kids figure out a solution to a problem, they work together to "design their own problems based on the knowledge they gained from the activity," says Schultz. "They come up with questions that are twice as hard as the ones any book, or I, could come up with!"
Schultz is amazed that so many of her students say they absolutely love math. They are also clear about what types of math activities excite them. Anything related to games and stories works well with the kids, but "they hate anything having to do with a rote sheet or opening the textbook." It's not a surprise, then, that Schultz seeks to find creative ways to meet the NCTM standards.
Many parents, however, learned math in a rote fashion themselves, and can be nervous when presented with problems and solutions they don't understand. To meet the problem, Schultz has planned a Family Math Night for this spring. "This will really open the parents' eyes to what their kids are doing."
Not knowing every answer can be OK even for a teacher, says Schultz. If students see their teacher as nervous and always looking for the one right answer, they will mimic her actions. "When you as a teacher don't know the answer or fail to come up with a different solution, celebrate it!" she asserts. "Students need to know that their teacher is human and doesn't know everything... Let students' knowledge lead the lesson."
Angela Andrews proves that it's possible for someone who hated math as a student to become an excellent and inspirational math teacher. A self-described "math anxious" student, Andrews had always found math irrelevant and incomprehensible. "My whole school career was filled with meaningless mathematics," she says. "My well-intentioned teachers answered my inquiries with comments such as: 'Yours is not to question why — just invert and multiply!' "
Andrews, a multi-grade teacher at the Scott Elementary School in Naperville, Illinois, knows that to answer children's questions with integrity, she must truly understand her subject. That's why she's always working to better understand elementary mathematics. Sometimes that means repairing the false assumptions that were passed on by her own teachers — who gave her the tools to solve a problem, but may not have fathomed the "whys" beyond the "how-tos." "I am constantly ... revisiting my early understanding of math and trying to make sense of it."
Andrews likens one of her favorite classroom activities to the Japanese "open approach" to problem solving. "At the beginning of the year, I ask my students to fill out a card with information about their families, hobbies, sports, talents, pets, and so on," she explains. "Then I use this information to design challenging math problems which are open-ended and solvable through a variety of methods." One example of such a problem: "Bobby got a $1.00 birthday check from his grandma. Before his mom would cash it, he had to tell her all the different ways she could give him $1.00 if she only had coins. How many ways can you think of?"
Students and teacher come to an agreement: The students will not ask for any help from parents or peers, and the teacher will not penalize them in any way for incorrect answers. "This arrangement builds an atmosphere of safety, and allows students the freedom to solve the problem in an unconventional manner," says Andrews.
On "Challenge Day," when the students must present their ideas, students and teacher spend the class hour discussing the problem and its possible solutions. Students commonly present several different strategies, including charts, drawings, or a "guess and check" method. Andrews sits back and allows the class to respond to the presentation. If they agree, they must offer a positive comment about the student's solution method. If they see an error, they must phrase their disagreement in the form of a question, such as, "Why did you multiply the length by 4 when you wanted to find the area?" Ensuing discussions shed light on where the student slipped off track in his or her computation, and lead to some wonderful math revelations.
Since Andrews has told the class that mistakes are avenues for understanding, any mistakes are welcomed and explored as much as correct solutions. Some students may hang back at first, but by the middle of the year they begin to see that this is a safe and interesting experience. Students often ask their teacher: "Can I share my mistake? It's really interesting!"
"I am constantly ... revisiting my early understanding of math and trying to make sense of it."
Math All Day and Everywhere
Everywhere you look in Jackie Howes's second-grade classroom, you see math. See is the operative word here, since Howes's classroom emphasizes the visual and spatial aspects of math — a tactic that inspires her students to think about math all day in innovative ways. A brightly-colored string of numbers above the blackboard skip-counts all the way to 30. Bulletin boards show objects grouped and labeled with price tags. Colorful math murals run up and down the hallways. Geometric shapes decorate any available wall space.
Howes, who has been teaching at the F.M. Kearns School in Granby, Connecticut for 15 years, begins the day with a math-oriented greeting. Children pair up to ask addition and subtraction questions, then share their results.
With the kids in a circle, Howes rolls out the "math ball" — a soccer ball with a math problem taped to each quadrant. The ball is rolled from one child to another, back and forth across the circle. Whatever problem one's thumb touches, one must solve that problem, then pass the ball to someone else. Already, the children's natural energy has been effectively channeled into active educational play. Says Howes, "Over the years many children have made comments such as, 'We didn't do any work this morning.' They are amazed when I remind them that they did math and reading all morning long!"
Outside the classroom, four math murals adorn the hallway. These murals, which feature the characters Matthew, Addie, and their pet, Minus the Dog, provide more than 85 different activities for the students. Skills built range from measurement to estimation to geometry. Howes will break up the class into groups, sending some kids to the murals with worksheets, while others turn to the Multicultural Stockroom — an interactive, holiday-themed bulletin board showing groups of objects such as Chinese dragons and Posada piÃƒÂƒÃ†Â’ÃƒÂ‚Ã‚Â±atas. Attached to the board with Velcro strips, these objects can be moved to present different mathematical challenges. At each station, the children puzzle over solutions and interact with each other. "I have a great strategy for this!" one child offers.
Later, children head to different centers around the room. Inside the "Kearns Travel Agency" — really a tent decorated with photos from around the world — a group of four plans excursions to Egypt and China, calculating costs for food, hotel prices, and airline fees. They chatter like kids playing house on the playground, but here the conversation is all about mathematics.
After work is complete at the centers, the children relax for some free time to do ... more math. "Our teacher," says one girl, "wants us to think that math is fun — like a special treat." In this case, the special treat is one in which students are immersed throughout the day, and yet they still want more. One boy, eager anticipation in his eyes, approaches another. He is holding a "math wheel" shaped like Minus the Dog — a device used to practice subtraction skills. "Hey, do you want to do some math thing with me?" he asks brightly. "Sure!" says his companion, and off they go.
"Our teacher wants us to think that math is fun — like a special treat."
—Jackie Howes's student
Cooperative Group Math
The actual bridges built in Lottie Mosher's sixth-grade math class may be made of tiny toothpicks, but the educational bridges that students build with the outside community, with each other, and with real-world experts are enduring. Mosher, who teaches at Pine Spring Elementary School in Fairfax, Virginia, designed her bridge-building project a few years ago to make math more meaningful. "Many students think of mathematics as something they are forced to do in school and will never need afterward," says this teacher of 24 years. "Of course, we just don't realize how math affects our lives."
In the project, student groups work together to plan, design, and construct a bridge, using simple materials such as glue and toothpicks. The bridges must be made to certain specifications; for example, the arch must be set at a minimum height, the bridge must be designed to scale on a computer program, and the final project must be sturdy enough to withstand strength tests. In addition, students have to budget for their building materials and "buy" them with a set amount of money. "Everybody has a particular job within the group," says Mosher. "One is the accountant, one is the architect, and one is the project director. In addition, there's a carpenter who's involved in building and a transportation chief who gets the materials from me."
Mosher uses a rubric to assess the final product, evaluating each person's contribution. For example, she might grade the bridge architect on the following points: Are the plans neatly done? Does the plan show three different views? Does the bridge come within the cost allowed? Can the boat go easily under the bridge?
At her former school, Mantua Elementary, Mosher was fortunate enough to have access to a distance learning lab. There, her students communicated in real time with bridge-building school groups overseas. They also received feedback from a group of New York civil engineers. Last year, Mosher won a grant from Curriculum Associates' Excellence in Teaching Cabinet Grant program (www.curriculumassociates.com) to continue and expand the bridge-building project. The grant also paid honorariums to the civil engineers involved. "I think that the advice and perspective of real-world experts really benefits the children," she says.
For Mosher, varied and lively teaching is a key concept. She keeps practice work to a minimum, instead choosing to focus on building enthusiasm and real-world skills through multidimensional projects. "If students can do three of the same type of problem, that's enough," she says.
To create a large group project, Mosher suggests that instructors work on teaching students how to get along and resolve conflicts. The teacher should make it clear what it takes to do a good job in a particular role, and to try to find for each student something he or she wants to do and can do. Each child can experience success as part of a team — learning skills that he or she will use all through life.
As a math teacher, says Mosher, it's important to keep your eye on the big picture. "What are the major things students need to know? They have to develop number sense; think critically; make connections; collect, understand, and analyze data; see geometric shapes in space and work with them; and so on. Mathematics is so much more than just computation."