From tangerines to algorithms.

"I can't imagine teaching math without manipulatives," says Louise Fiori, a teacher at Stanford Middle School in Long Beach, California. It's no secret that early elementary teachers across the country consider manipulative materials a godsend for reinforcing basic math concepts. But Fiori doesn't teach kindergarten or even second grade; she teaches seventh-grade math. "About half our students really benefit from using manipulatives," she adds. "They really need to see concrete representations of the abstract ideas,"

Fiori isn't the only teacher who tours the benefits of using manipulatives with upper-elementary and middle-school students. At the Spring Street School in Shrewsbury, Massachusetts, Cathy Ogren uses them to teach place value, multiplication, division, fractions, and basic geometry. "They're great for students who struggle with number sense," says Ogren. "The kids love them,"

Tim Bailey, a fifth-grade teacher at Escalante Elementary School in Salt Lake City, Utah, agrees. "Students are always eager to get their hands on something other than a pencil." Bailey believes that manipulatives allow students to "see" abstract ideas. "Students may forget the algorithms we teach them," he adds, "but if they can visualize and understand the basic concepts, they can develop a way to find a solution."

Who Uses Manipulatives?

"As you go up in grade, fewer and fewer manipulatives are used in math education," says Joe Porzio, of New York University's Metropolitan Center for Urban Education, "despite their usefulness at all grade levels."

Bob Krech, the K-5 math specialist for West Windsor-Plainsboro Schools in New Jersey, has noticed the same trend. "As mathematical ideas become more abstract in grades three and four, the mode of instruction tends to become more abstract."

According to Marilyn Burns, a nationally-recognized math education expert, abstract teaching techniques ignore the needs of many learners. Burns says, "Experiences with manipulatives help children see math as a subject to be understood, not memorized." She is convinced that when students have physical evidence of how their thinking process works, their understanding is more robust. (See "Marilyn's Manipulative Musts," on page 23.)

Seeing Is Believing for Teachers, Too

When Krech encounters upper-elementary teachers who eschew manipulatives as unsophisticated, he tries to show them the value of concrete materials by doing a few simple demonstrations. "Once teachers see for themselves how manipulatives can help students conceptualize more effectively, they become eager to use them," explains Krech.

Many of the upper-elementary teachers who are most enthusiastic about manipulatives saw their value as a learning tool while working with younger or special needs students. Denise Bryan has taught upper and lower elementary students and now teaches third grade at Miller Elementary School in Canton, Michigan. "During those years teaching first grade, I came to understand the concrete nature of young children's thinking," she says. "Some children retain a preference for a visual learning style well after their ability to think abstractly kicks in." That's why she uses manipulatives all the time.

Imaginative Manipulatives

While Bryan often uses the manipulatives that come with her math program, she has also developed many of her own. She uses tangerines to teach fractions, and drinking straws and paper clips to build polygons (see box below). She also uses Judy clocks to reach lapsed time, and pennies and number cards to practice making and reading decimals. "I can introduce the concept of 10 percent by having students move a penny, which represents the decimal point, to the left one place," she says. "Something about actually sliding that penny over is magical to understanding."

Fiori uses a deck of playing cards to teach integers. Bill Singer, who teaches grades three and four at the Santa Rose Charter School in Santa Rosa, California, uses his students themselves as manipulatives. When teaching coordinate graphing, he makes a giant x-y grid on the classroom floor, and the students stand on different points on the graph. Then they extend their arms to make a line that connects the points.

And when Singer's students complained that there weren't enough hooks in the classroom coat closet, new hooks became manipulatives. Singer had his students install the hooks. "They had to figure out exactly where the coathooks should go and put them in," recalls Singer. "They used measuring skills and division. They also used problem-solving skills. When they realized that screws on the opposite sides of a board were hitting each other, they had to make adjustments. Now the classroom is neater, and they have created something they are proud of." These are the kinds of lessons that last a lifetime.

Obstacles to Overcome

Despite the many benefits of manipulatives, even the most passionate supporters admit that using them involves challenges. "Fraction sets, unifix cubes, small clocks, and base ten blocks all cost money," says Bryan. "If the school district is unwilling or unable to lay out the funds, teachers are often left in the lurch. Making manipulatives is time consuming and can also be expensive."

Fiori has the greatest success when she shows the class what rheV will be doing first, and then has a couple of students come up and model the task. "That way the students know exactly what's expected of them," she says. "I usually have the students work in groups of three, and I move around the room to make sure everyone stays focused."

"You need to be organized when using manipulatives," says Dawn Sather, a fourth-grade teacher at Forest Avenue Elementary School in Hudson, Massachusetts. "I keep my snap cubes in bags of 100. When we are done with a lesson, the students size up their bag by having ten rows and ten columns of cubes in the bag. If any are missing, we look for them right away. There are usually one or two on the floor. We put them in the correct bag. Then we are all set." Bryan's students keep "tool kits,"

canvas bags donated by a local credit union, stored in their desks. Each kit contains a clock, a calculator, a ruler, a tape measure, pennies (in a film canister), a deck of cards, and a multiplication grid. Bryan stores base ten blocks, unifix cubes, straws, twist ties, geoboards, and fraction blocks in clear plastic bins with lids. She labels the outside of the bins and stacks them on book cases and in cabinets. "It takes creativity and organization," admits Bryan. "But manipulatives add so much concreteness, conceptual learning, and so much fun to math. They are well worth the trouble."

RELATED ARTICLE: Manipulative Magic

Teddy Bear Estimation

Here's an activity that gives children experience with that landmark number: 10! Give each group of students a container of small objects, such as teddy bears. Without counting first, each student tries to take a handful of 10 and then counts the items. The group records their numbers in columns labeled: Fewer than 10, Exactly 10, More than 10. Ask each group to discuss their results. How many times did they have exactly 10? What was their smallest number? Largest? What was the average number? Then have them pool their results and find the mean.

Fruity Fractions

Peel a tangerine. As you separate the sections, have the class count them aloud. Write the total number on the board. Give two pieces of fruit to one student. Explain that students can use a fraction to show how much fruit the student received. (The denominator is the total number of sections. The numerator is the number of sections you gave the student.) Pass out more sections and write more fractions on the board, making sure some students get more fruit than others. This way they can visually understand that 4/12 is greater than 2/12. Then try adding and subtracting fractions. You can also use other fruits, such as grapes or apple slices.

Super Straw Shapes

Challenge students to build polygon shapes using only lengths of drinking straws and small paper clips. Have students begin with constructing triangles (three straw lengths), then move on to rectangles (four straw lengths), and so on. Ask: How many angles does each structure have? Which structures are more stable?

MARILYN'S MANIPULATIVE MUSTS

Math education expert Marilyn Burns offers these helpful tips for incorporating manipulatives into your math lessons.

* INTRODUCE MANIPULATIVES: Talk with kids about why we use manipulatives: They help us to get our heads around--and hands on--math concepts. These discussions are essential for new users and are useful refreshers. * SET GROUND RULES: Explain that manipulatives are tools, not toys. We use them to help us learn new things and solve problems.

* ALLOW TIME FOR EXPLORATION: Encourage students to explore the materials and make their own discoveries. When students can satisfy their curiosity, they are less likely to become distracted from assigned tasks.

* STAY ORGANIZED: Post a chart in your classroom listing the names of the manipulative materials. Teach children how to use them and put them back.

* SET GROUND RULES: Explain that manipulatives are tools, not toys. We use them to help us learn new things and solve problems.

* LET PARENTS KNOW: Send home a letter that lets parents who may be unfamiliar with manipulatives know why their children are using them. Include manipulative math activities for home.

Melissa Stewart, the author of many books on science and technology, lives in Marlborough, Massachusetts.