LEBANON, Oregon—The bank of windows in Nancy Hunt’s classroom frames bucolic views of fallow fields, with an occasional John Deere kicking up dirt and sending crows flying. But inside the airy room at Pioneer School, the scene is anything but soporific. Twenty-two second- and third-graders race from one math exercise to another with the kind of energy usually reserved for recess.

Sprawled on a patch of blue carpet, a knot of boys and girls roll dice and neatly record the numbers in spaces reserved for the thousands, hundreds, tens, and ones. After adding the numbers across, they’ll tally all the sums with a calculator. Another group works at their desks doing Accelerated Math packets, which will be scanned into the computer later. A third cluster crowds around a table with Hunt, creating and illustrating story problems.

Later, when everyone is settled back at their desks, Hunt strides to the front of the room and announces, “I’m thinking of a two-digit number that’s less than 50. It’s an odd number and the difference between the digits is two.” There’s a flurry of activity as the students arrange numbered cards in their “place holders”—stiff cardboard pockets that look like overgrown name cards at a dinner party. “What number do you have?” asks Hunt.

Several students suggest 35. Hunt agrees and speculates that there may be more than one solution. An answer of 23 is rejected—the difference between the numbers is less than two—and 14 is thrown out after students realize it’s not an odd number. When students come up with 13 and 31, the class agrees that those fit the parameters of the problem.

Then they move on to another game that builds on an understanding of place value, seeing who can build the largest three-digit number by randomly drawing three cards.

With each activity, Hunt makes certain to use proper terminology. “Knowledge of math vocabulary is essential,” she believes. “Not only do kids need to know the vocabulary for the standardized tests, they need those words to communicate their ideas and to understand what others are telling them about math.”

The 90-minute lesson flies by, and there are groans when the children discover that it’s time for another subject. Eight-year-old Byron confides that math is one of his favorites because “you get to use tools.” He says, “I like to do it so I’ll keep doing it till I get a job.” Even then—when he becomes a tow-truck driver—he thinks math will come in handy to figure out mileage and how much gas it takes to get from one spot to another.

A Different Approach

Next door, Teresa Craig’s students do math to the beat of a different drummer—or, more accurately, to the rhythm of guitarists and pianists. The second-graders, sitting in a circle on the carpet, enthusiastically sing along to an upbeat tune on the CD player:

Let’s sort and classify ... We’re counting attributes!
Size and shape or color,
Use the weight or what it’s made of!

Craig enlivens her lessons with mathematics-themed music and games developed by California educator Kim Sutton, Creative Mathematics, and Ron Brown of Intelli-Tunes, sprinkling the exercises throughout the school day. “I start with a problem of the day when we’re serving breakfast and then I blend math with other subjects so kids do it all day long,” she says. Individually and in groups, her students work quietly and confidently on lessons tailored to their abilities—whether it’s doing decimals to the ten-thousandth place or trying to master simple subtraction problems.

Craig’s Montessori training and years spent as an environmental educator show up in how she approaches mathematics. “[As] a Montessori teacher, I always teach to the thousands,” she confides. “My colleagues think I’m strange!” Yet, professional development has validated that she’s on the right track. “A lot of teachers don’t see things the same way that I do, so it’s very encouraging when you go to a workshop and get confirmation that you’re doing the right things,” she states.

“What Can You Tell Me?”

Across town at Riverview Elementary, Marla Ernst gives her 25 first- and second-grade pupils a blank sheet of paper as they return from the playground. “Do a problem and then share it with the class,” she instructs them. There are no wallflowers as children vie to show their work. Garrett tells the class his problem: 8 ÷ 2 = 4. “Explain what you were thinking,” prompts Ernst. “You have two bags and you put four in each one.” “Why?” asks Ernst. “Because we know 4 + 4 = 8,” he answers. Ernst continues, “When I see the division sign, what do I do?” Aryah promptly suggests, “You have to put the same amount in both bags and keep going until you have 8 marbles.”

Ernst challenges the class with a trickier problem—(1/4 x 4) + 3 = 12 ÷ 3—and asks the class to draw out solutions and then “see what you can tell me.” Lakesha says, “You have a circle and you divide it in four, and then color in one part and another and another and another, because there’s four one-fourths and that equals one whole.” Armando says he made four circles and colored in one-fourth of each circle; when he added them all he got one. Reuben reports that he drew three bags and put four dots in each one, because three fours are 12.

Mason offers a problem on square roots and then Marcus wants to do one with negative numbers. When the class finishes sharing problems, Ernst distributes pattern blocks and traces on the overhead projector a square with a triangle adjacent to each side. She draws a second figure that has two squares side-by-side with six triangles jutting from the sides. How do the two figures compare? How many triangles would you have if there were three squares? Five squares? Twenty squares? The kids set up T charts and look for a pattern between the number of squares and the number of triangles. They nibble at the edges of the solution but just can’t make the jump to 20 squares. Ernst suggests, “Let’s think about it. We’ll come back to it tomorrow, put our tables together, and build it.” She smiles and gently adds, “Some problems can’t be solved in a day.”

“No Lake Wobegon”

You might think that, like Lake Wobegon, the children of Lebanon are all “above average.” At both Riverview and Pioneer, more than 95 percent of third-graders score “proficient or above” on Oregon’s state math achievement tests. But, this small farming town nestled in the foothills of the Cascade Mountains—90 miles south of Portland—faces many of the same problems as other rural communities: low family incomes, up to 80 percent of students on free or reduced-price lunch, and high rates of transience.

What, then, accounts for the stellar test scores and the high-level reasoning skills displayed in early elementary classrooms? Hunt, a 12-year veteran, thinks it comes down to good teaching which, in turn, is tied to good professional development. Hunt, Craig, and Ernst—and many of their K–2 colleagues—are graduates of the summer mathematics institute held during the past two years by the Northwest Regional Educational Laboratory. The intensive five-day program introduces teachers to instructional strategies to foster and support children’s number sense and algebraic reasoning. “We start with the premise that children come to school with some sense of numbers before they’re taught rules and formulas,” says Claire Gates, who organized the institutes (see “Voices”). “We look at their problem solving and build on that, stressing that teachers should present information in a way that helps students construct their own understanding and that encourages the children to listen to each other’s ideas.”

That philosophy has opened a whole new world for Raylene Sell, who teaches 27 rambunctious first-graders at Riverview. Before launching into a problem about how much porridge Goldilocks sampled, she reminds her class that there are story problems where the result (or total) is unknown and others where we don’t know the start or the change (for example, whether to add or subtract). The class works on each of those problem types—a tactic that researchers like Arthur Baroody of University of Illinois-Urbana say helps children become flexible and use different problem-solving methods.

Like her neighbor Marla Ernst, Sell constantly asks her students to explain their thinking. “For me, the most important thing I learned at the math institute was to ask all the different questions,” Sell admits—adding that she’s banished her boring old workbooks. “You can solve math with a variety of strategies and we need to ask the right questions to draw those strategies out.” Through questioning, the teacher gets to know students as individuals and can provide the experiences they need to increase their problem-solving skills.

An “Algebrafied” Approach

According to Professors Maria L. Blanton and James J. Kaput of the University of Massachusetts-Dartmouth, teachers can extend student thinking by using simple prompts: Tell me what you were thinking. Did you solve this in a different way? How do you know this is always true? Does this always work?

Blanton and Kaput advocate “algebrafying” the elementary school math curriculum: focusing on problem solving with answers that are student-generated generalizations. Using this approach, typical arithmetic problems with a single numerical answer are transformed into opportunities to make general statements about structure, properties, or relationships underlying mathematical ideas. Students learn how to express these generalizations in formal, symbolic ways and justify their mathematical claims.

The fact that this process can start at the preschool level is gaining wider acceptance through the research of Early Algebra Projects at the University of Wisconsin-Madison (www.wcer.wisc.edu/ncisla) and Australian Catholic University (www2.earlyalgebra.terc.edu/). The National Council of Teachers of Mathematics (NCTM) threw its weight behind the early algebra movement by declaring “Developing Algebraic Thinking: A Journey from Preschool to High School” as its professional development focus for 2004–2005. In announcing the theme, NCTM stated, “Algebra must be seen as more than a course that is offered at middle and high school. It is a way of thinking and reasoning about relationships that can begin as early as preschool and grow in complexity and sophistication through high school.”

Vernette Kittel, a pre-first grade teacher at Riverview who attended the NWREL math institute two years ago, knows firsthand that youngsters often are more capable than we think. “At the end of the school year, I wrote up equations and hung them around the classroom,” she recalls. “Then, one day I took them all down and put big stickers over one of the numbers [in each problem]. The kids knew right away what the hidden number was; they had absolutely no trouble with that. I said to them, ‘You know what? You’re doing algebra!’”

One of the main things that Kittel stresses in her classroom is the importance of the equal sign. A number of researchers emphasize the need for children to understand that the equal sign represents a relationship rather than simply an answer. Studies conducted by University of Wisconsin found that when sixth-grade students are given the problem 8 + 4 = ? + 5, 84 percent give the answer as “12.” Developing relational thinking helps students improve their computational skills and lays the groundwork for more advanced work.

By introducing algebraic reasoning early, the goal of “algebra for all” is more easily attained. As NCTM states, “By focusing on algebra across the grades, we will ensure that students develop the skills and ways of thinking and reasoning needed for success in high school and beyond.” In the process, teachers like those in Lebanon, Oregon, are discovering there are rewards not only for their students, but for themselves as well. Nancy Hunt reflects, “I used to hate teaching math because it was all out of a book that didn’t explain it very well. I never did much with algebraic reasoning because I thought this age couldn’t do it. But even little guys, if you give them counters, can figure it out. Now I love teaching math!”

“Children are really interested in math,” adds Ernst. “Math is a problem-solving kind of thing and we have to push it that way. Last year, my [student] Sam, who was seven, would walk by his brother who’s in eighth grade and say, ‘Hey, you’re doing algebra. Need any help?’ They have that kind of enthusiasm ... and think it’s really fun. Some days I can hardly contain myself!”

Note: NWREL’s K–2 Mathematics Institutes, focusing on number sense and algebraic reasoning in the primary grades, will be available to schools in 2006. The workshop is 30 hours and can be delivered in one week, six hours per day, or scheduled during several weeks. For more information, contact Claire Gates at NWREL (gatesc@nwrel.org), 800-547-6339, ext. 173. In addition, NWREL will offer the Institute in Portland during summer 2006. For more information, contact Vanessa Grayson (graysonv@nwrel.org) to be placed on a mailing list.

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