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Covering
All the Angles

Middle school geometry takes center stage in Idaho’s statewide math initiative.

Story by Rhonda Barton
Photos by Brad Talbutt

BOISE, Idaho—It’s the first week of August—prime vacation time—and the thermometer sizzles above 100 degrees in Idaho’s capital. Kayakers and rafters throng nearby rivers, and hundreds of tourists spill into the city’s historic quarter for a Basque celebration that’s held just once every five years. But for LaRue Lambert and more than 100 other middle school mathematics teachers, the late summer trip to Boise is no holiday. For five long days, they’ll sit in Boise State University classrooms, soaking up lessons on spatial relationships, coordinate graphing, and the Pythagorean Theorem.

Lambert, who teaches at the combined junior/senior high in the tiny ranching community of Mackay, listens intently as University of Idaho Professor Dave Thomas introduces Geometer’s Sketchpad, the most popular geometry computer program in the United States. As Lambert deftly uses her mouse to draw an equilateral triangle within two intersecting circles, Thomas gleefully exclaims, “If you play constructively with this tool for the next six months, you won’t be able to restrain yourself from sharing it with your kids!”

Targeting the Middle Years

Training teachers how to get middle-schoolers engaged and excited about mathematics is the whole point of the Idaho Math Academy. Now in its third year, the academy has reached more than 30 percent of the state’s fifth- through eighth-grade teachers, building their skills and their confidence in a subject that some fear. The rigorous five-day summer session targets the hormone-fueled middle school years because that’s a make-or-break period.

“More and more, the data tell us that something happens in those years when students begin to think of themselves in terms of what they think they’re good at and what they think they’re not,” State Superintendent Marilyn Howard informs academy participants. “It’s a tough age: no longer the fun time of elementary grades and not yet the memorable years of high school. Middle school needs a special approach.”

Middle school is also when more abstract concepts are introduced and math achievement begins to take a precipitous dip. The latest scores on the Idaho Standards Achievement Test (ISAT) in math show students dropping from 90.3 percent proficient and advanced in fourth grade to 69.4 percent in eighth grade. Results of the state’s Direct Math Assessment—requiring students to show their work in a timed test—tell a similar story: 61 percent of fourth-graders are proficient and advanced compared to only 46 percent of eighth-graders.

Aiming to boost those numbers, Superintendent Howard’s office teamed up with the governor’s staff four years ago on a math initiative that would incorporate more effective professional development. Two of the state’s largest employers—Hewlett-Packard and Micron—also came to the table. “Obviously they want an educated workforce and employees that are capable of doing the job,” points out Susan Harrington, the state math coordinator. “Also, they have children—so even though they’re coming at it as representatives of business, they’re parents, too.”

A math task force, backed by corporate funding, defined what needed to be done and set about developing materials and designing summer academies held at a different campus each year. Besides focusing on middle school teachers, the task force chose geometry and measurement as the first priorities. “When we were looking at the data, we saw those were the areas in most need of attention,” says Harrington.

Tangrams and Brownies

On day one of the academy, instructors don’t waste any time getting to the theoretical heart of the subject: van Hiele levels. In the 1950s, two Dutch math teachers—Pierre van Hiele and Dieke van Hiele-Geldof—observed their students and described five levels of geometrical reasoning. The levels are sequential and hierarchical, and progress depends more on the student’s mathematical experience than chronological age.

The van Hieles postulated that a student begins with a visual phase, recognizing basic shapes without attention to their parts or attributes. Next comes analysis: The student can recognize and name properties but doesn’t understand ordered relationships. This is followed by an abstract level, where properties are logically ordered and attached to meaningful definitions. The last two levels—deduction and rigor—are more appropriate to high school and college students who are able to construct proofs and compare mathematical systems and non-Euclidean systems.

As the academy progresses, the teachers are shown how to translate theory into practice. Through open-ended assessment and questioning in class, teachers should aim to identify each student’s reasoning skills and then use an array of problems and manipulatives to move the pupil to the next level.

The problem-solving tools can be as specialized as PowerPolygons or as prosaic as dessert. Use a pan of brownies to explore shapes, sizes, and proportions. Convert a two-dimensional sheet of paper into a three-dimensional prism by folding it in half and then eighths. Cut along one radius of a circle and overlap the edges to make a cone whose height changes as the diameter of the base changes. Incorporate tangrams, quilt squares, and pattern blocks into lessons on discovering relationships, developing formulas, and measuring angles. One after another, simple hands-on exercises demonstrate how to help students jump from visualization to analysis and beyond.

The lessons go deeper, though, than simply introducing teachers to a series of entertaining activities. A major consideration is tying instruction to Idaho mathematics standards. “Everything we do, whether it’s a computer lab or activity session, we make sure we’re not wasting the teachers’ time,” says Harrington. “They know that even though these activities may be fun and interesting, they’re presenting information that students need. Now, teachers have a way to present it that not only will help the students understand the concept, but will also help them do better on those tests that everyone is worried about.”

The activities also support standards from the National Council of Teachers of Mathematics (NCTM) that spell out what middle school students should be able to do:

•          Understand relationships among different two- and three-dimensional objects
•           Use two-dimensional representations of three-dimensional objects to visualize and solve problems
•           Examine the congruence, similarity, and symmetry of objects using transformations

According to the NCTM activity book Navigating Through Geometry in Grades 6–8, mastering these skills can help middle school students become aware that “from the alignment of the solar system to the structure of an atom, from rocks to crystals to flowers to rings on a snake, from architects to mechanics to artists to musicians, geometry pervades our world.”

Bringing It Home

Armed with both content and pedagogy—plus a bulging tote bag stuffed with computer programs, AlphaShapes, and MultiLink Cubes—academy participants are starting to apply their new knowledge back in the classroom this fall. “I find that this academy made me more aware of other ways to present materials which might reach some students better than the traditional methods,” says LaRue Lambert. That’s especially useful in a small district where Lambert’s biggest challenge is “bridging the gap between those who’ve already reached proficiency in basic skills and those who have not ... teaching to both ends of the spectrum.”

During the coming year, Lambert plans to take advantage of additional PLATO computer training offered by the state. Others at the summer session have signed up for an interactive online course at the University of Idaho that asks teachers to implement activities from the academy and report back on their results—critiquing each other’s lessons. Research suggests that sustaining professional development in this way is important. According to studies by Harold Wenglinsky, “the more extended the professional development, the more it encourages effective classroom practices.”

In the end, though, the academy’s most important lesson might be showing educators that teaching and learning mathematics is a balancing act: one that involves both direct instruction and group activities, memorization and discovery. As Superintendent Howard exhorted the teachers, “Your task is to ensure balance in your approach—memorization versus application, the usefulness of math against the appreciation for it, the theoretical application against the practicality. That balancing is one way you can help more and more youngsters begin to see themselves as good in math and eager to continue something they’re good at doing.”