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.” 
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