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Area math teachers share teaching ideas

By John Published: October 15, 2010
Jackie Harkness of Ellet High School finds a comfortable spot on the floor at the John S. Knight Center as she works on her program called Grab and Go Geometry that she will present during her session at the 60th annual Ohio Council of Teachers of Mathematics Conference on Thursda in Akron. Teachers from around the state of Ohio are attending the event. (Ed Suba Jr./Akron Beacon Journal)

John Higgins
Beacon Journal staff writer

Car mechanics, dressmakers and neurosurgeons all must be able to visualize how the parts of the engine, or the shapes cut from fabric or the lobes of the brain fit together when seen from different angles.

Yet American students aren't taught much about spatial reasoning compared with students in other countries, said Trish Koontz, who retired from Kent State University last month.

Her workshop on spatial sense Thursday was among the presentations at the 60th annual conference of the Ohio Council of Teachers of Mathematics, which concludes today at the John S. Knight Center in downtown Akron.

Math teachers from several area districts including Akron, Stow-Munroe Falls, Medina, Hudson, Rootstown, Woodridge and Wooster are making presentations.

The teachers who attended Koontz's session practiced working on puzzles using plastic bingo markers, scissors, strips of paper and masking tape.

They focused not so much on solving the puzzles but on the thinking both the successes and the dead-ends that goes into the attempt.

Spatial reasoning helps children think about fractions, geometry and even algebra, Koontz said. But it also teaches children that math takes time and requires perseverance.

In the paper chain puzzle, for example, students bend two equal strips of paper into interlinking loops.

They tape the links together where they cross and then estimate what shape they would get if they cut each loop of paper in half lengthwise through both links.

Spoiler alert for those who want to solve it on their own: Cutting loops that cross each other at perpendicular angles makes a square.

When Koontz was doing research with sixth- and seventh-graders, one boy asked if he could cut the loops in a way that would make a triangle. Koontz had no idea, but told him to try it out and share his results with the class.

He came back about six weeks later with a shopping bag jammed with failed attempts and a solution: a chain of three links could indeed be cut to form a triangle.

''The amazing thing is that it gets them thinking about what makes these shapes different from each other,'' Koontz said.

She was particularly impressed that he'd kept a journal describing each failed attempt.

''It was like trial 67 when he got it,'' Koontz said.

She said she is particularly interested in spatial sense because it's not emphasized in the United States as much as in other countries.

''We still keep going back to number as being the most important whereas measurement and spatial sense are just as important, if not more so,'' she said.

''In fact, some of the research earlier showed that males tended to do better on SAT/ACT type tests because they could finish it quicker. When they actually interviewed students, they found the reason males finished it quicker was because they solved many of the questions using spatial reasoning whereas females tended to use algebraic reasoning and it took longer. If they took the time limit off, they were equal.''

Spatial sense also helps students mentally flip and twist around a molecule to understand how it's put together.

''If you don't really have a lot of practice on spatial sense, much of the science you want to do is lost to you,'' Koontz said.

James Watson and Francis Crick discovered the double helix structure of the DNA molecule when they ''were able to fit a three-dimensional model to Rosalind Franklin's flat images of the molecule clearly a spatial task,'' according to an article in the current issue of the peer-reviewed journal Mind, Brain and Education.

Spatial sense is a product of human evolution that is present as a rudimentary ability in infants and can be improved, according to the article's authors, Nora S. Newcombe and Andrea Frick of the Temple University Psychology Department. Parents and teachers can encourage spatial thinking in the classroom and at home during play time.

Activities could include: solving jigsaw puzzles, using a map to find hidden treasure in a classroom and imagining where a ball will appear if it's dropped down into a twisting tube that opens up far away from the child, according to the article.

Koontz said children of all ages can benefit from practicing spatial skills. Children who have struggled with math gain great self confidence when they learn how to solve spatial puzzles.

''Sometimes this might be their first success and when they realize that this is really math, it's just one more way to turn them on,'' Koontz said.

John Higgins can be reached at 330-996-3792 or Read the education blog at



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