Putting Math Into Words
Putting Math Into Words
Learning mathematics at Glen Urquhart has always been about words. If that sounds illogical or counter-intuitive, Math Coordinator Maureen Twombly is quick to explain. “Language is essential for understanding mathematics and doing real world math,” she says. “Once you’ve learned the language, you don’t ever lose the skill.”
Since founding faculty member Merelyn Smith first created the unique GUS mathematics program in the 1980s, the emphasis on language has been one of its most distinguishing features. “Our mathematics program has three goals,” Maureen says. “To teach children the language behind the mathematics they are learning; to give them a conceptual understanding of the mathematics presented; and, finally, to provide them with the procedures that make mathematics most efficient.” According to Merelyn and Maureen, to get to the second two goals, and thereby develop deeper mathematical thinkers, the first goal must be achieved.
The emphasis on language is paired with the use of manipulatives that allow teachers and students to demonstrate problems and solutions concretely. In the lower grades, Cuisenaire rods are utilized. The rods provide both a physical representation of quantities and operations and a common mathematical language. In the upper school, algebra tiles are introduced because they can be used to represent algebraic equations. As early as kindergarten, students learn to express mathematical problems verbally and concretely. For example, children may learn that “four increased by three” can also be expressed as “four added to three” or “the sum of four and three,” and they can then represent the problem concretely with Cuisenaire rods. This provides students with a deep understanding of the concept which they can access in order to respond to different wordings of the same problem.
Applying concepts to real life situations is another hallmark of the GUS mathematics curriculum and students beginning in fifth grade are given five problems to solve every two weeks in addition to their regular homework (see sidebars for examples). “This is a way for them to use their math skills in unexpected ways,” Maureen explains. “They might use a procedure to solve it; they could use art to solve it. There is no prescribed way to solve these problems. Some can take ten minutes to solve; others an hour. Students can work collaboratively or individually, though they can only share ideas and not answers. Working collaboratively helps them develop problem solving skills and communication skills. That is often lost in teaching and learning math - how to communicate what you are doing in tackling or solving problems.” Maureen is amused and gratified by the many eighth graders who express how much easier the problems have gotten over the years. “They don’t realize that their skills have gotten so much better. It’s not that the problems have gotten easier!”
Finding textbooks that align with the GUS program was never going to be an easy - or, necessarily, desirable - endeavor. Consequently, there are no math textbooks for grades K-5. For grades six, seven, and eight, Merelyn found texts published by the University of Chicago that complement the GUS curriculum, though teachers have never used them in the same order as the books are arranged. “The textbooks are NOT our curriculum,” explains Maureen. “We use them to support our curriculum and as developmentally appropriate in our curriculum. They are great for practice and for language development. I let the kids use them as a resource after we’ve gone over something and as a review or for extra problems.”
Maureen, in her eighth year at GUS and third as Math Coordinator, continues to build on what Merelyn created. “Merelyn did an amazing job creating this program. She had such a passion for the way we teach mathematics.” She had great foresight, too. “Merelyn put in transformational geometry which other schools are just starting to do now,” says Maureen. As new research and ideas from mathematics schools become available, however, “We continually look at the program and evaluate it to make sure our students don’t graduate missing anything.” In addition, teachers are encouraged to come up with new ideas, as long as essential aspects of the GUS math curriculum are not eliminated. For example, there is now a library of 150 “flipped classroom” instructional videos (where students learn lessons at home and do “homework” in class) created by GUS faculty. In addition, Mahesh Sharma, the director of the Center for Teaching/Learning Mathematics and an early mentor to Merelyn, recently returned to GUS to present a workshop for teachers on new teaching methods.
When Maureen first started teaching mathematics at GUS, she was thrilled to develop a deeper understanding of the higher mathematics she studied in college because of the approach through language. Now, Maureen’s favorite part of her job is when she “can get a student to have an ‘aha’ moment about math. You don’t get an ‘aha’ moment when a procedure works; you get it when you’ve made a connection to mathematics, when you’ve developed an understanding of what you’re doing.” Needless to say, at GUS, those “aha” moments come pretty often.