| Geometry lite: |
Use the numbers that are given, put them into the formulas that are given, and multiply.
In Massachusetts last spring, more than 100 mathematics educators crammed into a standing-room-only meeting of the state board of education to weigh in on the state’s proposed new mathematics standards. Some of them carried placards. Others delivered impassioned testimony. News reporters furiously scribbled notes, but several confided that they didn’t really understand what the furor was about. What did the teachers mean when they said that these new standards favored memorization over “deep understanding”? Why do many educators continue to be concerned about the revised version of the new standards (known as the Mathematics Curriculum Framework), which were adopted by the board in July?
One way to understand what is at stake is to view it through the eyes of Ken, my neighbor’s 6th grade son.
Ken had received a perfect score on a geometry test. He had successfully found the area of a series of geometric figures for which he had been given the appropriate formulas. When Ken’s mom asked him to explain what he had done, however, he was only able to explain that you use the numbers that are given, put them into the formulas that are given, and multiply. What he could not do was answer why he was doing that, what it meant, and what would happen if he hadn’t been given the formula or the numbers to multiply. From Ken’s perspective, all he had to do was to get the right answer. Wasn’t that enough?
I am the resident mathematics teacher in my neighborhood, so I’m no stranger to having neighbors ask me to help their children with their math. This was, though, the first time I had been asked to tutor a child who had received 100 percent on a test. Yet, that’s what I did.
When I asked Ken to explain what he was supposed to do, he clearly articulated the need to find the area of the given geometric figures. He explained that he knew that “area” meant the space inside the figure, but that was as much as he could tell me. He could not explain why he used a particular formula for a particular figure, or why the formula for the triangle was one- half the area formula used for finding the area of the rectangle. Was that important? I think so.
I first asked Ken to find the area of a rectangle drawn on centimeter graph paper. Ken counted the square centimeters. When asked if there was another way to figure the area, he said he could multiply the length times the width.
Next, I asked Ken to show me how he could find the area of a parallelogram also drawn on graph paper. He asked to use a pair of scissors. He cut off one triangular piece and taped it to the opposite side to form a rectangle. At that point, he quickly recognized he could use the “length times width” formula again. When working with the triangles, he first drew a rectangle around the right triangle and immediately stated that it looked like half of the rectangle. He counted the square centimeters to be sure his observation was correct. His guided explorations continued for each figure, and he successfully made sense of what the formulas meant.
What does this all mean?
Does it matter that Ken now understands the concept of area? I think so. Certainly, Ken could have succeeded in a traditional middle-grades math program, plugging the numbers into formulas memorized for a test. But, at some point, his lack of understanding would come back to haunt him.
Ken returned home from our tutoring session with two surprises for his mom. The first was that he could explain exactly how each of the area formulas he had used correctly on his test had a relationship to each of the geometric figures. The second was his renewed enthusiasm for learning math.
The lessons I used with Ken are the same lessons used so successfully with my middle school students in the Connected Mathematics Project, a program that emphasizes student understanding of concepts as well as the development and use of efficient strategies for arriving at the correct answers.
Mathematics education in Massachusetts will suffer if districts drop successful programs—such as Connected Mathematics—that achieve a balance between understanding and computation.
This means a lot to educators. It is much more gratifying to help Ken understand and enjoy mathematics than to help him memorize formulas for a standardized test. It is better for Ken, too. After all, real-world problems do not appear as multiple-choice questions, but as complex situations that must be understood, analyzed, torn apart, and ultimately solved.
Anne M. Collins is the director of Partnerships Promoting Student Achievement in Mathematics at the Boston College Mathematics Institute in Boston. She is a former statewide mathematics coordinator for the Massachusetts Department of Education (1997-99).