The thing I enjoy most about working with practicing teachers is seeing them start to think about familiar mathematics in completely new ways. Bringing this about can be quite challenging in a room full of secondary-trained math teachers like myself. Because we are licensed to teach high school, we have studied a lot of undergraduate mathematics and we tend to value solutions according to the sophistication of the mathematical tools employed. It was hard for me to learn early in my teaching career that it would really be better if I knew a variety of unsophisticated strategies rather than a single sophisticated one. I was good at thinking like an undergraduate math major. I needed to learn to think like my students-not just like one of my students, but like any of my students. Learning to do this requires seeing familiar mathematics in a new light.
I was working on this in a professional development setting with seventh-grade teachers recently. Connected Mathematics (or the Connected Mathematics Project, CMP) is a National Science Foundation-funded middle school mathematics curriculum. The seventh-grade unit Stretching and Shrinking has student studying similarity and the relationships between similar figures-corresponding angles are congruent, while side lengths are proportional, for instance.
Stretching and Shrinking looks at similarity relationships as transformations. That is, if we have two similar figures, we think of one as the original and the other as the image. If the context suggests that one of these ought to be the original, we follow that. Otherwise, they may be arbitrarily chosen. For example, if we make a drawing of the car, the car is the original and the drawing is the image. If we make a drawing of a bacterium in a microscope, the bacterium is the original and the drawing is the image. As these two examples illustrate, the image may be smaller than the original, or it may be larger than the original.
In cases where original and image are not naturally defined, as with two similar triangles in a textbook, it is legitimate to consider either figure the original and the other the image. As students become familiar with the ideas of the unit, they tend to be more strategic about their choices; some prefer to have the smaller figure always be the original, while some prefer to have the figure with the most known side lengths be the original, for example.
The fundamental mathematical idea of the unit is scale factor. The scale factor between two figures is the number that the lengths of the original figure are multiplied by to make the lengths of the image. In the bacterium example, the image is larger than the original so the scale factor is greater than 1 (on a microscope, the scale factor is marked as the degree of magnification). In the car example, the image is smaller than the original so the scale factor from the car to the drawing is a fraction smaller than 1.
One way students are introduced to similar figures in the unit is through the coordinate plane. They draw a group of characters called the Wumps. The Wumps are charismatic little fellows (see below) who are defined by a small set of points in the coordinate plane. Mug Wump is the original Wump and students are provided a set of rules for transforming Mug into other Wumps. The trick is that some of these are not really Wumps, for to be a Wump you have to look exactly like Mug. You may be larger or smaller than Mug, but you must look just like him. So the rule (2x,2y) transforms Mug into Zug, who is a Wump. Zug is twice as wide and twice as tall as Mug and looks just like him. The rule (3x, y), though, transforms Mug into Lug, who is not a Wump. Lug is three times as wide as Mug and equally tall. Lug doesn’t really look like Mug at all; he is an impostor.
All of this is preliminary to an interesting experience I recently had while working with a group of seventh-grade CMP teachers. There are two parts to the experience. I’ll write about the first here, and the second part next week.
Once we have drawn the Wumps, we help them to get dressed. We start with hats. Like the Wumps, the Wump hats are defined by points on a coordinate grid, but fewer of them. What is new in the Wump hats problem is that the rules for transforming hats include addition and subtraction, not just multiplication. From the Wump problem, students know that similar figures have congruent corresponding angles and that rules of the form (ax, ay) will give images with side lengths a times as long as the original lengths. In the Wump hats problem, students consider what the effect will be of a rule like (x+1, y-2).
In many seventh grade classrooms, students come to the following two conclusions:
- adding and subtracting moves the original figure around, and
- multiplying changes the size of the figure
In my experience in professional development settings, teachers are usually happy to leave it at that as well. My dear friend Jim Mamer has asked me several times over the years about these conclusions, because they don’t tell the whole story. In particular, multiplying not only changes the size of the figure but also moves it. This is by necessity; if the points stay put, then the sides of the figure cannot get longer. I have never thought much about Jim’s questions on this because issues have not come up in my sessions.
But I made an instructional decision this time that brought the issue to the forefront. Continued in the next post.