Transformations in the Coordinate Plane, 3
Okay, so to continue with our regularly scheduled posting . . .
I mentioned here that when the topic of transformations comes up in late elementary and middle school mathematics, it is often presented as a kind of visuo-spatial training rather than as a mathematics topic, per se. That is, students are exposed to transformations as processes rather than as end-results (as I mentioned here), and thus they are expected to solve transformation problems using only physical or mental (or computer) modeling.
From the texts alone, it appears to me that students' experience with transformations mostly involves four to six years of sliding, flipping, and turning things.
One reason this is surprising to me is that here education passes up a golden opportunity to do something it loves to do and something of which I am not a huge fan—inductive teaching.
Suppose for instance that we have a point at (3, 2), as shown below, and we want to find the rotation of that point 90 degrees (counterclockwise) about the origin.
First, we can draw (or imagine) a "claw"--a horizontal line segment extending from the point of rotation (in this case, the origin) to the location of the point along the x-axis connected to a vertical line segment extending from the x-axis to the location of the point along the y-axis.
Then we can rotate the claw 90 degrees (counterclockwise) to find the new location of the point.
This is where, in my experience, most texts (and many state standards) call it quits. But we should notice something about this rotation.
The original x-arm of the claw is now the y-arm. And the original y-arm is now what we might describe as a negative x-arm. We can see these results not only on the grid, but also in the coordinates of the two points. (I use a dashed arrow to show that the sign of the original y-coordinate is reversed in the rotation.)
Let's rotate the point another 90 degrees counterclockwise.
We can, of course, see the same results. The x-coordinate of the original becomes the y-coordinate of the new, and the y-coordinate of the original becomes its opposite as the x-coordinate of the new.
You can see that the next rotation will produce the same results. There's no need for me to inadvertently draw a swastika at this site to demonstrate that.
More on this another time.
I mentioned here that when the topic of transformations comes up in late elementary and middle school mathematics, it is often presented as a kind of visuo-spatial training rather than as a mathematics topic, per se. That is, students are exposed to transformations as processes rather than as end-results (as I mentioned here), and thus they are expected to solve transformation problems using only physical or mental (or computer) modeling.
From the texts alone, it appears to me that students' experience with transformations mostly involves four to six years of sliding, flipping, and turning things.
One reason this is surprising to me is that here education passes up a golden opportunity to do something it loves to do and something of which I am not a huge fan—inductive teaching.
Suppose for instance that we have a point at (3, 2), as shown below, and we want to find the rotation of that point 90 degrees (counterclockwise) about the origin.
First, we can draw (or imagine) a "claw"--a horizontal line segment extending from the point of rotation (in this case, the origin) to the location of the point along the x-axis connected to a vertical line segment extending from the x-axis to the location of the point along the y-axis.
Then we can rotate the claw 90 degrees (counterclockwise) to find the new location of the point.
This is where, in my experience, most texts (and many state standards) call it quits. But we should notice something about this rotation.
The original x-arm of the claw is now the y-arm. And the original y-arm is now what we might describe as a negative x-arm. We can see these results not only on the grid, but also in the coordinates of the two points. (I use a dashed arrow to show that the sign of the original y-coordinate is reversed in the rotation.)
Let's rotate the point another 90 degrees counterclockwise.
We can, of course, see the same results. The x-coordinate of the original becomes the y-coordinate of the new, and the y-coordinate of the original becomes its opposite as the x-coordinate of the new.
You can see that the next rotation will produce the same results. There's no need for me to inadvertently draw a swastika at this site to demonstrate that.
More on this another time.
Labels: education, mathematics, textbooks