One thing I learned the hard way is that grades do not reflect an level of actual knowledge. In my eyes many grades measure how many tasks a student successfully completed. I am not trying to be dismissive of hard work. I have numerous students that work really, really, hard, some that I think work way too much and are on the verge of breakdown. I even experienced the struggle with this myself when I nearly flunked out of college.
But hard work isn’t enough to actually master a subject. Doing hard work means just that, you’ve worked really hard. Learning will always come from struggle, but struggle by itself doesn’t produce learning. I experience this for myself several years ago in graduate school, and watch it on so many of my students regularly.
Learning, real authentic learning that leads to mastery of something that is actually usable, comes in fits and spurts, in flashes of brilliance at inopportune times. We can’t schedule learning. We can schedule training, we can schedule task completion, but if our goal in education is to have students learn anything in any sort of meaningful manner, the schedule inherently works against us. No matter what questions I ask, resources I use, if a student comes into class obsessed about the recently failed Spanish test or whatever, no learning will occur. I might be able to get compliance and task completion out of a student, but unless if that student’s mind is clear and ready, learning will not happen. We think we can multi-task, but in reality all we are doing is switching between tasks, and if learning is the goal, then switching between tasks is detrimental.
I can think of one example that recently happened. While teaching rotations in Geometry this year I have been relying on the coordinate plane. I was showing the students how the x and y value of coordinates move when rotated 90 degrees, or multiples of 90 degrees. When I was asked about rotations that weren’t 90 degrees, I told them that they wouldn’t have to worry about them. Why would I do that? We had already talked about trigonometric ratios, so, with the benefit of hindsight, I realized that we could have done them. I thought it through a couple of days later and came away slightly miffed that I hadn’t thought of it earlier. Probably just dismissed it because the book didn’t have any examples.
That bothered me though. It bothered me because it should be possible to rotate something that wasn’t a multiple of 90 degrees, and fortunately I was able to figure out how to do that. I had the opportunity to quiz my former students on the subject, so I decided to let them take a stab at the non-90 degree rotation. First I showed them what I had covered in Geometry, how rotating 90 degrees is like moving legs on an ‘L’.
Then I showed them what I wanted.
A student came up with the idea of using the 53 degree rotation as a percentage of the known 90 degree rotation, then using the corresponding percentages to change the x and y values.
Which produced the following result.
Now the point A (3,4) includes an angle of approximately 53 degrees, so a rotation of 53 degrees is a total angle of 106 degrees, or a reference angle of 74 degrees in quadrant 2. Well, I checked the trig using the proportioned A prime, and came up with slightly less than 72 degrees.
Well, that’s not 74 degrees, nor is that anywhere near the realm of rounding error in this case. But the method seemingly made sense, if the percentage of the angle should be the percentage of the sides of the triangle. Why wasn’t it forming the same angle? Later I made a little table to investigate what was going on.
Clearly, when I was using percentages of the sides I wasn’t getting the same percentage of the 90 degree angle. The more I thought about it though, I started to realize that at zero percent the length is 8, but then at 100 percent the length would be 6. That doesn’t make sense. Later, I took one more crack at trying to figure out why the percentages were creating a different angle than a rotation and created this picture.
That’s when I finally saw it. A rotation implies a circular motion. Moving the point as a percentage of the x and y distance changes the distance from the center of the circular rotation. The bottom line of the right triangle is not broken into equal partitions because it is changing distance from the point of rotation. Using percentages the same percentages from taking 53 of the 90 degrees conflicts with taking the same percentages on a straight line distance of 7 units. The differences in the answers is because of the nature of the definition of rotation.
What did I learn from all this? About myself, I learned that my geometry skill is sorely lacking and very rusty. I also learned that I no longer have the trust to learn with the student in front of the students.
It also reinforced that the nature of school is not always conducive to learning. The students I were working with were capable of thinking of the answers. They even came up with a reasonable idea, but when they didn’t get the same answer as me they dropped the idea, writing it off to rounding error or just being wrong. This has nothing to do with capability. To learn is to devote every mental resource to a subject, to think, analyze, justify, and simply ponder. Our students are just simply too busy, they have to worry about 5 or more subjects, extracurriculars, college applications, part-time jobs, and any other myriad of activities. They just don’t have the time/interest/focus to clear their mind of all the other clutter to actually engage with math.
What the students want from me is clear, straight-forward, methods to find clear, straight-forward answers. They don’t have the time to think about open ended possibilities, there’s too much to do. But to really learn something, that’s what is needed, those open-ended, no solution, what do you notice type of questions that can spur a dialogue. How do I create that when all my audience really wants are the answers?