Learning Isn’t Easy

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?

Is This Where You Want to Be?

When I talk to other teachers about my school compared to theirs I often end up saying that I am so glad that I teach at a small school (30 to 40 kids per grade). Don’t get me wrong, there are perks to a big school. When I scan the #MTBoS, I see really cool things happening, but many times those take place  in larger schools. Usually at larger schools classes are more homogeneous when it comes to academic abilities. It allows the environment to cater towards a somewhat standard mindset. There are times when I dream about having a class of 20 some students who would willingly geek out and fully engage on math with me. I would even like teaching in an environment where students are simply ritualistically compliant, acknowledging the advance math they are learning will have no bearing beyond graduating high school. Unfortunately at a small school we often don’t have the opportunities to homogenize like that. I have had classroom of 11 students where one student had a tested IQ in the low 80’s and another had tested as cognitively gifted (IQ of at least 130 in our district). The unfortunate consequence of this is that I believe that I end up developing far too many strategically compliant students, and I personally detest the mindset of the strategically compliant, mostly because I was one. I personally have witnessed the hell that many of those students will go through, but perhaps I am engaging in one of humanities greatest follies, projecting my own image onto others. I think we do that far too often and I want to confront people when I see them do that, but I don’t because I am way to introverted and don’t feel comfortable without the protection and distance of a computer screen.

But I digress….

Why then do I stay at a small school when there appears to be positives to a larger school environment? It’s because of my introvertism. I fit in a small school environment much better because of my personality, whereas if I would succeed at a larger school it would be in spite of my personality. I once was told, “I get the impression that what happens behind closed doors is different than what I see,” by one of my past administrators. Ummmm…..yup, I don’t think there could be a truer statement. Here’s the thing, I interact with my administrators in a classroom environment for anywhere from 70 minutes to 190 minutes during the year, and because of our current revolving door with the administration it has maxed out at 300 minutes for an entire career. I interact with the majority of my students for a minimum of 16410 minutes, with a few of the students having interaction times as high as 40275 minutes.  I know that my administrators hold my job security in their hands, but I value the opinion of the students more. I really believe that they should have a larger say in the learning environment than the admin and legislatures.

Reflecting on my experiences in high school, I remember not feeling much respect towards the teachers  that treated us like children. When I began college I told myself I would start to think of high school kids differently. When I found myself working with high school students at the local YMCA in college I told myself I would think of them differently once I was student teaching. When I was student teaching I told myself I would think of my high school students in a different light once I graduated and obtained a full-time job. When I obtained a full-time job I told myself that I would think of high school kids differently when I had my own children. Now that I have been a parent for almost six years I have given up. I can’t think of my high school kids differently than I think of my coworkers, and I happened across some research to back that up.

Everyday that I enter a classroom I can’t help but see my students as equals. As long as we are talking about math I probably am superior, but that’s because of my experience with the subject. My authority is dictated by two things: one, my knowledge of the subject I teach, and two, my position as a teacher. The power I derive from knowledge is only confined to the realm of mathematics. When I discuss another topic with my students, they get the opportunity to claim power. But the power I derive from being a teacher is all based upon accepted societal pretense. Which is why I love teaching at a small school. In the thousands upon thousands of minutes I will spend with my students it is almost inevitable that the false power that the pretense of the student-teacher relationship is built upon will be obliterated. Once that power structure of a student-teacher relationship is gone I can truly get to work of education. Students will learn much more from me when they view me as an expert because of my knowledge and not because of my title.

Removing the power structure of the classroom also allows my students and I to separate math ability from character traits. We are able to acknowledge the IQ bridge that might exist between us that hinders instruction, but can guide learning anyway. (I really wished we lived in a society were we could rationally discuss the impact of IQ without shaming.) Obliterating the student-teacher power structure paves the way for students to form an opinion of me as a person aside from their opinion of the subject I teach. It allows me, as a teacher, to do the same for the student. It is why I want students in my class, even though it might not make the most sense for a particular student. It is why I want certain students in my class, even if math isn’t their strength. It is why I feel badly when I say scornful things in class. It is why I have students for whom I feel like I should have done more than teach trig functions of any angle. It is why I think I can have a long term impact on students. It is why I conflicted emotions about taking extracurricular duties. It is what allows me to describe my students as more than grades.

I once had a conversation where I was asked if this is the place I thought I should be.

Does the math instruction suffer in a small school environment under my watch? Probably, sometimes, maybe.

Do I get to have a bigger impact on the kind of person that leaves my classroom compared to a big school? Yes, definitely.

Am I happy here?

I couldn’t imagine it any other way.