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 School Really About Education?

Today, and the next few days, I hope to be able to just talk to my students in one of my classes. I plan on using the timing of losing many students to senior class trip, along with having to do a mandated Ohio Means Jobs lesson. Many of the lessons are rather basic, or those that do require a little upper level math feel rather forced, kind of like they were copied straight out of the textbook. Yet somehow it has more career connections because it came from the state website instead of a textbook. But, like usual I need to digress before I start to ramble into something I really didn’t intend to talk about.

I have been using my blog to write about some of the more transformative experiences throughout my education and I spent a good chunk of last night rereading some of them. This wasn’t my first attempt at making a personal website, it just changed from what I originally thought it would be. Originally I was going to make a site to supplement my class, a resource for mathematical information. However, I am a unitasking teacher, so I really didn’t need a website to explain all the different methods I am using. Providing mathematical information was kind of pointless because there are hundreds of websites out there to do that, all of them better than anything I could produce. Why have I stuck with writing this time?

I used to consider myself an educator who happened to use math as my medium. To steal a line from my pastor, my purpose was to, “comfort the afflicted and afflict the comfortable.” My goal was to salvage education for those on the brink, the perennial discipline problems, the helpless, and to push the honor roll students to their limits. I felt like I accomplished this goal during a couple of years, and now I find myself constantly chasing that nostalgic moment.

Several years ago I stumbled across a blog that laid out in rather blunt terms the social contract that exists in most schools. (I didn’t bookmark it at the time and cannot find it again, but I want to make it clear that while I agree with the premise that will follow, I did not originate it.) It laid out a vision of school that really resonated with me after I had a nervous breakdown in front of a couple of students. Authentic learning is an inefficient, messy endeavor that is not conducive to a typical educational setting. A classroom inherently relies on efficiency to educate the masses. The problem is that this education resembles training more than education. To be effectively trained, quiet obedience is necessary, but in-depth thinking and analysis is not. A contract develops between teachers and students in this environment, one where the students agree to be obedient and complacent, and the teachers agree to not really make students think, but rather rely on memorization. Students are willing to sacrifice freedom and opinions in exchange for not being challenged.

School becomes a place where an encyclopedia of examples is memorized, and we denote the ability to memorize with grades.

After I had my nervous breakdown in Calculus I, I started teaching differently. Well, teaching in a traditional sense wouldn’t be the correct description. I talked with my students, explained everything in excruciating detail. Since it was more conversational in nature two things happened. One, it was easier to get off task. Two, the questions in class changed. It was less, “How do you…,” and more, “Why did that happen?” Every so often we would actually lose track of time and class would end with nothing resembling any sort of closure, and simply resume the next day. Instead of intro and hooks, we opened the book, picked a problem and started mathing. As a teacher, I absolutely loved it. Every statement or action I did was directly in response to something the students did, and every statement or action they did was in direct response to something I did.

There was only one problem with this set up. How do I grade an open-ended discussion? What if I abandoned my end of the social contract? No more grades.

It worked better than I could have hoped. No more grades, no more contract, no more complacency, actual thought.

The next year I decided to try it for a full year rather than a quarter with my next Calculus I class. Same result, but with an added bonus. I started to realize that there is a huge difference between productivity and learning. It was after one of our off task conversations, it could have been about college athletics, school rules, or whatever else, but it left me with an odd feeling. By any normal definition of a typical classroom it was a wasted day. But it didn’t feel like that. I felt like something was learned because my students engaged in some level of thinking. Don’t get me wrong, I still knew how to set my foot down and decide we needed to do some math, but I stopped feeling guilty if every second of class wasn’t devoted to math.

Unfortunately the following year I did not have a Calculus I class. Additionally I had a Pre-Calculus class, a topic I hadn’t visited since my sophomore and junior year of high school. I was teaching Pre-Calc in a relatively traditional way, cover previous assignment, introduce topic, go through examples, release students to work independently. One day though, I had assigned the following problem from this book. It’s #18 on page 163.

A car leaves Oak Corners at 11:33 AM traveling south at 70 kmh. At the same time, another car is 65 km west of Oak Corners traveling east at 90 kmh.

a) Express the distance between the cars as a function of the time after the first car left Oak Corners.

b) Show that the cars are closest to each other at noon.

A student in class called me over to help her get started and another student joined in on the conversation. I became momentarily lost in the problem, probably a couple minutes elapsed, but when I looked up to talk to these two students I noticed every other single student had come over to observe. Right there it told me something wasn’t working. My students weren’t making the connections between the concepts I was teaching and the exercises that are supposed to enlighten those concepts. I immediately thought of my previous Calc class where I didn’t separate the concepts from the procedures and quickly sent out this poorly worded email.

I am looking for feedback on how I taught Calc I last year. Bascially, did the method of doing work in a small group and working through problems one at a time help or hinder your prepartation for whatever math, or attitude towards math, that you are encountering outside of high school? I ask because I have been burdened with trying to teach precalculus this year and I feel that my classes are creeping ever closer to the model that I used last year and the year before, just on a larger scale. If you guys feel that it actually helped your preparation I think I will try and do the same group work/pacing that we did with Calc. If it didn’t, I will stick with a more traditional model.

I know the sample size is tiny, but I received rather positive feedback. The closest to negative feedback I received was a student telling me he was on par with his classmates in the honors program where the students came from AP and IB classes. So I tried it with the larger group, and it worked surprisingly well. I had buy in from 12 of 14 students on a regular basis.

From these three years of experience I became comfortable admitting my own shortcomings in front of my students and learning with them at times. I accepted that I will never be able to embrace bell to bell productivity and always call it learning. I realized that the best learning is extremely difficult to pigeon hole into letter grades. Sometimes I would take a day off from math, but it never felt wasted because there is so much more to learn than what can be enlightened by mathematical procedures.

The next year I dropped many of the conventions found in the social contract of school. If the actions we were doing in class didn’t help enlighten mathematical knowledge, then I decided that that action was really about obedience. I stopped homework. I showed movies, played games, or just talked with my freshmen in Algebra I after they had mastered a set amount of material, which served the dual purpose of extrinsic motivation and allowed me to start to build personal connections. I completely eliminated the concept of a grade with my upper level electives and made the classes more about claiming authority over knowledge, rather than going over many different derivative rules.

There are things I can’t control in school, but for the first time I felt like I was actually teaching and the majority of my students were actually learning, instead of the usual dance around the burden of obedience. I had a purpose as an educator.

However,…

I no longer feel like I have a purpose as an educator who uses mathematics, but that I am now expected to be a provider of mathematical information, which makes be dependent on obedience. I’ve been told that students are liars (“they will just lie to protect you”). I’ve been told that students are not smart enough to engage with material (“they can’t be expected to push themselves like that”). I’ve been told that students are nothing but disrespectful and rude (“punch them in the face and tell them to shut it”). I could keep going, but I hope the picture is becoming clear. For the past three years, I feel like my work environment has been one that distrusts its most important stakeholders, its students, and places a premium on obedience and complacency.

That’s why I keep writing this time, because I’ve lost the autonomy to have these conversations about obedience with my students. If this was three years ago, I don’t think this blog would exist because it’s contents would exist between me and my students.

Having a Nervous Breakdown

During my second year of teaching Calculus I had a nervous breakdown in front of my students. I can’t remember exactly what topic I was trying to explain, I think it was the idea behind the Fundamental Theorem of Calculus, but I’m not sure. What I do remember was the feeling and how it started.

I was going over some procedure and explaining how to get the correct answer and one of the students asked me a simple question.

“Why did you do that?”

As soon as I thought about it I felt the sinking feeling of not knowing why. I hate that and have talked about it before. This time though, I didn’t respond with a command of just shut up and do the problem, and I did this for a couple of reasons. One, I was going through some graduate school classes that was completely rearranging my concept of knowledge. Two, the class consisted of only two students which had allowed me to develop more of a personal relationship than is typically involved in a classroom.

When I couldn’t explain the mathy stuff to my students beyond a just mimic me response, I stopped teaching. I literally stopped teaching and just sat there in class. After a few minutes I admitted that I had no idea what I was doing. It is blatantly obvious to most people that math that is used in school isn’t like math in reality, so if I can’t even explain what is happening, what’s the point of the entire endeavor?

I imagine that everyone has been in a class where they have thought to themselves, that the teacher has no clue what is going on, but I can’t imagine many people being in a class where the teacher came in on day one and said, “I don’t know what I’m doing…so, let’s get started.” While it wasn’t day one, that was essentially what I was doing in front of my students, admitting that I am clueless.

If I did that in class today I wonder how my students would respond? I fear they would use it as a justification to tune me out more than many of them already do. (He doesn’t know what’s going on why should I bother.) I fear that they would use it as a justification to complain about grades. (How can he give me a C when he doesn’t understand the stuff himself?) But when I admitted my cluelessness to these two Calculus students they didn’t pounce at the opportunity to take control of the class. I was met with empathy and sympathy, and it immediately transformed the class dynamic. No matter how comfortable I had felt with students in the past it was always centered around a me-them type relationship, but from my meltdown to the rest of the year, class took on a more of an us mentality. It was still a classroom, and I still had more mathematical knowledge than them, but I didn’t feel like the dispenser and controller of knowledge anymore. It felt more like I was talking with them rather than talking at them, as if we were exploring together.

What I learned from that moment on was that my classroom needs to be a place where vulnerability is acceptable, though I think I’ve lost that.  We finished out the rest of the year learning Calculus together. Some days were smooth, some days were messy, but it always felt as if it was together. Sometimes we talked about a concept for the entire period with no math written. Some days we worked on procedures from the book. Some days we did other non-math related stuff and embraced the human element of school. Looking back on the experience, it was probably the first time that I felt like authentic learning was occurring in my classroom and was finally finding a vision of what I want education to be.

That moment became the impetus that lead to what I consider the best two years of teaching in my life. Every class seemed to develop some sense of supportive community. I felt free to experiment with ideas and push the limits of my students. Unfortunately I have watched that environment whittle away that past three years. Why don’t I have that accepting, vulnerable, safe, welcoming classroom centered around togetherness? I’m not entirely sure, but I do know much has changed over the past three years. My schedule changed, students have changed, I have changed, administration has changed, technology has changed, testing has changed, all of this leaving me more disillusioned than I have been in a long time. Maybe I have to have one of those coming to Jesus type moments like I did way back in 2011 when I had a nervous breakdown. I don’t know.

All I know for certain is that right now, there are way more days that I leave work unfilled, like my presence at school has served no purpose. I was under no assumption that everyday would be a rewarding bed of roses when I started this profession, but I am tired of feeling like a piece of shit at the end of nearly every day.

Feeling Nostalgic Tonight

Two interesting things happened a few days ago. In one class, as I was trying to push some students with “why” type questions, a student pointed out that I seem to have disagreements with the American education system. In the next class, a student mentioned that she wasn’t sure where she was going to college. Well, I felt like a failure right then and there, but why? I am just a math teacher, my responsibility is to teach math, everything else is the responsibility of the student, right?

After the class of 2013 graduated, I decided the answer was no. The class of 2013 was the first group of freshman I had at my current job. I had many of them for three years of classes, and several for four. In a way I kind of grew up in my teaching along with them. Growing up with them meant that I formed countless inside jokes and built a level of comfort so that I could try and push their boundaries. In return, they pushed my boundaries as a teacher as well. This was the group that got to watch me experiment with different formats and styles. It was towards the end of their time with me that I began to regret that I didn’t demand more.

When I say demand more, I am not talking about academics. I have mentioned previously about how grad school affected my concept of knowledge, but it also made me realize that many more students are capable of achieving goals they never believed if put in the right environment. There was a segment of that graduating class that reminded me of myself. It was a group of students whose cognitive ability would allow them to pursue nearly anything they wanted, but they seemed to be going through the motions.

There was no subject or activity that invoked some sort of passion. If something did create a spark of interest, they were overwhelmed with worries about future career prospects. The cost of college frightened several to the point of not even trying.

They reminded me of myself because I was that student that was doing what I supposed to do. I had the good grades and test scores that got me the scholarships, but I didn’t have any real sort of passion. When I went to college I chose a major not based upon some sort of calling or desire, but on future career prospects. I chose my college largely based upon how much it would cost.

When I graduated I found myself wondering, now what? Six years into my career I found myself wondering, is this it? I felt like I had done everything society ever had told me to do, I got good grades, I was respectful, I stayed out of trouble, I participated in voluntary activities, I held down a part time job in school, got married, bought a house, had kids, and at 27 I felt like it was a big let down. Where’s my stability? Why don’t I have the enjoyable career? All that work for just a paycheck. Then I started to worry that I would take it out on my kids, literally and figuratively.

I don’t know what the future holds for the class of 2013, but there is a handful of students that come to mind that I wish I would have pushed more. I think I did an adequate job getting most ready for college math, or at least no worse than any other teacher, but I have this nagging feeling that I should be responsible for more than just an ACT score. My job should be to guide students down a path towards a rewarding livelihood, whether that be changing their current path or encouraging them on the one they are on. If that isn’t my job I should be replaced with an automated math program, like Khan Academy.

The next year I decided to make some changes. In a conversation with a coworker I mentioned that I decided to do what I thought was right. In earlier years I had learned that I cannot impact a student’s cognitive abilities any more than I can impact a student’s height. But I had learned that I can impact a student’s EQ. I had the most efficient, productive, fun, off-task Algebra I class I ever had to that point. I was fortunate to have a small group of dedicated seniors that let me push their limits (I cannot emphasize how much it was pushed), and I don’t solely mean academic. I proof read essays and talked about college with them. I got them to question assignments instead of just complying with the assignment. Seriously, coloring as as a senior? Not everyday was about math, but everyday felt productive.

I hope that when they left that something stuck, and I’m not necessarily talking about math rules. I hope that they developed a perseverance they didn’t have before. I hope that they developed an understanding of what it means to know something. I hope that when they left me, they learned not to be content with complacency. I know that sounds kind of hippieish, but I do consider myself an idealist at times. To me though, that’s what school should be about, not power-reduction rules.

So when that student said she didn’t know where she was going to college, my heart sank. I have known this student for four years. I have had a few opportunities to push her in the past, but this should have been the year that I had her, and a couple of other potential filled students, in a small group that would have allowed me to shift the focus of education away from math. It pains me because it is reminiscent of the class of 2013 all over again. They told me their plans, or lack of them, and kept my mouth shut. I’m back at that regret all over again. Things change and the good times can’t last forever, but there has to be some way for me to get at least closer to what I had. When I reminisce about those years I don’t know if I made a difference in those student’s lives, but I know I’m not making a difference now.

Twitter Math Makes Me Feel Dumb

Tonight has been weird. I left school with my mind fluctuating between anger, disappointment, and curiosity. I was curious about reflections and the thought of reflecting a triangle over a parabola came to mind, wondering if it would have some sort of fun house mirror effect. The more I thought about it on my commute home tonight I realized that derivatives would be involved, so clearly this wasn’t going to be something I could try to do in Geometry.

Then when I checked Twitter tonight I saw this.

<blockquote class=”twitter-tweet” data-lang=”en”><p lang=”en” dir=”ltr”>Reflecting over a circle <a href=”https://t.co/WaqQ04G5ji”>https://t.co/WaqQ04G5ji</a&gt; <a href=”https://t.co/1zzqMf1nKh”>pic.twitter.com/1zzqMf1nKh</a></p>&mdash; Christopher (@Trianglemancsd) <a href=”https://twitter.com/Trianglemancsd/status/837071141873164290″>March 1, 2017</a></blockquote>
//platform.twitter.com/widgets.js

Low and behold, nested within the Twitter stream was a question involving reflecting parabolas. I thought it was neat to watch the discussion unfold in front of me, but when I came across the term normal line, I felt stupid. I had no idea what a normal line is. I don’t know if I never was introduced to the term, or I had forgotten, but I do know it’s not that I was rusty. So rather than try to participate, I just kind of shut down because it made me feel dumb.

The problem I have is that I am the math zombie that became the teacher. I finished a math degree and didn’t really know math. I discovered I was a math zombie when I first began teaching Algebra I and I couldn’t answer a typical mixture problem. (If I had a book with me I probably could find the exact problem since it was that scarring.) Any math that I actually know has been more or less self taught with the aid of textbooks, YouTube, perseverance, an enormous debt of gratitude to the History professors at BGSU who challenged my conception of knowledge, and the students who drug me through. Hopefully that doesn’t mean I am dumb, but that does mean that my mathematical knowledge is extremely piecemeal and lacking in formality. Some of the reasoning I try to share with my high school students clearly lacks the rigor of proper mathematics, as has clearly been pointed out on occasion, but I can confidently say it is mine. Sometimes the responses sting though. I was so excited to share my explanation about rationalization with the world, but was dismissed by some because of canonical forms and the definition of radix. I had to look up canonical forms (which made me want to flip that guy the bird) and I’m still not sure what a radix is or how it impacts square roots. Kind of rips the confidence straight out from under me.

But as painful as times like these are, it helps remind me what it must be like for my students. I can empathize with ALMOST every single student in class to some extent, at least in the attitude towards academics, because I have been there. Math wasn’t, and still isn’t, always easy for me, I need moments like tonight to remind me of that. When I leave work angry and disappointed because of student work, it’s night’s like these that I remember what it was like…

to be worried about grades first and foremost.

to not wanting to focus because other assignments are due.

to just not being able to think about math because, well, just not today.

to struggle to try and remember all the steps in this witchcraft.

to look at a quiz and think, “We didn’t go over that!”

to wanting to just get by and get done.

In a perfect world all my students would come to me with amazing prerequisite knowledge and be highly motivated to learn. That’s not the world we live in though. Without empathy for all the situations our students find themselves in, to many of us wind up browbeating kids into obedient behavior, which just breeds a culture of compliance. My hope is that with some understanding and a little patience I can get a student to want to contemplate the reflection of a triangle over a parabola because…, well,…why not?