Thinking About Learning

After months and months of trying, it finally happened. A student asked me a question, specifically this question.

“Why do I understand this when you’re here, but when you leave I can’t do it?”

I find that this is often a conundrum that students encounter, especially when they dutifully take notes in class, look at their examples, and then get lost on the homework. When I teach something, or explain something, I am ultimately the one doing the thinking. The students just nod along and memorize what they have seen, and then are unable to duplicate the examples on their own because they have never actually THOUGHT about the process. The best description I have ever found for this is pseudoteaching (MIT physics and hunting monkeys are my favorite), and I believe it should be mandated reading for all teachers.

The problem as I see it, is that so many of our students, and people in general, detest thinking. We like to become familiar with information because when we become familiar with information we are usually able to recognize information, which often will get that hit of dopamine that comes with good grades. Do it enough and it becomes addicting. I frequently run into this behavior from students. It seems like so many of the students in front of me have forgotten nearly everything their previous teachers have taught them. So when I go to teach them, they are insanely driven by quick responses that are externally validated, because they want that satisfaction of being right. When I try to remove the external stimuli of immediate praise and grades, of mind numbing procedural duplication, I am often met with literal withdrawal symptoms. I am not joking about that whatsoever.

I had never really thought of this whole process of teaching an learning until one interaction with one student one day after school. It’s not as though I wasn’t aware of the process involved in mastery of an academic subject, I had just never contemplated what that looks like from a teacher’s perspective. A student came to my room after school to take a test that she had missed earlier and didn’t have a study hall to use. She was struggling. At first came the exasperation that she could remember covering the material, but didn’t remember how to do the problems. We cover information in class, but we seemingly forget so much of what was covered. Rarely do ever think about why that happens.

When students hit that point of struggle, specifically that point when they can acknowledge the familiarity of material, but fail in the execution of material, a dichotomy forms. Frequently students enter denial. We all can recognize the symptoms of denial, I’ve even participated in some of them before. We blame the teacher, saying, “You never covered this.” We sometimes blame our health, saying that I’m too sick. We question the worth of covering subject, asking ourselves, “Why do I have to do this?” We blame our classmates, saying they are too distracting. We might even blame ourselves and say, “I’m just not a math person.” Whatever the reason that is given, denial allows us to avoid confronting the limitations of our own ability and work ethic. Denial allows us to be in a state of mind where we can avoid actually THINKING and ENGAGING with academic material in any sort of significant way. When we then live in a state of denial, we internalize the mechanisms that allow our minds to get through the struggle of school, without learning much of anything, just waiting until we get to the stage where we can quit. (Hello Senioritis, my old friend.)

But back to my story about the girl working on a test after school. She didn’t just live in denial, she hit rock bottom, and in this case it manifested itself as bawling. I’ve had students get teary eyed during tests before, but it is usually tears of frustration and anger, tears that are symptoms of withdrawal. I am so used to students lashing out in frustration (“This is bullshit!”) that I have become almost numb to the symptoms of denial and withdrawal. But that bawling, it lives vividly in my mind because I have witnessed rock bottom so few times, and this was the first. So when she started bawling, I shut the door, pulled up a chair next to her and just talked. I took the test away and shared my own personal story of rock bottom, and we just talked for about an hour and a half. I didn’t know what else to do because hints and instruction at this point would not have been fruitful in any sort of way.

Not much else was accomplished that day, but it did change the nature of the typical student teacher relationship. It instantaneously showed me that no matter what assessment I give, what questions I ask, I will never be able to understand what actually happens inside students’ minds. All the things that I thought represented good student learning, really don’t necessarily mean students are learning anything. They do problems. They ask questions. They listen. But I can’t be sure if they are learning.

It also showed me that displaying your thought process is an incredibly vulnerable thing to do. As long as I stand in front of the room, making math appear easy, my students will almost always feel ashamed when they cannot duplicate the process as easily as me. That’s why my students so desperately want formulas and shortcuts. Because actually displaying their thought process is such a painful experience that most of them can’t handle in front of me out of a fear that they will be humiliated. (That happened when a student left class in tears because she thought I was laughing at her when she was struggling through working a problem.) I could go on and on about how comfort with vulnerability is essential to learning, but that should be something that rests entirely on its own merit. Besides, I tend to ramble enough already.

So, ever since that day of bawling, I have structured my classes to try an elicit rock bottom symptoms from my students. If a student is going to tune me out, fine tune me out. I would rather know a student is blatantly disengaged than be surprised when a student’s superficial engagement ultimately led to failure. It can be a struggle and a drain at times. And some kids don’t need it, but those complacent students living in denial, that have the potential to truly do anything they want, those are the ones that need to hit rock bottom. It finally happened this last Friday.

There was visible frustration as a student realized that she should be able to do this stuff, but couldn’t.

One of my students living in an illusion of superiority finally, finally, slowed down and worked through a problem.

And of course, “Why do I understand this when you’re here, but when you leave I can’t do it?”

It’s a start, but maybe some real education can actually begin.

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=””></a&gt; <a href=””></a></p>&mdash; Christopher (@Trianglemancsd) <a href=”″>March 1, 2017</a></blockquote>

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?


“This is stupid.”

That seems to be a common way that students will vent their frustrations with academics. Okay, maybe it isn’t verbatim, but every time that a subject’s legitimacy is questioned with a dumb, stupid, or pointless, the sentiment is always the same. It’s what leads my students to write poetry like this.


I don’t mind this, and I actually like the creativity behind it. Even now I have days where the last thing I want to do is go stand in front of 15-20 adolescences and talk about math. Usually though, the, “this is stupid,” sentiment comes from an incomplete understanding of the mathematical concepts being taught. Students know that the classes are important to their long term goals, but they realize that the math in the class is not important to their long term goal. Consequently, students become more obsessed with getting the correct answers, which leads them to ask, “how to do…” rather than the all important why. They want to get through class with the highest possible grade with the least amount of effort because the math itself isn’t important.

(My wife has a Doctorate of Chiropractic and has diagnosed some really cool stuff that I wouldn’t expect from chiropractors. But a few nights ago she couldn’t remember the formula for circumference of a circle. Yet she has a college transcript that shows she has successfully completed Calculus I. Needed the class to get into grad school, but not the math. She thinks math is stupid.)

Why this behavior manifests is a discussion I would love to have, but will do so in a different post. What I want to talk about in this one is what is class is like when everything is “stupid.” When my students experience this frustrations, I am more than empathetic to them because I have been in their place.

I enjoyed school throughout high school, but that enjoyment was based upon my success. When I started to struggle with school the enjoyment diminished. That mindset started to create a correlation, so the more I struggled, the more stupid I thought school was. Sometimes I would spend class time thinking about other assignments that needed to be completed, or about an upcoming work shift, or just spent time lost in my own contempt for all the students who seemed to get everything. The nice thing in college is that attendance is rarely required and majors can be changed. In high school though our students are stuck in that environment, they need Geometry, Algebra II, and maybe even Pre-Calculus just to gain the economic stability that comes with a high school diploma.

Looking back on my educational experience, I have noticed that my classes seemed to segregate based upon our attitudes towards school. There was me and the other future math teachers who would constantly complain about why we had to take classes we would never use, complained about the homework we couldn’t do, and looked upon our classmates that could engage with the professors with a mix of disdain and wonderment. Sure we were all math majors, but we weren’t a unified group. I never went to their study session, and mine, when they would exist usually turned into general venting sessions. Now that I am the teacher, I see this behavior manifest itself in my classes to some extent. We like to trumpet the positives of heterogeneity, but ultimately even our students know that homogeneity creates better learning conditions. All of my classes had an element of heterogeneity which allowed me to find the other students who also thought, “this is stupid.”

The first time I ever entered a classroom environment where true homogeneity existed was when I was 26, and entering grad school. The class was grueling work, especially with working full-time, buying a house, and starting a family. A typical workload was at least 1 book (non-fiction, dense, historical reading) that would be discussed each week, 1 book to present and discuss with the class about every three or four weeks, and 2-3 papers of original research during the semester. I don’t think I had read 5000 pages cumulative during my lifetime, let alone academic reading, in three and a half months. Then showing up once a week and discussing this for three plus hours, it was just too much.

But, as I quickly found out, I was the only one who felt like that.

That got real lonely, real quick. Instead of dropping out though, I slowly mimicked and embraced that behaviors that lead me to actually understand knowledge rather than just memorize answers. I started to speak up with original ideas, finding out even if they were dismissed I was not ridiculed. I began to ask questions in discussions and then participating in discussions. By the end of the class I was fully immersed in the subject matter, picking up the behaviors of my fellow classmates for which the subject was a real thing, and not just something stupid that is meant to be survived.

What I learned about myself was that I was part of that 80% of students who could achieve something (made me look back on my undergrad with regret), but could also become abject failures. I figured out that I was a product of my environment. If I was around highly motivated, inquisitive students, I became like those students, and if I was surrounded by students who wanted the grade, I became that grade motivated student who thinks classes are stupid. When I went back to the classroom I figured out that I am still part of that 80%, even though I am now a teacher. Essentially that means that a student will get out of me what they want. I don’t know if that is a good or bad thing, but it is reality.

One of the perks of working in a small school is that I get to see most of my students year after year and I know some of my students fall victim to me being unable to influence the environment inside my own classroom. I have watched some of the most inquisitive, creative minds stumble into a complacent stupor when surrounded by peers who just want the right answers and don’t really care why. I have also been surprised by watching some of the most indoctrinated students blossom, also when surrounded by the motivated knowledge driven students. I can try to influence students, but ultimately I am at the mercy of the dominant students in my classes, even if they are unaware they are the dominant influence.

I am empathetic to my students who are just trying to graduate. I just wish they could surround themselves with students who cared.


What I Learned About Knowledge From Dropping Out of Grad School

After moving to Ohio I found myself without a full-time job. I even had a little difficulty getting substituting positions just because the system was so different from that in Minnesota, and I didn’t find the area schools very helpful, with the exception of the secretaries at Upper Sandusky High School. On a complete side note, ODE was not helpful at all with getting my license transferred. That’s not really important, but it does give me another reason to complain about ODE.

I decided that since I was just substitute teaching I would trying applying to a graduate school program. I ended up choosing the History program at BGSU because of it’s location and the timing of the classes. My first couple of times in a graduate seminar I was a little lost. It didn’t represent anything like I was used to. I could best describe the setting as almost like being in a book club. We had assigned reading, and then we discussed the reading.

Admittedly, I was lost, and also a little star-struck, since my first professor I had instantly recognized from History Detectives.

I had read the book, but the discussion didn’t have the recall questions I was used to answering. I kept waiting for the professor to ask questions that would allow me to demonstrate that I had read the book, that I could show my classmates my superior intellect. But it never happened. He only kept asking these, “Why did the author use this,” or “What did you think about this,” kind of questions. The only time I had ever answered opinion questions throughout my educational experience, it was always a, “Yes, I liked it,” type of question. I asked for advice from some of my classmates and was informed that as long as I speak up a couple of times during class it would be fine. That really didn’t help since I didn’t know how to voice a comment during class without the fear of sounding stupid.

Part of the class involved reading a book individually and then presenting it to the class. I had chosen this book about the Dust Bowl. As I began rambling through my summary of the book I felt all those typical feelings of anxiety that comes when having to present in front of an authority figure. I first noticed the ubiquitous amounts of head nods as my report of the book was heavy on the summary, but light on analysis. I then mentioned something about the failure of the Russians to adapt corn to their climate, and an ensuing drought there, but I worded it as a question. When I looked at my professor he hadn’t a clue what I was talking about. This made me feel uneasy, so I kind of stumbled through the conclusion of my presentation and mentioned the American ethos.

Then the questions began.

I don’t remember any of the specific questions, but I do remember feeling caught off guard, especially the questions from the professor. There are two types of teacher questions. The first type is the one almost everyone is familiar with, the checking for comprehension question, the rhetorical question. Teachers already know the answer to these questions, we are only asking students to see if they know the answer as well. The second type of question is what I call a legitimate question. A legitimate question acknowledges the limitations of the questioner, and transfers authority and power to those being asked, and that’s what made it so scary for me.

It seems to me that most students seek affirmation of their correctness from the teacher, without much thought as to why something is correct. I see this all the time students volunteer an answer to a question and want to know if it is correct, but cannot explain how they came to their conclusion. Many times they will answer questions with an upward inflection in their voice, as if their answer is a question itself. Usually to save time, teachers, including myself, will either confirm or deny the educated guess from the students. This is a problem because the students’ concept of knowledge and truth is based upon affirmation of the authority figure.

Which is why my professor threw me for a loop when he asked me a legitimate question about the American ethos. He wanted to know more about the American ethos that the author was discussing, but he wasn’t testing me to make sure I read the book, he really wanted to know and was dependent upon me to provide him with information. Suddenly, I was an authority figure over my professor controlling his access to Worster’s paradigm of American ethos. My struggle happened because I had never developed the executive function necessary to regulate my own concept of knowledge. My definition of knowledge was like so many of my students’, dependent upon the affirmation of the teacher.

As the year progressed in the graduate course, I became more comfortable and started to understand how authoritative knowledge is formed. It started to impact my concept of mathematics and my concept of teaching. I have written about my struggles in school, whether it be in the classroom or as a teacher, but this post is ultimately about how a History class changed how I think about knowledge and power.

I started successfully adapting to History class when I started justifying my statements in class. If I was going to offer a comment I made sure I had a passage from the book or some other source ready to provide evidence. That way, no matter how my professor or classmates might respond I could reply with the proof of my statement. When I started to reflect upon the math I was teaching I became appalled at how much of my mathematical knowledge rested not on proof of knowledge, but how much had simply been affirmed by authority figures. I had just memorized many correct answers and procedures. I knew I was right because I was told I was right, and it showed in my teaching.

My teaching during the first four years of my career could be summed up as regurgitation. In more uncouth terms, it was like I was telling my students, “Here is the shit I had to learn in school, now it’s your turn.” Okay, maybe I hid behind some platitudes about critical thinking, or 21st century skills, but my whole concept of school had nothing to do with knowledge.

That’s how dropping out of grad school educated me. (I couldn’t handle the work load of full-time work, becoming a parent, and watching other areas of my life go to crap.) It enlightened me to the idea that knowledge and truth is not something that is owned by teachers. They try to make sense of the world and then share their understanding with us, but they do not  create and control knowledge. Yes, teachers are usually more of an expert in their fields than their students, but they control truth. Real power comes from being able to make sense of the knowledge around you independent of any other people. It made me feel like so much of my formal education was a waste.

School as we know it, isn’t set up to achieve knowledge. Authentic learning comes in fits and spurts, and is not easily confined to weekly assessments and standardized testing. Grades and test scores do not necessarily accompany knowledge. One of the proudest moments I have ever felt as a teacher was when a student remarked that he achieved a 96 on an economics test at the local community college. (I had taught economics to him in high school, a class I didn’t feel qualified to teach.) The grade wasn’t what made me proud, but what he said next, “I know it is a good grade, but I don’t feel like I really know anything. I would rather have an 80, but actual know something.” After years of classes with me it was finally clicking for him. Grades can make us delusional to our own abilities.

I was delusional. I graduated with honors from both high school and college, but struggled to explain Algebra I concepts. I essentially was exactly the same person that I was in junior high. I had never learned or mastered any academic subject. The only thing I had ever mastered was how to put down the right answers on tests to appease my teachers. And I didn’t realize this until I was 26.

Are high school students capable of mastering knowledge? I believe the answer is yes, but it is a near impossibility under the lock step current system we have. The only time I feel like I have had success convincing students the merits of mastery, rather than the merits of grades, have been in small homogeneous classes, or in regular after school sessions. Mastery of knowledge will lead to confidence.

Grade motivated students will eventually be exposed, one way or another. When smart students become motivated by grades they become complacent. Complacent students become stressed when pressed about their knowledge. Complacency breeds the anxiety that will eventually breed perpetual underachievement.

We preach creativity and mastery, but our actions tell students that all we really want from them is the right answers. We are so wrong.

What’s My Role?

Why am I here?

Yes, I need a job so that I can pay the bills. But why am I specifically sitting in a high school math classroom when I could be in so many other possible career locations?

I think society has set up three very incompatible goals for me to accomplish as a teacher and part of the confusion is simply how I define myself as a teacher.

Teacher as a Babysitter

School is compulsory. It is law that a child must be in some form of schooling under the age 18 (in Ohio). Essentially society is telling teenagers that we cannot handle them between 8:00 AM and 3:00 PM. While not many adults, let alone teachers, would admit it, I think many students would acknowledge that there is a daycare aspect to school. More negatively they might compare schools to prison. The sentiment is the same though, schools function as a warehouse facility to store students.

I honestly don’t have any problem with this image though. I genuinely like the vast majority of students. Maybe not as academics, but on a more interpersonal level. Recently at my school we had a reward day at the conclusion of testing. We were asked to devise activities to do with our students, not necessarily academic activities because the students could then sign up to work with any teacher. Basically, we were asked to hangout with our students. I absolutely relished the opportunity to drop the premise and rhetoric of instructional time and just do stuff. We made sushi and played games.

Teacher as Knowledge Disseminator

We send students to school to learn. In my case, I am there to teach the subject of mathematics. Whether the mathematics learned in school is of any real relevance to students is debatable at best. I really don’t think this is why I am here. I actually have trouble justifying my purpose in terms of knowledge dissemination. If a student really wants to learn math there are many more economical ways to be taught the subject, websites, books, software. Learning math in a public school setting is probably one of the least efficient uses resources imaginable.

Teacher as a Mentor

Frequently extracurricular activities seem to take precedence over academic activities in a school. The community at large sometimes seems more concerned with basketball scores and musical productions than they are with differential equation capabilities of their students. As a teacher I fit into this model by trying to help build a well rounded student who can work well with others, prioritize obligations, and manage emotions. My purpose under this model isn’t so much about teaching math, but teaching all of the little habits that lead to an effective math education.

Personally, I really like this view of education. It is the reason why I want to read essays for scholarships and other classes. It is why I go to softball games and track meets. It is why I want to discuss stories with students when they come to me from literature class. I am trying to foster an inclusive, supporting environment for learning, regardless of the particular subject. Success in math cannot come at the expense of other classes or activities in this model.



Is it really reasonable to expect us to do this all at the same time?