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.

I Used to Teach Algebra I

I used to teach Algebra I. Over time I had developed some eccentricities that matched my personality, and made my classroom fairly efficient. My current seniors are the last students that had me for Algebra I, and when they talk about it, often they will mention the movies they got to watch. HOLY LABEL MAKER BATMAN! I don’t want to give the impression that all we did was watch movies though. When most people recollect their math class experience the imagine, something like this.

And that’s what my class was like, for the majority of time. It started with some sort of homework review, introduction of new material, and then I would release the students to work on their assignment with roughly 10 to 20 minutes of class left, very much following the, “I do, we do, you do.” This wasn’t everyday, but it was the vast majority of them.

The last time I taught Algebra I though, it was different. I would simply start class by presenting the students with a question that would be familiar to them. Either something from the previous day or something that they had been taught the previous year. I had them show me their work on whiteboards right there so that I could give them feedback right there, instead of waiting until the next day.

This worked for me because of two reasons.

The first, and most important was consistency. The last time I taught Algebra I it was my fifth consecutive year teaching the class. With the exception of open enroll students, the pipeline was from the same teacher, so I knew what to expect in terms of prerequisite capabilities. The standards were the same, the state testing was the same. Teacher evaluations were the same. Utilization of special education resources were the same. All of the consistency meant that I taught using my schema, allowing me to devote every ounce of my working memory and fluid intelligence to provide feedback for my students. I think it takes me five years of teaching consistency to be a good teacher with a curriculum. It really makes a cycle; master curriculum to teach (this is different that getting answers to tests); find a good sequence of topics; properly pace the topics to align with state testing; analyze assessment choices; and then finally be an effective teacher.

Now I said there were two reasons that allowed me to teach Algebra I the way I wanted and I’ve already talked about the consistency of a schedule. The second reason was because of the degree of autonomy I was allowed. Basically, I was told to go teach math, and that was it. As long as math was taught, the how I taught wasn’t nearly that important. So I decided to make my class fit my personality. I dumped activities that seemed to represent more of an obedience (sorry, “on task”) component. I made a promise to my students that I would not have them do any activities that I felt were there solely for busy work. I stopped feeling guilty about providing my students with downtime. Every now and then I found myself mentally fried by the curriculum, especially that first year teaching Pre-Calculus, so I couldn’t imagine how it would be affecting the students, and I didn’t feel guilt acknowledging that I was stressed too.

That manifested itself in that first Pre-Calculus class in a manner where there were several discussions about learning and mastery in general because my students were stuck with a teacher who only a survivor when it came to his math background. Much of the math class was dedicated to trying to understand why things work because I was trying understand why they worked myself. Since I was so comfortable with Algebra I, I would look at student feedback and decide I was happy with where they were for the day, and occasionally notice that there was 10 to 15 minutes of class left. Remembering that I promised that I wouldn’t spend their time with busy work, I used the time to build relationships and share aspects of my life that I found important, and yes, that might manifest itself as movies. As my relationships with my students improved I noticed that learning became more natural, and more productive.

Then, rather suddenly it all changed. First, my schedule was altered, Algebra I, the class that I was so good with, was taken away going into my sixth year at my current school. This is what my schedule has been since then.

Year 1 – Algebra I, Geometry, Calculus I, 6th Grade math aide, junior high lunch duty, senior class adivisor

Year 2 – Algebra I, Geometry, Calculus I, junior high lunch duty

Year 3 – Algebra I, World History, Calculus I, Economics, Geography

Year 4 – Algebra I, World History, Economics, Geography, Pre-Calculus

Year 5 – Algebra I, Algbera II, Pre-Calculus, Calculus I, Math Intervention, Personal Business and Finance Math, senior class adivisor

Year 6 – Algebra II, Pre-Calculus, Personal Business and Finance Math, Statistics

Year 7 – Algebra II, Pre-Calculus, Statistics, junior high study hall/math intervention

Year 8 – 8th Grade Math, Geometry, combined Pre-Calc/Calc I

I was still excited to teach because I felt comfortable teaching how I wanted to, I still had that autonomy.  So I showed up the first day during year 6 with a stack of whiteboards, enthusiastic about how having the students work in class impacted the outcomes, only to be crushed when I shared that philosophy with the administration and that’s not how you should teach. I was pressed to defend myself (in writing) and referred to the experts at the local educational service center. I was even questioned about going out of order in the textbook.

Then came the day, during the first week of school, when I lost one of my Algebra II classes to a fundraiser meeting that came with no notice. I decided to take the opportunity to spend some significant time with the other Algebra II class not working on math, but building relationships that would make the rest of the year more productive and efficient. Of course, that would be the day that I got a walk through, my first experience with a “gotcha” moment, and was proceeded to be lectured about wasted time. I was told that this wasn’t an official walk through, but just checking to make sure I am using my time wisely.

In my Personal Business and Finance Math, another class that I was new to, I showed a video to the students about rationalization, and it just didn’t sink in. YouTube made the recommendation to show this Berenstain Bears video, so I tried it. It went perfect, the kids embraced the dorkiness of being high school students watching kids cartoons, and they seemed to grasp the concept of rationalization. But one of those educational service center experts walked by and I was later lectured on the inappropriateness of showing a cartoon, and then had to provide a written rationale for my choice.

Then came the day I gave a problem in Pre-Calculus that got me in trouble. We had spent weeks working on trig functions, especially transformations of trig graphs. I gave the students a problem in a worksheet that asked them to do the reverse, given a set of points, find a trig function. I was called down to the office and was lectured about how students aren’t capable to performing this task without being explicitly being shown how to do it first. It just goes on and on.

Novice learners were timed on problems to see how fast they could complete them.

I give out too many A’s.

No one learns anything in your class.

Students told me they didn’t care, they’re going to get a B.

There needs to be more ways to succeed in your classroom.

It is impossible to learn anything in your class.

You let the students do nothing.

More people would be complaining if the grades were lower.

Students will lie to defend you.

I want to tell them to shut it and punch them in the face.

If I were a student I don’t know what I would be learning.

There needs to be more grades in your class.

I’m not going to do it since it’s not graded.

On top of all those messages I have been receiving, the state has changed the end of year test. We have new standards to deal with. I’ve had to adapt to becoming a full inclusion classroom that doesn’t track students, meaning I have had classrooms with students with IQs in the 80’s have been in classes with gifted students. Now students are being pressured more than ever to get college credits while still in high school. Students and teachers are feeling intense pressure to get the most amount of academic achievement at an ever earlier age.

When we give students messages over and over and over again that they are dumb they start to internalize it and it becomes a self-fulling prophecy. The messages I’ve received the past three years, that my students are lie to me, that all they do is take advantage of me, that all they do is walk all over me, well, I start to internalize that too. So when they come to my class exhausted and stressed, then do not respond to my prodding questions with thought, quit from fatigue during complex tasks, I no longer meet them with sympathy. I just keep going because, well fuck them, I won’t let them take advantage of me anymore. If they are tuning me out it must be because they have already mastered the content. They can fail, their grades aren’t my problem. That’s the teacher I am now.

And here’s the bottom line, in this current environment, I am not the teacher anyone needs. I tried desperately to hold on to a few of my values, but slowly selling out one little piece at a time, bowing to the pressure from administration, students, parents, tests, has made me a bad teacher. I am a bad teacher because I got sucked into the spiral of my own paranoia. Instead of meeting my students fatigue, exhaustion, and confusion with sympathy and grace, I coldly pressed on. As it just became more confusing for them, more of them decided to just quit and I don’t blame them. Why should they stress out over math they won’t need other than to jump through some hoop to get a college degree? They have no incentive to master the topic. As long as they are getting a B or C, they’re good.

As I write this, I keep staring at the information about conic sections on my board that I used in Pre-Calc and thinking over and over to myself, this is not how it should be done. The more I look at it, the more appalled I am. It dumbs down our students and it dumbs down the math. It’s a result of me trying to hold on to three years ago, adapting to my new pressures, but producing an abomination.

That’s not education. If that’s what I am producing it’s time for me to go. I thought I knew what my calling in life was, but if this is all the more I am capable of making, this passion has just turned into a burdensome job, which means I am no good for anybody right now. I’m not teaching. I’m torturing.

I hope that I actually made a difference for a couple students along the way, because right now I shouldn’t be here.



Why do We Forget Everything that We do in Class?

My fourth year of teaching I really began to reflect upon the purpose of my educational experiences. Specifically, the purpose of taking so many college courses to become a teacher. (How does having Abstract Algebra help me teach Algebra I?) It was after I admitted that I really didn’t know the math I was teaching I began to question the whole purpose of school as we know it.

As educators, we like to toss around rhetorical statements about mastery of material, but the reality is that the vast majority of the students we see will quickly forget the material we taught them. I don’t mean kind of forgetting and becoming rusty with the material, but completely forgetting it, so that if they were to encounter the material in several years it will be as if it never happened. I had this happen at my in-laws over Christmas break a few years ago. I had given my Algebra I class a worksheet where they were asked to find solutions to systems of linear equations by graphing. I was in the basement correcting, and as a joke I decided to give it my brother-in-law who had never passed College Algebra. (He is a college grad because he ended up using a Statistics class for the math requirement, which prevented him from becoming a history teacher, which make any sense to me.) He couldn’t do anything on the worksheet. As the rest of the family made fun of him he offered to let them try. My in-laws have six members in the immediate family, five of the six are college grads of typical four year universities. Only one of the six could come even close so correctly solving a systems of equations, and it was the one member who only graduated high school.

Combined, my in-laws have at least 18 credits of college level math completed, yet were clueless when it came to something that was standard fare for 9th grade students at the time. That experience, combined with my own struggles with teaching mathematics, made me question the whole purpose of education as we know it. I often hear math being defended as a subject worthy of study because it teaches critical thinking and problem solving skills. But critical thinking skills cannot be taught outside of a context, and if the context is impermanent has anything really been learned? No content retained, no thinking retained, nothing learned. I started to view my college diploma not as an accomplishment, but as a receipt for time spent avoiding the realities of life.

I am enough a pragmatist to admit that not every student can be reached. I know that there will inevitably some students who slip through the cracks no matter what opportunities are presented to them. I also know that there are some students that will achieve tremendous things in spite of everything obstacle placed in their way. I know that there is a group of students who have their destiny already determined and are just surviving the hoops placed in front of them. But there is a group of students who need school to be something more. This group needs school to be a place where knowledge is gained and retained, and it will be used to push their limits. There is this group that needs to be broken out of the complacency of unquestioned honor rolls and 4.0s.

That group of students will never be served until we can unequivocally answer the question, “Why do we forget everything we learn in school?”

My epiphany occurred when I was teaching Algebra I in 2010. There was one problem the class wanted me to go over from the homework assignment. I asked for volunteers, which there were none. Probably yet another assignment that was either incomplete, copied, or just mindlessly filled in hopes of a completion grade, I thought to myself. The question came from this book, and was found on page 422. It’s number #47

In your chemistry class you have a bottle of 5% boric acid and a bottle of 2% boric acid solution. You need 60 milliliters of 3% boric acid solution for an experiment. How much of each solution do you need to mix together?

I couldn’t do it, couldn’t figure out the answer. I gave the answer that was in the teacher’s edition, but I didn’t have the worked out solutions manual and I had no clue how to get the answer. I have a BA in mathematics, taken courses such as Calculus I, II, and III, Ordinary Differential Equations, Elementary Statistics, Linear Algebra, Abstract Algebra, Physics I and II. I took three rounds of Chemistry classes for my science requirements. I graduated Cum Laude. I ….couldn’t do 9th grade math. That’s kind of humiliating, especially in front of freshmen.

At first I took the rust route of blame, “It’s been years since I’ve seen a problem like this.” That was my scapegoat for my struggles in Calculus I also. It kind of falls in line with that old cliche, “if you don’t use it, you lose it.” As I thought about that more and more, it just didn’t resonate very well with me. Instead of wondering why we forget everything we learned in school, I started a little thought experiment with myself.

What if that’s the point. What if we are supposed to forget everything we learn in school, unless we are explicitly using it. If we are supposed to forget, then what is the purpose of any class in the first place? The only logical conclusion I could reach was as some sort of gate keeping mechanism. Basically, as a society, we are finding out how much a person can temporarily withstand in pursuit of obtaining a long term goal. Once the goal has been achieved, the path to get there can be forgotten.

Want to be a doctor? Well, you’ll need to pass at least Calculus I. Why? Because I want to find out how bad you want to be a doctor. Once you’ve become a doctor, you can forget all that calc crap anyway. (I would venture that this a rather common sentiment, though I am basing it on my personal anecdotal evidence.) The only reason academics would exist then is to torture students, as a way of weeding out the weak.  Ghoulish images of evil old men devising ways to make students confused. “Quadratic Formula…Muwahahaha…”

Solely because of my principles, I refuse to believe that all of math was created as a means of inflicting pain on students. That might be the very real world outcome, but that can’t be the reason for the existence of academic subjects. This was a turning point for me, I either had to accept that the whole premise of school was to make students suffer through some kind of sorting mechanism, or I need to find a purpose behind the math I am teaching. Not only did there need to be a purpose for the math, I needed to find out why do we seemed doomed to forget everything we learn in school. Over the course of the past six years, here is what I believe causes us to seemingly forget so much of what we learn in school.

There are two large elephants that hang over public education that I don’t believe gets the level of discussion they deserve. One is determined largely upon genetics, and the other would require a massive change in society. This means that we should acknowledge them, but realize that they probably won’t change.

Cognitive Ability

The longer I have taught the more I believe that people get equal opportunity and equal outcomes confused. (If you’re not sure what I mean, the movie Ratatouille is a good example). There is such a stigma surrounding cognitive ability that I don’t know if we could ever design an education system that actually meets the needs of everyone involved. If I want to actually bring up cognitive ability in designing a curriculum or class schedule, I am at best written off as being an elitist or worse, thought of as being an inhumanely, cruel, dream crusher. Why? Because I don’t believe I can change someone’s cognitive ability any more than a basketball coach can change someone’s height. So when I am told another story about everyone achieving amazing results, it makes me think of every basket ball player dunking on a 7 foot hoop. Unfortunately, I believe that we have sacrificed so much of our students’ potential at the alter of equality. When we think and act like everyone is the same we decide we know what’s best, which leads me to…


We force students into school to take subjects they may or may not want to. We take this very heterogeneous group, force them into the meat grinder that is academia, and expect uniform results. There are countless analogies written about how school is like a prison, which to some extent are accurate. The problem with compulsion is that it forces people to do an activity, and when an activity is forced it will ultimately be of poor quality, whether or not that activity was enjoyed at one point. And if it wasn’t enough that we force students to go to school, we force them to take subjects that many in society view as largely useless. Then when we find students’ math skills lacking, we force them to take more, so they will be better prepared. It really is a vicious cycle.

I don’t think anything can be done to solve the problems posed by cognitive ability and compulsion, but at least acknowledging them would allow us to try and design an appropriate curriculum and structure, rather than the insanity we have now. But forcing students to do something they don’t want to is really going to impact…


Yes, they are forced to go to school, but what do they get out of class? Are they just trying to graduate? Do they need an ‘A’? Maybe they want to graduate with honors. It doesn’t matter, all of these are extrinsic motivators and are doomed to fail. Maybe the student will be fine in the long run, for example, the doctor who can’t remember linear relationships are modeled by y=mx+b, but nothing will remain in long term memory if extrinsic motivation was the reason. That’s because extrinsic motivation doesn’t produce results, just the opposite, they hinder results. Intrinsic motivation is the way to go. If students want to understand that tangent lines are perpendicular to radii of circles, they simply want to have to know WHY. The questions and problems have to be motivating enough, they need to be an end to themselves, not a means to an end. I might be able to convince a student that mathematics might provide a pathway to becoming an engineer, but I cannot make a student value mathematics for itself. I might be able to force compliance, but I just can’t make a student want to learn anything. And when students aren’t motivated to learn, they fall victim to…


If you are motivated, you are hard to distract. No motivation, easily distracted. The problem in a classroom is that distraction is not just limited to cell phones. If students are thinking about an upcoming Physics test, they are distracted, even though they might appear compliant. Overcoming distraction takes difficult, self-aware, personal work, and the ability to admit that multi-tasking doesn’t work. I will freely admit, that as a teacher that I do not try an eliminate all distractions for a couple of reasons. First, I firmly believe that limiting distractions is a personal endeavor and is best achieved through intrinsic means, not extrinsic. When students think, rather than rely on memory, distraction is difficult. Ironically, if students are thinking, distracting noises can actually be beneficial, as long as it’s not above typical human conversation, like sitting in a restaurant. When students are trying to memorize information for recall any sort of background noise can be distracting and detrimental. Which leads perfectly to…

Learned Helplessness

“I need help.”

“I don’t get it.”

“Is this right?”

As a teacher I have to acknowledge that I am somewhat an accomplice in this behavior. Students can only be told they are wrong so many times before they just start to assume anything they do will be wrong.  At that point math, or any subject, becomes some arbitrary set of rules to memorize, so students no longer have the capability of understanding their own work, which makes them reliant on the teacher for validation. When students encounter a problem many will start to try and recall previous examples. If they cannot find one similar enough to duplicate in their memory, they quit. They are helpless. They are helpless because students don’t actually like to think.

I don’t want to give the impression that all the responsibility is placed upon the students. Teachers have their role in memory retention also, which I feed into by…


This isn’t a scientifically researched topic as far as I know, but this post about pseudoteaching is one of the most influential I have ever read. I used to be a much more traditional teacher in format. I would spend several minutes going over previous homework, then I would spend several minutes going over new material, and finally give students several minutes to start their own assignment. The problem was that for the majority of the class it was only me doing any thinking, and then it wasn’t much. Even when I would present new material, I made sure to provide examples of everything that might appear on the homework, explicitly saying, “on this section you will see….” Pseudoteaching isn’t about methods, style or entertainment. It occurs when the teacher is the only one doing any thinking and the students nod along in agreement. They nod along because everything the teacher does makes sense. Then they try the homework or take a test and go, “What?!” So my goal is to try to create some controlled confusion, hopefully to make students uncomfortable. If students can embrace being uncomfortable, and differentiate their discomfort from being loss, then they are in the right environment for learning to occur. One thing I can do to try and cause some discomfort is to use…

The Worked Example Effect

The worked example effect is one part of cognitive load theory. Worked examples are one of the most efficient ways to learn a new task, however they pose a slippery slope. The best way to master a new concept or task is through goal free, open ended questions. But those types of questions pose a problem, one of efficiency. To increase efficiency, worked examples are used to guide students. If too many are used, if the tasks to be mastered are too similar though, worked examples actually have the effect of killing thought and creativity, which is why students end up relying on memorizing rather than thinking. My goal in class then is to use some worked examples. I might only use a couple and then make sure the tasks to be completed differ from the examples, or I might start, but not finish the example, forcing the students to complete it. The tough part for me as a teacher is trying to find the delicate balance between efficiency and mastery. Provide too many worked examples and I am contributing to learned helplessness, don’t provide enough and there is no semblance of efficiency. Worked examples are the primary medium in which I invest, but I also need to know…

Other Cognitive Theories

I need to know about the spacing effect and how to use it. I need to know about the expertise reversal effect and how to avoid it. I need to know about ways to reduce cognitive load. I need to know that learning styles, though they sound nice, basically have no evidence for their existence. I need to find a way to convince my students to overlearn. All these things will help students move what is learned into long-term memory. The goal is to force new information into a schema, which are large, framework like memories that allow us to interpret and analyze new information. If I can accomplish all this, and I find students willing to embrace it, maybe, just maybe, some sort of knowledge might last beyond the semester exam.


Please notice that nowhere did I talk about making learning interesting or relevant. Those are nice if they are available, but the purpose of this post is to discuss why we seem to forget everything we learn in school. Maybe that’s our destiny as a society, and until we stop using education certificates as economic gate keeping mechanisms, we will be stuck with an ever forgetting society. It kind of makes me sick that our education system is that, but it is what it is.


A Summary of Why We Forget What We Learned

Students come are forced to come to school and teachers are forced to teach certain topics. We both need to get over it. If we can’t let coercion component go, our motivation will always suffer. When we rely on punishment and rewards to motivate us, we never really do any action to the benefit of knowledge. All we ever do is try to avoid detentions and get stickers on our diplomas, the knowledge is actually pretty irrelevant. If we don’t care about the knowledge, we will turn our attention to something we actually care about, like Snapchat stories. Between our distracted attention and our willful ignorance of cognitive differences, we condition ourselves to dislike thinking, or at least thinking about academics. When we avoid thinking, we rely on memory because it is so much easier. Teachers provide step by step examples and students memorize them, meaning their knowledge is only, at best, an encyclopedia of examples, devoid of all meaning and context. It allows all students to succeed as defined by grades, but leaves us in the unfortunate position of creating a definition of book smart, which apparently doesn’t have anything to do with actual intelligence. When school is about book smarts, we are acknowledging the irrelevance of academic knowledge. We only perform tasks to get the grade, the test score, the scholarship, the degree, the paycheck, or the promotion. Once we get what we want, we don’t care. The memory is gone, poof, vanished.

This won’t change until we learn how to make ourselves care. It’s not about technology, movies, rewards, grades, tickets, 3 acts, projects, discovery, or anything else. It is about you. You control your care, and when you figure out how to care, you will see that you won’t forget.

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.

Generating a Genuine Mathematical Discussion

One of the most difficult tasks of a math teacher is fostering an authentic discussion about math. Every now and then it comes back momentarily in small groups, but I have trouble generating a real math discussion. I know there ideas out there in the internet ether, but I have found that as long as students are given prompting worksheets, think-pair-shares, they will always want to know what answers to put down so that they get the highest grade. When I ask a class to discuss for the sake of discussion, most of students will give me, at best, lip service, since the discussion won’t have any immediate impact on their grades.

I want my students to discuss math. I want them to discuss math because it is the most effective form of mathematical learning that I have encountered. In math teacher land there is often debate about finding the right balance between practicing procedural fluency and developing conceptual understanding. The procedural fluency camp usually follows a dogma of basic skills and will lament the “fuzzy” math of the 1980s and 1990s. The conceptualists worry about cookbook math and creating math zombies. Myself, I lean towards the conceptualist. However, I do rely on a lot of drill and kill during class. Procedures are great for immediate impact, but if I want long-term, flexible learning, I need to have high quality discussions.

In the past I have had one class where discussion has flourished. That has been my Caclulus I class. My Calc classes have always been small and have always been with students that I have had in previous classes. Because of this familiarity, I was able to make a bargain with my Calc I students. I would give up my power, in the form of grades, if they would give up their expectation of the reliance on examples. It worked beautifully for three years. There was absolutely no structure to the learning. When we would learn, we would just open the book and start reading and working. Some days math didn’t happen because, well, we didn’t want to. Some days we talked about other stuff, like college essays or homework assignments from other classes. Instead of viewing me as the authoritarian, or even authoritative teacher, my Calc students started to view me as more of first among equals, as more of a peer with extra experience. So, when we decided to math we did it because we wanted to, not because we had to.

Anything that was learned in that environment I really feel is more impactful, more powerful, and more portable than what is learned in a regular classroom. There is one story that I can think of that perfectly illustrates what I mean.

A student in Pre-Calc asks me, “Did you hear about Alex?” (Former Calc I student, name changed, who was then a freshman in college.)

“Umm….no. What happened?”

“He failed his Calc quiz.”

“Okay.” (I really think this student wanted me to make some sort scene in class, but I didn’t. Inside though, I was screaming WTF?!!!)

While my Calc I class is not for college credit or an AP class, I feel that I do enough that Calc I should be mostly review for my students when they get to college. Fortunately I ran into Alex around Christmas break and I felt compelled to ask about the failed quiz.

“So, I hear you failed one of your first quizzes.”

“Yeah, that was stupid. The quiz was about finding derivatives using the limit process, but I just used the power reduction rule.”

“Okay, whew. I was worried that I had really screwed up, but really it is about your inability to read directions.”

“Yeah. I met with the professor during his office hours and talked to him. I explained what happened and then talked to him about what I should be doing.”

It was reassuring to hear that he didn’t ask for extra credit, to redo the quiz, or fix his mistakes. He felt comfortable enough with the math I had taught him to go discuss it with his professor. Not only did he feel comfortable enough with math to discuss math, and not just demonstrate procedures, he felt that his knowledge granted him the authority to approach the professor. (I have wondered if this is a skill I was implicitly teaching during Calc and does it apply to subjects outside of math.)

That is what I want out of my Calc class, but this year my Calc and Pre-Calc classes are combined. I have figured out how to approach the topics so that I can teach both groups without giving too much subject material up, but I wasn’t sure how I was going to grade my Calc students compared to the Pre-Calc students. My Calc students know what my Calc classes in the past were like and have been wondering if they would get the grading leniency that I have shown in the past. I kept telling them I wasn’t sure, since they will be covering the same material as the Pre-Calc students.

This past Friday I gave my first quiz. I have already noticed a couple of interactions with my Calc I students that make them different than most of the Pre-Calc kids, but when the quiz was given they were the last ones working. Their approach to the problems were different than all but a few of the Pre-Calc students. Everything about how Friday went tells me that they are ready for how I run Calc I, but I know I can’t run my Pre-Calc class of 23 like I have run my Calc classes in the past.

I don’t know what to do.