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.

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.

Mindsets of School

Over the past couple of months I have been following a discussion about math zombies. It originated in the comments on Dan Meyer’s blog. It has been expanded upon with examples in several other blogs such as here, here, or here. I had previously been thinking about different mindsets in math class; how different people could sit in on the same class and come away with drastically different experiences. Now what I will be discussing in this post is the mindset of the person who sits in on the class, not the engagement of the person with the material of the class. For the purpose of this argument I am defining a mindset as how a person thinks about the material at hand and engagement would be how a person articulates that mindset.

From my personal experience here is how I would classify the mindsets of the students that come into my class.

  1. Wizards – Have you ever met that person that seems to “get” everything, that everything seems to come so easily? Wizards are those students that can perform a task that would make others struggle, to the point that sometimes it seems like magic. This is the student that usually craves projects or might want to initiate discussion. Every time a teacher asks questions about concepts, those “why” questions, the wizard is usually the first to volunteer an answer. (I got the idea for this name when my five-year-old grabbed a copy of our Pre-Calculus textbook and called it his wizard book. I thought it was appropriate since for some people math does seem magical, or at least like witchcraft.) For another good description view the difference between a good mathematician and a great mathematician.
  2. Survivors – The survivors are the students that do what they have to do to get through the class. They see much of their academic experience as gate keeping procedures.  An example from my experience would be a student who was deciding what math he should take his senior year and couldn’t decide between Statistics and Pre-Calculus. As we looked at different math requirements at various colleges for his planned accounting degree we noticed that the requirement ranged from College Algebra to Statistics to Calculus I. All of the graduates can be qualified to take the CPA exam, so what is the difference in the specific math requirements? A survivor has realized that these courses function for the purpose of weeding out those students who will not put in the requisite work. As such, the course material itself is irrelevant, putting the student in the mindset of wanting to pass the course with the least amount of effort possible. Understanding is inconsequential since the material will not be needed for the end result.
  3. The Lost – Everyone has probably seen the lost before. These are the students who seem to randomly be guessing all the time. These are the students where a slight change from procedure can flummox them. These are the students who seem to have never mastered procedures from other classes, that cause the teacher to wonder, “how did they pass?” When they complete an assessment, they have no idea whether if the outcome will be good or bad.
  4. Delusionals – A delusional student has many of the characteristics of a survivor, except they are not aware of the superficial aspect of their learning. I often find these students will create an image of themselves based upon a grade and not on any sort of comprehension. They measure success upon GPA, class rank, and  ACT scores. When they struggle in class, these students are often the first to ask for extra credit or blame the teacher for their struggles. If they achieve low ACT scores, they blame test anxiety or say that test scores don’t matter/ A delusional student will be eager to expunge the value and importance of a class, but then be unable to actually apply material to any sort of context. A delusional student dreads word problems, and will ask to do “actual math,” which means manipulate equations. They lack the ability to apply “school” math to the real world.

The problem with mindsets is that they are not easily measured. I find that I have to interact with my students to understand how they view math. The first thing I learned is that grades do not necessarily reflect a mindset. I have been fortunate enough to have been in every mindset on my list, albeit relying on the benefit of hindsight to come to that conclusion. I was delusional in high school, became lost during my sophomore year of college, slowly morphed into a survivor over the course of my junior year, and finally became a wizard after teaching for several years. This provides me the benefit to at least empathize with every student in my class to some extent.

Now as a teacher, I wonder how I could effectively reach each mindset. I wonder if I should be trying to encourage to students to change their mindset? I think of my presentation, am I modeling the type of mindset I want my students to have? What I have learned as a teacher is that I think heterogeneous classrooms strive for mediocrity. It’s not a numbers thing. I have had classes of two students and I have had classes of 25 students, and I have found that the most effective classrooms I have had the most homogeneous mindset. And I think that is because I can only effectively engage one mindset at a time, I can only teach from one perspective at a time. But homogeny ain’t cool, so what do I do?

Why I Teach (Part 3)

Why do I teach? A quick search of the internet provides a variety of reasons, one of the most frequent being that teaching is a way to change the world for the better. At this point in my career I truly believe that to be the case.

What exactly does that mean, though, to change the world for the better? My previous post referred to the sickening feeling that I got when I realized that my whole school career centered on authority and control. My teachers (authority) had the answers (knowledge) and used grades to reinforce certain behaviors (control). I don’t believe that they had malicious intent, but that they were a product of a system that produces this behavior.

I don’t want to be a part of it. I think it is the reason that we have students who can pass Algebra II, Calculus, and other college level math classes, but can’t manage to create and maintain a personal budget. I think it explains the phenomena of the student who is on honor roll, but is below average on the ACT and other standardized tests. I think it is the reason that there are phrases like, “He’s good at book learning, but has no common sense.” As a society, we have created a system that rewards obedience and complacency. Many of our students just become cogs in the giant mechanism. Remove the mechanism and they fail.

So, why do I teach?

  1. I am here to break the machine. I want to create students who are free and independent thinkers. I want to create students who question authority with the power of knowledge. This doesn’t mean that our schools can’t have the three R’s; rules, regulations, and rubrics. Everything in the lives of our students should have a purpose, and they should know that purpose. Nothing, absolutely nothing that we do in school should happen because, “it’s a rule,” or “because I am the authority.”

It is a struggle to turn students into free thinkers, but the rewards of the struggle are immense. That moment you realize you are not beholden to anyone for knowledge, that you have control over knowledge, you become empowered. And there are no words that can describe what living an empowered life is like. Unfortunately far too many students, far too many people, will never experience the empowerment that comes with knowledge.

  1. The second reason that I am a teacher is to have math make sense to my students. I want my students to move from a position of duplication and memorization to a place where they can reason through a problem. It’s a laudable goal, but I cannot emphasize the gulf that exists between #1 and #2 enough. Also, I hope it was noticed that there was nothing in #1 that specifically mentions math.
  2. I hope that students who go through my class and plan on attending college will at least avoid remedial classes.

As much as I hope that all my students will be in #1, I know that most are in the second and third reason that I teach, with a few not even in any. I care about all my students, I really do. I would never wish anything less than success for any of them. But for the few that fall into my first reason for being a teacher, you are the reason that I keep getting out of bed. You are the reason I keep coming back each year. You are what has kept teaching from becoming a “job.” The fact that I know at least a few of you exist will keep me coming back.

Why I Teach (Part 2)

Last time I wrote I laid out the foundation of why I teach, or at least what I thought those reasons were. With the benefit of hindsight I realize I was so very misguided. My moment of grand realization, this epiphany, occurred during the 2010-2011 school year. I had been teaching Geometry, Algebra I, and Calculus. I had noticed that my geometry had centered around memorizing a whole bunch of theorems. Not in the sense to just regurgitate them, but so that students could apply them to algebraic equations. I told myself that this made me a good teacher. Students criticized me for this style of teaching, but the criticism came in the form of complaints, so I usually brushed it off and kept moving forward.

One day, in my Calculus class, a student simply asked, “Why?” There was no complaining, no questioning of the premise of school, just, “why?” He could have just copied and memorized the procedure, like most students do, but in that moment he decided he wanted to know why the procedure worked. And guess how I answered, “I don’t know.”

That really frustrated me at the time. I had struggled teaching Calculus and I had simply blamed the problem on the fact that I had not spent time with the course for five years. But this time it was different. I started reflecting on the idea that math was just about all the procedures I had accumulated in class. I realized that I was able to DO most of the math problems that came my way, but I didn’t understand WHY the procedures I was using were working. As I spent more and more time reflecting, I came to the conclusion that for me, math was nothing more than a contrived system of rules for students to memorize and apply in the context of math class.

At the same time though, I knew that math was not invented in some vacuum away from reality. I knew that math was and is used by builders, engineers, scientists and many others, so why should it be viewed in school within the context of hoop jumping. Too many students, teachers, and people in general only view math as something that has to be done without any real value too it. I have heard far too many people describe my content as a way to weed out the lazy, stupid, and unmotivated. I didn’t want to be a part of that.

I also had a personal crisis. I had struggled to teach Calculus I. I had a BA in Mathematics. That thought caused some serious cognitive dissonance. What was the purpose of my math major if I couldn’t apply, let alone remember what I had done several years ago? At the same time I was thinking this I had just begun work on a History Masters, which has really transformed the way I view authority, power, and knowledge. I slowly began to rework my math understanding, moving into a realm where math had to make sense. It was no longer enough to get my answers to match the book because the book did not have authority of knowledge. The book only had authority of title. As math began to make sense I started to organically see the applications of the math subjects I would be teaching. I began to view complex math schema rather than a jumbled collection of memorized examples.

As math started to make sense to me, for the first time in my life really, I became very frustrated with my former self. I thought about all the opportunity that I had wasted striving for grades rather than understanding. To this very day I look back at my high school and college experience with regret because I was a label chaser, I wanted the GPA, I wanted the degrees. I achieved them, but knew nothing.

I never want my students to go through that.

Why do we have to do this?

At some point almost every teacher I know will get the dreaded, “Why do we have to do this stuff?” Today we were working through the verification of trigonometric identities and that question came out of the ether on a Monday morning. I have been thinking to myself, how should I answer this?

  1. I could just silence the student for insubordination, but that would require a misuse of authority. I distrust people who demand blind obedience.
  2. I could use the textbook line, “You can use trigonometric identities to rewrite trigonometric equations that model real-life situations. For instance, in exercise 58 on page 556, you can use trigonometric identities to simplify the equations that model the length of a shadow cast by a gnomon.” Wow, it says real-life, but then directs the reader back to the exercise problems. This had to be written to appease some bureaucrats that have no understanding of math. And how many people know what a gnomon is without looking it up.
  3. I could use the, “It is good for problem-solving and critical thinking skills.” But the flaw there is that problem-solving and critical thinking skills can be taught with other subjects, it’s not unique to math class, so why bother with the painful math?
  4. I could use the, “You will need this in college, work, etc.” However, do we really need the math as it is taught in school for college or work, or is it that those particular fields just use the class as a way to separate and classify people, making math class a filter? And if that is true, can I be content with my teaching knowing I am just a filter?
  5. I could say that, “The state requires this stuff, so there is nothing I can do.” That doesn’t sit well because now I am just passing the blame to someone else. Also, see what I said about authority in number one.

Honestly, I believe there is a kernel of truth in all of those statements, but I don’t believe alone, any one of those statements can be proper justification for the misery students feel in math class. So, what do I tell my students? What should I say when I really think their frustration stems from deeper issues of how we teach math and how shallow their understanding of mathematical concepts are? When I view trigonometric identities all I see are arithmetic expressions to be simplified. As long as I can do arithmetic, I can prove trigonometric identities. So a better question would be, “Why should I not verify this equation before me?”

Sitting in those desks though, trying to absorb the information, it doesn’t come across as relevant and pertinent. Our schools have created such a Pavlovian response to education that many students have lost the wonder of learning just for the sake of learning. Too many of them view education through the lens of correct answers and the quickest means to achieve them. Their minds have been trained to reward their emotions for extrinsic rewards like grades, meaning far too few of them ENJOY THE PROCESS.

And there I stand, at the front of the room, thoroughly enjoying the process of math that lay before me while they see a gauntlet that must be endured to achieve the grade. I’ve made up my mind, they’ve made up theirs. Our viewpoints are incompatible. We are just too different.