Learning Isn’t Easy

One thing I learned the hard way is that grades do not reflect an level of actual knowledge. In my eyes many grades measure how many tasks a student successfully completed. I am not trying to be dismissive of hard work. I have numerous students that work really, really, hard, some that I think work way too much and are on the verge of breakdown. I even experienced the struggle with this myself when I nearly flunked out of college.

But hard work isn’t enough to actually master a subject. Doing hard work means just that, you’ve worked really hard. Learning will always come from struggle, but struggle by itself doesn’t produce learning. I experience this for myself several years ago in graduate school, and watch it on so many of my students regularly.

Learning, real authentic learning that leads to mastery of something that is actually usable, comes in fits and spurts, in flashes of brilliance at inopportune times. We can’t schedule learning. We can schedule training, we can schedule task completion, but if our goal in education is to have students learn anything in any sort of meaningful manner, the schedule inherently works against us. No matter what questions I ask, resources I use, if a student comes into class obsessed about the recently failed Spanish test or whatever, no learning will occur. I might be able to get compliance and task completion out of a student, but unless if that student’s mind is clear and ready, learning will not happen. We think we can multi-task, but in reality all we are doing is switching between tasks, and if learning is the goal, then switching between tasks is detrimental.

I can think of one example that recently happened. While teaching rotations in Geometry this year I have been relying on the coordinate plane. I was showing the students how the x and y value of coordinates move when rotated 90 degrees, or multiples of 90 degrees. When I was asked about rotations that weren’t 90 degrees, I told them that they wouldn’t have to worry about them. Why would I do that? We had already talked about trigonometric ratios, so, with the benefit of hindsight, I realized that we could have done them. I thought it through a couple of days later and came away slightly miffed that I hadn’t thought of it earlier. Probably just dismissed it because the book didn’t have any examples.

That bothered me though. It bothered me because it should be possible to rotate something that wasn’t a multiple of 90 degrees, and fortunately I was able to figure out how to do that. I had the opportunity to quiz my former students on the subject, so I decided to let them take a stab at the non-90 degree rotation. First I showed them what I had covered in Geometry, how rotating 90 degrees is like moving legs on an ‘L’.

Then I showed them what I wanted.

A student came up with the idea of using the 53 degree rotation as a percentage of the known 90 degree rotation, then using the corresponding percentages to change the x and y values.

Which produced the following result.

Now the point A (3,4) includes an angle of approximately 53 degrees, so a rotation of 53 degrees is a total angle of 106 degrees, or a reference angle of 74 degrees in quadrant 2. Well, I checked the trig using the proportioned A prime, and came up with slightly less than 72 degrees.

Well, that’s not 74 degrees, nor is that anywhere near the realm of rounding error in this case. But the method seemingly made sense, if the percentage of the angle should be the percentage of the sides of the triangle. Why wasn’t it forming the same angle? Later I made a little table to investigate what was going on.

Clearly, when I was using percentages of the sides I wasn’t getting the same percentage of the 90 degree angle. The more I thought about it though, I started to realize that at zero percent the length is 8, but then at 100 percent the length would be 6. That doesn’t make sense. Later, I took one more crack at trying to figure out why the percentages were creating a different angle than a rotation and created this picture.

That’s when I finally saw it. A rotation implies a circular motion.  Moving the point as a percentage of the x and y distance changes the distance from the center of the circular rotation. The bottom line of the right triangle is not broken into equal partitions because it is changing distance from the point of rotation. Using percentages the same percentages from taking 53 of the 90 degrees conflicts with taking the same percentages on a straight line distance of 7 units. The differences in the answers is because of the nature of the definition of rotation.

What did I learn from all this? About myself, I learned that my geometry skill is sorely lacking and very rusty. I also learned that I no longer have the trust to learn with the student in front of the students.

It also reinforced that the nature of school is not always conducive to learning. The students I were working with were capable of thinking of the answers. They even came up with a reasonable idea, but when they didn’t get the same answer as me they dropped the idea, writing it off to rounding error or just being wrong. This has nothing to do with capability. To learn is to devote every mental resource to a subject, to think, analyze, justify, and simply ponder. Our students are just simply too busy, they have to worry about 5 or more subjects, extracurriculars, college applications, part-time jobs, and any other myriad of activities. They just don’t have the time/interest/focus to clear their mind of all the other clutter to actually engage with math.

What the students want from me is clear, straight-forward, methods to find clear, straight-forward answers. They don’t have the time to think about open ended possibilities, there’s too much to do. But to really learn something, that’s what is needed, those open-ended, no solution, what do you notice type of questions that can spur a dialogue. How do I create that when all my audience really wants are the answers?

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.

How Do I Get an “A” in Your Class?… Or How Failing Made Me a Better Teacher

It’s happened again. I have been accused of not teaching, by a student. It’s not that this particular student blatantly raised a voice during class and shouted, “You never teach us anything,” but it started with a couple of innocent statements.

“I think it would be better if you gave us notes.”

“Can you go over an example of an ‘A’ questions?”

“How do I get an ‘A’ in your class?”

The problem I have with all of these statements, no matter what the circumstances are surrounding them, is that they come from a mindset that I believe has infected education on far too many levels. Students come into my class operating on a training, recall, example laden mentality. The expectation from these students is that I will present the material as it will appear on assessments, and it is their responsibility to memorize the material presented, and the material will be identical. There is a subset of students, parents, administrators, and even the general public, that believe this is what education should be.

How does that happen? How did we get to a place in society where it is thought that education is the same as memorizing tasks?

Schools, both high schools and colleges, are under tremendous pressure to ensure students graduate. Graduation rates affect funding levels for schools. The higher percentage of students that pass the more money a school receives, or is less likely to lose.  If the graduation rate falls at my school, it will be endanger of losing funding. In addition, standards for graduation keep increasing, creating a perfect milieu for grade inflation.

Ahhh….grade inflation. That concept is not new. It has been the bane of education since the existence of grades. As long as there has been no standard definition of an “A” people have blamed others for inflating grades. But the idea of an inflated grade wouldn’t exist without someone finding out that the student who had that inflated “A” really wasn’t that smart. For that, we can blame employers, admissions departments, scholarships, and even teachers.

When people in authority use generic measurements, a GPA or transcript for example, as a gauge of intelligence it invites sympathetic teachers to inflate grades. We are to the point in society that if I were to truly hold a student accountable for mastery of a concept such as parabolic functions, it could represent that student’s ability to obtain a low skill job in the service industry. To me, that represents my incentive to make my class as easy as possible to pass, because I don’t want to be the person who tries to say with a straight face, “I’m sorry, but you can’t have a job bagging groceries because you cannot complete the square to find the vertex of a quadratic function in standard form.” (This is not meant to be an insult to grocery baggers of which I really appreciate the good ones. The statement was  there to try and point out the futility of connecting arbitrary education with work preparedness.) Every employer that has said a job applicant has had to have a high school diploma, without a thought as to what that student was subjected to learning, using a high school diploma as a rudimentary haphazard sorting device, causes an increase of grade inflation. It is because employers like that exist that my class is easy to pass.

But it’s not just low skill service jobs that use GPAs and transcripts as lazy sorting devices. Colleges and scholarships do so as well. Maybe they look beyond just obtaining a high school diploma and focus on certain classes and certain grades, but the concept is the same. When I have a student who is very capable of being a nurse, but they are encouraged to take Pre-Calculus because that is what is required of the college’s nursing program, I am incentivezed to make getting a decent grade relatively easy. I wonder how many doctors, not to mention nurses, could tell me what a conic section is, let alone describe the relationship of the sum and difference between foci that generates the different conic sections. The college won’t really care what she knows in regards to Pre-Calculus, only that the class shows up on her transcript with a certain grade by it.  As long as I have her prepared to take her one math class in college from the professor who is under pressure to make sure she passes, I feel like I have done my job. These students know the game of gatekeeping that is goes on at the different levels of education. It is why I try to make it relatively easy to get a “B” in my class. It might take dedication and work, but it is achievable by nearly all students who have a mediocre grasp of concepts learned in previous classes.

I make sure that “B” is achievable because anything less than a “B” must be justified. No one wants their child be the student that struggles, but I cannot assign a grade below a “B” without being able to document the behavior the student demonstrated that led to the low grade. I have to document how I tried to correct those behaviors. I have document all the interventions I tried for the student. I have never, NEVER, been asked to justify why a student has an “A.” By making a “B” relatively easy, I can defend my low grades with simple work ethic defenses and lack of prerequisite knowledge statements. (Those are legitimate issues, it just makes the administration of my class easier to have most students around a “B”.)

So far I have mentioned the money involved in education through the application of graduation rates  impacting a schools’ funding. I have also discussed the societal pressures to obtain a high school diploma or take certain class only for the label, without any regard to the content of those classes. These lead me, and probably other teachers, to ultimately reduce the rigor of their classes.

However, there is one other influence that shapes education into memorization. In my education classes about assessment in college we covered the concept of test validity. We were taught that for a test to be valid the material on the test must be explicitly taught. If the test material is not explicitly taught then the test is invalid. This was then interpreted as meaning teach what is on the test, though never said in that manner.  In class we provide students with every example they might see, with all the information that might be around, we provide study guides and review sheets, we play review games, and then we give a test. When students do well we congratulate ourselves and think our students are all above average. When they do poorly we point out all the places in the study guides or homework examples where the information was located. Even though we so often trumpet the mantra of, “don’t teach to the test,” we don’t listen to ourselves.

This is what my education was. When I entered college back in the fall of 2001, I had every indication that I should be successful. I had tested into the gifted program in elementary, I was accelerated in math in junior high, I had never placed below the 93rd percentile on any standardized test (Iowa Basics, ASVAB, PSAT, ACT), I took AP classes in high school for weighted grades and finished with above a 4.0 GPA. I finished my freshman year of college with nearly a 4.0 GPA, with a little struggle in the spring semester that I simply attributed to college being more difficult.

My sophomore year it all fell apart. I was failing classes. I dropped classes in a desperate attempt to salvage my grades in the remaining classes. I let the funk infect every aspect of my life. It ruined friendships and jobs. To this day I am not comfortable talking about my failure. Sure, I can mention it happened, but mentioning that failing happened is very different that coming to grips with the reality that my self-identity was a complete and utter lie. I visited depths of personal hell that I wouldn’t wish upon anyone. (Here is a link that describes it better than I can.)

I can remember sitting in classes, trying to take a test, and feeling like it was gibberish on the page in front of me. It’s the first time I can ever recall thinking, “He never taught this!”

Slowly, I started to develop the work ethic necessary to pass during the second semester of my sophomore year, but it was still an emotional period of my life. I still remember one of my moments of clarity during my embitterment. I went to pick up a quiz from our Quantitative Analysis professor. I had worked so hard for this quiz, I had put in so much effort trying to understand the examples and making sure the extra homework problems were correct and figuring out ways to evaluate them myself, but I still lacked confidence and was nervous. I got the quiz back and was ecstatic that it was a good grade. As I examined the work, I noticed one of the problems that I got correct was one that he had never covered in class. I let out a very loud, audible, “I got this right and he never even taught us this!” in front of his office door.

That’s the moment I considered a turning point in my college education. (I managed to finish with a 3.43 GPA after a semester on academic probation.) It’s when I realized the amount of work authentic learning requires. But it’s not the quantity of work learning takes, but the quality of the work that leads to success. I learned that I needed to generalize better.

When I was in high school I bore the label of being one of the smart students. Now, with the benefit of hindsight, I realize that it wasn’t that my GPA was any higher than my peers, it was that I had to work less than them to achieve it. My education amounted to me watching my teacher do some examples, easily memorizing them, and regurgitating them on a test. I was doing an academic binge and purge. Eventually it caught up to me. I never learned how to effectively learn, all I had ever done was memorized and thought I was a good learner. I even identified myself as a quick learner on job application when I should have been saying, “I am a really good at memorizing repetitive tasks, so I will be the perfect employee for Burger King.” Since I have become a teacher I have started to notice that several of my students share that mindset.

Unfortunately I feel and obligation to ensure that my students don’t feel that false sense of security. I don’t want a student to go through what I did. I can’t live with the thought of students who would leave my class thinking it would be easy, then fail their next math class. And that motivation, probably one of the earliest I ever adopted, has shaped my class in two distinct ways.

The first way my failure impacted my classes is that I have jumped on the teach less bandwagon. Teach less, be less helpful, productive struggle, productive stupidity, there are a plethora of blog posts, editorials, and even a few journal articles about the concept. Basically, it boils down to the idea that if I teach every possible example I have done all the thinking for my students. This is bad because then students never learn to think for themselves. From a progressive standpoint, I would say my students are discovering, and from a traditionalist view I probably have just gotten good scaffolding. But the point is the same, I lead kids, I prompt kids, but I never explain explicitly every step.

The second way that my failure impacted my class is on my assessments. Several of my students who are used to getting an “A” in class are undergoing a period of adjustment. They have trouble because they were me in high school, smart, but relying on the teachers to do all the thinking for them. I have modified my assessments so that questions that will warrant an “A” are never explicitly covered in class. I have covered concepts, but not specific examples. I am trying to use the training aspects of school to train my students to be prepared to answer unfamiliar questions.  I was working with a student after school last year on a quiz that she missed and the story almost perfectly illustrates the thinking that I am trying to avoid.

She was having trouble on the “A” question. “I don’t know what to do!”

I responded by saying, “Tell me what you’re thinking.”

“I’ve read the problem and it doesn’t look like any of the examples we’ve done in class. I tried to match the work to the problem like______(I forgot the one she mentioned, but it was about radians), but it doesn’t make any sense”

For years I had been trying to describe the point of not covering “A” questions in class, and now it made perfect sense. She read the problem on a surface level and scanned her memory for similar problems. She really didn’t comprehend that she needed to dissect the problem and pick out the concepts she knew and then apply those concepts to an unfamiliar question.

As I prompted her through the idea asking, “Do you know what this means?” over and over again, it finally seemed to click. She finally realized that she knew everything that was there.

I can’t prepare my students for every conceivable test question that might appear on an end of course exam, ACT, SAT, or whatever. So I purposely under prepare my students for my tests. When they grasp being under prepared, then they are really prepared for the tests they will have to take when I’m not around. When they embrace the mindset that comes with being under prepared, they will succeed in my class.

That’s how you get an “A” in my classroom.

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.

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?

It’s Not My Responsibility

When is a teacher free from the responsibility of teaching? I’ve heard the concept called release time. There might be other terminology for the concept, but I am not sure. So what is release time exactly?

As a teacher I provide explanations and examples to my students. I answer questions, walk through procedures, and attempt to ask thought provoking questions to stretch my students. I provide resources such as practice problems from a textbook or links to videos for further explanation. But when do I get to say, “I’ve done enough.”?

To me release time is when I get to release myself the responsibility and burden of educating a student. When release occurs the burden of education falls upon the student. However, there is not set standard for when this can be accomplished.

Is it a time thing? Are three class periods enough? Maybe a week?

Is it a certain number of exercises or worksheets?

Is it when students can successfully mimic the instruction in the class?

I find myself at an impasse. I think I have covered material thoroughly. I have explained all of my reasoning. I have used a multitude of examples of varying difficulty. Yet, in spite of my efforts, I feel like I have a large portion of my students who are willfully neglecting to learn.  I really don’t know what else I can do short of dumbing down the standard even more than I already have.

I feel like the students are being dismissal of my teaching efforts, a feeling that rarely happens on such a large scale. I know some would say it is the topic itself, that it isn’t interesting, or that it is just too difficult. It is confined to this one topic. Usually, it is a great group of kids, with great motivation, but I put the work in front of them and it’s a chorus of eye rolls, moans, and “I don’t get it(s).”

And so I am conflicted. Part of me wants to say, “screw ‘em, I’ve done my job,” but part of me is constantly trying to think of ways to make this make sense.  I can’t decide, is the shut down out lack of desire or is it because of lack of skills? If it is the latter, I can keep constructively working on improving skills, but if it is the former, there is nothing I can do. The more I watch, the more I try, the more I believe that it is the former, that they just don’t want to do this.

But I need to assess them, right? The purpose of assessment is to make sure the students understood the information. Can I give an assessment in good conscious when so many of my students refuse to participate? Maybe I should just forgo the assessment. Does the topic really matter in the all encompassing life of school? Or is that granting too much power to the students, to let them dictate the topics to be tested? I know that we always should think of the things that we can do better, but just this once is it okay to be selfish?

Can I say, just this once, that I’ve done enough?