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?

Winging It

I hate how time dictates the school day. I hate how we expect drastically different students to learn the same material, at the same age, at the same time of day, and even at the same pace. Sure some have drank the differentiation Kool-aid, but we haven’t yet differentiated high school as a whole. Same graduation requirements, same times, everything is the same.

From my experience, the majority of teaching takes the form of presentation of material, followed by some guided structure with the teacher, and then some independent time for the students to practice. This is the typical I do, We do, You do approach to education. When I prepared for this approach I would carefully think out my presentations. They wouldn’t be flashy, but I would take a substantial time to think about how I was going to talk about something, what I wanted students to notice, and what examples I want students to use. My goal wasn’t to entertain, but it was my goal to make the information clear. Though there are supporters of the entertainment aspect of education.

Are you not entertained?! When I used to show that to students they would remark that the class looks fun, that he makes the subject matter exciting. However, at the end of his courses, the attendance rates and failure rates were similar to other classes. So if the results were the same, what is the point of all that prep work to make the presentations exciting?

Maybe my presentations weren’t nearly as entertaining, but at least I was being clear on what I want accomplished. I started class telling the students what we’re going to do today. I wrote up clear definitions and gave clear examples with multiple steps shown. But the goal of education should be to create students with the ability to think, which involves a whole host of issues. Part of the problem with using clearly stated goals is that novice learners will only focus on the goals, most likely bypassing interesting and important connections along the way. Yes, I know that that study is talking about reading, but from my experience students do that with just about everything they do.

To put it another way, when I was presenting mathematical information I was covering information that is part of a complex tapestry of mathematics. However, my students only take in minor details, basically ignoring as much of my voice as they could, grasping for the bare minimum structure to be memorized so that they can correctly answer test questions. This picture eloquently summarizes what I think is going on in the journey from my mind to their minds, even though it really is about rubrics.

It is a visual representation of why so many of my students seem to think math is just a disjointed collection of random facts and procedures. When I thoroughly thought out my presentations, I made sure to highlight those red dots of importance, but in my mind those dots are just part of the whole picture. My students just pick up on the red dots though, which I often referred to them Charlie Browning me. My voice was the blue, my examples were the red, they copied the examples and heard this.

My good compliant complacent students were Charlie Brown. The had the appearance of listening, but really were just quietly searching out those red dots, those examples and steps to let them solve the next math question. My favorite are the students like Patty though. At least they weren’t pretending to care, yet an alarming amount of them are on the honor roll. They have internalized the process of hunting out those red dots, be it from examples in books, notes, online, or asking their friends, “How to do this?” They are obsessed with the how’s, but not the why’s?

To help try and combat this I changed my presentations. Instead of carefully planning out every individual step with concise, clear objectives, I started to wing it in class. It didn’t mean I wouldn’t lesson plan, it just means that my plans were a rough outline, a framework, that was then filled by the organic discussion in class. My goal was to make sure the students wouldn’t become fixated on the red dots. When I would be planning my presentations I would pick a topic, think of how it connected to the previous topics, and then try and use student questions and ideas to drive most of the presentation. When I know my students and my content I find this to be an enlightening experience. They start to finally see some of that blue background behind the red dots.

However, it does have a couple large drawbacks. It did give class a more organic feel, but students crave the conditioning that they have been experiencing for years and years. Charlie Browning is most prevalent in my honor roll students because it has allowed them to get success in the past, at least in terms of grades, with the least amount of mental effort. For most of my students, it takes a significant amount of time to overcome that conditioning, and some, unfortunately, never will.

It also gives the appearance that I am unprepared, but for me, it changed the hierarchy of my teaching prep. When I plan, I start with content from a teaching viewpoint, then worry about presentation and pacing, then worry about assessments, then worry about supporting activities, then worry about individual students. My ever changing schedule the past eight years has meant that I feel like I am perpetually stuck in my first hierarchy of teacher needs, focusing on content.

I guess I forever will be a rookie.

My Struggle with Homework

The math classroom I knew from school followed a typical pattern.

  1. Review/Collect/Correct previous assignment.
  2. Teacher introduces new concepts/topic(s).
  3. Teacher walks through several example problems.
  4. Students are given an assignment.
  5. Repeat process.

Some math teachers are quicker with a joke, or friendlier, or more strict, but ultimately I think the majority of classes follow this pattern most of the time. When I first started teaching I struggled with how to handle steps 1 and 4. Here is the story of how I came to my solution.

My first year teaching at my current school I followed this pattern fairly religiously. At first I collected homework assignment and tried to check every problem from every student. I quickly learned this is a nightmare. Between students not showing work, poor penmanship skills, and trying to decipher multiple approaches, it is just way too time consuming.

I next tried an approach I picked up student teaching. I wouldn’t correct every assignment, but I would choose them randomly. It was still time consuming to correct, but at least that time consumption was limited. However, there was still a flaw with this system. One day when I went to collect a homework assignment to grade I had a student approach me. He didn’t have the assignment done. Every other assignment was completed, but something came up and he didn’t get that assignment done. I liked the kid, he normally was everything a teacher wants from a student, so I decided to give him a break. The problem was that more and more students asked for breaks, and every now and then I would get the rare student that skipped every assignment, but just happened to complete the one that was collected. I felt like this system was just to coincidental and happenstance to represent some sort of accurate measure of knowledge.

And it was still time consuming. So I adopted something I saw during student teaching, instead of collecting entire assignments, I started collecting just a few specific questions from homework assignments. But the outcome was still largely the same, it just felt like the grades were coincidental and happenstance.

There was also one large problem with which I had an issue. Correcting homework for accuracy led a lot of students to blatantly copy a handful of students. This defeated the purpose of homework to me. I firmly believe homework is there for students to reflect and practice the skills covered in class.

I decided to handle this problem by making homework a participatory grade. About two times a chapter I would collect homework from the students and just check to make sure they did something. I used this system for three years. I liked it because it allowed me to distance myself from student responsibility. If students took the time to understand the homework, great, and if they just filled in their notebooks to get the participation points that was fine with me too because they would just get low test grades. They didn’t put in the effort to learn the material, so they would suffer the low grades. I didn’t feel bad because I was essentially offering 30% of their grade for free by making homework participatory.

Then, at the end of one school year, about 20% of my students failed. It came down to their homework. They weren’t making the connection between doing quality work and success in school. I heard reason after reason, excuse after excuse, as to why the homework wasn’t done. Some of them were legitimate and some weren’t, but that wasn’t the point. They saw me as an authority figure and the homework I assigned was about following directions, not an educational opportunity. I had already struggled with the cycle of detention, and I do fall on the side of the debate believing they don’t achieve the desired result. I kept second guessing myself, thinking maybe I should have assigned more detentions. But those detentions just would have reinforced the cycle of obedience for those students.

If my goal is to breed compliance and obedience in students there are much more effective ways than math homework and detentions. Actually, the more I think about it, math homework and grades are about the dumbest way to teach concepts of compliance, obedience, and following directions. A paycheck and a job are much more effective for that.

Sending 20% of my class to summer school or back to Algebra I again wasn’t enough to make me change my ways though. I spent one year teaching summer school and have had several students go through the process. Though summer school itself is largely unresearched, my personal experience is that it serves largely as a prolonged detention to avoid repeating a class. By sending kids to summer school, the homework wasn’t about obeying me, as a detention would have been, but the homework was about obedience in the system.

And I was perfectly okay with this set up until the end of the next school year. I didn’t have nearly as many students fail this time. Actually only a couple, but one stood out in my mind. It was the last day of class and I had a student who was sitting at around 50 some percent. His homework, 30% of his overall grade, was negligibly above a zero. I had always told students it’s not when you learn something, but rather that you learn it. Well, here it was, the last day of school. Simply do some of the homework and the student could pass the class. I knew the kid had the math ability, I had watched him do math during class before, he just needed to get enough participation points to pass the class.

When he claimed he didn’t have enough time to get the work done, one of his classmates offered the use of her old notebook and worksheets to copy. He still refused because, as he stated, he didn’t care. Homework still wasn’t having the impact I wanted it to. I decided I needed to change something for next year. I couldn’t keep going having so much of a student’s grade represent obedience.

I needed to devise a way to grade so that those grades represented math ability and not classroom obedience. I needed to get students to realize the work they do with homework is what led to success, not watching me give notes. Most of all, I needed to break the cycle where students defend their behaviors with, “…but I didn’t think you’d care.” If all we ever teach is school is to do things because the teachers care we haven’t really educated anybody.

Thinking About Learning

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I Used to Teach Algebra I

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

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

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

This worked for me because of two reasons.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I give out too many A’s.

No one learns anything in your class.

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

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

It is impossible to learn anything in your class.

You let the students do nothing.

More people would be complaining if the grades were lower.

Students will lie to defend you.

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

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

There needs to be more grades in your class.

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

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

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

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

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

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

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

 

 

Is School Really About Education?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

However,…

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

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

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…

Compulsion

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…

Motivation

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…

Distraction

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…

Pseudoteaching

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.